# Mathematics Standards -- K-12

• The AKS standards for each grade level spell out the essential concepts students are expected to know and skills they should acquire in Mathematics. The AKS offer a solid base on which teachers build rich learning experiences. Teachers use curriculum guides, technology, and instructional resources to teach the AKS and to make sure each and every student is learning to his or her potential.

The Academic Knowledge and Skills (AKS) standards were developed by our teachers, with input from our parents and community, in response to Gwinnett County Public Schools’ mission statement: The mission of Gwinnett County Public Schools is to pursue excellence in academic knowledge, skills, and behavior for each student resulting in measured improvement against local, national, and world-class standards.

• # A - Counting and Cardinality

• count to 100 by ones and tens
• count forward by ones, beginning from a given number within the known sequence (instead of having to begin at 1)
• write numerals from 0 to 20 and represent a number of objects with a written numeral 0 - 20, with 0 representing a count of no objects
• demonstrate the relationship between numbers and quantities to 20; connect counting to cardinality
• count objects by stating number names in the standard order, pairing each object with one, and only one, number name and each number name with one, and only one, object (one to one correspondence)
• demonstrate that the last number name said tells the number of objects counted (cardinality) ; the number of objects is the same regardless of their arrangement or the order in which they were counted
• demonstrate that each successive number name refers to a quantity that is one larger
• count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects; identify and be able to count pennies within 20.(use pennies as manipulatives in multiple mathematical contexts)
• compare two sets of objects and identify which set is equal to, more than, or less than the other using matching and counting strategies
• compare two numbers between 1 and 10 presented as written numerals
• identify coins by name and value: pennies, nickels, dimes, quarters, and dollar bills

• # B - Operations and Algebraic Thinking

• represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps) acting out situations, verbal explanations, expressions, or equations
• solve addition and subtraction word problems, and add and subtract within 10 (e.g., by using objects or drawings to represent the problem)
• decompose numbers less than or equal to 10 into pairs in more than one way (e.g., by using objects or drawing), and record each decomposition by a drawing or equations (e.g., 5 = 2 + 3 and 5 = 4 + 1)
• find the number that makes 10 when added to the given number, for any number from 1 to 9 (e.g., by using objects or drawings, and record the answer with a drawing or equation)
• add and subtract within 5 fluently
• identify, create, extend, and transfer patterns from one representation to another using actions, objects, and geometric shapes

• # C - Number and Operations in Base Ten

• compose and decompose numbers from 11 to 19 into ten ones and some further ones (e.g., by using objects or drawings), and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones

• # D - Measurement and Data

• describe several measurable attributes of an object, such as length or weight; for example an object may be described as heavy or light or long or short
• directly compare two objects on the basis of length (longer/shorter), capacity (more/less), height (taller/shorter), and weight (heavier/lighter) and describe the difference (e.g., directly compare the heights of two children and describe one child as taller/shorter)
• classify objects into given categories; count the numbers of objects in each category and sort the categories by count (limit category counts to be less than or equal to 10)

• # E - Geometry

• describe objects in the environment using names of shapes and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to
• name shapes correctly regardless of their orientations or overall size
• classify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid")
• analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/corners), and other attributes (e.g., having sides of equal length)
• model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes
• compose simple shapes to form larger shapes (e.g., "Can you join these two triangles with full sides touching to make a rectangle?")

• # A - Counting and Cardinality

• demonstrate the relationship between numbers and quantities to 20; connect counting to cardinality
• count objects by stating number names in the standard order, pairing each object with one and only one, number name and each number name with one, and only one, object (one to one correspondence)
• demonstrate that the last number name said tells the number of objects counted (cardinality); the number of objects is the same regardless of their arrangement or the order in which they were counted
• compare two sets of objects and identify which set is equal to, more than, or less than the other using matching and counting strategies
• compare two numbers between 1 and 10 presented as written numerals
• identify coins by name and value: pennies, nickels, dimes, quarters, and dollar bills

• # B - Operations and Algebraic Thinking

• decompose numbers less than or equal to 10 into pairs in more than one way (e.g., by using objects or drawing), and record each decomposition by a drawing or equations (e.g., 5 = 2 + 3 and 5 = 4 + 1)
• find the number that makes 10 when added to the given number, for any number from 1 to 9 (e.g., by using objects or drawings, and record the answer with a drawing or equation)
• identify, create, extend, and transfer patterns from one representation to another using actions, objects, and geometric shapes

• # C - Number and Operations in Base Ten

• compose and decompose numbers from 11 to 19 into ten ones and some further ones, and record each composition or decomposition by a drawing or equation; understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones

• # D - Measurement and Data

• directly compare two objects on the basis of length (longer/shorter), capacity (more/less), height (taller/shorter), and weight (heavier/lighter) and describe the difference (e.g., directly compare the heights of two children and describe one child as taller/shorter)

• # E - Geometry

• analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/corners), and other attributes (e.g., having sides of equal length)
• model shapes in the world by building shapes from components and drawing shapes
• compose simple shapes to form larger shapes (e.g., "Can you join these two triangles with full sides touching to make a rectangle?")

## A - Operations and Algebraic Thinking

• use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions (e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem)
• solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 (e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem)
• explore and apply properties of operations as strategies to add and subtract (e.g., If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (commutative property of addition). To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (associative property of addition)
• model and explain subtraction as an unknown-addend problem (e.g., subtract 10 - 8 by finding the number that makes 10 when added to 8)
• relate counting to addition and subtraction (e.g., by counting on 2 to add 2)
• add and subtract within 20, demonstrating fluency for addition and subtraction within Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13)
• model and explain the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. (e.g., which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2)
• determine the unknown whole number in an addition or subtraction equation relating to three whole numbers by using symbols (e.g., determine the unknown number that makes the equation true in each of the equations 8 + ? = 11; 5 = ? - 3; 6+ 6 = ?)

## B - Number and Operations in Base Ten

• count to 120, starting at any number less than In this range, read and write numerals and represent a number of objects with a written numeral
• model and explain that a two-digit number represents amounts of tens and ones
• explain that 10 can be thought of as a bundle of ten ones called a "ten"
• model the numbers 11 to 19 showing they are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones
• explain that the numbers 10, 20, 30, 40, 50, 60, 70, 80, and 90 refer to one, two, three, four, five, six, seven, eight, or nine tens and 0 ones
• compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <
• add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten
• using mental math strategies identify one more than, one less than, 10 more than, or 10 less than a given two-digit number explaining strategy used
• subtract multiples of 10 in the range 10 - 90 from multiples of 10 in the range 10 - 90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used (e.g., 70-30; 30-10; 60-60)
• exchange equivalent values of coins by making fair trades involving combinations of pennies, nickels, dimes, and quarters and count out a combination needed to purchase items less than a dollar

## C - Measurement and Data

• order the length of three objects; compare the lengths of two objects by using direct comparison or a third object
• express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps
• tell and write time to the nearest hour and half-hour using analog and digital clocks
• organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another

## D - Geometry

• distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes
• compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half- circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape and to compose new shapes from the composite shape. This is important for the future development of spatial relations which later connects to developing understanding of area, volume, and fractions
• partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares

## A - Operations and Algebraic Thinking

• use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions
• model and explain the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false (e.g., which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2)
• determine the unknown whole number in an addition or subtraction equation relating to three whole numbers by using symbols

## B - Number and Operations in Base Ten

• model and explain that a two-digit number represents amounts of tens and ones
• using mental math strategies identify one more than, one less than, 10 more than, or 10 less than a given two-digit number explaining strategy used
• subtract multiples of 10 in the range 10 - 90 from multiples of 10 in the range 10 - 90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used (e.g., 70-30; 30-10; 60-60)
• exchange equivalent quantities of coins by making fair trades involving combinations of pennies, nickels, dimes, and quarters, and count out a combination needed to purchase items less than a dollar

## C - Measurement and Data

• express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps
• tell and write time to the nearest hour and half-hour using analog and digital clocks
• organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another

## D - Geometry

• compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half- circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape and to compose new shapes from the composite shape; this is important for the future development of spatial relations which later connects to developing understanding of area, volume and fractions
• partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of; describe the whole as two of, or four of the shares; understand for these examples that decomposing into more equal shares creates smaller shares

## E - Understanding Place Value

• add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used; understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten

## A - Operations and Algebraic Thinking

• use addition and subtraction within 100 to solve one- and two-step word problems by using drawings and equations with a symbol for the unknown numbers to represent the problem. Problems include contexts that involve adding to, taking from, putting together, taking apart (part/part/whole), and comparing with unknowns in all positions
• fluently add and subtract within 20 using mental By the end of grade 2, know from memory all sums of two one-digit numbers
• determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends)
• apply the use of repeated addition (skip counting), model arrays up to 5 rows and 5 columns to determine a total number of objects, and write an equation to express the total as a sum of equal addends

## B - Number and Operations in Base Ten

• explain that the three digits of a three-digit number represent amounts of hundreds, tens, and ones (e.g., 706 equals 7 hundreds, 0 tens, and 6 ones)
• explain that 100 can be thought of as a bundle of ten tens, called a "hundred"
• explain the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones)
• count within 1000; skip-count by 5s, 10s, and 100s
• read, write, and represent numbers to 1000 using a variety of models, diagrams and base ten numerals, including standard and expanded form
• compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, = ,and < symbols to record the results of comparisons
• add and subtract fluently within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction
• add up to four two-digit numbers using strategies based on place value and properties of operations
• add and subtract within 1,000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds
• use mental math strategies to add and subtract 10 or 100 to a given number between 100-900
• explain why addition and subtraction strategies work using place value and the properties of operations

## C - Measurement and Data

• measure length by determining, selecting and using an appropriate tool (rulers, yardsticks, meter sticks, measuring tapes) and unit (in., ft., yd., cm, m)
• compare and explain the relationship of inches, feet, yards, centimeters and meters by measuring an object twice using different units. Understand the relative size of units in different systems of measurement (e.g., an inch is longer than a centimeter; but students are not expected to convert between systems of measurement)
• estimate lengths using units of inches, feet, yards, centimeters and meters, then measure to determine if estimations were reasonable
• measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit (relate addition and subtraction to length)
• solve word problems using addition and subtraction within 100 involving lengths of like units by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem
• represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram
• use analog and digital clocks to tell and write time to the nearest five minutes using AM and PM
• solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¢ symbols appropriately (e.g., if you have 2 dimes and 3 pennies, how many cents do you have?)
• generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object; show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units
• draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories; solve simple put-together, take-apart, and compare problems using information presented in a bar graph

## D - Geometry

• recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces and identify triangles, quadrilaterals, pentagons, hexagons, and cubes
• partition a rectangle into rows and columns of same-size squares and count to find the total number of them
• partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, , and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape

## A - Operations and Algebraic Thinking

• use addition and subtraction within 100 to solve one- and two-step word problems by using drawings and equations with a symbol for the unknown numbers to represent the problem; problems include contexts that involve adding to, taking from, putting together, taking apart (part/part/whole), and comparing with unknowns in all positions
• apply the use of repeated addition (skip counting), model arrays up to 5 rows and 5 columns to determine a total number of objects, and write an equation to express the total as a sum of equal addends

## B - Number and Operations in Base Ten

• explain that the three digits of a three-digit number represent amounts of hundreds, tens, and ones
• read, write, and represent numbers to 1000 using a variety of models, diagrams and base ten numerals including standard and expanded form
• compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >,=,and < symbols to record the results of comparisons
• add and subtract within 1,000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method; understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones, and sometimes it is necessary to compose or decompose tens or hundreds

## C - Measurement and Data

• measure length by determining, selecting and using an appropriate tool (rulers, yardsticks, meter sticks, measuring tapes) and unit (in., ft., yd., cm, m)
• compare and explain the relationship of inches, feet, yards, centimeters and meters by measuring an object twice using different units; understand the relative size of units in different systems of measurement
• estimate lengths using units of inches, feet, yards, centimeters and meters, then measure to determine if estimations were reasonable
• solve word problems using addition and subtraction within 100 involving lengths of like units by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem
• use analog and digital clocks to tell and write time to the nearest five minutes using AM and PM
• solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¢ symbols appropriately (e.g., If you have 2 dimes and 3 pennies, how many cents do you have?)
• draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories; solve simple put-together, take-apart, and compare problems using information presented in a bar graph

## D - Geometry

• partition a rectangle into rows and columns of same-size squares and count to find the total number of them
• partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, , and describe the whole as two halves, three thirds, four fourths; recognize that equal shares of identical wholes need not have the same shape

## A - Operations and Algebraic Thinking

• interpret products of whole numbers, [e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each (e.g., describe a context in which a total number of objects can be expressed as 5 x 7)]
• interpret whole-number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares; how many are in each group?), or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each; how many groups can you make? (e.g., describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8)
• apply multiplication and division within 100 (products or dividends 0 - 100) to solve word problems in situations involving equal groups, arrays and measurement quantities (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem)
• determine the unknown whole number in a multiplication or division equation relating three whole numbers using the inverse relationship of multiplication and division (e.g., determine the unknown number that makes the equation true in each of the equations 8 x ? = 48; 5 = ? ÷ 3, 6 x 6 = ?)
• apply commutative, associative, and distributive properties as strategies to multiply and divide (e.g., If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known (commutative property of multiplication); 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30 (associative property of multiplication), knowing that 8 x 5 = 40 and 8 x 2 = 16, then one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 (distributive property))
• understand division as an unknown-factor problem (e.g., find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8)
• fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5= 8, or properties of operations) By the end of Grade 3, know from memory all products of two one-digit numbers
• solve two-step word problems using the four Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to perform operations in the conventional order where there are no parentheses to specify a particular order (order of operations)
• identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations (e.g., observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends)

## B - Number and Operations in Base Ten

• use place value understanding to round whole numbers to the nearest 10 or 100
• add and subtract fluently within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction
• multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations

## C - Number and Operations: Fractions

• understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a fraction a/b as the quantity formed by a parts of size 1/b (e.g., 3/4 means there are three 1/4 parts, so 3/4 = 1/4+1/4+1/4)
• recognize a fraction as a number on the number line; represent fractions on a number line diagram
• represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into "b" equal parts; recognize that each part has size 1/b. Recognize that a unit fraction 1/b is located 1/b whole unit from 0 on the number line
• represent a non-unit fraction a/b on a number line diagram by marking off "a" lengths 1/b (unit fractions) from 0 and recognize that the resulting interval has size a/b and that its endpoint locates the non-unit fraction a/b on the number line
• explain equivalence of fractions through reasoning with visual fraction Compare fractions by reasoning about their size
• recognize two fractions as equivalent (equal) if they are the same size or the same point on a number line
• recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and (e.g., 1/2 = 2/4, 4/6 = 2/3); explain why the fractions are equivalent by using a visual fraction model
• express whole numbers as fractions and recognize fractions that are equivalent to whole numbers (e.g., express 3 in the form 3 = 6/2 (3 wholes is equal to six halves); recognize that 3/1 = 3; locate 4/4 and 1 at the same point of a number line diagram)
• compare two fractions with the same numerator or the same denominator by reasoning about their size; recognize that comparisons are valid only when the two fractions refer to the same whole and record the results of comparisons with the symbols >, =, or <

## D - Measurement and Data

• tell and write time to the nearest minute and measure elapsed time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes (e.g., by representing the problem on a number line diagram, drawing a pictorial representation of a clock face, etc.)
• measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units (e.g., by using drawings, such as a beaker with a measurement scale, to represent the problem)
• draw a scaled picture graph and a scaled bar graph to represent a data set with several categories; solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs (e.g., draw a bar graph in which each square in the bar graph might represent 5 pets)
• generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units - whole numbers, halves, or quarters
• recognize area as an attribute of plane figures and understand concepts of area measurement
• use words, pictures and/or numbers to show that "unit square" is a square with a side length of 1 unit, has an area of one square unit, and can be used to measure area of plane figures
• demonstrate that a plane figure which can be covered without gaps or overlaps by "n" unit squares is said to have an area of "n" square units
• measure areas by counting unit squares (e.g., square cm, square m, square in, square ft, and improvised units)
• relate area to the operations of multiplication and addition
• find the area of a rectangle with whole-number side lengths by tiling it and show that the area is the same as would be found by multiplying the side lengths
• multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real-world and mathematical problems and represent whole- number products as rectangular areas in mathematical reasoning
• use tiling to show in a concrete case that the area of a rectangle with whole- number side lengths a and b + c is the sum of a x b and a x c; use area models to represent the distributive property in mathematical reasoning
• solve real-world and mathematical problems involving the perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeters and different areas or the same areas with different perimeters

## E - Geometry

• understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories
• partition shapes into parts with equal Express the area of each part as a unit fraction of the whole (e.g., partition a shape into four parts with equal area, and describe the area of each part as 1/4 of the area of the shape)

## A - Operations and Algebraic Thinking

• interpret whole-number quotients of whole numbers, or as a number of shares when objects are partitioned into equal shares of objects; how many groups can you make?
• apply multiplication and division within 100 (products or dividends 0 - 100) to solve word problems in situations involving equal groups, arrays and measurement quantities (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem)
• apply commutative, associative, and distributive properties as strategies to multiply and divide (commutative property of multiplication); 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30 (associative property of multiplication), knowing that 8 x 5 = 40 and 8 x 2 = 16, then one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 (distributive property))
• solve two-step word problems using the four operations; represent these problems using equations with a letter standing for the unknown quantity; assess the reasonableness of answers using mental computation and estimation strategies including rounding; this standard is limited to problems posed with whole numbers and having whole number answers; students should perform operations in appropriate order according to the context of the problem
• identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations (e.g., observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends)

## B - Number and Operations in Base Ten

• add and subtract fluently within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction

## C - Number and Operations: Fractions

• understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a fraction a/b as the quantity formed by a parts of size 1/b; for example, 3/4 means there are three 1/4 parts, so 3/4 = 1/4+1/4+1/4
• represent a non-unit fraction a/b on a number line diagram by marking off “a” lengths 1/b (unit fractions) from 0 and recognize that the resulting interval has size a/b and that its endpoint locates the non-unit fraction a/b on the number line
• explain equivalence of fractions through reasoning with visual fraction models; compare fractions by reasoning about their size
• recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and 8.; explain why the fractions are equivalent by using a visual fraction model
• express whole numbers as fractions and recognize fractions that are equivalent to whole numbers (e.g., express 3 in the form 3 = 6/2 (3 wholes is equal to six halves); recognize that 3/1 = 3; locate 4/4 and 1 at the same point of a number line diagram)

## D - Measurement and Data

• tell and write time to the nearest minute and measure elapsed time intervals in minutes; solve word problems involving addition and subtraction of time intervals in minutes (e.g., by representing the problem on a number line diagram, drawing a pictorial representation of a clock face. etc.)
• draw a scaled picture graph and a scaled bar graph to represent a data set with several categories; solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs (e.g., draw a bar graph in which each square in the bar graph might represent 5 pets)
• use tiling to show in a concrete case that the area of a rectangle with whole- number side lengths a and b + c is the sum of a x b and a x c; use area models to represent the distributive property in mathematical reasoning
• solve real-world and mathematical problems involving the perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeters and different areas or the same areas with different perimeters

## E - Geometry

• understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category ; recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories

## A - Operations and Algebraic Thinking

• explain that a multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity; interpret a multiplication equation as a comparison (e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5); represent verbal statements of multiplicative comparisons as multiplication equations
• solve multiplication and division word problems involving multiplicative comparison using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison
• solve multi-step word problems with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding
• find all factor pairs for a whole number in the range 1 - Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 - 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite
• generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Explain informally why the numbers will continue to alternate in this way (e.g., given the rule "ADD 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers)

## B - Number and Operations in Base Ten

• explain that in a multi-digit whole number, a digit in any one place represents ten times what it represents in the place to its right (e.g., recognize that 700 ÷ 70 = 10 by applying concepts of place value and division)
• read and write multi-digit whole numbers using base ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons
• use place value understanding to round whole numbers to any place using tools such as a number line and/or charts
• add and subtract multi-digit whole numbers fluently using the standard algorithm
• multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of Illustrate and explain multiplication calculations by using equations, rectangular arrays, and/or area models
• find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models

## C - Number and Operations: Fractions

• explain why two or more fractions are equivalent to a fraction (n x a/n x b) ex: 1/4 = (3 x 1)/(3 x 4) by using visual fraction models. Focus attention on how the number and size of the parts differ even though the fractions themselves are the same size; use this principle to recognize and generate equivalent fractions
• compare two fractions with different numerators and different denominators (e.g., by using virtual fraction models, by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2); recognize that comparisons are valid only when the two fractions refer to the same whole; record the results of comparisons with symbols >, =, or <
• recognize that a fraction a/b with a > 1 as a sum of unit fractions 1/b
• model and explain addition and subtraction of fractions as joining and separating parts referring to the same whole
• decompose a fraction, by using a visual fraction model, into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation and justify reasoning using visual fraction models (e.g., 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8; 8/8 = 7/8 + 1/8)
• add and subtract mixed numbers with like denominators (e.g., by replacing each mixed number with an equivalent fraction and/or by using properties of operations and the relationship between addition and subtraction)
• solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators by using visual fraction models and equations to represent the problem
• apply and extend previous understanding of multiplication to multiply a fraction by a whole number (e.g., by using a visual such as a number line or area model)
• recognize a fraction a/b as a multiple of 1/b (e.g., use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4= 5 x (1/4))
• understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number (e.g., use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5; (In general, n x (a/b) = (n x a)/b))
• solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. (e.g., if each person at a party will eat 3/8 of a pound of roast beef and there will be five people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?)
• express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100 (e.g., express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100)
• use decimal notation for fractions with denominators 10 or 100 (e.g., rewrite 0.62 as 62/100; describe a length as 62 meters; locate 0.62 on a number line diagram)
• compare two decimals to hundredths by reasoning about their Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by using a visual model)

## D - Measurement and Data

• know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Understand the relationship between gallons, cups, quarts, and Express larger units in terms of smaller units within the same measurement system. Record measurement equivalents in a two column table
• use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale
• apply the area and perimeter formulas for rectangles in real-world and mathematical problems (e.g., find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor)
• make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8) Solve problems involving addition and subtraction of fractions with common denominators by using information presented in line plots (e.g., from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection)

## E - Geometry

• recognize angles as geometric shapes that are formed wherever two rays share a common endpoint and understand concepts of angle measurement
• recognize that an angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle; an angle that turns through 1/360 of a circle is called a "one-degree angle", and can be used to measure angles
• recognize that an angle that turns through "n" one-degree angles is said to have an angle measure of "n" degrees
• measure and draw angles using tools such as a protractor or angle ruler
• recognize angle measure as additive. When an angle is decomposed into non- overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems (e.g., by using an equation with a symbol or letter for the unknown angle measure)
• recognize area as additive; find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying the technique to solve real-world problems
• draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines and identify these in two-dimensional figures
• classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles
• recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line- symmetric figures and draw lines of symmetry

## A - Operations and Algebraic Thinking

• solve multi-step word problems with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted; represent these problems using equations with a symbol or letter standing for the unknown quantity; assess the reasonableness of answers using mental computation and estimation strategies, including rounding
• find all factor pairs for a whole number in the range 1-100; recognize that a whole number is a multiple of each of its factors; determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number; determine whether a given whole number in the range 1-100 is prime or composite

## B - Number and Operations: Fractions

• compare two fractions with different numerators and different denominators (e.g., by using visual fraction models, by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2); recognize that comparisons are valid only when the two fractions refer to the same whole; record the results of comparisons with symbols >, =, or <, and justify the conclusions
• add and subtract mixed numbers with like denominators (e.g., by replacing each mixed number with an equivalent fraction and/or by using properties of operations and the relationship between addition and subtraction)
• solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators by using visual fraction models and equations to represent the problem
• understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number (e.g., use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5 (In general, n x (a/b) = (n x a)/b)
• solve word problems involving multiplication of a fraction by a whole number (e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?)

## C - Measurement and Data

• know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, ; l, ml; hr, min, sec; understand the relationship between gallons, cups, quarts and pints; express larger units in terms of smaller units within the same measurement system; record measurement equivalents in a two column table
• use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit; represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale
• make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8); solve problems involving addition and subtraction of fractions with common denominators by using information presented in line plots (e.g., from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection)
• recognize area as additive; find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying the technique to solve real world problems

## D - Geometry

• classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size; recognize right triangles as a category, and identify right triangles

## A - Operations and Algebraic Thinking

• use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols
• write simple expressions that record calculations with numbers and interpret numerical expressions without evaluating them (e.g., express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7)) and recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product)
• generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms by completing a function table or input/output Using the terms created, form, and graph ordered pairs on a coordinate plane

## B - Number and Operations in Base Ten

• recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left
• explain patterns in the number of zeros of the product when multiplying a number by powers of 10 and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10; use whole-number exponents to denote powers of 10
• read, write, order, and compare place value of decimals to thousandths using base ten numerals, number names, and expanded form (e.g., 392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000))
• compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons
• use place value understanding to round decimals to any place
• multiply multi-digit whole numbers fluently using the standard algorithm (or other strategies demonstrating understanding of multiplication) up to a 3 digit by a 2 digit factor
• find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations or concrete models (e.g., rectangular arrays and/or area models)
• add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used

## C - Number and Operations: Fractions

• add and subtract fractions and mixed numbers with unlike denominators by finding a common denominator and equivalent fractions to produce like denominators
• solve word problems involving addition and subtraction of fractions including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem); use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g., recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2)
• interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b); solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem (e.g., 3/5 can be interpreted as 3 divided by 5 and as 3 shared by 5)
• apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction
• apply and use the understanding of multiplication to multiply a fraction or whole number by a Examples: (a/b) x q as (a/b) x (q/1) and (a/b) x (c/d) = ac/bd
• find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths
• relate the principle of fraction equivalence, a/b = (n x a)/(n x b), to the effect of multiplying a/b by 1
• interpret multiplication as scaling by comparing the size of the product to the sizes of the factors without multiplying
• explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number and why multiplying a given number by a fraction less than 1 results in a product smaller than the given number
• solve real-world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem
• interpret division of a unit fraction by a non-zero whole number and compute such quotients (e.g., create a story context for (1/3) ÷ 4 and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3)
• apply and extend previous understanding of division to interpret the quotient of a whole number by a unit fraction and compute such quotients (e.g., create a story context for 4 ÷ (1/5) and use a visual fraction model to show the Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4)
• solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem, (e.g., how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3- cup servings are in 2 cups of raisins?)

## D - Measurement and Data

• convert among different-sized standard measurement units (mass, weight, length, time, etc.) within a given measurement system (customary and metric), and use these conversions in solving multi-step, real-world problems (e.g., convert 5 cm to 0.05 m)
• make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8) and solve problems using the line plot data (e.g., given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally)
• use words, pictures, or numbers to show a cubic unit is represented by a cube in which each edge has a length of one unit
• apply concepts of volume measurement to explain volume as an attribute of solid figures packed without gaps or overlaps using "n" unit cubes
• measure volume as cubic centimeters, cubic meters, cubic inches, cubic feet and improvised units
• relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume
• find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base and represent threefold number products as volumes; associative property
• estimate, derive and apply the formula(V= l x w x h and V= b x h) for the volume of a cube and a right rectangular prism using manipulatives and relate volume to the operations of multiplication and addition to solve real-world and mathematical problems
• recognize and calculate volume as additive when volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non- overlapping parts, applying this technique to solve real-world problems

## E - Geometry

• create, label, and use a coordinate grid system
• represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation
• demonstrate that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category (e.g., all rectangles have four right angles and squares are rectangles so all squares have four right angles)
• classify two-dimensional figures in a hierarchy based on properties (polygons, triangles, and quadrilaterals)

## A - Number and Operations in Base Ten

• find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division; illustrate and explain the calculation by using equations or concrete models (e.g., rectangular arrays and/or area models)
• add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used

## B - Number and Operations: Fractions

• find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths
• solve word problems involving addition and subtraction of fractions including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem); use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g., recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2)
• solve real world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem
• solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions e.g., by using visual fraction models and equations to represent the problem, (e.g., how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3- cup servings are in 2 cups of raisins?)

## C - Measurement and Data

• estimate, derive and apply the formula(V= l x w x h and V= b x h) for the volume of a cube and a right rectangular prism using manipulatives and relate volume to the operations of multiplication and addition to solve real world and mathematical problems
• recognize and calculate volume as additive when volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non- overlapping parts, applying this technique to solve real world problems

## D - Geometry

• represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation
• demonstrate that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category (e.g., all rectangles have four right angles and squares are rectangles so all squares have four right angles)
• classify two-dimensional figures in a hierarchy based on properties (polygons, triangles, and quadrilaterals)

• # 6th Grade Mathematics -- Accelerated Mathematics

## A - Ratio and Proportional Relationships

• explain the meaning of and use ratio language to describe a ratio relationship between two quantities (e.g., the ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak)
• explain the concept of a unit rate a/b associated with a ratio a:b with b not equal to 0, and use rate language in the context of a ratio relationship (e.g., this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar; we paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger)
• use ratio and proportional reasoning to solve real world and mathematical problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, and equations
• use ratio and proportional reasoning to make tables of equivalent ratios, find missing values in the tables, and plot the pairs of values on the coordinate plane
• solve unit rate problems including those involving unit pricing and constant speed (e.g., if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?)
• find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); given a percent, solve problems involving finding the whole given a part and the part given the whole
• given a conversion factor, use ratio reasoning to convert measurement units within one system of measurement and between two systems of measurements (i.e., customary and metric); manipulate and transform units appropriately when multiplying or dividing quantities (e.g., given 0.5 inch = 1.27 cm, how many centimeters are in 6.5 inches?)
• compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units (e.g., if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour)
• recognize and represent proportional relationships between quantities
• determine whether two quantities are in a proportional relationship (e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin)
• identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships
• represent proportional relationships by equations (e.g., if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn)
• explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate
• use proportional relationships to solve multistep ratio and percent problems (e.g., simple interest, tax, markups and markdowns, gratuities and commissions, and fees)

## B - The Number System

• interpret and compute quotients of fractions to solve word problems involving division of fractions by fractions, including reasoning strategies such as visual fraction models and equations to represent the problem [e.g., find the width of a rectangular strip of land with length 3/4 mile and area 1/2 square mile; create a story context for (2/3)/(3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3)/(3/4) = 8/9 because 3/4 of 8/9 is 2/3 (i.e., (a/b)/(c/d) = ad/bc)]
• fluently divide multi-digit numbers using the standard algorithm
• find the common multiples of two whole numbers less than or equal to 12 and the common factors of two whole numbers less than or equal to 100; find the greatest common factor of 2 whole numbers and use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor [e.g., express 36 + 8 as 4(9 + 2)]; apply the least common multiple of two whole numbers less than or equal to 12 to solve real world problems
• understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation
• recognize that a rational number is a point on the number line; extend number line diagrams from previous grades to represent rational numbers on the number line (i.e., positive and negative fractions, decimals, and whole numbers); extend previous understanding of plotting integer points in Quadrant I to plotting points in all four quadrants of the coordinate plane
• recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself (e.g., -(-3) = 3, and that 0 is its own opposite); show that a number and its opposite have a sum of 0 (i.e., zero pairs)
• understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes
• find and position integers and other rational numbers on a horizontal or vertical number line diagram (i.e., represent positive and negative fractions, decimals, and whole numbers on a number line); find and position pairs of integers and other rational numbers in all four quadrants of the coordinate plane [e.g., plot the following ordered pairs on a coordinate plane: (-2, 3), (1/2, -4), (-5.75, -1.25)]
• understand ordering and absolute value of rational numbers (e.g., order the following numbers from least to greatest 1.7, -2, -11/4, -0.5, 2/3)
• interpret statements of inequality as statements about the relative position of two numbers on a number line diagram (e.g., interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right)
• write, interpret, and explain statements of order for rational numbers in real-world contexts (e.g., write -3° C is warmer than - 7° C)
• recognize the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation (e.g., for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars)
• distinguish comparisons of absolute value from statements about order
• solve real world and mathematical problems by graphing points in all four quadrants of the coordinate plane; include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate
• apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram
• show that a number and its opposite have a sum of 0 (are additive inverses). Describe situations in which opposite quantities combine to make For example, your bank account balance is -\$25.00. You deposit \$25.00 into your account. The net balance is \$0.00.
• identify p + q as the number located a distance (|q|) from p, in the positive or negative direction depending on whether q is positive or negative; interpret sums of rational numbers by describing real-world contexts
• identify subtraction of rational numbers as adding the additive inverse, p - q = p + (- q); show that the distance between two rational numbers on the number line is the absolute value of their difference and apply this principle in real-world contexts
• apply properties of operations as strategies to add and subtract rational numbers
• apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers
• recognize that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for
• recognize that integers can be divided, provided that the divisor is not zero and every quotient of integers (with non-zero divisor) is a rational number (e.g., if p and q are integers, then -(p/q) = (-p)/q = p/(-q)); interpret quotients of rational numbers by describing real world contexts
• apply properties of operations as strategies to multiply and divide rational numbers
• convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats
• solve real-world and mathematical problems involving the four operations with rational numbers
• analyze and apply previous understanding of the operations of multi-digit decimals to fluently add, subtract, multiply, and divide non-negative rational numbers in real- world contexts

## C - Expressions and Equations

• write and evaluate numerical expressions involving whole-number exponents by applying order of operations
• read, write, evaluate, and compare expressions with variables
• write expressions that record operations with numbers and variables (e.g., express the calculation " Subtract y from 5" as 5 - y)
• identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity (e.g., describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms)
• evaluate expressions with specific non-negative rational numbers for the variable (i.e., fractions, decimals, and whole numbers); include expressions that arise from formulas in real-world problems; perform arithmetic operations, including those involving whole-number exponents, using order of operations when there are no parentheses (e.g., use the formulas V = s³ and A = 6s² to find the volume and surface area of a cube with sides of length s = 1/2)
• apply the properties of operations to generate equivalent expressions [e.g., apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y; apply the distributive property and combine like terms to the expression 2(x+4) + 3x - 7 to produce the equivalent expression 5x + 1]
• identify when two expressions are equivalent (e.g., when the two expressions name the same number regardless of which value is substituted into them)
• understand solving an equation or inequality as a process of answering a question (e.g., which values from a specified set, if any, make the equation or inequality true?); use substitution to determine whether a given number in a specified set makes an equation or inequality true
• use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set
• solve real world and mathematical problems by writing and solving equations of the form x + p = q, x - p = q, px = q and x/p = q for cases in which the solution of the equation results in a nonnegative rational number (i.e., positive fraction, decimal, or whole number)
• write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem; recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams
• use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable; analyze and interpret the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation (e.g., in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time)
• apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients
• understand that rewriting an expression in different forms in a problem context can clarify the problem and how the quantities in it are related (e.g., a + 0.05a means that "increase by 5" is the same as "multiply by 1.05")
• solve multi-step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals) by applying properties of operations as strategies to calculate with numbers, converting between forms as appropriate; and assessing the reasonableness of answers using mental computation and estimation strategies (e.g., If a woman making \$25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or \$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation)
• use variables to represent quantities in a real world or mathematical problem and construct simple equations and inequalities to solve problems by reasoning about the quantities
• solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers and solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution identifying the sequence of the operations used in each approach (e.g., the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?)
• solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, r are specific rational numbers; graph the solution set of the inequality and interpret it in the context of the problem(e.g., as a salesperson, you are paid \$50 per week plus \$3 per sale. This week you want your pay to be at least \$100. Write an inequality for the number of sales you need to make, and describe the solutions)
• solve real-world and mathematical problems by writing and solving equations of the form x+p=q and px=q in which p and q are rational numbers

## D - Geometry

• find the area of right triangles, other triangles, quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving multi-step real world and mathematical problems
• find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths (1/2 u), and show that the volume is the same as would be found by multiplying the edge lengths of the prism; apply the formulas V = (length) x (width) x (height) and V = (area of base) x (height) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real world and mathematical problems
• draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate; apply these techniques in the context of solving real-world and mathematical problems
• represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures; apply these techniques in the context of solving real-world and mathematical problems
• solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale

## E - Statistics and Probability

• recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers (e.g., "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages)
• describe and analyze a set of data collected to answer a statistical question based on measures of central tendency, identifying the center, spread, and overall shape of the distribution
• contrast a measure of center with a measure of variation for a numerical set; determine and explain the most appropriate measure of center and measure of variability, based on the shape of a given data and the context of a problem
• display, read, and analyze data using appropriate graphs, including box-and-whisker plots, scatter plots, histograms, and line plots
• summarize numerical data sets in relation to their context such as by reporting the number of observations; describing the nature of the attribute under investigation, including how it was measured and its units of measurement; giving quantitative measures of center (median and/or mean) and variability (interquartile range) as well as relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data was gathered

• # 6th Grade Mathematics -- Mathematics Connection

## A - Ratio and Proportional Relationships

• explain the meaning of and use ratio language to describe a ratio relationship between two quantities
• explain the concept of a unit rate a/b associated with a ratio a:b with b not equal to 0, and use rate language in the context of a ratio relationship
• use ratio and rate reasoning to solve real world and mathematical problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, or equations
• find a percent of a quantity as a rate per 100; given a percent, solve problems involving finding the whole given a part and the part given the whole

## B - The Number System

• find the common multiples of two whole numbers less than or equal to 12 and the common factors of two whole numbers less than or equal to 100; find the greatest common factor of two whole numbers and use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor; apply the least common multiple of the two whole numbers less than or equal to 12 to solve real world problems
• interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, including reasoning strategies such as visual fraction models and equations to represent the problem
• understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation
• interpret statements of inequality as statements about the relative position of two numbers on a number line diagram
• recognize the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation
• solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane; include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate

## C - Expressions and Equations

• evaluate expressions at specific values for their variables (include expressions that arise from formulas in real-world problems); perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order
• apply the properties of operations to generate equivalent expressions
• use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set
• solve real-world and mathematical problems by writing and solving one-step linear equations, in the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers
• write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem; recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams
• use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable; analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation

## D - Statistics and Probability

• contrast a measure of center with a measure of variation for a numerical set
• display, read, and analyze data using appropriate graphs, including box-and-whisker plots, scatter plots, histograms, and line plots
• summarize numerical data sets in relation to their context, such as by reporting the number of observations; describing the nature of the attribute under investigation, including how it was measured and its units of measurement; giving quantitative measures of center (median and/or mean) and variability (interquartile range) as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data was gathered; relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

## A - Ratio and Proportional Relationships

• explain the meaning of and use ratio language to describe a ratio relationship between two quantities (e.g., the ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak)
• explain the concept of a unit rate a/b associated with a ratio a:b with b not equal to 0, and use rate language in the context of a ratio relationship (e.g., this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar; we paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger)
• use ratio and proportional reasoning to solve real world and mathematical problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, and equations
• use ratio and proportional reasoning to make tables of equivalent ratios, find missing values in the tables, and plot the pairs of values on the coordinate plane
• solve unit rate problems including those involving unit pricing and constant speed (e.g., if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?)
• find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); given a percent, solve problems involving finding the whole given a part and the part given the whole
• given a conversion factor, use ratio reasoning to convert measurement units within one system of measurement and between two systems of measurements (i.e., customary and metric); manipulate and transform units appropriately when multiplying or dividing quantities (e.g., given 0.5 inch = 1.27 cm, how many centimeters are in 6.5 inches?)

## B - The Number System

• interpret and compute quotients of fractions to solve word problems involving division of fractions by fractions, including reasoning strategies such as visual fraction models and equations to represent the problem [e.g., find the width of a rectangular strip of land with length 3/4 mile and area 1/2 square mile; create a story context for (2/3)/(3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3)/(3/4) = 8/9 because 3/4 of 8/9 is 2/3 (i.e., (a/b)/(c/d) = ad/bc)]
• fluently divide multi-digit numbers using the standard algorithm
• find the common multiples of two whole numbers less than or equal to 12 and the common factors of two whole numbers less than or equal to 100; find the greatest common factor of 2 whole numbers and use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor [e.g., express 36 + 8 as 4(9 + 2)]; apply the least common multiple of two whole numbers less than or equal to 12 to solve real world problems
• understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation
• recognize that a rational number is a point on the number line; extend number line diagrams from previous grades to represent rational numbers on the number line (i.e., positive and negative fractions, decimals, and whole numbers); extend previous understanding of plotting integer points in Quadrant I to plotting points in all four quadrants of the coordinate plane
• recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself (e.g., -(-3) = 3, and that 0 is its own opposite); show that a number and its opposite have a sum of 0 (i.e., zero pairs)
• understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes
• find and position integers and other rational numbers on a horizontal or vertical number line diagram (i.e., represent positive and negative fractions, decimals, and whole numbers on a number line); find and position pairs of integers and other rational numbers in all four quadrants of the coordinate plane [e.g., plot the following ordered pairs on a coordinate plane: (-2, 3), (1/2, -4), (-5.75, -1.25)]
• understand ordering and absolute value of rational numbers (e.g., order the following numbers from least to greatest 1.7, -2, -11/4, -0.5, 2/3)
• interpret statements of inequality as statements about the relative position of two numbers on a number line diagram (e.g., interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right)
• write, interpret, and explain statements of order for rational numbers in real-world contexts (e.g., write -3° C is warmer than - 7° C)
• recognize the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation (e.g., for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars)
• distinguish comparisons of absolute value from statements about order
• solve real world and mathematical problems by graphing points in all four quadrants of the coordinate plane; include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate
• analyze and apply previous understanding of the operations of multi-digit decimals to fluently add, subtract, multiply, and divide non-negative rational numbers in real- world contexts

## C - Expressions and Equations

• write and evaluate numerical expressions involving whole-number exponents by applying order of operations
• read, write, evaluate, and compare expressions with variables
• write expressions that record operations with numbers and variables (e.g., express the calculation " Subtract y from 5" as 5 - y)
• identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity (e.g., describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms)
• evaluate expressions with specific non-negative rational numbers for the variable (i.e., fractions, decimals, and whole numbers); include expressions that arise from formulas in real-world problems; perform arithmetic operations, including those involving whole-number exponents, using order of operations when there are no parentheses (e.g., use the formulas V = s³ and A = 6s² to find the volume and surface area of a cube with sides of length s = 1/2)
• apply the properties of operations to generate equivalent expressions [e.g., apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y; apply the distributive property and combine like terms to the expression 2(x+4) + 3x - 7 to produce the equivalent expression 5x + 1]
• identify when two expressions are equivalent (e.g., when the two expressions name the same number regardless of which value is substituted into them)
• understand solving an equation or inequality as a process of answering a question (e.g., which values from a specified set, if any, make the equation or inequality true?); use substitution to determine whether a given number in a specified set makes an equation or inequality true
• use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set
• solve real world and mathematical problems by writing and solving equations of the form x + p = q, x - p = q, px = q and x/p = q for cases in which the solution of the equation results in a nonnegative rational number (i.e., positive fraction, decimal, or whole number)
• write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem; recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams
• use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable; analyze and interpret the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation (e.g., in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time)

## D - Geometry

• find the area of right triangles, other triangles, quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving multi-step real world and mathematical problems
• find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths (1/2 u), and show that the volume is the same as would be found by multiplying the edge lengths of the prism; apply the formulas V = (length) x (width) x (height) and V = (area of base) x (height) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real world and mathematical problems
• draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate; apply these techniques in the context of solving real-world and mathematical problems
• represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures; apply these techniques in the context of solving real-world and mathematical problems

## E - Statistics and Probability

• recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers (e.g., "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages)
• describe and analyze a set of data collected to answer a statistical question based on measures of central tendency, identifying the center, spread, and overall shape of the distribution
• contrast a measure of center with a measure of variation for a numerical set; determine and explain the most appropriate measure of center and measure of variability, based on the shape of a given data and the context of a problem
• display, read, and analyze data using appropriate graphs, including box-and-whisker plots, scatter plots, histograms, and line plots
• summarize numerical data sets in relation to their context such as by reporting the number of observations; describing the nature of the attribute under investigation, including how it was measured and its units of measurement; giving quantitative measures of center (median and/or mean) and variability (interquartile range) as well as relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data was gathered

## A - The Number System

• know that numbers that are not rational are called Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number
• use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions

## B - Expressions and Equations

• apply and know the properties of integer exponents to generate equivalent numerical expressions
• use square root and cube root symbols to represent solutions to equations. Recognize that x² = p (where p is a positive rational number) has two solutions and x³ = p (where p is a negative or positive rational number) has one Evaluate square roots of perfect squares up to and including 625 and cube roots of perfect cubes up to and including 1000 and -1000.
• use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other
• add, subtract, multiply, and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are Interpret and use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading) (Interpret scientific notation that has been generated by technology; e.g., calculators)
• graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways (e.g., compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed)
• determine the meaning of slope by using similar right triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate Derive and graph linear equations in slope intercept form y = mx+ b
• derive and graph linear equations in slope intercept form y = mx + b
• give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers)
• solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms
• analyze and solve pairs of simultaneous linear equations
• understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously
• solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations; solve simple cases by inspection (e.g., 3x + 2y = 5 and 3x +2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• solve real world mathematical problems leading to two linear equations in two variables (e.g., given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair)

## C - Functions

• understand that a function is a rule that assigns to each input exactly one The graph of a function is the set of ordered pairs consisting of an input and the corresponding output
• describe functions in a variety of representations, including the graph of a function that is the set of ordered pairs consisting of an input and the corresponding output
• compare properties of two functions each represented among verbal, tabular, graphic and algebraic representations of functions
• interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear (e.g., the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line)
• construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values
• describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally
• simplify, add, subtract, multiply, and divide radical expressions to include rationalizing denominators

## D - Geometry

• explore various geometric shapes with given focus on creating triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle
• describe the two-dimensional figures (cross sections) that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms, right rectangular pyramids, cones, and spheres
• given the formulas for the area and circumference of a circle, use them to solve problems; give an informal derivation of the relationship between the circumference and the area of a circle.
• write and solve equations for an unknown angle in a figure using facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem
• solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms
• verify experimentally the congruence properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length Angles are taken to angles of the same measure c. Parallel lines are taken to parallel lines
• understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations; given two congruent figures, describe a sequence that exhibits the congruence between them
• describe the effect of dilations, translations, rotations and reflections on two- dimensional figures using coordinates
• understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them
• use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of (e.g., arrange three copies of the same triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so)
• explain a proof of the Pythagorean Theorem and its converse
• apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions
• explain and apply the distance formula as an application of the Pythagorean Theorem
• solve real world and mathematical problems involving the volume of cylinders, cones and spheres

## E - Statistics and Probability

• understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that Understand that random sampling tends to produce representative samples and support valid inferences
• generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or Draw inferences from a random sample about a population with an unknown characteristic of interest. Compare and contrast multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions (e.g., estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be)
• compare and contrast the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the medians by expressing it as a multiple of the interquartile range
• use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations (e.g., decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book)
• understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event
• approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative predict the approximate relative frequency given the probability (e.g., when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times)
• develop a probability model and use it to find probabilities of Compare experimental and theoretical probabilities of events. if the probabilities are not close, explain possible sources of the discrepancy
• develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events (e.g., if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected)
• develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process (e.g., find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?)
• determine the probability of compound simple events using organized lists, tables, tree diagrams, and simulation
• explain that a compound event is the fraction of outcomes in the sample space for which the compound event occurs
• represent sample spaces using tree diagrams, lists, simulations, and tables to identify the outcomes in the sample space which compose the event; for an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event
• design and use simulation to generate frequencies for compound events
• construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association
• know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line
• apply the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting slope and intercept (e.g., in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height)
• recognize that patterns of association can be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same Use relative frequencies calculated for rows or columns to describe possible association between the two variables (e.g., collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?)

• # 7th Grade Mathematics -- 7th Mathematics Connection

## A - Ratio and Proportional Relationships

• identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships
• represent proportional relationships by equations
• explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate
• use proportional relationships to solve multi-step ratio and percent problems

## B - The Number System

• identify p + q as the number located a distance (|q|) from p, in the positive or negative direction depending on whether q is positive or negative; interpret sums of rational numbers by describing real-world contexts
• identify subtraction of rational numbers as adding the additive inverse, p - q = p + (- q); show that the distance between two rational numbers on the number line is the absolute value of their difference and apply this principle in real-world contexts
• recognize that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers; interpret products of rational numbers by describing real-world contexts
• recognize that integers can be divided, provided that the divisor is not zero and every quotient of integers (with a non-zero divisor) is a rational number (if p and q are integers, then -(p/q) = (-p)/q = p/(-q)); interpret quotients of rational numbers by describing real world contexts
• solve real-world and mathematical problems involving the four operations with rational numbers

## C - Expressions and Equations

• solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals) by applying properties of operations as strategies to calculate with numbers, converting between forms as appropriate and assessing the reasonableness of answers using mental computation and estimation strategies
• use variables to represent quantities in a real world or mathematical problem and construct simple equations to solve problems by reasoning about the quantities
• solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px=q in which p and q are rational numbers
• solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, r are specific rational numbers; graph the solution set of the inequality and interpret it in the context of the problem

## D - Geometry

• solve problems involving scale drawings of geometric figure, including computing actual lengths and areas from scale drawing and reproducing a scale drawing at a different scale

## E - Statistics and Probability

• develop a probability model and use it to find probabilities of events; compare experimental and theoretical probabilities of events; if the probabilities are not close, explain possible sources of the discrepancy
• develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of event
• develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process
• represent sample spaces using tree diagrams, lists, simulations, and tables to identify the outcomes in the sample space which compose the event; for an event described in everyday language, identify the outcomes in the sample space which compose the event
• design, use and explain ways to set up a simulation to generate frequencies for compound events
• generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions; draw inferences from a random sample about a population with an unknown characteristic of interest; compare and contrast multiple samples of the same size to gauge the variation in estimates or predictions
• compare and contrast the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the medians by expressing it as a multiple of the interquartile range

## A - Ratio and Proportional Relationships

• compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units (i.e., if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour)
• recognize and represent proportional relationships between quantities
• determine whether two quantities are in a proportional relationship (i.e., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin)
• identify the constant of proportionality (i.e., unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships
• represent proportional relationships by equations (i.e., if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn)
• explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate
• use proportional relationships to solve multi-step ratio and percent problems (i.e., simple interest, tax, markups and markdowns, gratuities and commissions, and fees)
• solve problems involving scale drawings of geometric figure, including computing actual lengths and areas from scale drawing and reproducing a scale drawing at a different scale

## B - The Number System

• apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram
• show that a number and its opposite have a sum of 0 (are additive inverses); describe situations in which opposite quantities combine to make 0 (i.e., your bank account balance is -\$25.00; you deposit \$25.00 into your account; the net balance is \$0.00)
• identify p + q as the number located a distance (|q|) from p, in the positive or negative direction depending on whether q is positive or negative; interpret sums of rational numbers by describing real-world contexts
• identify subtraction of rational numbers as adding the additive inverse, p - q = p + (- q); show that the distance between two rational numbers on the number line is the absolute value of their difference and apply this principle in real-world contexts
• apply properties of operations as strategies to add and subtract rational numbers
• apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers
• recognize that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers; interpret products of rational numbers by describing real-world contexts
• recognize that integers can be divided, provided that the divisor is not zero and every quotient of integers (with a non-zero divisor) is a rational number (if p and q are integers, then -(p/q) = (-p)/q = p/(-q)); interpret quotients of rational numbers by describing real world contexts
• apply properties of operations as strategies to multiply and divide rational numbers
• convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats
• solve real-world and mathematical problems involving the four operations with rational numbers
• compare and contrast rational and irrational numbers; understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number
• use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line and estimate the value of expressions

## C - Expressions and Equations

• apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients
• understand that rewriting an expression in different forms in a problem context can clarify the problem and how the quantities in it are related (i.e., a + 0.05a means that "increase by 5%" is the same as "multiply by 1.05")
• solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals) by applying properties of operations as strategies to calculate with numbers, converting between forms as appropriate and assessing the reasonableness of answers using mental computation and estimation strategies (i.e., if a woman making \$25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or \$2.50, for a new salary of \$27.50; if you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation)
• use variables to represent quantities in a real world or mathematical problem and construct simple equations to solve problems by reasoning about the quantities
• solve word problems leading to equations of the form px + q = r and p(x+q) = r, where p, q, and r are specific rational numbers and solve equations of these forms fluently; compare an algebraic solution to an arithmetic solution identifying the sequence of the operations used in each approach (i.e., the perimeter of a rectangle is 54 cm; its length is 6 cm; what is its width?)
• solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, r are specific rational numbers; graph the solution set of the inequality and interpret it in the context of the problem (i.e., as a salesperson, you are paid \$50 per week plus \$3 per sale; this week you want your pay to be at least \$100; write an inequality for the number of sales you need to make, and describe the solutions)
• solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px=q in which p and q are rational numbers
• solve linear equations both algebraically and graphically, including examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions (e.g., equivalent equations of the form x = a, a = a, or a = b results where a and b are different numbers)
• solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms
• graph proportional relationships, interpreting the unit rate as the slope of the graph; compare two different proportional relationships represented in graphs, tables, equations, and verbal descriptions (e.g., compare a distance-time graph to a distance-time equation to determine which of the two moving objects has greater speed)
• apply and know the properties of integer exponents to generate equivalent numerical expressions
• use square root and cube root symbols to represent solutions to equations; recognize that x² = p (where p is a positive rational number) has two solutions and x³ = p (where p is a negative or positive rational number) has one solution; evaluate square roots of perfect squares up to an including 625 and cube roots of perfect cubes up to and included 1000 and -1000

## D - Geometry

• explore various geometric shapes with given conditions; focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle
• describe the two-dimensional figures (cross sections) that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms, right rectangular pyramids, cones and spheres
• given the formulas for the area and circumference of a circle, use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle
• write and solve equations for an unknown angle in a figure using facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem
• solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms

## E - Statistics and Probability

• understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population; understand that random sampling tends to produce representative samples and support valid inferences
• generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions; draw inferences from a random sample about a population with an unknown characteristic of interest; compare and contrast multiple samples of the same size to gauge the variation in estimates or predictions (i.e., estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data; gauge how far off the estimate or prediction might be)
• compare and contrast the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the medians by expressing it as a multiple of the interquartile range
• use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations (i.e., decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book)
• understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring; larger numbers indicate greater likelihood; a probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event
• approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency; predict the approximate relative frequency given the probability (i.e., when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times)
• develop a probability model and use it to find probabilities of events; compare experimental and theoretical probabilities of events; if the probabilities are not close, explain possible sources of the discrepancy
• develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events (i.e., if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected)
• develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process (i.e., find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down; do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?)
• determine the probability of compound simple events using organized lists, tables, tree diagrams, and simulation
• explain that a compound event is the fraction of outcomes in the sample space for which the compound event occurs
• represent sample spaces using tree diagrams, lists, simulations, and tables to identify the outcomes in the sample space which compose the event; for an event described in everyday language (i.e., "rolling double sixes"), identify the outcomes in the sample space which compose the event
• design, use and explain ways to set up a simulation to generate frequencies for compound events (i.e., if 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?)

## A - The Number System

• know that numbers that are not rational are called irrational; understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number
• use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions
• add, subtract, and multiply polynomial expressions
• explain why the sum or product of rational numbers is rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational

## B - Expressions and Equations

• apply and know the properties of integer exponents to generate equivalent numerical expressions
• use square root and cube root symbols to represent solutions to equations; recognize that x² = p (where p is a positive rational number) has two solutions and x³ = p (where p is a negative or positive rational number) has one solution; evaluate square roots of perfect squares up to and including 625 and cube roots of perfect cubes up to and including 1000 and -1000
• express and use numbers in scientific notation to estimate very large or very small quantities; express how many times one is compared to the other
• add, subtract, multiply, and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are used; interpret and use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading); interpret scientific notation that has been generated by technology (e.g., calculators)
• graph proportional relationships, interpreting the unit rate as the slope of the graph; compare two different proportional relationships represented in different ways (e.g., compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed)
• determine the meaning of slope by using similar right triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive and graph linear equations in slope intercept form y = mx + b
• solve linear equations both algebraically and graphically, including examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions (e.g., equivalent equation of the form x = a, a = a, or a = b results where a and b are different numbers)
• solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms
• solve systems of equations algebraically and estimate solutions by graphing the equations
• understand the point of intersection for a system of linear equations satisfies both equations simultaneously and represents the solution to the system of linear equations
• solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations; solve simple cases by inspection (e.g., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• solve real world mathematical problems leading to two linear equations in two variables (e.g., given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair)
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (e.g., rearrange Ohm's law V=IR to highlight resistance R)
• show and explain why the elimination method works to solve a system of two- variable equations

## C - Functions

• understand that a function is a rule that assigns to each input exactly one output; the graph of a function is the set of ordered pairs consisting of an input and the corresponding output
• compare properties of two functions each represented among verbal, tabular, graphic, and algebraic representations of functions (e.g., given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change)
• interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear (e.g., the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line)
• construct a function to model a linear relationship between two quantities; determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph; interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values
• describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear); sketch a graph that exhibits the qualitative features of a function that has been described verbally
• simplify, add, subtract, multiply, and divide radical expressions to include rationalizing denominators
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function); represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 < x = 10})
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts as determined by the function or by context
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima as determined by the function or by context
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• factor any quadratic expression to reveal the zeros of the function it defines, utilizing the zero product property
• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (e.g., compare and contrast quadratic functions in standard, vertex, and intercept forms)

## D - Geometry

• verify experimentally the congruence properties of rotations, reflections, and translations (i.e., lines are taken to lines, and line segments to line segments of the same length, angles are taken to angles of the same measure, parallel lines are taken to parallel lines)
• understand and explain that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations; given two congruent figures, describe a sequence that exhibits the congruence between them
• describe the effect of dilations, translations, rotations and reflections on two- dimensional figures using coordinates
• recognize that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations and dilations; describe a sequence of transformations, that when given, proves similarity between two figures
• use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles (e.g., arrange three copies of the same triangle so that the three angles appear to form a line, and give an argument in terms of transversals why this is so)
• explain a proof of the Pythagorean Theorem and its converse
• apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions
• apply the Pythagorean Theorem to find the distance between two points in a coordinate system
• solve real world and mathematical problems involving the volume of cylinders, cones, and spheres

## E - Statistics and Probability

• construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities; describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association
• know that straight lines are widely used to model relationships between two quantitative variables; for scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line
• apply the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting slope and intercept (e.g., in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height)
• recognize that patterns of association can be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table; construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects; use relative frequencies calculated for rows or columns to describe possible association between the two variables (e.g., collect data from your students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home; is there evidence that those who have a curfew also tend to have chores?)

• # 8th Grade Mathematics -- Mathematics Connection

## A - Algebra

• add, subtract, and multiply polynomials
• interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors
• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solution; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 < x = 10})
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• graph the solutions to a linear inequality in two variables as a half plane, excluding the boundary in the case of a strict inequality
• graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
• represent constraints by equations or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• represent constraints by systems of equations and/or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• solve systems of linear equations exactly (i.e., algebraically) and approximately (i.e., with graphs), focusing on pairs of linear equations in two variables; solve simple cases by inspection (i.e., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• show and explain why the elimination method works to solve a system of two- variable equations
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (i.e., reveal the zeros, minimum, or maximum)
• factor any quadratic expression to reveal the zeros of the function it defines
• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)
• create quadratic equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation; graph these quadratic equations on coordinate axes with appropriate labels and scales
• create exponential equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, graph these exponential equations on coordinate axes with appropriate labels and scales

## B - Statistics and Probability

• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (i.e., outliers)
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (i.e., including joint, marginal, and conditional relative frequencies); recognize possible associations and trends in the data
• represent data on two quantitative variables on a scatter plot and describe how the variables are related
• fit a function to bivariate data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear, quadratic, and exponential models

## C - Functions

• understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)
• interpret key features of linear functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing); sketch graphs showing these key features when given a verbal description of the relationship
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 < x = 10})
• write arithmetic sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (i.e., compare and contrast quadratic functions in standard, vertex, and intercept forms)
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; asymptote; and end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• write geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect geometric sequences to exponential functions
• prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals; this can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
• graph linear, quadratic, and exponential functions algebraically and show key features of the graph by hand and by using technology
• interpret key features of linear, quadratic, and exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function

## D - Number and Quantity

• rewrite expressions involving radicals (i.e., simplify and/or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)

• # 8th Grade Mathematics -- Algebra I

## A - Algebra

• add, subtract, and multiply polynomials
• interpret parts of an expression, such as terms, factors, and coefficients in context
• interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors
• create linear equations and inequalities in one variable and use them to solve problems
• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solutions; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., {x | 2 &lt; x =10})
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (i.e., rearrange Ohm’s law V=IR to highlight resistance R)
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• demonstrate that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane
• graph the solutions to a linear inequality in two variables as a half plane, excluding the boundary in the case of a strict inequality
• graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
• represent constraints by equations or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• represent constraints by systems of equations and/or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• solve systems of linear equations exactly (i.e., algebraically) and approximately (i.e., with graphs), focusing on pairs of linear equations in two variables; solve simple cases by inspection (i.e., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• show and explain why the elimination method works to solve a system of two- variable equations
• explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately (i.e., using technology to graph the functions, make tables of values, or find successive approximations)
• use the structure of an expression to rewrite it in different equivalent forms [i.e., see x4 - y4 as ((x²) -(y²))², thus recognizing it as a difference of squares that can be factored as (x²-y²)(x²+y²)]
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (i.e., reveal the zeros, minimum, or maximum)
• factor any quadratic expression to reveal the zeros of the function it defines
• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from ax² + bx + c = 0
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)
• create quadratic equations in one variable and use them to solve problems
• create quadratic equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation; graph these quadratic equations on coordinate axes with appropriate labels and scales
• create exponential equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, graph these exponential equations on coordinate axes with appropriate labels and scales
• create exponential equations in one variable and use them to solve simple equations

## B - Statistics and Probability

• represent data with plots on the real number line (i.e., dot plots, histograms, and box plots)
• use statistics appropriate to the shape of the data distribution to compare center (i.e., median, mean) and spread (i.e., interquartile range, mean absolute deviation) of two or more different data sets
• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (i.e., outliers)
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (i.e., including joint, marginal, and conditional relative frequencies); recognize possible associations and trends in the data
• represent data on two quantitative variables on a scatter plot and describe how the variables are related
• fit a function to bivariate data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear, quadratic, and exponential models
• determine and interpret the slope (i.e., rate of change) and the intercept (i.e., constant term) of a linear model in the context of the data
• compute (using technology) and interpret the correlation coefficient of a linear fit (i.e., by looking at a scatter plot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the “r” value after calculating the line of best fit using technology, describe how strong the goodness of fit of the regression is using ”r”)
• explain the difference between correlation and causation

## C - Functions

• understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)
• interpret key features of linear functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing); sketch graphs showing these key features when given a verbal description of the relationship
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 &lt; x =10})
• write a function that describes a linear relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
• write arithmetic sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions
• recognize that arithmetic sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; experiment with cases and illustrate an explanation of the effects on the graph using technology; include recognizing even and odd functions from their graphs and algebraic expressions
• use second differences to write a quadratic function that describes a relationship between two quantities
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (i.e., compare and contrast quadratic functions in standard, vertex, and intercept forms)
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; asymptote; and end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• recognize that geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
• write a function that describes an exponential relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
• write geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect geometric sequences to exponential functions
• prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals; this can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another (i.e., linear)
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another (i.e., exponential)
• construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, table, a description of a relationship, or two input-output pairs
• interpret the parameters in a linear (i.e., f(x) = mx + b) or exponential function (i.e., f (x)=a·dx) in terms of a context (i.e., in the provided functions, “m” and “b” are the parameters of the linear function, and “a” and “d” are the parameters of the exponential function); in context, students should describe what these parameters mean in terms of change and starting value
• graph linear, quadratic, and exponential functions algebraically and show key features of the graph by hand and by using technology
• interpret key features of linear, quadratic, and exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function

## D - Number and Quantity

• rewrite expressions involving radicals (i.e., simplify and/or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)
• explain why the sum or product of rational numbers is rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational
• use units of measure (linear, area, capacity, rates, and time) as a way to understand problems; identify, use, and record appropriate units of measure within context, within data displays, and on graphs; convert units and rates using dimensional analysis (English to English and Metric to Metric without conversion factor provided and between English and Metric with conversion factor); use units within multi-step problems and formulas; interpret units of input and resulting units of output
• define appropriate quantities for the purpose of descriptive modeling; given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation
• choose a level of accuracy appropriate to limitations on measurement when reporting quantities (e.g., money situations are generally reported to the nearest hundredth; also, an answers’ precision is limited to the precision of the data given)

• # 8th Grade Mathematics -- Algebra I Strategies

## A - Algebra

• add, subtract, and multiply polynomials
• interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors
• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solution; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 &lt; x =10})
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• graph the solutions to a linear inequality in two variables as a half plane, excluding the boundary in the case of a strict inequality
• graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
• represent constraints by equations or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• represent constraints by systems of equations and/or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• solve systems of linear equations exactly (i.e., algebraically) and approximately (i.e., with graphs), focusing on pairs of linear equations in two variables; solve simple cases by inspection (i.e., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• show and explain why the elimination method works to solve a system of two- variable equations
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (i.e., reveal the zeros, minimum, or maximum)
• factor any quadratic expression to reveal the zeros of the function it defines
• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)
• create quadratic equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation; graph these quadratic equations on coordinate axes with appropriate labels and scales
• create exponential equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, graph these exponential equations on coordinate axes with appropriate labels and scales

## B - Statistics and Probability

• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (i.e., outliers)
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (i.e., including joint, marginal, and conditional relative frequencies); recognize possible associations and trends in the data
• represent data on two quantitative variables on a scatter plot and describe how the variables are related
• fit a function to bivariate data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear, quadratic, and exponential models

## C - Functions

• understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)
• interpret key features of linear functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing); sketch graphs showing these key features when given a verbal description of the relationship
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 &lt; x =10})
• write arithmetic sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (i.e., compare and contrast quadratic functions in standard, vertex, and intercept forms)
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; asymptote; and end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• write geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect geometric sequences to exponential functions
• prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals; this can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
• graph linear, quadratic, and exponential functions algebraically and show key features of the graph by hand and by using technology
• interpret key features of linear, quadratic, and exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function

## D - Number and Quantity

• rewrite expressions involving radicals (i.e., simplify and/or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)

• # High School Mathematics -- Algebra I

## A - Algebra

• represent constraints by systems of equations and/or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solutions; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., {x | 2 &lt; x =10})
• create quadratic equations in one variable and use them to solve problems
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• show and explain why the elimination method works to solve a system of two-variable equations
• create exponential equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, graph these exponential equations on coordinate axes with appropriate labels and scales
• factor any quadratic expression to reveal the zeros of the function it defines
• graph the solutions to a linear inequality in two variables as a half plane, excluding the boundary in the case of a strict inequality
• create quadratic equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation; graph these quadratic equations on coordinate axes with appropriate labels and scales
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)
• represent constraints by equations or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• solve systems of linear equations exactly (i.e., algebraically) and approximately (i.e., with graphs), focusing on pairs of linear equations in two variables; solve simple cases by inspection (i.e., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• add, subtract, and multiply polynomials
• interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors
• create exponential equations in one variable and use them to solve simple equations
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (i.e., rearrange Ohm’s law V=IR to highlight resistance R)
• explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately (i.e., using technology to graph the functions, make tables of values, or find successive approximations)
• create linear equations and inequalities in one variable and use them to solve problems
• demonstrate that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane
• use the structure of an expression to rewrite it in different equivalent forms [i.e., see x4 - y4 as ((x²) -(y²))², thus recognizing it as a difference of squares that can be factored as (x²-y²)(x²+y²)]
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (i.e., reveal the zeros, minimum, or maximum)
• interpret parts of an expression, such as terms, factors, and coefficients in context
• use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from ax² + bx + c = 0

## B - Statistics and Probability

• explain the difference between correlation and causation
• compute (using technology) and interpret the correlation coefficient of a linear fit (i.e., by looking at a scatter plot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the “r” value after calculating the line of best fit using technology, describe how strong the goodness of fit of the regression is using ”r”)
• use statistics appropriate to the shape of the data distribution to compare center (i.e., median, mean) and spread (i.e., interquartile range, mean absolute deviation) of two or more different data sets
• represent data on two quantitative variables on a scatter plot and describe how the variables are related
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (i.e., including joint, marginal, and conditional relative frequencies); recognize possible associations and trends in the data
• represent data with plots on the real number line (i.e., dot plots, histograms, and box plots)
• fit a function to bivariate data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear, quadratic, and exponential models
• determine and interpret the slope (i.e., rate of change) and the intercept (i.e., constant term) of a linear model in the context of the data
• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (i.e., outliers)

## C - Functions

• understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]
• interpret key features of linear, quadratic, and exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals; this can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (i.e., compare and contrast quadratic functions in standard, vertex, and intercept forms)
• graph linear, quadratic, and exponential functions algebraically and show key features of the graph by hand and by using technology
• interpret key features of linear functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing); sketch graphs showing these key features when given a verbal description of the relationship
• construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, table, a description of a relationship, or two input-output pairs
• write arithmetic sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions
• write a function that describes an exponential relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 &lt; x =10})
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; asymptote; and end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• write a function that describes a linear relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
• write geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect geometric sequences to exponential functions
• show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)
• use second differences to write a quadratic function that describes a relationship between two quantities
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another (i.e., exponential)
• interpret the parameters in a linear (i.e., f(x) = mx + b) or exponential function (i.e., f(x)=a·dx) in terms of a context (i.e., in the provided functions, “m” and “b” are the parameters of the linear function, and “a” and “d” are the parameters of the exponential function); in context, students should describe what these parameters mean in terms of change and starting value
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; experiment with cases and illustrate an explanation of the effects on the graph using technology; include recognizing even and odd functions from their graphs and algebraic expressions
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another (i.e., linear)
• recognize that geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
• recognize that arithmetic sequences are functions, sometimes defined recursively, whose domain is a subset of the integers

## D - Number and Quantity

• define appropriate quantities for the purpose of descriptive modeling; given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation
• choose a level of accuracy appropriate to limitations on measurement when reporting quantities (e.g., money situations are generally reported to the nearest hundredth; also, an answers’ precision is limited to the precision of the data given)
• rewrite expressions involving radicals (i.e., simplify and/or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)
• explain why the sum or product of rational numbers is rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational
• use units of measure (linear, area, capacity, rates, and time) as a way to understand problems; identify, use, and record appropriate units of measure within context, within data displays, and on graphs; convert units and rates using dimensional analysis (English to English and Metric to Metric without conversion factor provided and between English and Metric with conversion factor); use units within multi-step problems and formulas; interpret units of input and resulting units of output

• # High School Mathematics -- Accelerated Algebra I

## A - Algebra

• demonstrate that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane
• add, subtract, and multiply polynomials
• create exponential equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, graph these exponential equations on coordinate axes with appropriate labels and scales
• represent constraints by systems of equations and/or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• factor any quadratic expression to reveal the zeros of the function it defines
• create exponential equations in one variable and use them to solve simple equations
• graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
• create linear equations and inequalities in one variable and use them to solve problems
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)
• interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors
• use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions; derive the quadratic formula from ax² + bx + c = 0
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations)
• create quadratic equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation; graph these quadratic equations on coordinate axes with appropriate labels and scales
• represent constraints by equations or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solutions; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., {x | 2 &lt; x = 10})
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• solve systems of linear equations exactly (i.e., algebraically) and approximately (i.e., with graphs), focusing on pairs of linear equations in two variables; solve simple cases by inspection (e.g., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• graph the solutions to a linear inequality in two variables as a half plane, excluding the boundary in the case of a strict inequality
• use the structure of an expression to rewrite it in different equivalent forms [i.e., see x4 - y4 as ((x²) -(y²))², thus recognizing it as a difference of squares that can be factored as (x²-y²)(x²+y²)]
• show and explain why the elimination method works to solve a system of two-variable equations
• create quadratic equations in one variable and use them to solve problems
• interpret parts of an expression, such as terms, factors, and coefficients in context
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (e.g., rearrange Ohm's law V=IR to highlight resistance R)
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (e.g., reveal the zeros, minimum, or maximum)

## B - Statistics and Probability

• use statistics appropriate to the shape of the data distribution to compare center (i.e., median, mean) and spread (i.e., interquartile range, mean absolute deviation) of two or more different data sets
• explain the difference between correlation and causation
• fit a function to bivariate data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear, quadratic, and exponential models
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (i.e., including joint, marginal, and conditional relative frequencies); recognize possible associations and trends in the data
• represent data on two quantitative variables on a scatter plot and describe how the variables are related
• determine and interpret the slope (i.e., rate of change) and the intercept (i.e., constant term) of a linear model in the context of the data
• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (i.e., outliers)
• compute (using technology) and interpret the correlation coefficient of a linear fit (e.g., by looking at a scatter plot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the "r" value; after calculating the line of best fit using technology, describe how strong the goodness of fit of the regression is using "r")
• represent data with plots on the real number line (e.g., dot plots, histograms, and box plots)

## C - Geometry

• use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent
• know and apply the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc
• understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles
• given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software; specify a sequence of transformations that will carry a given figure onto another
• given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides
• develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments
• explain how the criteria for triangle congruence (i.e., ASA, SAS, SSS, HL, AAS) follow from the definition of congruence in terms of rigid motions
• describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself
• use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent
• prove theorems about lines and angles (i.e., vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints)
• use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures
• make formal geometric constructions with a variety of tools and methods (e.g., compass and straightedge, string, reflective devices, paper folding, dynamic geometric software); copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line
• use the properties of similarity transformations to establish the AA criterion for two triangles to be similar
• construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle
• explain and use the relationship between the sine and cosine of complementary angles
• prove theorems about triangles (i.e., measures of the interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point)
• prove theorems about triangles (i.e., a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity)
• verify experimentally the properties of dilations given by a center and a scale factor (i.e., a dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged; the dilation of a line segment is longer or shorter according to the ratio given by the scale factor)
• describe transformations as function that take points in the plane as inputs and give other points as outputs; compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch)
• prove theorems about parallelograms (i.e., opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals)
• use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

## D - Functions

• interpret key features of linear, quadratic, and exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; symmetries; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another (i.e., exponential)
• write arithmetic sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• interpret the parameters in a linear or exponential function in terms of a context; in context, students should describe what these parameters mean in terms of change and starting value
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• recognize that geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
• write geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect geometric sequences to exponential functions
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• write a function that describes an exponential relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 < x = 10})
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• recognize that arithmetic sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
• use second differences to write a quadratic function that describes a relationship between two quantities
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another (i.e., linear)
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)
• show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (e.g., compare and contrast quadratic functions in standard, vertex, and intercept forms)
• <font face="Calibri, sans-serif"><span style="font-size: 16px;">write a function that describes a linear relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context</span></font><br>
• identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; experiment with cases and illustrate an explanation of the effects on the graph using technology; include recognizing even and odd functions from their graphs and algebraic expressions
• construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, table, a description of a relationship, or two input-output pairs
• understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]
• graph linear, quadratic, and exponential functions algebraically and show key features of the graph by hand and by using technology
• interpret key features of linear functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing); sketch graphs showing these key features when given a verbal description of the relationship
• prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals; this can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• use units of measure (linear, area, capacity, rates, and time) as a way to understand problems; identify, use, and record appropriate units of measure within context, within data displays, and on graphs; convert units and rates using dimensional analysis (English to English and Metric to Metric without conversion factor provided and between English and Metric with conversion factor); use units within multi-step problems and formulas; interpret units of input and resulting units of output
• choose a level of accuracy appropriate to limitations on measurement when reporting quantities (e.g., money situations are generally reported to the nearest hundredth; also, an answers' precision is limited to the precision of the data given)
• rewrite expressions involving radicals (i.e., simplify and /or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)
• explain why the sum or the product of rational numbers is rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational
• define appropriate quantities for the purpose of descriptive modeling; given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation

• # High School Mathematics -- Geometry

## A - Geometry

• use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent
• use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)
• use coordinates to prove simple geometric theorems algebraically (i.e., prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that a point lies on a circle centered at the origin and containing a given point), including quadrilaterals, circles, right triangles, and parabolas
• use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula)
• understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles
• prove theorems about triangles (i.e., measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point)
• given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software; specify a sequence of transformations that will carry a given figure onto another
• prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point)
• use volume formulas for cylinders, pyramids, cones, and spheres to solve problems
• explain and use the relationship between the sine and cosine of complementary angles
• explain how the criteria for triangle congruence (i.e., ASA, SAS, SSS, HL, AAS) follow from the definition of congruence in terms of rigid motions
• know and apply the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc
• make formal geometric constructions with a variety of tools and methods (e.g., compass and straightedge, string, reflective devices, paper folding, dynamic geometric software); copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line
• describe transformations as functions that take points in the plane as inputs and give other points as outputs; compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch)
• construct the inscribed and circumscribed circle of a triangle
• use informal arguments to establish facts about the angle sum and exterior angles of triangles and about the angles created when parallel lines are cut by a transversal
• give informal arguments for geometric formulas (i.e., informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments; informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri's principle)
• apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)
• derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector
• prove theorems about lines and angles (i.e., vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints)
• apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real- world and mathematical problems in three dimensions
• prove properties of angles for a quadrilateral inscribed in a circle
• give an informal argument using Cavalieri's principle for the formulas of the volume of a sphere and other solid figures
• construct a tangent line from a point outside a given circle to the circle
• use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent
• find the point on a directed line segment between two given points that partitions the segment in a given ratio
• develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments
• prove theorems about triangles (i.e., a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity)
• informally prove the Pythagorean Theorem and its converse geometrically (i.e., using area model)
• describe the rotations and reflections that carry a rectangle, parallelogram, trapezoid, or regular polygon onto itself
• explain and apply the distance formula as an application of the Pythagorean Theorem
• given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides
• verify experimentally the properties of dilations given by a center and a scale factor (i.e., a dilation of a line not passing through the center of the dilation results in a parallel line and leaves a line passing through the center unchanged; the dilation of a line segment is longer or shorter according to the ratio given by the scale factor)
• prove theorems about parallelograms (i.e., opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals)
• derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation
• construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle
• prove that all circles are similar (i.e., using transformations; ratio of circumference to the diameter is a constant)
• identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects
• use the properties of similarity transformations to establish the AA criterion for two triangles to be similar
• use similarity criteria for triangles to solve problems and to prove relationships in geometric figures
• use congruence criteria for triangles to solve problems and to prove relationships in geometric figures
• use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
• identify and describe relationships among inscribed angles, radii, chords, tangents, and secants (i.e., the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle)
• describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates
• apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios)

## B - Statistics and Probability

• find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in context
• construct and interpret two-way frequency tables of data when two categories are associated with each object being classified; use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities (e.g., collect data from a random sample of students in your school on their favorite subject among math, science, and English; estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade; do the same for other subjects and compare the results)
• understand the conditional probability of A given B as P(A and B)/P(B); interpret independence of A and B in terms of conditional probability (i.e., the conditional probability of A given B is the same as the probability of A and the conditional probability of B given A is the same as the probability of B)
• apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in context
• recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations (e.g., compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer)
• understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent
• describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (i.e., "or," "and," "not")

• # High School Mathematics -- Accelerated Geometry

## A - Algebra

• create equations and inequalities in one variable and use them to solve problems (i.e., create equations in one variable that describes simple rational functions, exponential functions, )
• know and apply that the Binomial Theorem gives the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined using Pascal’s Triangle
• interpret expressions that represent a quantity in terms of its context
• create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (i.e., create and graph equations in two variables to describe radical functions, rational functions, etc.)
• interpret parts of an expression such as terms, factors, and coefficients, in context
• solve quadratic equations by inspection (e.g.,x² = -49), taking square roots, factoring, completing the square, and using the quadratic formula, as appropriate to the initial form of the equation; recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b
• solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise
• solve quadratic equations in one variable
• know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x)
• derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems (e.g., calculate mortgage payments)
• prove polynomial identities and use them to describe numerical relationships (e.g., the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples)
• given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations
• represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods)
• explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g (x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately (e.g., using a graph, find the solution to a system of equations where f(x) and/or g(x) are rational functions)
• understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions
• use the structure of an expression to rewrite it in different equivalent forms (e.g., recognize x4 - y4as (x²)²- (y²)², thus recognizing it as a difference of squares that can be factored as (x²- y²)(x²+ y²))
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
• identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial
• use the properties of exponents to transform expressions for exponential functions
• rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b (x), using inspection, long division, or, for the more complicated examples, a computer algebra system
• understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials

## B - Function

• use the properties of exponents to interpret expressions for exponential functions
• graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function)
• graph exponential and logarithmic functions, showing intercepts and end behavior
• graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior
• using tables, graphs, equations, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features, including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior
• graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions
• read values of an inverse function from a graph or a table, given that the function has an inverse
• write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases
• write a function that describes a relationship between two quantities (e.g., quadratic, polynomial, rational, radical, exponential, logarithmic)
• compose functions
• solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse (e.g., f(x) = 2(x^3) or f(x) = (x+1)/(x-1) for x?1)
• identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; experiment with cases and illustrate an explanation of the effects on the graph using technology; include recognizing even and odd functions from their graphs and algebraic expressions for them
• verify by composition that one function is the inverse of another
• find inverse functions
• express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology
• combine standard function types using arithmetic operations
• understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph

## C - Geometry

• apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios)
• use volume formulas for cylinders, pyramids, cones, and spheres to solve problems
• give an informal argument using Cavalieri's principle for the formulas of the volume of a sphere and other solid figures
• derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector
• identify and describe relationships among inscribed angles, radii, chords, tangents, and secants (i.e., the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle)
• use coordinates to prove simple geometric theorems algebraically (i.e., prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that a point lies on a circle centered at the origin and containing a given point), including quadrilaterals, circles, right triangles, and parabolas
• use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)
• identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects
• prove that all circles are similar (i.e., using transformations; ratio of circumference to the diameter is a constant)
• construct a tangent line from a point outside a given circle to the circle
• give informal arguments for geometric formulas (i.e., informal arguments for the formulas of the circumference of a circle and area of a circle using dissection arguments and informal limit arguments; informal arguments for the formula of the volume of a cylinder, pyramid, and cone using Cavalieri's principle)
• prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point)
• derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation
• apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)
• construct the inscribed and circumscribed circle of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle
• find the point on a directed line segment between two given points that partitions the segment in a given ratio
• use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula)

## D - Numbers

• find the conjugate of a complex number; use conjugates to find quotients of complex numbers
• know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials
• extend polynomial identities to the complex numbers (e.g., rewrite x² + 4 as (x + 2i)(x - 2i))
• solve quadratic equations with real coefficients that have complex solutions explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents
• rewrite expressions involving radicals and rational exponents using the properties of exponents
• use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers
• know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real

## E - Statistics and Probability

• find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in context
• construct and interpret two-way frequency tables of data when two categories are associated with each object being classified; use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities (e.g., collect data from a random sample of students in your school on their favorite subject among math, science, and English; estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade; do the same for other subjects and compare the results)
• describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (i.e., "or," "and," "not")
• recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations (e.g., compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer)
• understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent
• apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in context
• understand the conditional probability of A given B as P(A and B)/P(B); interpret independence of A and B in terms of conditional probability (i.e., the conditional probability of A given B is the same as the probability of A and the conditional probability of B given A is the same as the probability of B)

• # High School Mathematics -- Algebra II

## A - Algebra

• explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately (e.g., using a graph, find the solution to a system of equations where f(x) and/or g(x) are rational functions)
• represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods)
• understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials
• identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial
• rewrite simple rational expressions in different forms using inspection, long division, or a computer algebra system; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x)
• use the properties of exponents to transform expressions for exponential functions
• solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise
• create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (i.e., create and graph equations in two variables to describe radical functions, rational functions, etc.)
• understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions
• solve quadratic equations in one variable
• derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems (e.g., calculate mortgage payments)
• use the structure of an expression to rewrite it in different equivalent forms [e.g., see x4 - y4 as ((x²) -(y²))², thus recognizing it as a difference of squares that can be factored as (x²-y²)(x²+y²)]
• interpret parts of an expression such as terms, factors, and coefficients
• create equations and inequalities in one variable and use them to solve problems (i.e., create equations in one variable that describes simple rational functions, exponential functions, etc.)
• know and apply that the Binomial Theorem gives the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined using Pascal’s Triangle
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations
• interpret expressions that represent a quantity in terms of its context
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
• know and apply the Remainder Theorem: for a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x)
• prove polynomial identities and use them to describe numerical relationships (e.g., the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate Pythagorean triples)
• interpret complicated expressions by viewing one or more of their parts as a single entity (e.g., interpret P(1+r) as the product of P and a factor not depending on P)
• solve quadratic equations by inspection (e.g., x²= -49), taking square roots, factoring, completing the square, and using the quadratic formula, as appropriate to the initial form of the equation; recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

## B - Statistics and Probability

• use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages; recognize that there are data sets for which such a procedure is not appropriate; use calculators, spreadsheets, and tables to estimate areas under the normal curve
• decide if a specified model is consistent with results from a given data-generating process (e.g., a model says a spinning coin falls heads up with probability 5; would a result of 5 tails in a row cause you to question the model?)
• evaluate reports based on data
• recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each
• understand statistics as a process for making inferences about population parameters based on a random sample from that population
• use statistics appropriate to the shape of the data distribution to compare center (i.e., median, mean) and spread (i.e., interquartile, range, standard deviation) of two or more different data sets
• use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant
• use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling

## C - Functions

• identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them
• graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases
• verify by composition that one function is the inverse of another
• graph exponential and logarithmic functions, showing intercepts and end behavior
• combine standard function types using arithmetic operations (e.g., build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model)
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context
• find inverse functions
• read values of an inverse function from a graph or a table, given that the function has an inverse
• express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology
• write a function that describes a relationship between two quantities (e.g., quadratic, polynomial, rational, radical, exponential, logarithmic)
• understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• compose functions
• use the properties of exponents to interpret expressions for exponential functions
• solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse (e.g., f(x) = 2(x^3) or f(x) = (x+1)/(x-1) for x?1)
• graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function)
• using tables, graphs, equations, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features, including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior
• graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions
• write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function

## D - Number and Quantity

• know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real
• find the conjugate of a complex number; use conjugates to find quotients of complex numbers
• know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials
• rewrite expressions involving radicals and rational exponents using the properties of exponents
• explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents
• solve quadratic equations with real coefficients that have complex solutions
• use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers
• extend polynomial identities to the complex numbers (e.g., rewrite x² + 4 as (x + 2i)(x - 2i))

• # High School Mathematics -- Pre-calculus

## A - Algebra

• represent a system of linear equations as a single matrix equation in a vector variable
• find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater)
• solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically

## B - Functions

• graph trigonometric functions, showing period, midline, and amplitude
• explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle
• produce an invertible function from a non-invertible function by restricting the domain
• use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi - x, pi + x, and 2pi - x in terms of their values for x, where x is any real number
• use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions
• find inverse functions
• understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle
• use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context
• understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed
• choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline
• graph functions expressed algebraically and show key features of the graph both by hand and by using technology
• prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems
• using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity
• prove the Pythagorean identity (sin A)² + (cos A)² = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle

## C - Geometry

• derive the equation of a parabola given a focus and
• prove the Laws of Sines and Cosines and use them to solve problems
• derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side
• derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant
• understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces)

## D - Number and Quantity

• recognize vector quantities as having both magnitude and direction; represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes
• solve problems involving velocity and other quantities that can be represented by vectors
• understand vector subtraction v - w as v + (-w), where (-w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction; represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise
• understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers; the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse
• represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number
• calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints
• work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
• find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers
• add, subtract, and multiply matrices of appropriate dimensions
• given two vectors in magnitude and direction form, determine the magnitude and direction of their sum
• understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties
• add vectors end-to-end, component-wise, and by the parallelogram rule; understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes
• multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector; work with matrices as transformations of vectors
• represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation
• find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point
• compute the magnitude of a scalar multiple cv using ||cv|| = |c|v; compute the direction of cv knowing that when |c|v not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0)
• represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, (e.g., as c(vx, vy) = (cvx, cvy))
• multiply matrices by scalars to produce new matrices, (e.g., as when all of the payoffs in a game are doubled)
• multiply a vector by a scalar
• use matrices to represent and manipulate data, (e.g., to represent payoffs or incidence relationships in a network)

## E - Statistics and Probability

• evaluate and compare strategies on the basis of expected values (e.g., compare a high- deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident)
• analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game)
• develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value (e.g., find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household; how many TV sets would you expect to find in 100 randomly selected households?)
• calculate the expected value of a random variable; interpret it as the mean of the probability distribution
• develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value (e.g., find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes)
• find the expected payoff for a game of chance (e.g., find the expected winnings from a state lottery ticket or a game at a fast-food restaurant)
• weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values
• apply the general Multiplication Rule in a uniform probability model, P(A and B) = [P(A)]x[P(B|A)]

=[P(B)]x[P(A|B)], and interpret the answer in terms of the model

• define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions

• use probabilities to make fair (equally likely) decisions (e.g., drawing by lots, using a random number generator)

• use permutations and combinations to compute probabilities of compound events and solve problems

• # High School Mathematics -- Accelerated Pre-calculus

## A - Algebra

• graph and identify characteristics of simple polar equations including lines, circles, cardioids, limaçons and roses
• convert between Cartesian and parametric form
• represent a system of linear equations as a single matrix equation in a vector variable
• use mathematical induction to find and prove formulae for sums of finite series
• express coordinates of points in rectangular and polar form
• describe parametric representations of plane curves
• establish and utilize trigonometric identities to simplify expressions and verify equivalence statements (e.g., double angle, half angle, reciprocal, quotient, pythagorean, even, and odd)
• find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3x3 or greater)
• solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically
• graph equations in parametric form showing direction and endpoints where appropriate

## B - Functions

• understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed
• graph trigonometric functions, showing period, midline, and amplitude
• use special triangles to determine geometrically the values of sine, cosine, tangent, cosecant, secant, cotangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, tangent, cosecant, secant, and cotangent for pi - x, pi + x, and 2pi - x in terms of their values for x, where x is any real number
• use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context
• graph functions expressed algebraically and show key features of the graph both by hand and by using technology
• explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle
• using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities; sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity
• explore the continuity of functions of two independent variables in terms of the limits of such functions as (x,y) approaches a given point in the plane
• choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline
• prove the Pythagorean identity (sin A)² + (cos A)² = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle
• prove the addition, subtraction, and double angle formulas for sine, cosine, and tangent and use them to solve problems
• find inverse functions
• use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions
• produce an invertible function from a non-invertible function by restricting the domain
• understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle

## C - Geometry

• prove the Laws of Sines and Cosines and use them to solve problems
• derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant
• derive the equation of a parabola given a focus and directrix
• derive the formula A = (1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side
• understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces)

## D - Number and Quantity

• multiply a vector by a scalar
• multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector; work with matrices as transformations of vectors
• work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
• represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation
• understand vector subtraction v - w as v + (-w), where (-w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction; represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise
• understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse
• recognize vector quantities as having both magnitude and direction; represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v)
• calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints
• use matrices to represent and manipulate data, (e.g., to represent payoffs or incidence relationships in a network)
• multiply matrices by scalars to produce new matrices, (e.g., as when all of the payoffs in a game are doubled)
• represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, (e.g., as c(vx, vy) = (cvx, cvy))
• find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers
• add, subtract, and multiply matrices of appropriate dimensions
• understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties
• compute the magnitude of a scalar multiple cv using ||cv|| = |c|v; compute the direction of cv knowing that when |c|v not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0)
• represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number
• solve problems involving velocity and other quantities that can be represented by vectors
• add vectors end-to-end, component-wise, and by the parallelogram rule; understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes
• given two vectors in magnitude and direction form, determine the magnitude and direction of their sum
• find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point

## E - Statistics and Probability

• develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value (e.g., find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household; how many TV sets would you expect to find in 100 randomly selected households?)
• calculate the expected value of a random variable; interpret it as the mean of the probability distribution
• understand statistics as a process for making inferences about population parameters based on a random sample from that population
• recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each
• weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values
• analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game)
• find the expected payoff for a game of chance (e.g., find the expected winnings from a state lottery ticket or a game at a fast-food restaurant)
• apply the general Multiplication Rule in a uniform probability model, P(A and B)= [P(A)]x[P(B|A)]=[P(B)]x[P(A|B)], and interpret the answer in terms of the model
• use permutations and combinations to compute probabilities of compound events and solve problems
• evaluate reports based on data
• use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling
• use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator)
• develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value (e.g., find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes)
• use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread(interquartile, range, standard deviation) of two or more different data sets
• decide if a specified model is consistent with results from a given data-generating process, (e.g., using simulation; for example, a model says a spinning coin falls heads up with probability 0.5; would a result of 5 tails in a row cause you to question the model?)
• use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant
• use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages; recognize that there are data sets for which such a procedure is not appropriate; use calculators, spreadsheets, and tables to estimate areas under the normal curve
• evaluate and compare strategies on the basis of expected values (e.g., compare a high- deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident)
• define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions

• # High School Mathematics -- Calculus

## A - Process Skills

• solve problems that arise in mathematics and in other areas
• create and use pictures, manipulatives, models, and symbols to organize, record, and communicate mathematical ideas
• use representations to model and interpret physical, social, and mathematical phenomena
• organize and consolidate mathematics thinking
• select and use various types of reasoning and methods of proof
• communicate mathematical thinking coherently to peers, teachers, and others
• build new mathematical knowledge through problem-solving
• analyze and evaluate the mathematical thinking and strategies of others
• recognize and apply mathematics in contexts outside of mathematics
• recognize reasoning and proof (evidence) as fundamental aspects of mathematics
• use the terminology and language of mathematics to express mathematical ideas precisely
• investigate, develop, and evaluate mathematical arguments and proofs
• monitor and reflect on the process of mathematical problem-solving
• make and investigate mathematical conjectures
• use appropriate technology to solve mathematical problems
• select, apply, and translate among mathematical representations to solve problems
• recognize and use connections among mathematical ideas
• explain how mathematical ideas interconnect and build on one another to produce a coherent whole
• apply and adapt a variety of appropriate strategies to solve problems

## B - Functions

• apply concepts of functions including domain, range, intercepts, symmetry, asymptotes, zeros, odd, even, and inverse
• identify and apply properties of algebraic, trigonometric, piecewise, absolute value, exponential, and logarithmic functions
• apply the algebra of functions by finding sum, product, quotient, composition, and inverse, where they exist

## C - Limits and Continuity

• evaluate limits of functions and apply properties of limits, including one-sided limits
• indicate where a function is continuous and where it is discontinuous
• estimate limits from graphs or tables of data
• apply the definition of continuity to a function at a point
• describe asymptotic behavior in terms of limits involving infinity
• identify types of discontinuities graphically and analytically
• calculate limits using algebra

## D - Derivatives

• apply the rules of differentiation to trigonometric functions, such as product, quotient, and chain rules, including successive derivatives
• determine if a function is differentiable over an interval
• determine where a function fails to be differentiable
• interpret derivative as a rate of change in the context of speed, velocity, and acceleration
• apply the chain rule to composite functions, implicitly defined relations, and related rates of change
• define the derivative of a function in various ways: the limit of the difference quotient, the slope of the tangent line at a point, instantaneous rate of change, and the limit of the average rate of change
• apply the rules of differentiation, such as product and quotient rules, to algebraic functions, including successive derivatives

## E - Applications of Derivatives

• solve optimization problems
• apply the extreme value theorem to problem situations
• use the relationships between f(x), f'(x), and f"(x) to determine the increasing/decreasing behavior of f(x); determine the critical point(s) of f(x); determine the concavity of f(x) over an interval; and determine the point(s) of inflection of f(x)
• apply the derivative to determine: the slope of a curve at a point, the equation of the tangent line to a curve at a point, and the equation of the normal line to a curve at a point
• given various pieces of information, sketch of graph(s) of f(x), f'(x), and f"(x)
• apply Rolle's Theorem and the Mean Value Theorem
• find absolute (global) and relative (local) extrema
• model rates of change involved with related rates problems

## F - Integrals

• calculate area by a definite integral of Riemann sums over equal subdivisions
• evaluate integrals by substitution of variables (including change of limits for definite integrals)
• calculate areas by evaluation sums using sigma notation
• relate the definite integral to the concept of the area under a curve; define and apply the properties of the definite integral
• define the antiderivative and apply its properties to problems such as distance and velocity from acceleration with initial condition, growth, and decay
• compute Riemann sums using left, right, and midpoint evaluations and trapezoids
• identify and use the Fundamental Theorem of Calculus in evaluation of definite integrals
• evaluate integrals following directly from derivatives of basic functions

## G - Applications of the Integral

• evaluate the area between curves using integration formulas
• evaluate the volume of a solid of revolutions using the disk or washer method
• apply the integral to the average or mean value of a function on an interval
• evaluate the volume of a solid using known cross-sections

## H - Reading Across the Curriculum

• read and discuss mathematical material to establish context for subject matter, develop mathematical vocabulary, and develop an awareness of current research

• # High School Mathematics -- Advanced Calculus II

## A - Process Skills

• use appropriate technology to solve mathematical problems
• select, apply, and translate among mathematical representations to solve problems
• use representations to model and interpret physical, social, and mathematical phenomena
• apply and adapt a variety of appropriate strategies to solve problems
• make and investigate mathematical conjectures
• build new mathematical knowledge through problem-solving
• select and use various types of reasoning and methods of proof
• explain how mathematical ideas interconnect and build on one another to produce a coherent whole
• use the terminology and language of mathematics to express mathematical ideas precisely
• monitor and reflect on the process of mathematical problem-solving
• recognize reasoning and proof (evidence) as fundamental aspects of mathematics
• investigate, develop, and evaluate mathematical arguments and proofs
• recognize and apply mathematics in context outside of mathematics
• communicate mathematical thinking coherently to peers, teachers, and others
• analyze and evaluate the mathematical thinking and strategies of others
• solve problems that arise in mathematics and in other areas
• organize and consolidate mathematics thinking
• recognize and use connections among mathematical ideas
• create and use pictures, manipulatives, models, and symbols to organize, record, and communicate mathematical ideas

## B - Integrals

• evaluate integrals using integration by parts
• evaluate integrals of rational functions using partial fractions
• evaluate improper integrals

## C - Functions

• apply concepts of functions including domain, range, intercepts, and symmetry

## D - Geometry

• define parabolas, ellipses, and hyperbolas geometrically
• explain the role of conic sections in the reflection of light and sound
• express curves in parametric form
• use Pappus's Theorem on Surface Area to find area of a surface
• calculate the area of a surface generated by revolution
• sketch curves in polar coordinates
• calculate arc length using integrals
• calculate the area of a region with boundary given in polar coordinates
• find equations of tangents to curves given parametrically

## E - Sequences

• write a given sequence in sigma notation
• determine the convergence of improper integrals
• calculate limits of sequences with indeterminate forms
• state the convergence or divergence of a sequence
• find least upper bounds and greatest lower bounds
• determine the limit of a sequence
• calculate the limit of a sequence using L'Hospital's Rule

## F - Series

• find the Lagrange form of the remainder of a series and use it to test for accuracy of the polynomial
• find the interval of convergence for a power series
• differentiate and integrate power series
• evaluate the sum of series
• test for convergence of MacLaurin series
• determine if a series converges or diverges using the integral test, basic comparison test, limit comparison test, root test, and the ratio test
• find a Taylor polynomial for given functions and use them to estimate function values
• apply Taylor's Theorem to find the Taylor polynomial of a given function for a given value
• test for absolute and conditional converges of alternating series

## G - Vectors

• find the volume of parallelepipeds
• find unit normal vectors for a plane
• find equations for a plane
• calculate distance between points in 3-space
• find points of intersections of intersecting vectors
• write an equation for spheres with given conditions
• calculate distance from a point to a plane
• calculate the angle between two vectors
• find vector parameterizations for lines
• find direction angles of a vector
• determine whether vectors are parallel, skew, or intersecting
• find the angle between two planes
• determine the co-planarity of vectors
• find a set of scalar parametric equations for lines formed by the intersection of planes
• calculate the norm of a vector
• find a unit vector for a given vector
• calculate dot products
• calculate cross products
• find projection vectors

## H - Vector Calculus

• find the angular speed and the magnitude of the acceleration of a particle moving along a curve
• integrate vectors
• sketch curves defined by vectors
• find the tangent vector at a given point
• calculate the arc length of a curve defined in vector form
• calculate acceleration vectors
• apply the rules of differentiation to find the derivative of vectors

• # High School Mathematics -- Advanced Mathematical Decision Making

## A - Number and Operations

• extend the understanding of proportional reasoning, ratios, rates, and percents by applying them to various settings to include business, media, and consumerism; use proportional reasoning to solve problems involving ratios; analyze and use averages, weighted averages, and indices; solve problems involving large quantities that are not easily measured; explain how identification numbers, such as UPCs, are created and verified

## B - Algebra

• use a variety of network models to organize data in quantitative situations, make informed decisions, and solve problems; solve problems represented by a vertex-edge graph, and find critical paths, Euler paths, and minimal spanning trees; construct, analyze, and interpret flow charts to develop an algorithm to describe processes such as quality control procedures; investigate the scheduling of projects using PERT; consider problems that can be resolved by coloring graphs
• use vectors and matrices to organize and describe problem situations; represent situations and solve problems using vectors in areas such as transportation, computer graphics, and the physics of force and motion; represent geometric transformations and solve problems using matrices in fields such as computer animations
• create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money; use exponential functions to model change in a variety of financial situations; determine, represent, and analyze mathematical models for income, expenditures, and various types of loans and investments
• analyze and evaluate the mathematics behind various methods of voting and selection; evaluate various voting and selection processes, which include polling biases voting to determine an appropriate method for a given situation; apply various ranking algorithms to determine an appropriate method for a given situation

## C - Geometry

• solve geometric problems involving inaccessible distances using basic trigonometric principles, including the Law of Sines and the Law of Cosines
• create and use two- and three-dimensional representations of authentic situations

## D - Data Analysis and Probability

• apply statistical methods to design, conduct, and analyze statistical studies
• build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays
• determine probability and expected value to inform everyday decision making; determine conditional probabilities and probabilities of compound events to make decisions in problem situations; use probabilities to make and justify decisions about risks in everyday life; calculate expected value to analyze mathematical fairness, payoff, and risk
• use functions to model problem situations in both discrete and continuous relationships; determine whether a problem situation involving two quantities is best modeled by a discrete relationship; use linear, exponential, logistic, piecewise and sine functions to construct a model

• # High School Mathematics -- Algebra I Strategies

## A - Algebra

• complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (i.e., reveal the zeros, minimum, or maximum)
• create exponential equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, graph these exponential equations on coordinate axes with appropriate labels and scales
• add, subtract, and multiply polynomials
• solve systems of linear equations exactly (i.e., algebraically) and approximately (i.e., with graphs), focusing on pairs of linear equations in two variables; solve simple cases by inspection (i.e., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6)
• graph the solutions to a linear inequality in two variables as a half plane, excluding the boundary in the case of a strict inequality
• graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes
• solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (i.e., solve multi-step linear equations with one solution, infinitely many solutions, or no solution; extend this reasoning to solve compound linear inequalities and literal equations); express solution sets to inequalities using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 &lt; x =10})
• factor any quadratic expression to reveal the zeros of the function it defines
• represent constraints by systems of equations and/or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers; justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties
• create linear equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation, including writing equations when given a slope and a y-intercept or slope and a point; graph these linear equations on coordinate axes with appropriate labels and scales
• show and explain why the elimination method works to solve a system of two-variable equations
• represent constraints by equations or inequalities, and interpret data points as possible (i.e., a solution) or not possible (i.e., a non-solution) under the established constraints
• interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors
• create quadratic equations in two variables to represent relationships between quantities expressed in a table of values or verbal representation; graph these quadratic equations on coordinate axes with appropriate labels and scales
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions)

## B - Statistics and Probability

• represent data on two quantitative variables on a scatter plot and describe how the variables are related
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (i.e., including joint, marginal, and conditional relative frequencies); recognize possible associations and trends in the data
• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (i.e., outliers)
• fit a function to bivariate data; use functions fitted to data to solve problems in the context of the data; use given functions or choose a function suggested by the context; emphasize linear, quadratic, and exponential models

## C - Functions

• recognize situations in which one quantity changes at a constant rate per unit interval relative to another
• graph quadratic functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts, maxima, and minima (as determined by the function or by context)
• interpret key features of linear functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts, positive, negative; intervals where the function is increasing, decreasing); sketch graphs showing these key features when given a verbal description of the relationship
• evaluate functions for inputs in their domains using function notation and interpret statements that use function notation in terms of a context
• understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range [i.e., if f is a function, x is the input (an element of the domain), and f(x) is the corresponding output (an element of the range); the graph of the function is the set of ordered pairs consisting of an input and the corresponding output]
• graph linear functions expressed algebraically in slope-intercept and standard form by hand and by using technology; show and interpret key features including slope and intercepts (as determined by the function or by context)
• compare properties of two functions each represented algebraically, graphically, numerically in tables, and/or by a verbal description
• interpret key features of linear, quadratic, and exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; asymptotes; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• interpret key features of quadratic functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• graph linear, quadratic, and exponential functions algebraically and show key features of the graph by hand and by using technology
• prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals; this can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals
• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes [e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function]; represent the domain and range using both interval notation (e.g., (2, 10]) and set notation (e.g., {x|2 &lt; x =10})
• graph exponential functions expressed algebraically by hand and by using technology; show and interpret key features including intercepts and end behavior
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph
• write arithmetic sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context (i.e., compare and contrast quadratic functions in standard, vertex, and intercept forms)
• interpret key features of exponential functions represented in graphs, tables, equations, and verbal descriptions (i.e., intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; asymptote; and end behavior); sketch graphs showing these key features when given a verbal description of the relationship
• write geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms; connect geometric sequences to exponential functions
• show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function

## D - Number and Quantity

• rewrite expressions involving radicals (i.e., simplify and/or use the operations of addition, subtraction, multiplication, and division with radicals within algebraic expressions limited to square roots)

• # High School Mathematics -- Algebra II Strategies

## A - Algebra

• solve quadratic equations in one variable
• identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial
• create equations and inequalities in one variable and use them to solve problems; include equations arising from linear and quadratic functions, and simple rational and exponential functions
• derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems
• interpret complicated expressions by viewing one or more of their parts as a single entity
• rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations
• use the properties of exponents to transform expressions for exponential functions
• prove polynomial identities and use them to describe numerical relationships
• solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation; recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b; connect the solutions to the graph and real life application
• represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context
• create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales
• interpret expressions that represent a quantity in terms of its context

## B - Statistics and Probability

• evaluate reports based on data
• decide if a specified model is consistent with results from a given data-generating process
• use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling

## C - Functions

• relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes
• use the properties of exponents to interpret expressions for exponential functions
• interpret key features of graphs and tables in terms of the quantities for a function that models a relationship between two quantities, and sketch graphs showing key features given a verbal description of the relationship; key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity
• compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)
• compose polynomial functions
• use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context
• identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; experiment with cases and illustrate an explanation of the effects on the graph using technology; include recognizing even and odd functions from their graphs and algebraic expressions for them
• understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents
• express as a logarithm for exponential models the solution to ab raised to the (ct) power equals d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology
• graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases
• calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval; estimate the rate of change from a graph; consider all types of functions
• combine standard function types using arithmetic operations

## D - Number and Quantity

• extend polynomial identities to the complex numbers
• use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers
• solve quadratic equations with real coefficients that have complex solutions; connect complex solutions to the graphs and examples using the quadratic formula
• rewrite expressions involving radicals and rational exponents using the properties of exponents

• # High School Mathematics -- Applications of Linear Algebra in Computer Science

## A - Functions and Linear Equations

• review function notation, domain/co-domain, identity/associativity, inverse/invertibility connecting to the computer science concept of perfect secrecy (i.e., encryption)
• review Python psuedo random number generation, calculating probability distribution and interpreting probability events through application of Caesar's Cypher and other examples of cryptosystems

## B - Python Programming

• understand how to program and utilize modules and control statements (e.g., loops, conditionals, grouping) in Python
• utilize sets, lists, dictionaries, comprehensions, indexing, and tuples in Python
• program input and output features to read from and write to files in Python

## C - The Complex Field

• perform operations of complex number numbers (e.g., absolute value, adding, multiplying) and understand how they produce different transformations
• understand how complex numbers connect to the unit circle and represent them in polar form; perform transformations in polar form and utilize Euler's formula/The First Law of Exponentiation to understand these transformations; connect to performing image transformation on a computer graphic program
• work with the Galois Field to understand further concepts in perfect secrecy and network coding (i.e., providing efficiency in streaming services)

## D - The Vector

• connect use of vectors in Galois Field(2) by applying concepts of perfect secrecy, all-or-nothing secret sharing, and programming/solving lights out games
• perform vector operations in R(n) including addition, scalar multiplication, and dot product; review concepts of convex and affine combinations; represent and perform these operations using dictionaries and the Vec.py class in Python
• find the distance, its unit vector in the same or opposite direction, the projection of a vector onto a given vector or vector space, dot product, inner product, cross product, and angle between two vectors in Euclidean space
• solve triangular systems of linear equations using upper-triangular systems, backward substitution, and other algorithms
• use the dot product to display the concept of simple authentication schemes and interacting with them, and performing a senator voting record analysis

## E - Vector Spaces

• determine if a given set of vectors in a vector space is a spanning set for that vector space and if they are linearly independent
• define and discuss uses of linear combinations and understand how to solve for coefficients or linear combinations, connect to programmed/solved lights out game
• determine if a linear combination is an affine combination and determine if an affine space exists by translating a vector space, represent and affine space as a solution set to a linear system
• find whether a vector is a linear combination of a given finite set of vectors in a vector space and provide this linear combination
• determine whether a provided subset of a vector space is a subspace and find the dimension of a subspace
• define span and what it means for linear combinations to be a span of vectors; connect span to simple authentication schemes; understand the geometric depiction of the span of vectors over R and the geometry of solution sets of homogenous linear equations and systems; understand the geometric interpretations of R^2 and R^3

## F - Matrices

• define and understand what the null space is by connecting to concepts of homogenous linear systems/matrix equations, and error correcting codes such as linear codes and Hamming's code
• factor a given matrix into the product of two elementary matrices, find the adjoint of a matrix and use it to find the inverse of the matrix (understanding the conditions for invertibility) or solve a system of linear equations
• generate an augmented coefficient matrix from a system of linear equations
• program transformations in 2D geometry using Python and concepts of matrix operations
• perform matrix operations including transpose, addition, scaler multiplication, dot product, and multiplication; compute the inner and outer product
• program error correcting code concepts such as Hamming's code using matrix operations
• review the structure of a matrix and composition of the identity matrix, determine the size, transpose, inverse, rank, and LU-factorization of a matrix; interpret matrices as vectors

## G - The Basis and Dimension

• determine whether a given set of vectors in a vector space forms a basis for that vector space and recognize standards bases in the vector spaces nth dimensional Euclidean space, the set of all polynomials of degree greater than or equal to n
• understand if a linear function is invertible utilizing the concept of dimension and determine if a function is onto or one-to-one; discuss in connection with Kernal-Image Theorem and Rank- Nullity Theorem, demonstrate using checksums
• discuss rank theorem and demonstrate its use via the Simple Authentication Schema in computer science
• connect the Exchange Lemma to the concept of camera image perspective rendering in a Python lab
• define the coordinate representation of a basis and connect to lossy compression in computer science; find a basis for the column or row space of a matrix, and find a basis for and the dimension of the nullspace of a matrix
• define what it means for vectors to be linearly dependent and linearly independent and define the Superfluous-Vector Lemma; perform tests of linear dependence
• utilize direct sum to add vector spaces and find the basis for the direct sum and understand if two subspaces are complementary
• review the minimum spanning forest problem in GF(2) in connection with the Grow and Shrink Algorithms and how to formulate the problem in linear algebra
• find the transition matrix from one basis to another (i.e., change of basis)
• define and determine the dimension and rank of a basis (and therefore vector space); use it to prove the Morphing Lemma and prove the Superset-Basis Lemma
• demonstrate that every vector space has a basis and any finite set of vectors contains a basis for its span (e.g., Subset-Basis Lemma)

## H - Gaussian Elimination

• solve systems of linear equations and finding the basis for a Null space by use of Gaussian Elimination, Gauss-Jordan Elimination, LU factorization, and Cramer's Rule; show how the simple authentication scheme can be attacked/improved over GF(2) using Guassian Elimination
• understand how Threshold Secret Sharing works in conjunction with Gaussian Elimination through a programming lab in Python
• understand how factoring integers can be performed using Euclid's algorithm and utilizing prime set factors in Python
• use elementary row operations to create matrices in row-echelon and reduced row-echelon form

## I - Orthogonalization

• define and perform QR factorization of a matrix to compute solutions to the matrix equation Ax=b; use to perform the application of least squares to find the line or curve of best fit (linear/quadratic) to approximate data in the industrial espionage problem/sensor node problem/machine learning problem
• find an orthogonal basis for a given basis/subspace/inner product space by applying the Gram- Schmidt orthonormalization process
• given the solution space of a homogenous system of linear equations, find an orthonormal basis
• determine if two given vectors/sets of vectors(complements)/subspaces are orthogonal, parallel, or neither; find the orthogonal component of a given subspace
• use orthogonalization to find the closest point in the span of many vectors, compute a basis/subset basis, direct sums of complements

## J - Special Bases

• utilize compression by suppression to find the closest k-sparse vector coordinate representation in terms of an orthogonal basis
• understand how images and sounds can be represented as wavelets and the bases representation of wavelets as well as wavelet transformation, implementation, and decomposition, perform Python lab on using wavelets to perform file compression
• define and demonstrate the Fourier transform connecting how a sound is stored as a sequence of amplitude samples and how the Fast Fourier Transform Algorithm is utilized/derived/coded

## K - The Eigenvalue/Eigenvector

• connect use of the determinant and eigenvectors to code functionality of Google's PageRank search engine in Python
• discuss how Markov chains work to model various concepts such as population movement, dance patterns, literary documents, and Google's search engine PageRank
• utilize eigenvalues/vectors and single value decomposition to program face recognition software (Eigenfaces)
• determine an orthogonal matrix that diagonalizes a given matrix
• find a nonsingular matrix(D) for a given matrix (if one exists) such that D^-1AD is diagonal; find a basis for the domain of a linear transformation such that the matrix of the linear transformation relative to the basis is diagonal
• find the determinant, minors, and cofactors of a given matrix and use the determinant to find whether a given matrix is singular/non-singular; use determinant properties to characterize eigenvalues
• find the eigenvalues of a given symmetric matrix and find the dimension of the corresponding eigenspace
• find the characteristic equation and eigenvalues/corresponding eigenvectors of a given matrix and determine if the matrix is diagonizable/symmetric/orthogonal
• verify the eigenvalue/eigenvector of a given matrix while understanding the geometric interpretation and coordinate representation; connect to the Internet Worm case of 1988; perform eigen theorem proofs

## L - Linear Programming

• perform a Python lab to explore concepts of linear programming
• explore samples of linear programming cases including the diet problem, the vertices of polyhedra (polyhedral combinatorics), the simplex algorithm, game theory, nonzero-sum games, and compressed sensing for MRI imaging
• perform a machine learning lab on a large set of health care data that incorporates concepts of linear programming

• # High School Mathematics -- College Readiness Mathematics

## A - Algebra

• justify the steps of a simple, one-solution equation using algebraic properties and the properties of real numbers. Justify own steps, or, if given two or more steps of an equation, explain the progression from one step to the next using the properties
• create equations and inequalities in one variable and use them to solve problems in context; include equations arising from linear, quadratic simple rational, and exponential functions- integer inputs only; identify variables and describe their relationship in context
• interpret expressions that represent a quantity in terms of its context
• interpret parts of an expression, such as, but not limited to, terms, factors and coefficients, in context
• analyze and solve linear equations in one variable
• use graphs, tables, or successive approximations and use technology to explain and show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same
• given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors
• graph proportional relationships, interpreting the unit rate as the slope of the compare two different proportional relationships represented in different ways
• factor any quadratic expression to reveal the zeros of the function defined by the expression
• solve systems of linear equations exactly and approximately in context, focusing on pairs of linear equations in two variables
• solve and graph linear equations and inequalities in one variable, including equations with coefficients represented by letters
• represent constraints by equations or inequalities, and by systems of equations and/or inequalities; interpret data points as viable or non-viable under the established constraints
• use the structure of an expression to rewrite it in different equivalent forms
• solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation limited to real number solutions
• use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions; derive the quadratic formula from ax² + bx + c = 0
• solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically
• graph and analyze the solution set to a linear inequality in two variables in context
• solve quadratic equations in one variable
• choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
• use right triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx+ b for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b
• rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations
• show and explain why the elimination method works to solve a system of two-variable equations
• solve simple rational and radical equations in one variable, demonstrate understanding, and give examples showing how extraneous solutions may arise
• convert a quadratic expression to vertex form to reveal the maximum or minimum value of the function defined by the expression; complete the square, use the axis of symmetry, or use the midpoint of the x-intercepts
• create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales

## B - Number and Quantity

• use units within multi-step problems and formulas; interpret units of input and resulting units of output in context
• convert units and rates, reasoning quantitatively and using dimensional analysis
• use units of measure (linear, area, capacity, rates, and time) as a way to understand problems
• define appropriate quantities for the purpose of descriptive given a situation, context, or problem, determine, identify, use, and justify appropriate quantities for representing the situation
• identify , use, and record appropriate units of measure within context, within data displays, and on graphs

## C - Functions

• identify and compare the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs; experiment with cases and illustrate an explanation of the effects on the graph using technology; include recognizing even and odd functions from their graphs and algebraic expressions for them
• construct a function to model a linear relationship between two Determine the rate of change and initial value of the function from a description of a relationship from two (x, y) values, including reading these from a table or from a graph; interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values; interpret functions that arise in application in terms of the context
• compare properties of two functions each represented among verbal, tabular, graphic, and algebraic representations of functions
• construct arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms; connect arithmetic sequences to linear functions and geometric sequences to exponential functions
• compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)
• construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)
• show and explain, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function
• determine an explicit expression and recursive process (steps for calculation) from context
• graph linear and quadratic functions and show intercepts, maxima, and minima as determined by the function or by context
• distinguish between situations that can be modeled with linear functions and with exponential functions
• graph functions expressed algebraically and show key features of the graph, both by hand and by using technology
• recognize situations in which one quantity changes at a constant rate per unit interval relative to another
• interpret the key characteristics of a function that models the relationship between two quantities, using tables, graphs, and verbal descriptions; sketch and analyze a graph showing key features, including intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity
• understand that a function is a rule that assigns to each input exactly one output; the graph of a function is the set of ordered pairs consisting of an input and corresponding output; represent domain and range using interval and set notation
• compare and contrast quadratic functions in standard, vertex, and intercept forms, using the process of factoring and completing the square in a quadratic function to show zeros, extreme values and symmetry of the graph, and interpret these in terms of a context
• recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another
• write a function defined by an expression in different, but equivalent, forms to reveal and explain different properties of the function
• graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude
• use the properties of exponents to interpret expressions for exponential functions and classify them as representing exponential growth and decay
• explain that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals
• construct a function that models a relationship between two quantities or contexts
• interpret the parameters in a linear function and an exponential function in terms of context; in context, describe what these parameters mean in terms of change and starting value
• interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear

## D - Geometry

• use volume formulas for cylinders, pyramids, cones, and spheres to solve problems in context
• apply concepts of density based on area and volume in modeling situations
• give an informal argument for the formulas for the circumference of a circle, for the area of a circle, and for the volume of a cylinder, pyramid, and cone, using dissection arguments, Cavalieri's principle, and informal limit arguments
• interpret and use coordinates to compute perimeters of polygons and areas of triangles and rectangles
• interpret and use coordinates to prove simple geometric theorems algebraically
• apply geometric methods to solve design problems with multiple representations

## E - Statistics and Probability

• recognize and explain the difference between correlation and causation
• interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data
• recognize and analyze the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each
• decide which type of function is most appropriate by observing graphed data or charted data, or by analysis of context; emphasize linear, quadratic and exponential models
• summarize categorical data for two categories in two-way frequency tables; interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies); analyze possible associations and trends in the data
• use statistics appropriate to the shape of the data distribution to compare and describe center (i.e., median, mean) and spread (i.e., interquartile range, mean absolute deviation, standard deviation) in context of two or more different data sets
• understand statistics as a process for making inferences about population parameters in context, based on a random sample from that population
• using a linear association model based on given or collected bivariate data, fit a linear function for a scatter plot that suggests a linear association
• represent and interpret data with plots on the real number line (i.e., dot plots, histograms, and box plots)
• interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers)
• represent data on two quantitative variables on a scatter plot, and describe how the variables are related in context
• compute (using technology) and interpret the correlation coefficient "r" of a linear fit; after calculating the line of best fit using technology, students should be able to describe how strong the goodness of fit of the regression is, using "r"

• # High School Mathematics -- Differential Equations

## A - First Order Differential Equations

• draw direction fields containing solution curves for first order differential equations by hand and using modeling software
• classify differential equation by type (i.e. ordinary/partial), order, and linearity
• solve first order differential equations and initial value problems using integrating factors
• partially differentiate functions of multiple variables as it pertains to Exact Equations for first order differential equations
• solve first order differential equations that apply to various real-world models including falling bodies, mixtures, population and the Logistic equation, continuously compounded interest, and other physics application
• draw and interpret real world solutions to first order differential equations using modeling software
• solve first order Exact Equations
• use modeling software to solve more complex first order differential equations
• solve separable differential equations for general solutions and initial value problems

## B - Second and Higher Order Differential Equations

• recognize the structure of solution sets to higher order linear differential equations, the basic Existence and Uniqueness Theorem, and the generalization of the Wronskian for higher order
• recognize the existence and uniqueness of solutions for second order linear differential equations and a fundamental set of solutions; verify that two solutions form a fundamental set by taking the Wronskian
• recognize systems of differential equations and the basic existence and uniqueness results for the corresponding initial value problems
• solve second order linear homogeneous and non-homogeneous equations by finding characteristic equations, using the method of undetermined coefficients and variation of parameters
• when given a solution to a non-homogeneous second order equation, find a second linearly dependent solution using reduction of order
• solve special case non-homogeneous second order ODE’s including Cauchy-Euler Equations
• solve second order differential equations that apply to various real-world models such as mass- spring systems, electric circuits, and economic growth
• solve higher order constant coefficient homogeneous equations
• determine whether a first or second order differential equation has a unique solution over a given interval by working with the Existence and Uniqueness Theorem
• use vector function notation when discussing the structure of solutions sets for homogeneous systems as it pertains to the Wronskian

## C - Systems of Differential Equations

• determine which non-linear systems are locally linear, and identify the systems’ behavior about each critical point
• apply various population models derived from locally linear systems including Lotka-Volterra, competition and cooperation models
• solve constant coefficient homogeneous systems using eigenvalues and eigenvectors; solve systems with real, distinct eigenvalues, as well as those with repeated and imaginary eigenvalues
• plot locally linear systems by hand and using modeling software
• draw Phase Portraits for solutions of homogeneous systems with constant coefficients by hand and using a modeling software
• solve non-homogeneous systems of ODE’s using the method of undetermined coefficients and variation of parameters

## D - Laplace Transforms

• recognize the general uniqueness and existence of solutions for Step functions, and will use the Laplace transform to find solutions to Step functions.
• find the Laplace transform of the Dirac Delta function
• discuss the main properties of the Laplace transform which make it useful for solving initial value problems
• use the integral definition to perform Laplace transforms for functions, such as, but not limited to polynomials, exponentials, and trigonometric functions; use a Laplace table to accurately and efficiently identify Laplace transforms, such as, but not limited to, the transforms for polynomials, exponentials, and trigonometric functions, and the product of these functions
• solve linear systems of differential equations using Laplace transforms
• write piecewise functions as compositions of Step (Heaviside) functions
• solve first and second order differential equations using Laplace transforms that apply to real world fields such as Electrical and Mechanical Engineering
• perform inverse Laplace transforms using a variety of techniques, such as but not limited to, algebraic manipulation partial fraction decomposition

## E - Series Solutions

• review Power Series as an introduction to series solutions of differential equations
• recognize ordinary points, recurrence relations, and changing indexes as it relates to series solutions to ODE’s
• find series solutions to first and second order non-linear initial value problems

## F - Mathematical Connections

• identify and describe the contribution of several key mathematicians and scientists to the field of differential equations

• # High School Mathematics -- Engineering Calculus

## A - Multidimensional Engineering Analysis

• learn to evaluate matrices and apply their properties to solve engineering problems; calculate determinants of matrices; express systems of linear equations in matrix equation form; use Gaussian elimination to compute solution sets of linear systems
• investigate functions of two and three independent variables to model engineering systems; compute limits of scalar and vector-valued functions; identify, interpret and graph level curves of multivariate functions; calculate regions of continuity of such functions
• apply knowledge of mathematics, science, and engineering design to solve problems; determine the equations of lines and surfaces using vectors and 3D graphing; apply dot and cross products of vectors to express equations of planes, parallelism, perpendicularity, angles; describe the role of vectors in engineering applications, such as modeling the velocity of moving objects or static forces on structures and objects
• use visual and written communication to express basic design elements in the appropriate mathematics notation; demonstrate fundamentals of technical sketching using computer- generated visuals by using the appropriate mathematics scale; present a technical design, using computer-generated model, for an assigned design project utilizing the appropriate scientific units (US standards and SI units)

## B - Differentiation In Engineering

• evaluate and apply partial differentiation of multivariable functions with two or more independent variables; compute the first and second partial derivatives of a function; use the general chain rule to determine the partial derivatives of composite functions; compute and apply the gradient of multivariable functions; solve engineering optimization problems by applying partial differentiation or Lagrange multipliers; utilize partial derivatives in developing the appropriate system balances (e.g., mass balance) in engineering problems

## C - Multidimensional Integration in Engineering Systems

• apply the techniques of double and triple integration to multivariable scalar- and vector-valued functions; manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates; evaluate and apply line integrals that are independent of path; apply properties of integrals to calculate and represent area, volume, or mass; use integrals of vectors to define and apply the gradient, divergence, or the curl e. Interpret the theorems of Green, Stokes, or Gauss and apply them to the study of real-world phenomena

• # High School Mathematics -- Geometry Strategies

## A - Geometry

• use congruence criteria for triangles to solve problems and to prove relationships in geometric figures
• use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent
• prove theorems about lines and angles (i.e., vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints)
• know and apply the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc
• understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles
• describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates
• use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder)
• informally prove the Pythagorean Theorem and its converse geometrically (i.e., using area model)
• use volume formulas for cylinders, pyramids, cones, and spheres to solve problems
• prove theorems about parallelograms (i.e., opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals)
• prove that all circles are similar (i.e., using transformations; ratio of circumference to the diameter is a constant)
• use informal arguments to establish facts about the angle sum and exterior angles of triangles and about the angles created when parallel lines are cut by a transversal
• explain and use the relationship between the sine and cosine of complementary angles
• use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems
• apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot)
• prove theorems about triangles (i.e., measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point)
• explain how the criteria for triangle congruence (i.e., ASA, SAS, SSS, HL, AAS) follow from the definition of congruence in terms of rigid motions
• use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula)
• prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point)
• prove theorems about triangles (i.e., a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity)
• use similarity criteria for triangles to solve problems and to prove relationships in geometric figures
• apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios)

## B - Statistics and Probability

• construct and interpret two-way frequency tables of data when two categories are associated with each object being classified; use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities (e.g., collect data from a random sample of students in your school on their favorite subject among math, science, and English; estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade; do the same for other subjects and compare the results)
• apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in context
• find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in context

• # High School Mathematics -- Mathematics of Finance

## A - Process Skills

• create and use pictures, manipulatives, models, and symbols to organize, record, and communicate mathematical ideas
• apply and adapt a variety of appropriate strategies to solve problems
• use the terminology and language of mathematics to express mathematical ideas precisely
• recognize reasoning and proof (evidence) as fundamental aspects of mathematics
• use appropriate technology to solve mathematical problems
• select and use various types of reasoning and methods of proof
• use representations to model and interpret physical, social, and mathematical phenomena
• recognize and apply mathematics in contexts outside of mathematics
• explain how mathematical ideas interconnect and build on one another to produce a coherent whole
• solve problems that arise in mathematics and in other areas
• recognize and use connections among mathematical ideas
• select, apply, and translate among mathematical representations to solve problems
• analyze and evaluate the mathematical thinking and strategies of others
• investigate, develop, and evaluate mathematical arguments and proofs
• monitor and reflect on the process of mathematical problem-solving
• make and investigate mathematical conjectures
• communicate mathematical thinking coherently to peers, teachers, and others
• organize and consolidate mathematics thinking
• build new mathematical knowledge through problem-solving

## B - Numbers and Operations

• use fractions, percents, and ratios to solve problems related to stock transactions, credit cards, taxes, budgets, automobile purchases, fuel economy, Social Security, Medicare, retirement planning, checking and saving accounts, and other related finance applications
• apply percent increase and decrease, ratios, and proportions

## C - Geometry

• apply the distance formula to trip planning
• apply the concepts of area, volume, scale factors, and scale drawings to planning for housing
• apply the properties of angles and segments in circles to accident investigation data

## E - Algebra

• apply linear, quadratic, and cubic functions
• apply greatest integer and piecewise functions
• use basic functions to solve and model problems related to stock transactions, banking and credit, employment and taxes, rent and mortgages, retirement planning, and other related finance applications
• understand domain and range when limited to a financial problem situation
• represent data and solve banking and retirement planning problems using matrices
• apply rational and square root functions
• evaluate investments in banking and retirement planning using simple and compound interest, and future and present value formulas
• apply limits as end behavior of modeling functions
• apply exponential and logarithmic functions

## F - Data Analysis and Probability

• use probability, the Monte Carlo method, and expected value model and predict outcomes related to the stock market, retirement planning, insurance, and investing
• draw conclusions about applied problems using decision theory
• recognize and interpret trends related to the stock market, retirement planning, insurance, car purchasing, and home rental or ownership using data displays including bar graphs, line graphs, stock bar charts, candlestick charts, box-and-whisker plots, stem and leaf plots, circle graphs, and scatter plots
• investigate data found in the stock market, retirement planning, transportation, budgeting, and home rental or ownership using measures of central tendency
• use linear, quadratic, and cubic regressions as well as the correlation coefficient to evaluate supply and demand, revenue, profit, and other financial problem situations

• # High School Mathematics -- Mathematics of Industry and Government

## A - Deterministic Decision Making

• determine optimal locations and use them to make appropriate decisions
• use advanced linear programming to make decisions
• determine optimal paths and use them to make appropriate decisions

## B - Probabilistic Decision Making

• use properties of other distributions (e.g., binomial, geometric, Poisson) to make decisions about optimization and efficiency
• make connections among mathematical ideas and to other disciplines
• solve problems (using appropriate technology)
• reason and evaluate mathematical arguments
• use computer simulations to make decisions
• represent mathematics in multiple ways
• use properties of normal distributions to make decisions about optimization and efficiency
• communicate mathematically
• will use other probabilistic models to make decisions

• # High School Mathematics -- Multivariable Calculus

## A - Algebra

• recognize and apply properties of matrices; find the determinants of 2-by-2 and 3-by-3 matrices; represent a 3-by-3 system of linear equations as a matrix and solve the system in multiple ways the inverse matrix, row operations, and Cramer's Rule; apply properties of similar and orthogonal matrices to prove statements about matrices; find and apply the eigenvectors and eigenvalues of a 3-by-3 matrix; determine if a given set is a vector space; determine whether a vector v is a linear combination of the vectors in S; express a vector in a linearly independent set as a linear combination of the vectors in the set; determine whether a given set of vectors span; determine whether a set of vectors is linearly independent or linearly dependent; show that a set of vectors is a basis for a vector space; find a basis for the null space, row space, and column space of a matrix; find the rank and nullity of a matrix
• investigate the relationship between points, lines, and planes in three-dimensions; represent equations of lines in space using vectors; express analytic geometry of three dimensions (equations of planes, parallelism, perpendicularity, angles) in terms of the dot product and cross product of vectors; . recognize conic sections and identify quadric surfaces
• explore functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0; evaluate such functions at a point in the plane; graph the level curves of such functions.; determine points or regions of discontinuity of such functions

## B - Derivatives

• explore the continuity of functions of two independent variables in terms of the limits of such functions as (x, y) approaches a given point in the plane
• explore, find, use, and apply partial differentiation of functions of two independent variables of the form z = f(x, y) and implicit functions of the form f(x, y, z) = 0; approximate the partial derivatives at a point of a function defined by a table of data; find expressions for the first and second partial derivatives of a function; define and apply the total differential to approximate real-world phenomena; represent the partial derivatives of a system of two functions in two variables using the Jacobian; find the partial derivatives of the composition of functions using the general chain rule; apply partial differentiation to problems of optimization, including problems requiring the use of the Lagrange multiplier; investigate the differential, tangent plane, and normal lines
• define and apply the gradient, the divergence, and curl in terms of differential vector operations

## C - Integration

• apply and interpret the theorems of Green, Stokes, and a. Apply line and surface integrals to functions representing real-world phenomena. b. Recognize, understand, and use line integrals that are independence of path. c. Define and apply the gradient, the divergence, and the curl in terms of integrals of vectors
• integrate functions of the form z = f(x, y) or w = f(x, y, z); define, use, and interpret double and triple integrals in terms of volume and mass; represent integrals of vectors as double and triple integrals; integrate functions through various techniques such as changing the order of integration, substituting variables, or changing to polar coordinates

## D - Differential Equations

• use, apply, and solve linear first-order differential equations; solve linear first-order differential equations of the form y' + p(x)y = q(x) with an integrating factor; solve homogeneous linear first- order differential equations using a variable substitution; solve Clairaut equations; explore the concepts of families of solutions and envelopes; write linear first-order differential equations that represent real-world phenomena and solve them, such as those arising from Kirchhoff's Law and mixing problems; students will solve linear second-order differential equations of the form y''+ p(x)y' + q(x)y = c using the characteristic equation where the characteristic equation has two real roots, one real root, or no real roots

• # High School Mathematics -- Number Theory

## A - Discrete Mathematics

• solve problems using concepts in graph theory including directed and undirected graphs, the Handshaking Theorem, isomorphisms, paths and path-connectedness, as well as Euler and Hamilton Paths
• apply counting principles, such as recurrence relations, Polya’s Enumeration Theorem, inclusion- exclusion, and the Pigeonhole principle
• apply game theory including Nash Equilibrium and two player zero sum games

## B - Logic

• apply quantifiers, conditionals, negations, contrapositives, converses, and inverses to determine the validity of logic statements
• apply quantifiers, conditionals, negations, contrapositives, converses, and inverses to determine the truth value of logical propositions, including, but not limited to, whether a proposition is a tautology, contradiction, or neither.
• represent logical operators such as AND,OR, NOT, NOR, and XOR in symbolic notation and use truth tables and in assessing logical equivalence
• apply modus ponens and modus tollens to determine the validity of logical arguments involving conditionals
• determine truth tables for sentences and use Venn diagrams to illustrate the relationships represented by these truth tables

## C - Set Theory

• determine if a relation is an equivalence relation on two sets by showing that the relation satisfies reflexive, symmetric, and transitive properties
• understand that equivalence classes form a partition on a set
• prove set relations, including DeMorgan’s Laws, proving a set is a subset of another set, and proving set equivalence
• recognize that a partition of a set is a collection of pairwise disjoint subsets
• describe sets using set builder notation; define, use notation of, and pictorially represent set theory components, including union, intersection, difference, element of, cardinality, complement, subset, and proper subset; define and determine the power set of a given set
• calculate the union, intersection, difference, and Cartesian product and Power of sets
• recognize that a function is a bijective (injective and surjective) relation on two sets, be able to prove or disprove that a relation is a function, and be able to determine the inverse of a function if it exists

## D - Proof Methods

• prove previously recognized mathematical theorems, such as but not limited to the Pythagorean Theorem, the Minimax Theorem, the Binomial Theorem, and Cantor’s Theorem
• differentiate between mathematical axioms, postulates, and theorems
• write theorems containing a hypothesis and conclusion; prove previously recognized mathematical theorems from various Set Theory and Number Theory concepts
• recognize and utilize appropriate methods of proof: direct proof, proof by mathematical induction (including the Principle of Mathematical Induction and the Second Principle of Mathematical Induction), proof by contradiction, proof by contraposition, proofs involving conditional and biconditional statements, proofs involving universal and existential quantifiers, and proof by counterexample

## E - Number Theory

• prove statements involving properties of numbers; prove that the square of any odd integer can be expressed at 8k+1 for some integer k; prove there are infinitely many primes; prove that the square root of 2 is irrational
• execute various primality tests to determine if large integers are prime; recognize certain prime numbers as Fermat Numbers or Mersenne Primes
• derive formulas for sums and products of series; derive definitions for various mathematical sequences, such as but not limited to triangular numbers and the Fibonacci numbers
• determine integral solutions to linear Diophantine equations
• apply Divisibility rules to base b number systems
• use mathematical induction to prove results about the natural numbers
• determine the modular inverse of a given integer for any positive integer modulus, if it exists
• define and utilize the greatest integer function to write rules to represent sequences
• find solutions to linear, polynomial, simultaneous, and systems of congruences, and prove results involving congruences and modular arithmetic
• use Fermat Factorization, Pollard Rho Factorization, and Pollard (p-1) Factorization to determine the GCD of two integers
• convert integers between a variety of number systems with different bases, including decimal, octal, binary, and hexadecimal
• apply the Chinese Remainder Theorem
• prove results involving divisibility and the greatest common divisor
• use congruences to prove Fermat’s Little Theorem and Wilson’s Theorem
• analyze basic cryptology including Character ciphers, Block ciphers, Hill ciphers, Stream ciphers, Exponentiation ciphers, and Knapsack ciphers
• define and explore concepts involving pseudoprimes
• apply congruences to several real world situations, including but not limited to creating a perpetual calendar, error detection in bit strings, and various types of hashing functions
• apply modular arithmetic concepts; apply the “divides” (a|b) relation to the natural numbers and “a (mod m)” for integers a and m
• explore Public Key Cryptography including the RSA cryptosystem
• explore various Prime conjectures, such as but not limited to Bertrand’s Conjecture, the Twin Prime Conjecture, the Legendre Conjecture, and the n²+1 Conjecture
• apply the Euclidean algorithm to determine the GCD of two integers

• # High School Mathematics -- Statistical Reasoning

## A - Statistics

• implement a reasonable random method for selecting a sample or for assigning treatments in an experiment
• randomly assign treatments to experimental subjects or objects
• analyze associations between variables and make predictions from one variable to another
• recognize that randomization reduces bias where bias occurs when certain outcomes are systematically more likely to appear
• using simulation, determine the appropriate model to decide if there is a significant difference between two treatment effects
• ask if the difference between two population parameters (or two treatment effects) is due to random variation or if the difference is statistically significant
• recognize that random selection from a population plays a different role than random assignment in an experiment
• describe the distribution for quantitative and categorical data
• analyze associations between two variables
• recognize a population distribution has fixed values of its parameters that are usually unknown
• implement a simple random sample
• formulate questions to clarify the problem at hand and formulate one (or more) questions that can be answered with data
• describe and interpret any outliers or gaps in the distribution
• make predictions and draw conclusions from two-variable data based on data displays
• distinguish between association and causation
• determine the type of study design appropriate for answering a statistical question
• analyze data by selecting appropriate graphical and numerical methods and using these methods to analyze the data
• describe and interpret patterns that exist for the distribution
• describe and interpret the measures of center for the distribution
• use distributions to identify the key features of the data collected
• create an appropriate simulated sampling distribution (using technology) and develop a alpha- value
• create two-way tables for two-variable categorical data
• create an appropriate simulated sampling distribution (using technology) and develop a margin of error
• use distributions to compare two or more groups
• determine if there are significant differences between two population parameters or treatment effects
• use graphical and numerical attributes of distributions to make comparisons between distributions
• identify the difference between categorical and quantitative (numerical) data
• analyze patterns and trends in data displays
• interpret results by interpreting the analysis and relating the interpretation to the original question
• create a sampling distribution of a statistic by taking repeated samples from a population (either hands-on or by simulation with technology)
• determine the type of data used to produce a given graphical display
• describe and interpret the shape of the distribution
• identify the three types of distributions
• distinguish the roles of randomization and sample size with designing studies
• distinguish between the role of randomness and the role of sample size with respect to using a statistic from a sample to estimate a population parameter
• identify whether the data are categorical or quantitative (numerical)
• distinguish between a population distribution, a sample data distribution, and a sampling distribution
• describe and interpret the modal category for the distribution
• create simulated sampling distributions and understand how to use the sampling distribution to make predictions about a population parameter(s) or the difference in treatment effects
• apply the statistical method to real-world situations
• create a sample data distribution by taking a sample from a defined population and summarizing the data in a distribution
• construct appropriate graphical displays of distributions
• create sample data distributions and a sampling distribution
• recognize a sample data distribution is taken from a population distribution and the data distribution is what is seen in practice hoping it approximates the population distribution
• determine the appropriate scope of inference for the study design used
• understand that randomness should be incorporated into a sampling or experimental procedure
• recognize a sampling distribution is the distribution of a sample statistic (such as a sample mean or a sample proportion) obtained from repeated samples; the sampling distribution provides the key for determining how close to expect a sample statistic approximates the population parameter
• understand that when randomness is incorporated into a sampling or experimental procedure, probability provides a way to describe the "long-run" behavior of a statistic as described by its sampling distribution
• distinguish between the three types of study designs for collecting data (i.e., sample survey, experiment, and observational study) and will know the scope of the interpretation for each design type
• compare two or more groups by analyzing distributions
• determine if an association exists between two variables (e.g., pattern or trend in bivariate data) and use values of one variable to predict values of another variable
• create scatter plots for two-variable numerical data
• describe and interpret the patterns in variability for the distribution
• using simulation, determine the appropriate model to decide if there is a significant difference between two populations
• recognize that sample size impacts the precision with which estimates of the population parameters can be made (i.e., larger the sample size the more precision)
• determine the appropriate graphical display for each type of data
• collect data by designing a plan to collect appropriate data and employ the plan to collect the data