# 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.*

## Kindergarten Mathematics

- Kindergarten Mathematics
- Kindergarten Mathematics -- Mathematics Specials
- Kindergarten - Enriched Mathematics

## Kindergarten Mathematics

## A - Numerical Reasoning - counting, money, place value, numbers to 20, addition, subtraction, and fluency

- demonstrate and explain the relationship between numbers and quantities up to 20; connect counting to cardinality (the last number counted represents the total quantity in a set)
- use count sequences within 100 to count forward and backward in sequence
- use place value understanding to compose and decompose numbers from 11-19
- identify, write, and represent numbers to 20; compare two sets of up to 10 objects each
- explain the concepts of addition, subtraction, and equality and use these concepts to solve real-life problems within 10

## B - Patterning & Algebraic Reasoning - repeating patterns and time

- explain, extend, and create repeating patterns with a repetition, not exceeding four; describe patterns involving the passage of time

## C - Measurement & Data Reasoning - attributes of objects, classifying objects

- observe, describe, and compare the physical and measurable attributes of objects and analyze graphical displays of data to answer relevant questions

## D - Geometric & Spatial Reasoning - 2D and 3D shapes, relative locations, attributes

- identify, describe, and compare basic shapes encountered in the environment; form two-dimensional shapes and three-dimensional figures

## Kindergarten Mathematics -- Mathematics Specials

## A - Numerical Reasoning - counting, money, place value, numbers to 20, addition, subtraction, and fluency

- demonstrate and explain the relationship between numbers and quantities up to 20; connect counting to cardinality (the last number counted represents the total quantity in a set)
- use count sequences within 100 to count forward and backward in sequence
- use place value understanding to compose and decompose numbers from 11-19
- identify, write, and represent numbers to 20; compare two sets of up to 10 objects each
- explain the concepts of addition, subtraction, and equality and use these concepts to solve real-life problems within 10

## B - Patterning & Algebraic Reasoning - repeating patterns and time

- explain, extend, and create repeating patterns with a repetition, not exceeding four; describe patterns involving the passage of time

## C - Measurement & Data Reasoning - attributes of objects, classifying objects

- observe, describe, and compare the physical and measurable attributes of objects and analyze graphical displays of data to answer relevant questions

## Kindergarten - Enriched Mathematics

## 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?")

## 1st Grade -- Mathematics

## 1st Grade Mathematics

## A - Numerical Reasoning - counting, numbers, equality, place value, addition, subtraction

- extend the count sequence to 120; read, write, and represent numerical values to 120; compare numerical values to 100
- solve real-life addition and subtraction problems within 20; explain the relationship between addition and subtraction and apply the properties of operations

## B - Patterning & Algebraic Reasoning - repeating patterns, growing patterns, and shrinking patterns

- identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns found in real-life situations

## C - Geometric & Spatial Reasoning - shapes, attributes, partitions of circles and rectangles

- identify and compose shapes, analyze the attributes of shapes, and relate their parts to the whole

## D - Numerical Reasoning - base ten structure, addition and subtraction within 100

- use concrete models, the base ten structure, and properties of operations to add and subtract within 100

## E - Measurement & Data Reasoning - length, time, money

- use appropriate tools to measure, order, and compare intervals of length and time, as well as denominations of money to solve real-life, mathematical problems and analyze graphical displays of data to answer relevant questions

## 1st Grade Mathematics -- Mathematics Specials

## A - Numerical Reasoning - counting, numbers, equality, place value, addition, subtraction

- extend the count sequence to 120; read, write, and represent numerical values to 120; compare numerical values to 100
- solve real-life addition and subtraction problems within 20; explain the relationship between addition and subtraction and apply the properties of operations

## B - Patterning & Algebraic Reasoning - repeating patterns, growing patterns, and shrinking patterns

- identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns found in real-life situations

## D - Numerical Reasoning - base ten structure, addition and subtraction within 100

- use concrete models, the base ten structure, and properties of operations to add and subtract within 100

## E - Measurement & Data Reasoning - length, time, money

- use appropriate tools to measure, order, and compare intervals of length and time, as well as denominations of money to solve real-life, mathematical problems and analyze graphical displays of data to answer relevant questions

## 2nd Grade -- Mathematics

## 2nd Grade Mathematics

## A - Numerical Reasoning - counting within 1000, place value, addition and subtraction, fluency to 20, developing multiplication through arrays

- use the place value structure when exploring the count sequences to represent, read, write, and compare numerical values to 1000; describe basic place-value relationships and structures
- apply multiple part-whole strategies, properties of operations and place value understanding to solve real-life, mathematical problems involving addition and subtraction within 1000
- work with equal groups to gain foundations for multiplication through real-life, mathematical problems

## B - Patterning & Algebraic Reasoning - patterns up to 20 and addition and subtraction within 1,000

- identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns

## C - Measurement & Data Reasoning - length, distance, time, and money

- estimate and measure the lengths of objects and distance to solve problems found in real-life using standard units of measurement, including inches, feet, and yards and analyze graphical displays of data to answer relevant questions
- solve real-life problems involving time and money

## D - Geometric & Spatial Reasoning - sorting shapes, lines of symmetry, partitioning circles and rectangles

- draw and partition shapes and other objects with specific attributes and conduct observations of everyday items and structures to identify how shapes exist in the world

## 2nd Grade Mathematics -- Mathematics Specials

## A - Numerical Reasoning - counting within 1000, place value, addition and subtraction, fluency to 20, developing multiplication through arrays

- use the place value structure when exploring the count sequences to represent, read, write, and compare numerical values to 1000; describe basic place-value relationships and structures
- apply multiple part-whole strategies, properties of operations and place value understanding to solve real-life, mathematical problems involving addition and subtraction within 1000

## B - Patterning & Algebraic Reasoning - patterns up to 20 and addition and subtraction within 1,000

- identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns

## C - Measurement & Data Reasoning - length, distance, time, and money

- estimate and measure the lengths of objects and distance to solve problems found in real-life using standard units of measurement, including inches, feet, and yards and analyze graphical displays of data to answer relevant questions

## D - Geometric & Spatial Reasoning - sorting shapes, lines of symmetry, partitioning circles and rectangles

- draw and partition shapes and other objects with specific attributes and conduct observations of everyday items and structures to identify how shapes exist in the world

## 3rd Grade -- Mathematics

## 3rd Grade Mathematics

## A - Numerical Reasoning - base ten numerals and place value up to 10,000, and rounding up to 1000

- use place value reasoning to represent, read, write, and compare numerical values up to 10,000 and round whole numbers up to 1000

## B - Patterning & Algebraic Reasoning - fluency, addition and subtraction within 10,000, multiplication and division within 100, equality, properties of operations

- use part-whole strategies to represent and solve real-life problems involving addition and subtraction with whole numbers within 10,000
- use part-whole strategies to solve real life mathematical problems involving multiplication and division with whole numbers within 100

## C - Numerical Reasoning - unit fractions, equivalent fractions, fractions greater than 1

- represent fractions with denominators of two, three, four, six, and eight in multiple ways within a framework using visual models (i.e., area models, set models, linear models)

## D - Measurement & Data Reasoning - elapsed time, liquid volume, mass, lengths in half and fourth of an inch, and data

- solve real-life mathematical problems involving length, liquid volume, mass, and time and analyze graphical displays of data to answer relevant questions

## E - Geometric & Spatial Reasoning - polygons, parallel line segments, perpendicular line segments, right angles, lines of symmetry, area, and perimeter

- identify the attributes of polygons, including parallel segments, perpendicular segments, right angles, and symmetry
- identify area as a measurable attribute of rectangles and determine the area of a rectangle presented in real-life mathematical problems
- determine the perimeter of a polygon presented in real-life mathematical problems

## 3rd Grade Mathematics -- Mathematics Specials

## B - Patterning & Algebraic Reasoning - fluency, addition and subtraction within 10,000, multiplication and division within 100, equality, properties of operations

- use part-whole strategies to represent and solve real-life problems involving addition and subtraction with whole numbers within 10,000
- use part-whole strategies to solve real life mathematical problems involving multiplication and division with whole numbers within 100

## C - Numerical Reasoning - unit fractions, equivalent fractions, fractions greater than 1

- represent fractions with denominators of two, three, four, six, and eight in multiple ways within a framework using visual models (i.e., area models, set models, linear models)

## D - Measurement & Data Reasoning - elapsed time, liquid volume, mass, lengths in half and fourth of an inch, and data

- solve real-life mathematical problems involving length, liquid volume, mass, and time and analyze graphical displays of data to answer relevant questions

## E - Geometric & Spatial Reasoning - polygons, parallel line segments, perpendicular line segments, right angles, lines of symmetry, area, and perimeter

- identify area as a measurable attribute of rectangles and determine the area of a rectangle presented in real-life mathematical problems

## 4th Grade -- Mathematics

## 4th Grade Mathematics

## A - Numerical Reasoning - place value, rounding, comparisons with multi-digit numbers, addition and subtraction, multiplicative comparisons, multiplication, and division involving whole numbers

- recognize patterns within the base ten place value system with quantities presented in real-life situations to compare and round multi-digit whole numbers through the hundred-thousands place
- solve problems involving addition and subtraction through the hundred-thousands place, as well as multiplication and division of multi-digit whole numbers presented in real-life, mathematical situations using part-whole strategies

## B - Patterning & Algebraic Reasoning - patterns, input-output tables, factors, multiples, composite numbers, prime numbers

- generate and analyze patterns, including those involving shapes, input/output diagrams, factors, multiples, prime numbers, and composite numbers

## C - Numerical Reasoning - fraction equivalence, comparison of fractions, and addition and subtraction of fractions with like denominators

- solve real-life problems involving addition, subtraction, equivalence, and comparison of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100 using part-whole strategies and visual models
- solve real-life problems involving addition, equivalence, comparison of fractions with denominators of 10 and 100, and comparison of decimal numbers as tenths and hundredths using part-whole strategies and visual models

## D - Measurement & Data Reasoning - time, metric measurements, distance, elapsed time, liquid volume, mass, and length

- measure time and objects that exist in the world to solve real-life, mathematical problems and analyze graphical displays of data to answer relevant questions

## E - Geometric & Spatial Reasoning - polygons, points, lines, line segments, rays, angles, perpendicular lines, area, perimeter

- investigate the concepts of angles and angle measurement to estimate and measure angles
- identify and draw geometric objects, classify polygons based on properties, and solve problems involving area and perimeter of rectangular figures

## 4th Grade Mathematics -- Mathematics Specials

## A - Numerical Reasoning - place value, rounding, comparisons with multi-digit numbers, addition and subtraction, multiplicative comparisons, multiplication, and division involving whole numbers

- recognize patterns within the base ten place value system with quantities presented in real-life situations to compare and round multi-digit whole numbers through the hundred-thousands place
- solve problems involving addition and subtraction through the hundred-thousands place, as well as multiplication and division of multi-digit whole numbers presented in real-life, mathematical situations using part-whole strategies

## C - Numerical Reasoning - fraction equivalence, comparison of fractions, and addition and subtraction of fractions with like denominators

- solve real-life problems involving addition, subtraction, equivalence, and comparison of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100 using part-whole strategies and visual models
- solve real-life problems involving addition, equivalence, comparison of fractions with denominators of 10 and 100, and comparison of decimal numbers as tenths and hundredths using part-whole strategies and visual models

## D - Measurement & Data Reasoning - time, metric measurements, distance, elapsed time, liquid volume, mass, and length

- measure time and objects that exist in the world to solve real-life, mathematical problems and analyze graphical displays of data to answer relevant questions

## E - Geometric & Spatial Reasoning - polygons, points, lines, line segments, rays, angles, perpendicular lines, area, perimeter

- identify and draw geometric objects, classify polygons based on properties, and solve problems involving area and perimeter of rectangular figures

## 5th Grade -- Mathematics

## 5th Grade Mathematics

## A - Numerical Reasoning - place value, multiplying by powers of 10, multiplication and division of multi-digit numbers, fractions, decimal numbers, numerical expressions

- use place value understanding to solve real-life, mathematical problems
- multiply and divide multi-digit whole numbers to solve relevant, mathematical problems
- describe fractions and perform operations with fractions to solve relevant, mathematical problems using part-whole strategies and visual models
- read, write, and compare decimal numbers to the thousandths place; round decimal numbers to the hundredths place; add and subtract with decimal numbers to the hundredths place to solve relevant, mathematical problems
- write, interpret, and evaluate numerical expressions within authentic problems

## B - Patterning & Algebraic Reasoning - generating patterns, plotting ordered pairs in the first quadrant

- solve relevant problems by creating and analyzing numerical patterns using the given rule(s)

## C - Measurement & Data Reasoning - measurements within the metric system, measurement conversions and time as a unit of measurement

- solve problems involving customary measurements, metric measurements, and time and analyze graphical displays of data to answer relevant questions

## D - Geometric & Spatial Reasoning - properties of polygons and rectangular prisms, classify polygons

- examine properties of polygons (e.g., triangles, quadrilaterals including kites, trapezoids, rectangles, squares, rhombuses, other parallelograms, pentagons, hexagons, octagons) and rectangular prisms; classify polygons by their properties; discover volume of right rectangular prisms

## 5th Grade Mathematics -- Mathematics Specials

## A - Numerical Reasoning - place value, multiplying by powers of 10, multiplication and division of multi-digit numbers, fractions, decimal numbers, numerical expressions

- use place value understanding to solve real-life, mathematical problems
- multiply and divide multi-digit whole numbers to solve relevant, mathematical problems
- describe fractions and perform operations with fractions to solve relevant, mathematical problems using part-whole strategies and visual models
- read, write, and compare decimal numbers to the thousandths place; round decimal numbers to the hundredths place; add and subtract with decimal numbers to the hundredths place to solve relevant, mathematical problems

## D - Geometric & Spatial Reasoning - properties of polygons and rectangular prisms, classify polygons

- examine properties of polygons (e.g., triangles, quadrilaterals including kites, trapezoids, rectangles, squares, rhombuses, other parallelograms, pentagons, hexagons, octagons) and rectangular prisms; classify polygons by their properties; discover volume of right rectangular prisms

## 6th Grade -- Mathematics

- 6th Grade Mathematics
- 6th Grade Mathematics -- Accelerated Mathematics
- 6th Grade Mathematics -- Mathematics Connection

## 6th Grade Mathematics

## A - Numerical Reasoning

- solve relevant, mathematical problems involving operations with whole numbers, fractions, and decimal numbers
- design, collect, model, and analyze data distributions to answer statistical questions and solve relevant, mathematical problems
- solve a variety of problems involving integers; model rational numbers on a number line to describe problems presented in relevant, mathematical situations
- solve a variety of contextual problems involving ratios, unit rates, equivalent ratios, percentages, and conversions within measurement systems using proportional reasoning

## B - Geometric & Spatial Reasoning

- solve relevant, mathematical problems involving area of polygons, surface area of 3-D figures, and volume of right rectangular prisms

## C - Patterning & Algebraic Reasoning

- identify, write, evaluate, and interpret numerical and algebraic expressions as mathematical models to explain relevant, mathematical problems
- write and solve one-step equations and inequalities to explain authentic, realistic scenarios
- graph rational numbers as points on the coordinate plane to represent and solve contextual, mathematical problems; draw polygons using the coordinates for their vertices and find the length of a side of a polygon

## 6th Grade Mathematics -- Accelerated Mathematics

## A - Numerical Reasoning

- solve relevant, mathematical problems involving operations with whole numbers, fractions, and decimal numbers (e.g., proper fractions, improper fractions, mixed numbers)
- design, collect, model, and analyze data distributions to answer statistical questions and solve relevant, mathematical problems
- solve a variety of problems involving integers; model rational numbers on a number line to describe problems presented in relevant, mathematical situations
- solve a variety of contextual problems involving ratios, unit rates, equivalent ratios, percentages, and conversions within measurement systems using proportional reasoning
- solve relevant, mathematical problems involving the four operations with rational numbers in any form (i.e., integers, percentages, fractions, and decimal numbers)

## B - Geometric & Spatial Reasoning

- solve relevant, mathematical problems involving area of polygons, surface area of 3-D figures, and volume of right rectangular prisms

## C - Patterning & Algebraic Reasoning

- identify, write, evaluate, and interpret numerical and algebraic expressions as mathematical models to explain relevant, mathematical problems
- write and solve one-step equations and inequalities to explain authentic, realistic scenarios
- graph rational numbers as points on the coordinate plane to represent and solve contextual, mathematical problems; draw polygons using the coordinates for their vertices and find the length of a side of a polygon
- use properties of operations to generate equivalent expressions and interpret the expressions to explain relevant situations
- represent authentic situations using equations and inequalities with variables; solve equations and inequalities symbolically, using the properties of equality
- recognize proportional relationships in relevant, mathematical problems; represent, solve, and explain these relationships with tables, graphs, verbal descriptions, and equations

## 6th Grade Mathematics -- Mathematics Connection

## A - Numerical Reasoning

- solve relevant, mathematical problems involving operations with whole numbers, fractions, and decimal numbers
- design, collect, model, and analyze data distributions to answer statistical questions and solve relevant, mathematical problems
- solve a variety of problems involving integers; model rational numbers on a number line to describe problems presented in relevant, mathematical situations
- solve a variety of contextual problems involving ratios, unit rates, equivalent ratios, percentages, and conversions within measurement systems using proportional reasoning

## B - Geometric & Spatial Reasoning

- solve relevant, mathematical problems involving area of polygons, surface area of 3-D figures, and volume of right rectangular prisms

## C - Patterning & Algebraic Reasoning

- identify, write, evaluate, and interpret numerical and algebraic expressions as mathematical models to explain relevant, mathematical problems
- write and solve one-step equations and inequalities to explain authentic, realistic scenarios
- graph rational numbers as points on the coordinate plane to represent and solve contextual, mathematical problems; draw polygons using the coordinates for their vertices and find the length of a side of a polygon

## 7th Grade -- Mathematics

- 7th Grade Mathematics
- 7th Grade Mathematics -- 7th Grade Accelerated Mathematics
- 7th Grade Mathematics -- 7th Mathematics Connection

## 7th Grade Mathematics

## A - Numerical Reasoning

- solve relevant, mathematical problems involving the four operations with rational numbers in any form (i.e., integers, percentages, fractions, and decimal numbers), including problems that involve multiple steps

## B - Patterning & Algebraic Reasoning

- use properties of operations to generate equivalent expressions and interpret the expressions to explain relevant situations
- represent authentic situations using equations and inequalities with variables; solve equations and inequalities symbolically, using the properties of equality
- recognize proportional relationships in relevant, mathematical problems; represent, solve, and explain these relationships with tables, graphs, verbal descriptions, and equations
- design, collect, model, and analyze data from a representative sample to answer statistical questions and solve relevant, mathematical problems

## C - Geometric & Spatial Reasoning

- solve practical problems involving angle measurement, circles, area of circles, surface area of prisms and cylinders, and volume of cylinders and prisms composed of cubes and right prisms

## D - Probability Reasoning

- investigate chance processes and develop, evaluate, and use probability models to find probabilities of simple events presented in authentic situations

## 7th Grade Mathematics -- 7th Grade Accelerated Mathematics

## A - Numerical Reasoning

- solve relevant, multi-step mathematical problems involving the four operations with rational numbers in any form (i.e., integers, percentages, fractions, and decimal numbers)

## B - Patterning & Algebraic Reasoning

- use properties of operations to generate equivalent expressions and interpret the expressions to explain relevant situations
- create and interpret expressions within realistic situations; create, interpret, and solve linear equations and linear inequalities in one variable to model and explain real-life phenomena
- recognize proportional relationships in relevant, mathematical problems; represent, solve, and explain these relationships with tables, graphs, verbal descriptions, and equations
- show and explain the connections between proportional and non-proportional relationships and linear equations; create and interpret graphical, mathematical models and use the graphical, mathematical model to explain real-life phenomena represented in the graph
- design, collect, model, and analyze data from a representative sample to answer statistical questions and solve relevant, mathematical problems

## C - Functional & Graphical Reasoning

- describe the properties of functions to define, evaluate, and compare relationships; model and explain real-life phenomena using linear functions
- use bivariate quantitative data to solve relevant, linear problems

## D - Geometric & Spatial Reasoning

- solve practical problems involving angle measurement, circles, area of circles, surface area of prisms and cylinders, and volume of cylinders and prisms composed of cubes and right prisms
- solve problems involving the Pythagorean Theorem and the volume of cones, cylinders, and spheres to explain real-life phenomena

## E - Probability Reasoning

- investigate chance processes and develop, evaluate, and use probability models to find probabilities of simple events presented in authentic situations

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

## A - Numerical Reasoning

- solve relevant, mathematical problems involving the four operations with rational numbers in any form (i.e., integers, percentages, fractions, and decimal numbers), including problems that involve multiple steps

## B - Patterning & Algebraic Reasoning

- represent authentic situations using equations and inequalities with variables; solve equations and inequalities symbolically, using the properties of equality
- design, collect, model, and analyze data from a representative sample to answer statistical questions and solve relevant, mathematical problems

## C - Geometric & Spatial Reasoning

- solve practical problems involving angle measurement, circles, area of circles, surface area of prisms and cylinders, and volume of cylinders and prisms composed of cubes and right prisms

## D - Probability Reasoning

- investigate chance processes and develop, evaluate, and use probability models to find probabilities of simple events presented in authentic situations

## 8th Grade -- Mathematics

- 8th Grade Mathematics
- 8th Grade Mathematics -- Mathematics Connection
- 8th Grade Mathematics -- Enhanced Algebra: Concepts & Connections
- 8th Grade Mathematics -- Foundations of Algebra

## 8th Grade Mathematics

## A - Numerical Reasoning

- solve mathematical problems involving irrational numbers and rational approximations of irrational numbers to explain realistic applications
- solve relevant, mathematical problems involving radicals and integer exponents; apply place value understanding with scientific notation and use scientific notation to explain real-life phenomena

## B - Patterning & Algebraic Reasoning

- create and interpret expressions within realistic situations; create, interpret, and solve linear equations and linear inequalities in one variable to model and explain real-life phenomena
- show and explain the connections between proportional and non-proportional relationships and linear equations; create and interpret graphical, mathematical models and use the graphical, mathematical model to explain real-life phenomena represented in the graph

## C - Functional & Graphical Reasoning

- describe the properties of functions to define, evaluate, and compare relationships; model and explain real-life phenomena using linear functions
- use bivariate quantitative data to solve relevant, linear problems
- justify and use various strategies to solve systems of linear equations to model and explain realistic phenomena

## D - Geometric & Spatial Reasoning

- solve problems involving the Pythagorean Theorem and the volume of cones, cylinders, and spheres to explain real-life phenomena

## 8th Grade Mathematics -- Mathematics Connection

## A - Numerical Reasoning

- solve mathematical problems involving irrational numbers and rational approximations of irrational numbers to explain realistic applications
- solve relevant, mathematical problems involving radicals and integer exponents; apply place value understanding with scientific notation and use scientific notation to explain real-life phenomena

## B - Patterning & Algebraic Reasoning

- create and interpret expressions within realistic situations; create, interpret, and solve linear equations and linear inequalities in one variable to model and explain real-life phenomena
- show and explain the connections between proportional and non-proportional relationships and linear equations; create and interpret graphical, mathematical models and use the graphical, mathematical model to explain real-life phenomena represented in the graph

## C - Functional & Graphical Reasoning

- describe the properties of functions to define, evaluate, and compare relationships; model and explain real-life phenomena using linear functions
- use bivariate quantitative data to solve relevant, linear problems
- justify and use various strategies to solve systems of linear equations to model and explain realistic phenomena

## D - Geometric & Spatial Reasoning

- solve problems involving the Pythagorean Theorem and the volume of cones, cylinders, and spheres to explain real-life phenomena

## 8th Grade Mathematics -- Enhanced Algebra: Concepts & Connections

## A - Patterning & Algebraic Reasoning

- create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena
- build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations
- create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations

## B - Functional & Graphical Reasoning

- construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena; use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and non-linear functions using parent graphs
- justify and use various strategies to solve systems of linear equations to model and explain realistic phenomena
- construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a mathematically applicable situation for which the graph serves as a model
- construct and analyze the graph of an exponential function to explain a mathematically applicable situation for which the graph serves as a model; compare exponential with linear and quadratic functions
- use bivariate quantitative data to solve relevant, linear problems

## C - Numerical Reasoning

- solve mathematical problems involving irrational numbers and rational approximations of irrational numbers to explain realistic applications
- solve relevant, mathematical problems involving radicals and integer exponents; apply place value understanding with scientific notation and use scientific notation to explain real-life phenomena
- investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots

## D - Data & Statistical Reasoning

- collect, analyze, and interpret univariate quantitative data to answer statistical investigative questions that compare groups to solve real-life problems; represent bivariate data on a scatter plot and fit a function to the data to answer statistical questions and solve real-life problems

## E - Geometric & Spatial Reasoning

- solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena

## 8th Grade Mathematics -- Foundations of Algebra

## A - Numerical Reasoning

- solve relevant, mathematical problems involving operations with whole numbers, fractions, and decimal numbers
- solve relevant, mathematical problems involving the four operations with rational numbers in any form (i.e., integers, percentages, fractions, and decimal numbers), including problems that involve multiple steps

## B - Patterning & Algebraic Reasoning

## C - Functional & Graphical Reasoning

## D - Geometric & Spatial Reasoning

## High School Mathematics

- High School Mathematics -- Accelerated Geometry: Concepts & Connections
- High School Mathematics -- Advanced Algebra: Concepts & Connections
- High School Mathematics -- Advanced Algebra: Concepts & Connections Strategies
- High School Mathematics -- Advanced Calculus II
- High School Mathematics -- Advanced Financial Algebra
- High School Mathematics -- Advanced Mathematical Decision Making
- High School Mathematics -- Algebra: Concepts & Connections
- High School Mathematics -- Algebra: Concepts & Connections Strategies
- High School Mathematics -- Applications of Linear Algebra in Computer Science
- High School Mathematics -- Calculus (Non-AP)
- High School Mathematics -- College Readiness Mathematics (Mathematics Capstone Course)
- High School Mathematics -- Differential Equations
- High School Mathematics -- Engineering Calculus
- High School Mathematics -- Enhanced Advanced Algebra and Precalculus: Concepts and Connections
- High School Mathematics -- Geometry: Concepts & Connections
- High School Mathematics -- Geometry: Concepts & Connections Strategies
- High School Mathematics -- Mathematics of Industry and Government
- High School Mathematics -- Multivariable Calculus
- High School Mathematics -- Number Theory
- High School Mathematics -- Precalculus
- High School Mathematics -- Statistical Reasoning

## High School Mathematics -- Accelerated Geometry: Concepts & Connections

## A - Patterning & Algebraic Reasoning

- interpret the structure of and perform operations with polynomials within a geometric context

## B - Geometric & Spatial Reasoning

- experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena
- establish facts between angle relations and generate valid arguments to prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena
- use center and scale factor to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena
- examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena
- explore the concept of a radian measure and special right triangles
- examine and apply theorems involving circles; describe and derive arc length and area of a sector; model and explain real-life frameworks involving circles
- develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena

## C - Probabilistic Reasoning

- solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions

## D - Data & Statistical Reasoning

- examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena
- communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data

## E - Algebraic & Geometric Reasoning

- manipulate, prove, and apply trigonometric identities and equations to solve contextual, mathematical problems

## High School Mathematics -- Advanced Algebra: Concepts & Connections

## A - Data & Statistical Reasoning

- communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data

## B - Functional & Graphical Reasoning

- explore and analyze structures, patterns, and inverse relationships for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena
- explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena
- extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-life phenomena
- analyze the behaviors of rational functions to model applicable, mathematical problems

## F - Patterning & Algebraic Reasoning

- represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications

## G - Geometric & Spatial Reasoning

- develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle

## High School Mathematics -- Advanced Algebra: Concepts & Connections Strategies

## A - Data & Statistical Reasoning

- communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data

## B - Functional & Graphical Reasoning

- explore and analyze structures, patterns, and inverse relationships for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena
- explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena
- extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-life phenomena
- analyze the behaviors of rational functions to model applicable, mathematical problems

## C - Patterning & Algebraic Reasoning

- represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications

## D - Geometric & Spatial Reasoning

- develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle

## High School Mathematics -- Advanced Calculus II

## A - Abstract & Quantitative Reasoning

- interpret integrals of functions of one independent variable to solve contextual situations and explain real-life phenomena

## B - Geometric & Spatial Reasoning

- apply Calculus to polar and parametric equations within a geometric context to explain real-life phenomena
- express functional relationships with vectors in two dimensions and use these relationships to explain real-life phenomena
- express spatial relationships with vectors and planes in three dimensions and use these relationships to explain real-life phenomena
- interpret vector functions using contextual situations to explain real-life phenomena

## C - Patterning & Algebraic Reasoning

- express spatial and functional relationships with infinite sequences
- express spatial and functional relationships with infinite series

## High School Mathematics -- Advanced Financial Algebra

## A - Quantitative Reasoning

- utilize fractions, decimals, percents, and ratios to write and solve a variety of financial problems

## B - Functional & Graphical Reasoning

- explore and apply functions (i.e., linear, exponential, quadratic, cubic, rational, square root, greatest integer, piecewise) to model and explain real-life phenomena and to solve complex problems in business and financial contexts

## C - Patterning & Algebraic Reasoning

- explore, evaluate, and rearrange formulas applicable to business and financial contexts
- write and solve systems of equations and inequalities in context of financial applications

## E - Geometric & Spatial Reasoning

- apply properties of polygons, circles, and trigonometry to model and explore real-world applications

## F - Data & Statistical Reasoning

- collect, analyze, interpret, summarize, and construct displays of data to make predictions within real-world applications
- conduct investigative research to solve real-life problems and answer statistical questions involved in business and financial decision-making

## High School Mathematics -- Advanced Mathematical Decision Making

## A - Quantitative and Proportional Reasoning

- make decisions and solve problems using ratios, rates, and percents in a variety of real-world applications
- predict potential outcomes by analyzing averages and indices of large data sets through investigations of real-world contexts

## B - Patterning and Algebraic Reasoning

- develop methods or algorithms to analyze discrete situations
- create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money
- use vectors and matrices to model and solve real-life situations
- make informed decisions and solve problems with a variety of network models in quantitative situations

## C - Probabilistic Reasoning

- analyze the probability of success or failure in order to make decisions
- rationalize decisions based on probabilities and risk of loss and gain of real-life situations

## D - Data and Statistical Reasoning

- conduct investigative research to solve real-life problems and answer statistical investigative questions involved in business and financial decision-making

## E - Functional and Graphical Reasoning

- use functions to model problem situations in both discrete and continuous relationships

## F - Geometric and Spatial Reasoning

- use functions to model trigonometric problems

## High School Mathematics -- Algebra: Concepts & Connections

## A - Functional & Graphical Reasoning

- construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena; use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and non-linear functions using parent graphs
- construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a mathematically applicable situation for which the graph serves as a model
- construct and analyze the graph of an exponential function to explain a mathematically applicable situation for which the graph serves as a model; compare exponential with linear and quadratic functions

## B - Geometric & Spatial Reasoning

- solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena

## C - Patterning & Algebraic Reasoning

- create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena
- build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations
- create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations

## D - Numerical Reasoning

- investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots

## E - Data & Statistical Reasoning

- collect, analyze, and interpret univariate quantitative data to answer statistical investigative questions that compare groups to solve real-life problems; represent bivariate data on a scatter plot and fit a function to the data to answer statistical questions and solve real-life problems

## High School Mathematics -- Algebra: Concepts & Connections Strategies

## A - Functional & Graphical Reasoning

- construct and interpret arithmetic sequences as functions, algebraically and graphically, to model and explain real-life phenomena; use formal notation to represent linear functions and the key characteristics of graphs of linear functions, and informally compare linear and non-linear functions using parent graphs
- construct and interpret quadratic functions from data points to model and explain real-life phenomena; describe key characteristics of the graph of a quadratic function to explain a mathematically applicable situation for which the graph serves as a model
- construct and analyze the graph of an exponential function to explain a mathematically applicable situation for which the graph serves as a model; compare exponential with linear and quadratic functions

## B - Geometric & Spatial Reasoning

- solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena

## C - Patterning & Algebraic Reasoning

- create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena
- build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations
- create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations

## D - Numerical Reasoning

- investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots

## E - Data & Statistical Reasoning

- collect, analyze, and interpret univariate quantitative data to answer statistical investigative questions that compare groups to solve real-life problems; represent bivariate data on a scatter plot and fit a function to the data to answer statistical questions and solve real-life problems

## 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 -- Calculus (Non-AP)

## A - Functional & Graphical Reasoning

- apply limit notation and characteristics of continuity to analyze behaviors of functions

## B - Algebraic & Graphical Reasoning

- relate limits and continuity to the derivative as a rate of change and apply it to a variety of situations including modeling contexts
- apply derivatives to situations in order to draw conclusions including curve analysis and modeling rates of change in applications

## C - Geometric & Algebraic Reasoning

- analyze the relationship between the derivative and the integral using the Fundamental Theorem of Calculus

## D - Patterning & Algebraic Reasoning

- apply the definite integral and indefinite integral to contextual situations

## High School Mathematics -- College Readiness Mathematics (Mathematics Capstone Course)

## A - Numerical & Quantitative Reasoning

- utilize exact and approximate calculations to quantify real-world phenomena and solve problems

## B - Patterning & Algebraic Reasoning

- construct expressions, equations, and inequalities, and use them to represent and solve problems by choosing appropriate procedures and interpreting solutions in context

## C - Functional & Graphical Reasoning

- define, build, and interpret functions that arise in various contexts by applying knowledge of the characteristics of the different families of functions, and analyze the effects of parameters

## D - Geometric & Spatial Reasoning

- reason deductively and inductively about figures and their properties and make sense of geometric situations using measurements in real-world contexts

## E - Data & Statistical Reasoning

- make sense of and reason about variation in data using graphs, tables, and probability models to solve problems and draw appropriate conclusions from solutions

## High School Mathematics -- Differential Equations

## A - Abstract Reasoning

- solve contextual, mathematical problems involving first-order differential equations to explain real-life phenomena
- solve contextual, mathematical problems involving second-order and higher-order differential equations to explain real-life phenomena
- solve contextual, mathematical problems involving systems of differential equations to explain real-life phenomena
- solve contextual, mathematical problems using Laplace transforms to explain real-life phenomena
- approximate solutions to differential equations using power series and apply the approximations to real-life phenomena

## 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 -- Enhanced Advanced Algebra and Precalculus: Concepts and Connections

## A - Data & Statistical Reasoning

## B - Functional & Graphical Reasoning

- explore and analyze structures, patterns, and inverse relationships for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena
- explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena
- extend exploration of quadratic solutions to include real and non-real numbers and explore how these numbers behave under familiar operations and within real-world situations; create polynomial expressions, solve polynomial equations, graph polynomial functions, and model real-life phenomena
- analyze the behaviors of rational functions to model applicable, mathematical problems
- analyze the behaviors of rational and piecewise functions to model contextual mathematical problems
- utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions

## C - Patterning & Algebraic Reasoning

- represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications
- demonstrate how geometric sequences and series apply to mathematical models in real-life situations

## D - Geometric & Spatial Reasoning

- develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle
- analyze the behaviors of conic sections and polar equations to model contextual, mathematical problems

## E - Algebraic & Geometric Reasoning

- manipulate, prove, and apply trigonometric identities and equations to solve contextual, mathematical problems
- represent and model vector quantities to solve problems in contextual situations

## High School Mathematics -- Geometry: Concepts & Connections

## A - Patterning & Algebraic Reasoning

- interpret the structure of and perform operations with polynomials within a geometric context

## B - Geometric & Spatial Reasoning

- experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena
- establish facts between angle relations and generate valid arguments to prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena
- use center and scale factor to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena
- examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena
- explore the concept of a radian measure and special right triangles
- examine and apply theorems involving circles; describe and derive arc length and area of a sector; model and explain real-life frameworks involving circles
- develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena

## C - Probabilistic Reasoning

- solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions

## D - Data & Statistical Reasoning

- examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena

## High School Mathematics -- Geometry: Concepts & Connections Strategies

## A - Patterning & Algebraic Reasoning

- interpret the structure of and perform operations with polynomials within a geometric context

## B - Geometric & Spatial Reasoning

- experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena
- establish facts between angle relations and generate valid arguments to prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena
- use center and scale factor to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena
- examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena
- explore the concept of a radian measure and special right triangles
- examine and apply theorems involving circles; describe and derive arc length and area of a sector; model and explain real-life frameworks involving circles
- develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena

## C - Probabilistic Reasoning

- solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions

## D - Data & Statistical Reasoning

- examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena

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

## A - Abstract Reasoning & Deterministic Decision-Making

- solve contextual, mathematical problems involving linear programming and use the mathematics as a model to make decisions about real-life phenomena
- solve contextual, mathematical problems involving optimal locations and use the mathematics as a model to make decisions about real-life phenomena
- solve contextual, mathematical problems involving optimal paths and use the mathematics as a model to make decisions about real-life phenomena

## B - Abstract Reasoning & Probabilistic Decision-Making

- solve contextual, mathematical problems with normal distributions to make appropriate decisions
- solve contextual, mathematical problems using other distributions (e.g., binomial, geometric, and Poisson) as well as simulations to make appropriate decisions
- use simulations to make appropriate decisions
- using quantitative reasoning, determine fair methods to reflect the wishes of a larger population with representatives

## C - Probabilistic Reasoning

- use probabilistic models to make appropriate decisions

## High School Mathematics -- Multivariable Calculus

## A - Patterning & Algebraic Reasoning

- express spatial and functional relationships with vectors and matrices, functions, and analytic geometry in three dimensions, and use these relationships to solve contextual, mathematical problems

## B - Abstract & Quantitative Reasoning

- define, describe, and represent the differentiation of functions of two independent variables and differential vectors to solve contextual, mathematical problems and to explain real-life phenomena

## C - Abstract & Quantitative Reasoning

- interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain real-life phenomena

## High School Mathematics -- Number Theory

## A - Logical Reasoning

- interpret, represent, and communicate logical arguments to explain reasoning and justify thinking when solving problems and to include real-life phenomena
- apply methods of proof to prove or disprove mathematical statements; explain reasoning and justify thinking through mathematical induction when formulating mathematical arguments

## B - Abstract and Quantitative Reasoning

- use sets to describe relationships and equivalence when solving contextual, mathematical problems used to explain real-life phenomena

## C - Numerical Reasoning

- apply properties of numbers to uncover mathematical patterns
- explore and solve problems using properties of integer divisibility
- explore and apply mathematical congruences in real-life phenomena
- explore and apply mathematical theorems and conjectures related to prime numbers in real-life phenomena

## High School Mathematics -- Precalculus

## A - Functional & Graphical Reasoning

- analyze the behaviors of rational and piecewise functions to model contextual, mathematical problems
- utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions

## B - Algebraic & Geometric Reasoning

- manipulate, prove, and apply trigonometric identities and equations to solve contextual, mathematical problems

## C - Geometric & Spatial Reasoning

- analyze the behaviors of conic sections and polar equations to model contextual, mathematical problems

## D - Algebraic & Graphical Reasoning

- represent and model vector quantities to solve problems in contextual situations

## E - Patterning & Algebraic Reasoning

- demonstrate how geometric sequences and series apply to mathematical models in real-life situations

## High School Mathematics -- Statistical Reasoning

## A - Data & Statistical Reasoning

- formulate statistical investigative questions of interest that can be answered with data
- collect data by designing and implementing a plan to address the formulated statistical investigative question
- analyze data by selecting and using appropriate graphical and numerical methods
- interpret the results of the analysis, making connections to the formulated statistical investigative question