Georgia Math Standards - GSE

1291 standards, Kindergarten through AP.

Georgia Standards of Excellence (GSE), adopted in 2015, organize math around four reasoning strands: Numerical, Geometric and Spatial, Patterning and Algebraic, and Measurement and Data. This strand based approach centers the underlying mathematical thinking rather than discrete skill lists.

Kindergarten

K.GSR Geometric & Spatial Reasoning – 2D and 3D Shapes, Relative Locations, Attributes

K.GSR.8 Identify, describe, and compare basic shapes encountered in the environment, and form two-dimensional shapes and three-dimensional figures.

K.GSR.8.1Identify, sort, classify, analyze, and compare two-dimensional shapes and three-dimensional figures, in different sizes and orientations, using informal language to describe their similarities, differences, number of sides and vertices, and other attributes.

K.GSR.8.2Describe the relative location of an object using positional words.

K.GSR.8.3Use basic shapes to represent specific shapes found in the environment by creating models and drawings.

K.GSR.8.4Use two or more basic shapes to form larger shapes.

K.MDR Measurement & Data Reasoning – Attributes Of Objects, Classifying Objects

K.MDR.7 Observe, describe, and compare the physical and measurable attributes of objects and analyze graphical displays of data.

K.MDR.7.1Directly compare, describe, and order common objects, using measurable attributes (length, height, width, or weight) and describe the difference.

K.MDR.7.2Classify and sort up to ten objects into categories by an attribute; count the number of objects in each category and sort the categories by count.

K.MDR.7.3Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to solve problems relevant to everyday life.

K.NR Numerical Reasoning – Counting, Money, Place Value, Numbers To 20, Addition, Subtraction and Fluency

K.NR.1 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).

K.NR.1.1Count up to 20 objects in a variety of structured arrangements and up to 10 objects in a scattered arrangement.

K.NR.1.2When counting objects, explain that the last number counted represents the total quantity in a set (cardinality), regardless of the arrangement and order.

K.NR.1.3Given a number from 1-20, identify the number that is one more or one less.

K.NR.1.4Identify pennies, nickels, and dimes and know their name and value.

K.NR.2 Use count sequences within 100 to count forward and backward in sequence.

K.NR.2.1Count forward to 100 by tens and ones and backward from 20 by ones.

K.NR.2.2Count forward beginning from any number within 100 and count backward from any number within 20.

K.NR.3 Use place value understanding to compose and decompose numbers from 11–19.

K.NR.3.1Describe numbers from 11 to 19 by composing (putting together) and decomposing (breaking apart) the numbers into ten ones and some more ones.

K.NR.4 Identify, write, represent, and compare numbers up to 20.

K.NR.4.1Identify written numerals 0-20 and represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

K.NR.4.2Compare two sets of up to 10 objects and identify whether the number of objects in one group is more or less than the other group, using the words “greater than,” “less than,” or “the same as”.

K.NR.5 Explain the concepts of addition, subtraction, and equality and use these concepts to solve real-life problems within 10.

K.NR.5.1Compose (put together) and decompose (break apart) numbers up to 10 using objects and drawings.

K.NR.5.2Represent addition and subtraction within 10 from a given authentic situation using a variety of representations and strategies.

K.NR.5.3Use a variety of strategies to solve addition and subtraction problems within 10.

K.NR.5.4Fluently add and subtract within 5 using a variety of strategies to solve practical, mathematical problems.

K.PAR Patterning & Algebraic Reasoning – Repeating Patterns and Time

K.PAR.6 Explain, extend, and create repeating patterns with a repetition, not exceeding 4 and describe patterns involving the passage of time.

K.PAR.6.1Create, extend, and describe repeating patterns with numbers and shapes, and explain the rationale for the pattern.

K.PAR.6.2Describe patterns involving the passage of time using words and phrases related to actual events.

Grade 1

1.GSR Geometric & Spatial Reasoning – Shapes, Attributes, Partitions Of Circles and Rectangles

1.GSR.4 Compose shapes, analyze the attributes of shapes, and relate their parts to the whole.

1.GSR.4.1Identify common two-dimensional shapes and three-dimensional figures, sort and classify them by their attributes and build and draw shapes that possess defining attributes.

1.GSR.4.2Compose two-dimensional shapes (rectangles, squares, triangles, half-circles, and quarter-circles) and three-dimensional figures (cubes, rectangular prisms, cones, and cylinders) to create a shape formed of two or more common shapes and compose new shapes from the composite shape.

1.GSR.4.3Partition circles and rectangles into two and four equal shares.

1.MDR Measurement & Data Reasoning – Length, Time, Money

1.MDR.6 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 answer relevant questions.

1.MDR.6.1Estimate, measure, and record lengths of objects using non-standard units, and compare and order up to three objects using the recorded measurements. Describe the objects compared.

1.MDR.6.2Tell and write time in hours and half-hours using analog and digital clocks, and measure elapsed time to the hour on the hour using a predetermined number line.

1.MDR.6.3Identify the value of quarters and compare the values of pennies, nickels, dimes, and quarters.

1.MDR.6.4Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to compare and order whole numbers.

1.NR.1.1Count within 120, forward and backward, starting at any number. In this range, read and write numerals and represent a number of objects with a written numeral.

1.NR.1.2Explain that the two digits of a 2-digit number represent the amounts of tens and ones.

1.NR.1.3Compare and order whole numbers up to 100 using concrete models, drawings, and the symbols >, =, and <.

1.NR.2.1Use a variety of strategies to solve addition and subtraction problems within 20.

1.NR.2.2Use pictures, drawings, and equations to develop strategies for addition and subtraction within 20 by exploring strings of related problems.

1.NR.2.3Recognize the inverse relationship between subtraction and addition within 20 and use this inverse relationship to solve authentic problems.

1.NR.2.4Fluently add and subtract within 10 using a variety of strategies.

1.NR.2.5Use the meaning of the equal sign to determine whether equations involving addition and subtraction are true or false.

1.NR.2.6Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers.

1.NR.2.7Apply properties of operations as strategies to solve addition and subtraction problem situations within 20.

1.NR.5.1Use a variety of strategies to solve applicable, mathematical addition and subtraction problems with one- and two-digit whole numbers.

1.NR.5.2Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

1.NR.5.3Add and subtract multiples of 10 within 100.

1.NRa Numerical Reasoning – Counting, Numbers, Equality, Place Value, Addition, Subtraction

1.NRa.1Extend the count sequence to 120. Read, write, and represent numerical values to 120 and compare numerical values to 100.

1.NRa.2Explain the relationship between addition and subtraction and apply the properties of operations to solve real-life addition and subtraction problems within 20.

1.NRb Numerical Reasoning – Base Ten Structure, Addition and Subtraction Within 100

1.NRb.5Use concrete models, the base ten structure, and properties of operations to add and subtract within 100.

1.NRc Numerical Reasoning – Base Ten Structure, Addition and Subtraction Within 101

1.NRc.5Use concrete models, the base ten structure, and properties of operations to add and subtract within 100.

1.NRd Numerical Reasoning – Base Ten Structure, Addition and Subtraction Within 102

1.NRd.5Use concrete models, the base ten structure, and properties of operations to add and subtract within 100.

1.PAR Patterning & Algebraic Reasoning – Repeating Patterns, Growing, Patterns, and Shrinking Patterns

1.PAR.3 Identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns found in real-life situations.

1.PAR.3.1MA about repeating patterns with a core of up to 3 elements resulting from repeating an operation, as a series of shapes, or a number string.

1.PAR.3.2Identify, describe, and create growing, shrinking, and repeating patterns based on the repeated addition or subtraction of 1s, 2s, 5s, and 10s.

Grade 2

2.GSR Geometric & Spatial Reasoning – Sorting Shapes, Lines Of Symmetry, Partitioning Circles and Rectangles

2.GSR.7 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.

2.GSR.7.1Describe, compare and sort 2-D shapes including polygons, triangles, quadrilaterals, pentagons, hexagons, and 3-D shapes including rectangular prisms and cones, given a set of attributes.

2.GSR.7.2Identify at least one line of symmetry in everyday objects to describe each object as a whole.

2.GSR.7.3Partition circles and rectangles into two, three, or four equal shares. Identify and describe equal-sized parts of the whole using fractional names (“halves,” “thirds,” “fourths”, “half of,” “third of,” “quarter of,” etc.).

2.GSR.7.4Recognize that equal shares of identical wholes may be different shapes within the same whole.

2.MDR Measurement & Data Reasoning – Length, Distance, Time, and Money

2.MDR.5 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.

2.MDR.5.1Construct simple measuring instruments using unit models. Compare unit models to rulers.

2.MDR.5.2Estimate and measure the length of an object or distance to the nearest whole unit using appropriate units and standard measuring tools.

2.MDR.5.3Measure to determine how much longer one object is than another and express the length difference in terms of a standard-length unit.

2.MDR.5.4Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to solve problems relevant to everyday life.

2.MDR.5.5Represent whole-number sums and differences within a standard unit of measurement on a number line diagram.

2.MDR.6 Solve real-life problems involving time and money.

2.MDR.6.1Tell and write time from analog and digital clocks to the nearest five minutes, and estimate and measure elapsed time using a timeline, to the hour or half hour on the hour or half hour.

2.MDR.6.2Find the value of a group of coins and determine combinations of coins that equal a given amount that is less than one hundred cents, and solve problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

2.NR Numerical Reasoning – Counting Within 1000, Place Value, Addition and Subtraction, Fluency To 20, Developing Multiplication Through Arrays

2.NR.1 Using the place value structure, explore the count sequences to represent, read, write, and compare numerical values to 1000 and describe basic place-value relationships and structures.

2.NR.1.1Explain the value of a three-digit number using hundreds, tens, and ones in a variety of ways.

2.NR.1.2Count forward and backward by ones from any number within 1000. Count forward by fives from multiples of 5 within 1000. Count forward and backward by 10s and 100s from any number within 1000. Count forward by 25s from 0.

2.NR.1.3Represent, compare, and order whole numbers to 1000 with an emphasis on place value and equality. Use >, =, and < symbols to record the results of comparisons.

2.NR.2 Apply multiple part-whole strategies, properties of operations and place value understanding to solve real-life, mathematical problems involving addition and subtraction within 1,000.

2.NR.2.1Fluently add and subtract within 20 using a variety of mental, part-whole strategies.

2.NR.2.2Find 10 more or 10 less than a given three-digit number and find 100 more or 100 less than a given three-digit number.

2.NR.2.3Solve problems involving the addition and subtraction of two-digit numbers using part-whole strategies.

2.NR.2.4Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

2.NR.3 Work with equal groups to gain foundations for multiplication through real-life, mathematical problems.

2.NR.3.1Determine whether a group (up to 20) has an odd or even number of objects. Write an equation to express an even number as a sum of two equal addends.

2.NR.3.2Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

2.PAR.4 2.PAR.4

2.PAR.4.1Identify, describe, and create a numerical pattern resulting from repeating an operation such as addition and subtraction.

2.PAR.4.2Identify, describe, and create growing patterns and shrinking patterns involving addition and subtraction up to 20.

2.PARa Patterning & Algebraic Reasoning – Patterns Up To 20 and Addition and Subtraction Within 1,000

2.PARa.4Identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns.

2.PARb Patterning & Algebraic Reasoning – Patterns Up To 20 and Addition and Subtraction Within 1,001

2.PARb.4Identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns.

Grade 3

3.GSR Geometric & Spatial Reasoning – Polygons, Parallel Line Segments, Perpendicular Line Segments, Right Angles, Lines Of Symmetry, Area, Perimeter

3.GSR.6 Identify the attributes of polygons, including parallel segments, perpendicular segments, right angles, and symmetry.

3.GSR.6.1Identify perpendicular line segments, parallel line segments, and right angles, identify these in polygons, and solve problems involving parallel line segments, perpendicular line segments, and right angles.

3.GSR.6.2Classify, compare, and contrast polygons, with a focus on quadrilaterals, based on properties. Analyze specific 3-dimensional figures to identify and describe quadrilaterals as faces of these figures.

3.GSR.6.3Identify lines of symmetry in polygons.

3.GSR.7 Identify area as a measurable attribute of rectangles and determine the area of a rectangle presented in real-life, mathematical problems.

3.GSR.7.1Investigate area by covering the space of rectangles presented in realistic situations using multiple copies of the same unit, with no gaps or overlaps, and determine the total area (total number of units that covered the space).

3.GSR.7.2Determine the area of rectangles (or shapes composed of rectangles) presented in relevant problems by tiling and counting.

3.GSR.7.3Discover and explain how area can be found by multiplying the dimensions of a rectangle.

3.GSR.8 Determine the perimeter of a polygon presented in real-life, mathematical problems.

3.GSR.8.1Determine the perimeter of a polygon and explain that the perimeter represents the distance around a polygon. Solve problems involving perimeters of polygons.

3.GSR.8.2Investigate and describe how rectangles with the same perimeter can have different areas or how rectangles with the same area can have different perimeters.

3.MDR Measurement & Data Reasoning – Elapsed Time, Liquid Volume, Mass, Lengths In Half and Fourth Of An Inch, Data

3.MDR.5 Solve real-life, mathematical problems involving length, liquid volume, mass, and time.

3.MDR.5.1Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to solve problems relevant to everyday life.

3.MDR.5.2Tell and write time to the nearest minute and estimate time to the nearest fifteen minutes (quarter hour) from the analysis of an analog clock.

3.MDR.5.3Solve meaningful problems involving elapsed time, including intervals of time to the hour, half hour, and quarter hour where the times presented are only on the hour, half hour, or quarter hour within a.m. or p.m. only.

3.MDR.5.4Use rulers to measure lengths in halves and fourths (quarters) of an inch and a whole inch.

3.MDR.5.5Estimate and measure liquid volumes, lengths and masses of objects using customary units. Solve problems involving mass, length, and volume given in the same unit, and reason about the relative sizes of measurement units within the customary system.

3.NR.1.1Read and write multi-digit whole numbers up to 10,000 using base-ten numerals and expanded form.

3.NR.1.2Use place value reasoning to compare multi-digit numbers up to 10,000, using >, =, and < symbols to record the results of comparisons.

3.NR.1.3Use place value understanding to round whole numbers up to 1000 to the nearest 10 or 100.

3.NR.4.1Describe a unit fraction and explain how multiple copies of a unit fraction form a non-unit fraction. Use parts of a whole, parts of a set, points on a number line, distances on a number line and area models.

3.NR.4.2Compare two unit fractions by flexibly using a variety of tools and strategies.

3.NR.4.3Represent fractions, including fractions greater than one, in multiple ways.

3.NR.4.4Recognize and generate simple equivalent fractions.

3.NRa Numerical Reasoning – Base Ten Numerals and Place Value Up To 10,000, and Rounding Up To 1,000

3.NRa.1Use place value reasoning to represent, read, write, and compare numerical values up to 10,000 and round whole numbers up to 1,000.

3.NRb Numerical Reasoning – Unit Fractions, Equivalent Fractions, Fractions Greater Than 1

3.NRb.4Represent fractions with denominators of 2, 3, 4, 6 and 8 in multiple ways within a framework using visual models.

3.PAR Patterning & Algebraic Reasoning – Fluency, Addition and Subtraction Within 10,000, Multiplication and Division Within 100, Equality, Properties Of Operations

3.PAR.2 Use part-whole strategies to represent and solve real-life problems involving addition and subtraction with whole numbers within 10,000.

3.PAR.2.1Fluently add and subtract within 1000 to solve problems.

3.PAR.2.2Apply part-whole strategies, properties of operations and place value understanding, to solve problems involving addition and subtraction within 10,000. Represent these problems using equations with a letter standing for the unknown quantity. Justify solutions.

3.PAR.3 Use part-whole strategies to solve real-life, mathematical problems involving multiplication and division with whole numbers within 100.

3.PAR.3.1Describe, extend, and create numeric patterns related to multiplication. Make predictions related to the patterns.

3.PAR.3.2Represent single digit multiplication and division facts using a variety of strategies. Explain the relationship between multiplication and division.

3.PAR.3.3Apply properties of operations (i.e., commutative property, associative property, distributive property) to multiply and divide within 100.

3.PAR.3.4Use the meaning of the equal sign to determine whether expressions involving addition, subtraction, and multiplication are equivalent.

3.PAR.3.5Use place value reasoning and properties of operations to multiply one-digit whole numbers by multiples of 10, in the range 10-90.

3.PAR.3.6Solve practical, relevant problems involving multiplication and division within 100 using part-whole strategies, visual representations, and/or concrete models.

3.PAR.3.7Use multiplication and division to solve problems involving whole numbers to 100. Represent these problems using equations with a letter standing for the unknown quantity. Justify solutions.

Grade 4

4.GSR Geometric & Spatial Reasoning – Polygons, Points, Lines, Line Segments, Rays, Angles, Perpendicular Lines, Area, Perimeter

4.GSR.7 Investigate the concepts of angles and angle measurement to estimate and measure angles.

4.GSR.7.1Recognize angles as geometric shapes formed when two rays share a common endpoint. Draw right, acute, and obtuse angles based on the relationship of the angle measure to 90 degrees.

4.GSR.7.2Measure angles in reference to a circle with the center at the common endpoint of two rays. Determine an angle’s measure in relation to the 360 degrees in a circle through division or as a missing factor problem.

4.GSR.8 Identify and draw geometric objects, classify polygons based on properties, and solve problems involving area and perimeter of rectangular figures.

4.GSR.8.1Explore, investigate, and draw points, lines, line segments, rays, angles (right, acute, obtuse), perpendicular lines, parallel lines, and lines of symmetry. Identify these in two-dimensional figures.

4.GSR.8.2Classify, compare, and contrast polygons based on lines of symmetry, the presence or absence of parallel or perpendicular line segments, or the presence or absence of angles of a specified size and based on side lengths.

4.GSR.8.3Solve problems involving area and perimeter of composite rectangles involving whole numbers with known side lengths.

4.MDR Measurement & Data Reasoning – Time, Metric Measurements, Distance, Elapsed Time, Liquid Volume, Mass, and Length

4.MDR.6 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.

4.MDR.6.1Use the four operations to solve problems involving elapsed time to the nearest minute, intervals of time, metric measurements of liquid volumes, lengths, distances, and masses of objects, including problems involving fractions with like denominators, and also problems that require expressing measurements given in a larger unit in terms of a smaller unit, and expressing a smaller unit in terms of a larger unit based on the idea of equivalence.

4.MDR.6.2Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to solve problems relevant to everyday life.

4.MDR.6.3Create dot plots to display a distribution of numerical (quantitative) measurement data.

4.NR.1.1Read and write multi-digit whole numbers to the hundred-thousands place using base-ten numerals and expanded form.

4.NR.1.2Recognize and show that a digit in one place has a value ten times greater than what it represents in the place to its right and extend this understanding to determine the value of a digit when it is shifted to the left or right, based on the relationship between multiplication and division.

4.NR.1.3Use place value reasoning to represent, compare, and order multi-digit numbers, using >, =, and < symbols to record the results of comparisons.

4.NR.1.4Use place value understanding to round multi-digit whole numbers.

4.NR.2.1Fluently add and subtract multi-digit numbers to solve practical, mathematical problems using place value understanding, properties of operations, and relationships between operations.

4.NR.2.2Interpret, model, and solve problems involving multiplicative comparison.

4.NR.2.3Solve relevant problems involving multiplication of a number with up to four digits by a 1-digit whole number or involving multiplication of two two-digit numbers using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NR.2.4Solve authentic division problems involving up to 4-digit dividends and 1-digit divisors (including whole number quotients with remainders) using strategies based on place-value understanding, properties of operations, and the relationships between operations.

4.NR.2.5Solve multi-step problems using addition, subtraction, multiplication, and division involving whole numbers. Use mental computation and estimation strategies to justify the reasonableness of solutions.

4.NR.4.1Using concrete materials, drawings, and number lines, demonstrate and explain the relationship between equivalent fractions, including fractions greater than one, and explain the identity property of multiplication as it relates to equivalent fractions. Generate equivalent fractions using these relationships.

4.NR.4.2Compare two fractions with the same numerator or the same denominator by reasoning about their size and recognize that comparisons are valid only when the two fractions refer to the same whole.

4.NR.4.3Compare two fractions with different numerators and/or different denominators by flexibly using a variety of tools and strategies and recognize that comparisons are valid only when the two fractions refer to the same whole.

4.NR.4.4Represent whole numbers and fractions as the sum of unit fractions.

4.NR.4.5Represent a fraction as a sum of fractions with the same denominator in more than one way, recording with an equation.

4.NR.4.6Add and subtract fractions and mixed numbers with like denominators using a variety of tools.

4.NR.5.1Demonstrate and explain the concept of equivalent fractions with denominators of 10 and 100, using concrete materials and visual models. Add two fractions with denominators of 10 and 100.

4.NR.5.2Represent, read, and write fractions with denominators of 10 or 100 using decimal notation, and decimal numbers to the hundredths place as fractions, using concrete materials and drawings.

4.NR.5.3Compare two decimal numbers to the hundredths place by reasoning about their size. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

4.NRa Numerical Reasoning – Place Value, Rounding, Comparisons With Multi-Digit Numbers, Addition and Subtraction, Multiplicative Comparisons, Multiplication, and Division Involving Whole Numbers

4.NRa.1Recognize 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.

4.NRa.2Using part-whole strategies, 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.

4.NRb Numerical Reasoning – Fraction Equivalence, Comparison Of Fractions, and Addition and Subtraction Of Fractions With Like Denominators

4.NRb.4Solve 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.

4.NRb.5Solve 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.

4.PAR Patterning & Algebraic Reasoning – Patterns, Input-Output Tables, Factors, Multiples, Composite Numbers, Prime Numbers

4.PAR.3 Generate and analyze patterns, including those involving shapes, input/output diagrams, factors, multiples, prime numbers, and composite numbers.

4.PAR.3.1Generate both number and shape patterns that follow a provided rule.

4.PAR.3.2Use input-output rules, tables, and charts to represent and describe patterns, find relationships, and solve problems.

4.PAR.3.3Find factor pairs in the range 1–100 and find multiples of single-digit numbers up to 100.

4.PAR.3.4Identify composite numbers and prime numbers and explain the relationship with the factor pairs.

Grade 5

5.GSR Geometric & Spatial Reasoning – Properties Of Polygons and Rectangular Prisms, Classify Polygons

5.GSR.8 Examine properties of polygons and rectangular prisms, classify polygons by their properties, and discover volume of right rectangular prisms.

5.GSR.8.1Classify, compare, and contrast polygons based on properties.

5.GSR.8.2Determine, through exploration and investigation, that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

5.GSR.8.3Investigate volume of right rectangular prisms by packing them with unit cubes without gaps or overlaps. Then, determine the total volume to solve problems.

5.GSR.8.4Discover and explain how the volume of a right rectangular prism can be found by multiplying the area of the base times the height to solve authentic, mathematical problems.

5.MDR Measurement & Data Reasoning – Measurements Within The Metric System, Measurement Conversions and Time As A Unit Of Measurement

5.MDR.7 Solve problems involving customary measurements, metric measurements, and time and analyze graphical displays of data to answer relevant questions.

5.MDR.7.1Explore realistic problems involving different units of measurement, including distance, mass, weight, volume, and time.

5.MDR.7.2Ask questions and answer them based on gathered information, observations, and appropriate graphical displays to solve problems relevant to everyday life.

5.MDR.7.3Convert among units within the metric system and then apply these conversions to solve multi-step, practical problems.

5.MDR.7.4Convert among units within relative sizes of measurement units within the customary measurement system.

5.NR Numerical Reasoning – Place Value, Multiplying By Powers Of 10, Multiplication and Division Of Multi-Digit Numbers, Fractions, Decimal Numbers, Numerical Expressions

5.NR.1 Use place value understanding to solve real-life, mathematical problems.

5.NR.1.1Explain 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.

5.NR.1.2Explain patterns in the placement of digits when multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10, up to 10^3.

5.NR.2 Multiply and divide multi-digit whole numbers to solve relevant, mathematical problems.

5.NR.2.1Fluently multiply multi-digit (up to 3-digit by 2-digit) whole numbers to solve authentic problems.

5.NR.2.2Fluently divide multi-digit whole numbers (up to 4-digit dividends and 2-digit divisors no greater than 25) to solve practical problems.

5.NR.3 Describe fractions and perform operations with fractions to solve relevant, mathematical problems using part-whole strategies and visual models.

5.NR.3.1Explain the meaning of a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

5.NR.3.2Compare and order up to three fractions with different numerators and/or different denominators by flexibly using a variety of tools and strategies.

5.NR.3.3Model and solve problems involving addition and subtraction of fractions and mixed numbers with unlike denominators.

5.NR.3.4Model and solve problems involving multiplication of a fraction and a whole number.

5.NR.3.5Explain why multiplying a whole number by a fraction greater than one results in a product greater than the whole number, and why multiplying a whole number by a fraction less than one results in a product less than the whole number and multiplying a whole number by a fraction equal to one results in a product equal to the whole number.

5.NR.3.6Model and solve problems involving division of a unit fraction by a whole number and a whole number by a unit fraction.

5.NR.4 Read, write, and compare decimal numbers to the thousandths place, and round and perform operations with decimal numbers to the hundredths place to solve relevant, mathematical problems.

5.NR.4.1Read and write decimal numbers to the thousandths place using base-ten numerals written in standard form and expanded form.

5.NR.4.2Represent, compare, and order decimal numbers to the thousandths place based on the meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NR.4.3Use place value understanding to round decimal numbers to the hundredths place.

5.NR.4.4Solve problems involving addition and subtraction of decimal numbers to the hundredths place using a variety of strategies.

5.NR.5 Write, interpret, and evaluate numerical expressions within authentic problems.

5.NR.5.1Write, interpret, and evaluate simple numerical expressions involving whole numbers with or without grouping symbols to represent actual situations.

5.PAR Patterning & Algebraic Reasoning – Generating Patterns, Plotting Ordered Pairs In The First Quadrant

5.PAR.6 Solve relevant problems by creating and analyzing numerical patterns using the given rule(s).

5.PAR.6.1Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms by completing a table.

5.PAR.6.2Represent problems by plotting ordered pairs and explain coordinate values of points in the first quadrant of the coordinate plane.

Grade 6

6.GSR Geometric & Spatial Reasoning – Area Of Polygons, Volume Of Right Rectangular Prisms, Surface Area Of 3-D Figures

6.GSR.5 Solve relevant problems involving area, surface area, and volume.

6.GSR.5.1Explore area as a measurable attribute of triangles, quadrilaterals, and other polygons conceptually by composing or decomposing into rectangles, triangles, and other shapes. Find the area of these geometric figures to solve problems.

6.GSR.5.2Given the net of three-dimensional figures with rectangular and triangular faces, determine the surface area of these figures.

6.GSR.5.3Calculate the volume of right rectangular prisms with fractional edge lengths by applying the formula, V = (area of base) x (height).

6.NR Numerical Reasoning – Multiplication and Division Of Whole Numbers and Fractions, and All Four Operations With Decimal Numbers

6.NR.1 Solve relevant, mathematical problems involving operations with whole numbers, fractions, and decimal numbers.

6.NR.1.1Fluently add and subtract any combination of fractions to solve problems.

6.NR.1.2Multiply and divide any combination of whole numbers, fractions, and mixed numbers using a student-selected strategy. Interpret products and quotients of fractions and solve word problems.

6.NR.1.3Perform operations with multi-digit decimal numbers fluently using models and student-selected strategies.

6.NR.2 Apply operations with whole numbers, fractions and decimals within relevant applications.

6.NR.2.1Describe and interpret the center of the distribution by the equal share value (mean).

6.NR.2.2Summarize categorical and quantitative (numerical) data sets in relation to the context: display the distributions of quantitative (numerical) data in plots on a number line, including dot plots, histograms, and box plots and display the distribution of categorical data using bar graphs.

6.NR.2.3Interpret numerical data to answer a statistical investigative question created. Describe the distribution of a quantitative (numerical) variable collected, including its center, variability, and overall shape.

6.NR.2.4Design simple experiments and collect data. Use data gathered from realistic scenarios and simulations to determine quantitative measures of center (median and/or mean) and variability (interquartile range and range). Use these quantities to draw conclusions about the data, compare different numerical data sets, and make predictions.

6.NR.2.5Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

6.NR.2.6Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Create data displays using a dot plot or box plot to examine this impact.

6.NR.3 Solve a variety of problems involving whole numbers and their opposites; model rational numbers on a number line to describe problems presented in relevant, mathematical situations.

6.NR.3.1Identify and compare integers and explain the meaning of zero based on multiple authentic situations.

6.NR.3.2Order and plot integers on a number line and use distance from zero to discover the connection between integers and their opposites.

6.NR.3.3Recognize and explain that opposite signs of integers indicate locations on opposite sides of zero on the number line; recognize and explain that the opposite of the opposite of a number is the number itself.

6.NR.3.4Write, interpret, and explain statements of order for rational numbers in authentic, mathematical situations. Compare rational numbers, including integers, using equality and inequality symbols.

6.NR.3.5Explain the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as distance for a positive or negative quantity in a relevant situation.

6.NR.3.6Distinguish comparisons of absolute value from statements about order.

6.NR.4 Solve a variety of contextual problems involving ratios, unit rates, equivalent ratios, percentages, and conversions within measurement systems using proportional reasoning.

6.NR.4.1Explain the concept of a ratio, represent ratios, and use ratio language to describe a relationship between two quantities.

6.NR.4.2Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.NR.4.3Solve problems involving proportions using a variety of student-selected strategies.

6.NR.4.4Describe the concept of rates and unit rate in the context of a ratio relationship.

6.NR.4.5Solve unit rate problems including those involving unit pricing and constant speed.

6.NR.4.6Calculate a percent of a quantity as a rate per 100 and solve everyday problems given a percent.

6.NR.4.7Use ratios to convert within measurement systems (customary and metric) to solve authentic problems that exist in everyday life.

6.PAR Patterning & Algebraic Reasoning – Numerical and Algebraic Expressions, Factors, Multiples, Algebraic Expressions, Plotting Points In All Four Quadrants, Rational Numbers On A Number Line, Polygons In The Coordinate Plane

6.PAR.6 Identify, write, evaluate, and interpret numerical and algebraic expressions as mathematical models to explain authentic situations.

6.PAR.6.1Write and evaluate numerical expressions involving rational bases and whole-number exponents.

6.PAR.6.2Determine greatest common factors and least common multiples using a variety of strategies to make sense of applicable problems.

6.PAR.6.3Write and read expressions that represent operations with numbers and variables in realistic situations.

6.PAR.6.4Evaluate expressions when given values for the variables, including expressions that arise in everyday situations.

6.PAR.6.5Apply the properties of operations to identify and generate equivalent expressions.

6.PAR.7 Write and solve one-step equations and inequalities as mathematical models to explain authentic, realistic situations.

6.PAR.7.1Solve one-step equations and inequalities involving variables when values for the variables are given. Determine whether an equation and inequality involving a variable is true or false for a given value of the variable.

6.PAR.7.2Write one-step equations and inequalities to represent and solve problems; explain that a variable can represent an unknown number or any number in a specified set.

6.PAR.7.3Solve problems by writing and solving equations of the form x + p = q, px = q and x/p = q for cases in which p, q and x are all nonnegative rational numbers.

6.PAR.7.4Recognize and generate inequalities of the form x > c, x ≥ c, x < c, or x ≤ c to explain situations that have infinitely many solutions; represent solutions of such inequalities on a number line.

6.PAR.8 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.

6.PAR.8.1Locate and position rational numbers on a horizontal or vertical number line; find and position pairs of integers and other rational numbers on a coordinate plane.

6.PAR.8.2Show and explain that signs of numbers in ordered pairs indicate locations in quadrants of the coordinate plane and determine how two ordered pairs may differ based only on the signs.

6.PAR.8.3Solve 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 x-coordinate or the same y-coordinate.

6.PAR.8.4Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same x-coordinate or the same y-coordinate.

Grade 7

7.GSR Geometric & Spatial Reasoning – Vertical, Adjacent, Complementary, and Supplementary Angles, Circumference and Area Of Circles, Area and Surface Area, Volume Of Cubes, Right Prisms, and Cylinders

7.GSR.5 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.

7.GSR.5.1Measure angles in whole non-standard units.

7.GSR.5.2Measure angles in whole number degrees using a protractor.

7.GSR.5.3Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve equations for an unknown angle in a figure.

7.GSR.5.4Explore and describe the relationship between pi, radius, diameter, circumference, and area of a circle to derive the formulas for the circumference and area of a circle.

7.GSR.5.5Given the formula for the area and circumference of a circle, solve problems that exist in everyday life.

7.GSR.5.6Solve realistic problems involving surface area of right prisms and cylinders.

7.GSR.5.7Describe the two-dimensional figures (cross sections) that result from slicing three-dimensional figures, as in the plane sections of right rectangular prisms, right rectangular pyramids, cones, cylinders, and spheres.

7.GSR.5.8Explore volume as a measurable attribute of cylinders and right prisms. Find the volume of these geometric figures using concrete problems.

7.NR Numerical Reasoning – Integers, Percentages, Fractions, Decimal Numbers

7.NR.1 Solve relevant, mathematical problems, including multi-step problems, involving the four operations with rational numbers and quantities in any form (integers, percentages, fractions, and decimal numbers).

7.NR.1.1Show that a number and its opposite have a sum of 0 (are additive inverses). Describe situations in which opposite quantities combine to make 0.

7.NR.1.10Convert rational numbers between forms to include fractions, decimal numbers and percentages, using understanding of the part divided by the whole. Know that the decimal form of a rational number terminates in 0s or eventually repeats.

7.NR.1.11Solve multi-step, contextual problems involving rational numbers, converting between forms as appropriate, and assessing the reasonableness of answers using mental computation and estimation strategies.

7.NR.1.2Show and explain 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 applicable situations.

7.NR.1.3Represent addition and subtraction with rational numbers on a horizontal or a vertical number line diagram to solve authentic problems.

7.NR.1.4Show and explain 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 contextual situations.

7.NR.1.5Apply properties of operations, including part-whole reasoning, as strategies to add and subtract rational numbers.

7.NR.1.6Make sense of multiplication of rational numbers using realistic applications.

7.NR.1.7Show and explain that integers can be divided, assuming the divisor is not zero, and every quotient of integers is a rational number.

7.NR.1.8Represent the multiplication and division of integers using a variety of strategies and interpret products and quotients of rational numbers by describing them based on the relevant situation.

7.NR.1.9Apply properties of operations as strategies to solve multiplication and division problems involving rational numbers represented in an applicable scenario.

7.PAR Patterning & Algebraic Reasoning – Linear Expressions With Rational Coefficients, Complex Unit Rates, Proportional Relationships

7.PAR.2 Use properties of operations, generate equivalent expressions and interpret the expressions to explain relevant situations.

7.PAR.2.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.PAR.2.2Rewrite an expression in different forms from a contextual problem to clarify the problem and show how the quantities in it are related.

7.PAR.3 Represent authentic situations using equations and inequalities with variables; solve equations and inequalities symbolically, using the properties of equality.

7.PAR.3.1Construct algebraic equations to solve practical problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Interpret the solution based on the situation.

7.PAR.3.2Construct algebraic inequalities to solve problems, leading to inequalities of the form px ± q > r, px ± q < r, px ± q ≤ r, or px ± q ≥ r, where p, q, and r are specific rational numbers. Graph and interpret the solution based on the realistic situation that the inequalities represent.

7.PAR.4 Recognize proportional relationships in relevant, mathematical problems; represent, solve, and explain these relationships with tables, graphs, and equations.

7.PAR.4.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units presented in realistic problems.

7.PAR.4.10Predict characteristics of a population by examining the characteristics of a representative sample. Recognize the potential limitations and scope of the sample to the population.

7.PAR.4.11Analyze sampling methods and conclude that random sampling produces and supports valid inferences.

7.PAR.4.12Use data from repeated random samples to evaluate how much a sample mean is expected to vary from a population mean. Simulate multiple samples of the same size.

7.PAR.4.2Determine the unit rate (constant of proportionality) in tables, graphs (1, r), equations, diagrams, and verbal descriptions of proportional relationships to solve realistic problems.

7.PAR.4.3Determine whether two quantities presented in authentic problems are in a proportional relationship.

7.PAR.4.4Identify, represent, and use proportional relationships.

7.PAR.4.5Use context to 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.

7.PAR.4.6Solve everyday 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.

7.PAR.4.7Use similar triangles to explain why the slope, m, is the same between any two distinct points on a non-vertical line in the coordinate plane.

7.PAR.4.8Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

7.PAR.4.9Use proportional relationships to solve multi-step ratio and percent problems presented in applicable situations.

7.PR Probability Reasoning – Likelihood, Theoretical and Experimental Probability

7.PR.6 Using mathematical reasoning, investigate chance processes and develop, evaluate, and use probability models to find probabilities of simple events presented in authentic situations.

7.PR.6.1Represent the probability of a chance event as a number between 0 and 1 that expresses the likelihood of the event occurring. Describe that 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.

7.PR.6.2Approximate the probability of a chance event by collecting data on an event and observing its long-run relative frequency will approach the theoretical probability.

7.PR.6.3Develop a probability model and use it to find probabilities of simple events. Compare experimental and theoretical probabilities of events. If the probabilities are not close, explain possible sources of the discrepancy.

7.PR.6.4Develop a uniform probability model by assigning equal probability to all outcomes and use the model to determine probabilities of events.

7.PR.6.5Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

7.PR.6.6Use appropriate graphical displays and numerical summaries from data distributions with categorical or quantitative (numerical) variables as probability models to draw informal inferences about two samples or populations.

Grade 8

8.FGR Functional & Graphical Reasoning – Relate Domain To Linear Functions, Rate Of Change, Linear Vs. Nonlinear Relationships, Graphing Linear Functions, Systems Of Linear Equations, Parallel and Perpendicular Lines

8.FGR.5 Describe the properties of functions to define, evaluate, and compare relationships, and use functions and graphs of functions to model and explain real phenomena.

8.FGR.5.1Show and explain that a function is a rule that assigns to each input exactly one output.

8.FGR.5.2Within realistic situations, identify and describe examples of functions that are linear or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

8.FGR.5.3Relate the domain of a linear function to its graph and where applicable to the quantitative relationship it describes.

8.FGR.5.4Compare properties (rate of change and initial value) of two functions used to model an authentic situation each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

8.FGR.5.5Write and explain the equations y = mx + b (slope-intercept form), Ax + By = C (standard form), and (y – y_1) = m(x-x_1) (point-slope form) as defining a linear function whose graph is a straight line to reveal and explain different properties of the function.

8.FGR.5.6Write a linear function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

8.FGR.5.7Construct 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.

8.FGR.5.8Explain the meaning of 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.

8.FGR.5.9Graph and analyze linear functions expressed in various algebraic forms and show key characteristics of the graph to describe applicable situations.

8.FGR.6 Solve practical, linear problems involving situations using bivariate quantitative data.

8.FGR.6.1Show that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, visually fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line of best fit.

8.FGR.6.2Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercepts.

8.FGR.6.3Explain the meaning of the predicted slope (rate of change) and the predicted intercept (constant term) of a linear model in the context of the data.

8.FGR.6.4Use appropriate graphical displays from data distributions involving lines of best fit to draw informal inferences and answer the statistical investigative question posed in an unbiased statistical study.

8.FGR.7 Justify and use various strategies to solve systems of linear equations to model and explain realistic phenomena.

8.FGR.7.1Interpret and solve relevant mathematical problems leading to two linear equations in two variables.

8.FGR.7.2Show and explain that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because the points of intersection satisfy both equations simultaneously.

8.FGR.7.3Approximate solutions of two linear equations in two variables by graphing the equations and solving simple cases by inspection.

8.FGR.7.4Analyze and solve systems of two linear equations in two variables algebraically to find exact solutions.

8.FGR.7.5Create and compare the equations of two lines that are either parallel to each other, perpendicular to each other, or neither parallel nor perpendicular.

8.GSR Geometric & Spatial Reasoning – Pythagorean Theorem and Volume Of Triangles, Rectangles, Cones, Cylinders, and Spheres

8.GSR.8 Solve geometric problems involving the Pythagorean Theorem and the volume of geometric figures to explain real phenomena.

8.GSR.8.1Explain a proof of the Pythagorean Theorem and its converse using visual models.

8.GSR.8.2Apply the Pythagorean Theorem to determine unknown side lengths in right triangles within authentic, mathematical problems in two and three dimensions.

8.GSR.8.3Apply the Pythagorean Theorem to find the distance between two points in a coordinate system in practical, mathematical problems.

8.GSR.8.4Apply the formulas for the volume of cones, cylinders, and spheres and use them to solve in relevant problems.

8.NR Numerical Reasoning – Rational and Irrational Numbers, Decimal Expansion, Integer Exponents, Square and Cube Roots, Scientific Notation

8.NR.1 Solve problems involving irrational numbers and rational approximations of irrational numbers to explain realistic applications.

8.NR.1.1Distinguish between rational and irrational numbers using decimal expansion. Convert a decimal expansion which repeats eventually into a rational number.

8.NR.1.2Approximate irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.

8.NR.2 Solve problems involving radicals and integer exponents including relevant application situations; apply place value understanding with scientific notation and use scientific notation to explain real phenomena.

8.NR.2.1Apply the properties of integer exponents to generate equivalent numerical expressions.

8.NR.2.2Use square root and cube root symbols to represent solutions to equations. Recognize that x^2 = p (where p is a positive rational number and |x| ≤ 25) has two solutions and x^3 = p (where p is a negative or positive rational number and |x| ≤ 10) has one solution. Evaluate square roots of perfect squares ≤ 625 and cube roots of perfect cubes ≥ -1000 and ≤ 1000.

8.NR.2.3Use numbers expressed in scientific notation to estimate very large or very small quantities, and to express how many times as much one is than the other.

8.NR.2.4Add, subtract, multiply and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Interpret scientific notation that has been generated by technology (e.g., calculators or online technology tools).

8.PAR Patterning & Algebraic Reasoning – Expressions, Linear Equations, and Inequalities

8.PAR.3 Create and interpret expressions within relevant situations. Create, interpret, and solve linear equations and linear inequalities in one variable to model and explain real phenomena.

8.PAR.3.1Interpret expressions and parts of an expression, in context, by utilizing formulas or expressions with multiple terms and/or factors.

8.PAR.3.2Describe and solve linear equations in one variable with one solution (x = a), infinitely many solutions (a = a), or no solutions (a = b). 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).

8.PAR.3.3Create and solve linear equations and inequalities in one variable within a relevant application.

8.PAR.3.4Using algebraic properties and the properties of real numbers, justify the steps of a one-solution equation or inequality.

8.PAR.3.5Solve linear equations and inequalities in one variable with coefficients represented by letters and explain the solution based on the contextual, mathematical situation.

8.PAR.3.6Use algebraic reasoning to fluently manipulate linear and literal equations expressed in various forms to solve relevant, mathematical problems.

8.PAR.4 Show and explain the connections between proportional and non-proportional relationships, lines, and linear equations; create and interpret graphical mathematical models and use the graphical, mathematical model to explain real phenomena represented in the graph.

8.PAR.4.1Use the equation y = mx (proportional) for a line through the origin to derive the equation y = mx + b (non-proportional) for a line intersecting the vertical axis at b.

8.PAR.4.2Show and explain that the graph of an equation representing an applicable situation in two variables is the set of all its solutions plotted in the coordinate plane.

Algebra: Concepts & Connections

A.DSR Data & Statistical Reasoning – Univariate Data and Single Quantitative Variables; Bivariate Data

A.DSR.10 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.

A.DSR.10.1Use statistics appropriate to the shape of the data distribution to compare and represent center (median and mean) and variability (interquartile range, standard deviation) of two or more distributions by hand and using technology.

A.DSR.10.2Interpret differences in shape, center, and variability of the distributions based on the investigation, accounting for possible effects of extreme data points (outliers).

A.DSR.10.3Represent data on two quantitative variables on a scatter plot and describe how the variables are related.

A.DSR.10.4Interpret the slope (predicted rate of change) and the intercept (constant term) of a linear model based on the investigation of the data.

A.DSR.10.5Calculate the line of best fit and interpret the correlation coefficient, r, of a linear fit using technology. Use r to describe the strength of the goodness of fit of the regression. Use the linear function to make predictions and assess how reasonable the prediction is in context.

A.DSR.10.6Decide which type of function is most appropriate by observing graphed data.

A.DSR.10.7Distinguish between correlation and causation.

A.FGR.2 A.FGR.2

A.FGR.2.1Use mathematically applicable situations algebraically and graphically to build and interpret arithmetic sequences as functions whose domain is a subset of the integers.

A.FGR.2.2Construct and interpret the graph of a linear function that models real-life phenomena and represent key characteristics of the graph using formal notation.

A.FGR.2.3Relate the domain and range of a linear function to its graph and, where applicable, to the quantitative relationship it describes. Use formal interval and set notation to describe the domain and range of linear functions.

A.FGR.2.4Use function notation to build and evaluate linear functions for inputs in their domains and interpret statements that use function notation in terms of a mathematical framework.

A.FGR.2.5Analyze the difference between linear functions and nonlinear functions by informally analyzing the graphs of various parent functions (linear, quadratic, exponential, absolute value, square root, and cube root parent curves).

A.FGR.7 A.FGR.7

A.FGR.7.1Relate the domain and range of a quadratic function to its graph and, where applicable, to the quantitative relationship it describes.

A.FGR.7.2Identify the effect on the graph generated by a quadratic function when replacing f(x) with 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.

A.FGR.7.3Graph and analyze the key characteristics of quadratic functions.

A.FGR.7.4Relate the domain and range of a quadratic function to its graph and, where applicable, to the quantitative relationship it describes.

A.FGR.7.5Rewrite a quadratic function representing a mathematically applicable situation to reveal the maximum or minimum value of the function it defines. Explain what the value describes in context.

A.FGR.7.6Create quadratic functions in two variables to represent relationships between quantities; graph quadratic functions on the coordinate axes with labels and scales.

A.FGR.7.7Estimate, calculate, and interpret the average rate of change of a quadratic function and make comparisons to the average rate of change of linear functions.

A.FGR.7.8Write a function defined by a quadratic expression in different but equivalent forms to reveal and explain different properties of the function.

A.FGR.7.9Compare characteristics of two functions each represented in a different way.

A.FGR.9 A.FGR.9

A.FGR.9.1Use function notation to build and evaluate exponential functions for inputs in their domains and interpret statements that use function notation in terms of a context.

A.FGR.9.2Graph and analyze the key characteristics of simple exponential functions based on mathematically applicable situations.

A.FGR.9.3Identify the effect on the graph generated by an exponential function when replacing f(x) with f(x) + k, and k f(x), for specific values of k (both positive and negative); find the value of k given the graphs.

A.FGR.9.4Use mathematically applicable situations algebraically and graphically to build and interpret geometric sequences as functions whose domain is a subset of the integers.

A.FGR.9.5Compare characteristics of two functions each represented in a different way.

A.FGRa Functional & Graphical Reasoning – Function Notation, Modeling Linear Functions, Linear Vs. Nonlinear Comparisons

A.FGRa.2Construct 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.

A.FGRb Functional & Graphical Reasoning – Quadratic Functions

A.FGRb.7Construct 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.

A.FGRc Functional & Graphical Reasoning – Exponential Functions

A.FGRc.9Construct 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.

A.GSR Geometric & Spatial Reasoning – Distance, Midpoint, Slope, Area, and Perimeter

A.GSR.3 Solve problems involving distance, midpoint, slope, area, and perimeter to model and explain real-life phenomena.

A.GSR.3.1Solve real-life problems involving slope, parallel lines, perpendicular lines, area, and perimeter.

A.GSR.3.2Apply the distance formula, midpoint formula, and slope of line segments to solve real-world problems.

A.MM Mathematical Modeling

A.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

A.MM.1.1Explain applicable, mathematical problems using a mathematical model.

A.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities domains.

A.MM.1.3Use units of measure (linear, area, capacity, rates, and time) as a way to make sense of conceptual problems; identify, use, and record appropriate units of measure within the given framework, within data displays, and on graphs; convert units and rates using proportional reasoning given a conversion factor; use units within multi-step problems and formulas; interpret units of input and resulting units of output.

A.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

A.MM.1.5Define appropriate quantities for the purpose of descriptive modeling.

A.NR Numerical Reasoning - Rational and Irrational Numbers, Square Roots and Cube Roots

A.NR.5 Investigate rational and irrational numbers and rewrite expressions involving square roots and cube roots.

A.NR.5.1Rewrite algebraic and numeric expressions involving radicals.

A.NR.5.2Using numerical reasoning, show and explain that the sum or product of rational numbers is rational, the sum of a rational number and an irrational number is irrational, and the product of a nonzero rational number and an irrational number is irrational.

A.PAR.4 A.PAR.4

A.PAR.4.1Create and solve linear inequalities in two variables to represent relationships between quantities including mathematically applicable situations; graph inequalities on coordinate axes with labels and scales.

A.PAR.4.2Represent constraints of linear inequalities and interpret data points as possible or not possible.

A.PAR.4.3Solve systems of linear inequalities by graphing, including systems representing a mathematically applicable situation.

A.PAR.6 A.PAR.6

A.PAR.6.1Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context.

A.PAR.6.2Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression.

A.PAR.6.3Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena.

A.PAR.6.4Represent constraints by quadratic equations and interpret data points as possible or not possible in a modeling framework.

A.PAR.8 A.PAR.8

A.PAR.8.1Interpret exponential expressions and parts of an exponential expression that represent a quantity in terms of its framework.

A.PAR.8.2Create exponential equations in one variable and use them to solve problems, including mathematically applicable situations.

A.PAR.8.3Create exponential equations in two variables to represent relationships between quantities, including in mathematically applicable situations; graph equations on coordinate axes with labels and scales.

A.PAR.8.4Represent constraints by exponential equations and interpret data points as possible or not possible in a modeling environment.

A.PARa Patterning & Algebraic Reasoning – Linear Inequalities and Systems Of Linear Inequalities

A.PARa.4Create, analyze, and solve linear inequalities in two variables and systems of linear inequalities to model real-life phenomena.

A.PARb Patterning & Algebraic Reasoning – Quadratic Expressions & Equations

A.PARb.6Build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations.

A.PARc Patterning & Algebraic Reasoning – Exponential Expressions and Equations

A.PARc.8Create and analyze exponential expressions and equations to represent and model real-life phenomena; solve exponential equations in mathematically applicable situations.

Advanced Algebra (Algebra II): Concepts and Connections

AA.DSR Data & Statistical Reasoning – Descriptive and Inferential Statistics

AA.DSR.2 Communicate descriptive and inferential statistics by collecting, critiquing, analyzing, and interpreting real-world data.

AA.DSR.2.1Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Distinguish between primary and secondary data and how it affects the types of conclusions that can be drawn.

AA.DSR.2.2When collecting and considering data, critically evaluate ethics, privacy, potential bias, and confounding variables along with their implications for interpretation in answering a statistical investigative question. Implement strategies for organizing and preparing big data sets.

AA.DSR.2.3Distinguish between population distributions, sample data distributions, and sampling distributions. Use sample statistics to make inferences about population parameters based on a random sample from that population and to communicate conclusions using appropriate statistical language.

AA.DSR.2.4Calculate and interpret z-scores as a measure of relative standing and as a method of standardizing units.

AA.DSR.2.5Given a normally distributed population, estimate percentages using the Empirical Rule, z-scores, and technology.

AA.DSR.2.6Model sample-to-sample variability in sampling distributions of a statistic using simulations taken from a given population.

AA.DSR.2.7Given a margin of error, develop and compare confidence intervals of different models to make conclusions about reliability.

AA.DSR.2.8Summarize and evaluate reports based on data for appropriateness of study design, analysis methods, and statistical measures used.

AA.FGR.3 AA.FGR.3

AA.FGR.3.1Find the inverse of exponential and logarithmic functions using equations, tables, and graphs, limiting the domain of inverses where necessary to maintain functionality, and prove by composition or verify by inspection that one function is the inverse of another.

AA.FGR.3.2Analyze, graph, and compare exponential and logarithmic functions.

AA.FGR.3.3Use the definition of a logarithm, logarithmic properties, and the inverse relationship between exponential and logarithmic functions to solve problems in context.

AA.FGR.3.4Create exponential equations and use logarithms to solve mathematical, applicable problems for which only one variable is unknown.

AA.FGR.3.5Create and interpret logarithmic equations in one variable and use them to solve problems.

AA.FGR.3.6Create, interpret, and solve exponential equations to represent relationships between quantities and analyze the relationships numerically with tables, algebraically, and graphically.

AA.FGR.3.7Create, interpret, and solve logarithmic equations in two or more variables to represent relationships between quantities.

AA.FGR.4 AA.FGR.4

AA.FGR.4.1Rewrite radical expressions as expressions with rational exponents. Extend the properties of integer exponents to rational exponents.

AA.FGR.4.2Solve radical equations in one variable, and give examples showing how extraneous solutions may arise.

AA.FGR.4.3Analyze and graph radical functions.

AA.FGR.4.4Create, interpret and solve radical equations with one unknown value and use them to solve problems that model real-world situations.

AA.FGR.4.5Create, interpret, and solve radical equations in two or more variables to represent relationships between quantities.

AA.FGR.5 AA.FGR.5

AA.FGR.5.1Graph and analyze quadratic functions in contextual situations and include analysis of data sets with regressions.

AA.FGR.5.10Use the structure of an expression to factor polynomials, including the sum of cubes, the difference of cubes, and higher-order polynomials that may be expressed as a quadratic within a quadratic.

AA.FGR.5.11Using all the zeros of a polynomial function, list all the factors and multiply to write a multiple of the polynomial function in standard form.

AA.FGR.5.2Define complex numbers i such that i^2 = -1 and show that every complex number has the form a + bi where a and b are real numbers and that the complex conjugate is a - bi.

AA.FGR.5.3Use the relation i^2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

AA.FGR.5.4Use the structure of an expression to factor quadratics.

AA.FGR.5.5Write and solve quadratic equations and inequalities with real coefficients and use the solution to explain a mathematical, applicable situation.

AA.FGR.5.6Solve systems of quadratic and linear functions to determine points of intersection.

AA.FGR.5.7Create and analyze quadratic equations to represent relationships between quantities as a model for contextual situations.

AA.FGR.5.8Identify the number of zeros that exist for any polynomial based upon the greatest degree of the polynomial and the end behavior of the polynomial by observing the sign of the leading coefficient.

AA.FGR.5.9Identify zeros of polynomial functions using technology or pre-factored polynomials and use the zeros to construct a graph of the function defined by the polynomial function. Analyze identify key features of these polynomial functions.

AA.FGR.8 AA.FGR.8

AA.FGR.8.1Rewrite simple rational expressions in equivalent forms.

AA.FGR.8.2Add, subtract, multiply and divide rational expressions, including problems in context and express rational expressions in irreducible form.

AA.FGR.8.3Graph rational functions, identifying key characteristics.

AA.FGR.8.4Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

AA.FGRa Functional & Graphical Reasoning – Exponential and Logarithmic Functions

AA.FGRa.3Explore and analyze structures and patterns for exponential and logarithmic functions and use exponential and logarithmic expressions, equations, and functions to model real-life phenomena.

AA.FGRb Functional & Graphical Reasoning – Radical Functions

AA.FGRb.4Explore and analyze structures and patterns for radical functions and use radical expressions, equations, and functions to model real-life phenomena.

AA.FGRc Functional & Graphical Reasoning – Polynomial Functions

AA.FGRc.5Extend 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-world phenomena.

AA.FGRd Functional & Graphical Reasoning – Rational Functions

AA.FGRd.8Analyze the behaviors of rational functions to model applicable, mathematical problems.

AA.GSR Geometric & Spatial Reasoning – Trigonometry and The Unit Circle

AA.GSR.7 Develop an introductory understanding of the unit circle; solve trigonometric equations using the unit circle.

AA.GSR.7.1Define the three basic trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane.

AA.GSR.7.2Apply understanding of the angle measures and coordinates of the unit circle to solve practical, real-life problems involving trigonometric equations.

AA.MM Mathematical Modeling

AA.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

AA.MM.1.1Explain applicable, mathematical problems using a mathematical model.

AA.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities domains.

AA.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a mathematical, applicable situation.

AA.MM.1.4Use various mathematical representations and structures to represent and solve real-life problems.

AA.PAR Patterning & Algebraic Reasoning – Linear Algebra and Matrices

AA.PAR.6 Represent data with matrices, perform mathematical operations, and solve systems of linear equations leading to real-world linear programming applications.

AA.PAR.6.1Use matrices to represent data, and perform mathematical operations with matrices and scalars, demonstrating that some properties of real numbers hold for matrices, but that others do not.

AA.PAR.6.2Rewrite a system of linear equations using a matrix representation.

AA.PAR.6.3Use the inverse of an invertible matrix to solve systems of linear equations.

AA.PAR.6.4Utilize linear programming to represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as solutions or non-solutions under the established constraints in real-world problems.

Advanced Financial Algebra

AFA.DSR AFA.DSR

AFA.DSR.7 AFA.DSR.7

AFA.DSR.7.1Interpret measures of central tendency (mean, median, mode) and spread (range, interquartile range, variance, standard deviation) to analyze contextualized data sets.

AFA.DSR.7.2Construct and interpret common data displays (bar graphs, line graphs, stock bar charts, candlestick charts, box and whisker plots, stem and leaf plots, and circle graphs) to recognize and interpret trends.

AFA.DSR.7.3Construct and interpret scatterplots to recognize and interpret trends.

AFA.DSR.7.4Use technology to find, interpret, and graph linear, quadratic, and exponential regression equations to make predictions about the corresponding context.

AFA.DSR.7.5Use technology to determine the correlation coefficient of linear, quadratic, and exponential regression curves.

AFA.DSR.7.6Distinguish between causation and correlation for bivariate data.

AFA.DSR.7.7Create and analyze discrete probability distributions.

AFA.DSR.7.8Apply the Arithmetic Average Formula to calculate and interpret a d-day simple moving average given a set of n data points, p(1), p(2), p(3), ..., p(n-1), p(n).

AFA.DSR.8 AFA.DSR.8

AFA.DSR.8.1Identify a contextual, real-life problem that can be answered using investigative research.

AFA.DSR.8.2Develop statistical questions that can help solve a real-life problem involved in business and financial decision-making.

AFA.DSR.8.3Create a statistical study using sound methodology to answer statistical questions and to solve the real-life problem.

AFA.DSR.8.4Explain how the sample size impacts the precision with which estimates of the population parameters can be made.

AFA.DSR.8.5Recognize that random selection from a population plays a different role than random assignment in an experiment.

AFA.DSR.8.6Incorporate random designs in data collection.

AFA.DSR.8.7Describe ways in which “big data” can be used to make decisions in various business enterprises and in the context of business and financial decision-making.

AFA.DSR.8.8Use distributions to identify the key features of the data collected.

AFA.DSR.8.9Interpret results and make connections to the original research question.

AFA.DSRa Data & Statistical Reasoning – Data Displays

AFA.DSRa.7Collect, analyze, interpret, summarize, and construct displays of data to make predictions within real-world applications.

AFA.DSRb Data & Statistical Reasoning – Investigative Research

AFA.DSRb.8Conduct investigative research to solve real-life problems and answer statistical questions involved in business and financial decision-making.

AFA.FGR Functional & Graphical Reasoning – Linear, Exponential, Quadratic, Cubic, Rational, Square Root, Greatest Integer, and Piecewise Functions

AFA.FGR.3 Explore and apply functions to model and explain real-life phenomena and to solve complex problems in business and financial contexts.

AFA.FGR.3.1Examine and identify the key characteristics of functions that model financial situations given the parameters of the context.

AFA.FGR.3.10Recognize real-world situations where square root, cubic, or rational functions apply.

AFA.FGR.3.11Create and use inequalities to define domains when creating algebraic expressions and functions.

AFA.FGR.3.2Solve financial problems given the parameters of the applicable context using a variety of functions.

AFA.FGR.3.3Describe the meaning of functions and how to determine if a relation is a function or not.

AFA.FGR.3.4Utilize function notation to represent a functional relation and to evaluate functions.

AFA.FGR.3.5Create, apply, and interpret linear functions to model real-world financial problems.

AFA.FGR.3.6Create, apply, and interpret exponential functions of the form y = ab^x and classify them as exponential decay (when 0 < b < 1) or as exponential growth (when b > 1).

AFA.FGR.3.7Create, apply, and interpret quadratic functions to model real-world financial applications.

AFA.FGR.3.8Create, apply, and interpret the greatest integer function in real-world financial applications.

AFA.FGR.3.9Create, apply, and interpret piecewise functions in real-world financial applications.

AFA.GSR Geometric & Spatial Reasoning – Polygons, Circles, and Trigonometry

AFA.GSR.6 Apply properties of polygons, circles, and trigonometry to model and explore real-world applications.

AFA.GSR.6.1Apply concepts of area, volume, and scale factors to a variety of real-world financial applications.

AFA.GSR.6.2Use factors of dilations to draw to scale in contextual situations.

AFA.GSR.6.3Use sectors and central angles of a circle to depict proportional categories on a pie chart when given categorical information.

AFA.GSR.6.4Solve problems using the Pythagorean Theorem and trigonometric functions and their inverses in context.

AFA.MM Mathematical Modeling

AFA.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

AFA.MM.1.1Explain contextual, mathematical problems using a mathematical model.

AFA.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

AFA.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

AFA.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

AFA.NR Numerical (Quantitative) Reasoning – Fractions, Decimals, Percents, and Ratios

AFA.NR.2 Utilize fractions, decimals, percents, and ratios to write and solve a variety of financial problems.

AFA.NR.2.1Use fractions, decimals, percents, and ratios to solve problems related to budgets, income tax rates, payroll deductions, pie charts, percent yield, sales tax, percent populations, rent increase, cost savings, debt-to-income ratios, stock splits, floor plans and scale models, trigonometric calculations, banking services, and other business and financial applications.

AFA.NR.2.2Convert numerical quantities of one form (fractions, decimals, percents) to another within financial applications.

AFA.NR.2.3Calculate and interpret percent of increase and decrease.

AFA.NR.2.4Construct, solve, and interpret algebraic ratios and proportions.

AFA.PAR AFA.PAR

AFA.PAR.4 AFA.PAR.4

AFA.PAR.4.1Use and rearrange formulas applicable to real-world contexts.

AFA.PAR.4.10Utilize the monthly payment formula to assist in calculating the total interest paid (finance charge) when using credit. Compare the total of monthly payments to the original (cash) price.

AFA.PAR.4.11Interpret and use sigma notation.

AFA.PAR.4.12Explore and identify how the elements of the present value of a single deposit formula and the periodic deposit investment formula relate to the compound interest formula.

AFA.PAR.4.13Utilize the present and future value of a periodic investment formulas to make calculations regarding long-term investments and retirement planning.

AFA.PAR.4.2Investigate the impact of changing the value of the different variables in financial formulas to compare the resulting financial outcomes.

AFA.PAR.4.3Write algebraic formulas for use in spreadsheets and utilize technology to perform both iterate and formulaic calculations.

AFA.PAR.4.4Use the simple interest formula, I = Prt, and inverse operations to solve for specified variables in banking services applications and other interest problems.

AFA.PAR.4.5Demonstrate by iteration (both with technology and without) that the compounding process pays “interest on your interest.”

AFA.PAR.4.6Derive the compound interest formula, A = P(1 + r/t), by using patterns and inductive reasoning, then compute compound interest with and without the formula.

AFA.PAR.4.7Explore the concept of limits of rational functions in discovering the compound continuous formula. Use technology to investigate and verify what happens as the number of compounds approaches infinity.

AFA.PAR.4.8Apply the natural base e in the continuous compounding formula, A = Pe^rt.

AFA.PAR.4.9Use the monthly payment formula to calculate payment amounts in a variety of circumstances.

AFA.PAR.5 AFA.PAR.5

AFA.PAR.5.1Write, graph, solve, and interpret systems of linear equations given an applicable financial situation.

AFA.PAR.5.2Write, graph, solve, and interpret systems of equations containing one linear and one quadratic equation, given an applicable financial situation.

AFA.PAR.5.3Write, graph, and interpret systems of equations containing one linear and one exponential equation, given an applicable financial situation.

AFA.PAR.5.4Write, graph, and interpret systems of a linear and a piecewise function.

AFA.PAR.5.5Solve linear systems of equations and inequalities to identify points of intersection and define domains in the context of the problem situation.

AFA.PARa Patterning & Algebraic Reasoning – Formulas

AFA.PARa.4Explore, evaluate, and rearrange formulas applicable to business and financial contexts.

AFA.PARb Patterning & Algebraic Reasoning – Systems Of Equations and Inequalities

AFA.PARb.5Write and solve systems of equations and/or inequalities in context of financial applications.

Advanced Finite Mathematics

AFM.AQR AFM.AQR

AFM.AQR.5 AFM.AQR.5

AFM.AQR.5.1Find the union, intersection, difference, complement, and Cartesian product of sets, and classify sets as equal, subsets, and power sets.

AFM.AQR.5.2Justify whether the union of subsets of a set is a partition of that set.

AFM.AQR.5.3Given a relation on two sets, determine whether the relation is a function and find its inverse relation, if it exists.

AFM.AQR.5.4Determine the equivalence classes given an equivalence relation on a set; determine whether the union of equivalence classes of a set is a partition of that set.

AFM.AQR.5.5Prove set relations, including DeMorgan’s Laws and equivalence relations.

AFM.AQR.5.6Prove statements in Boolean algebra.

AFM.AQR.5.7Simplify Boolean algebra expressions using Karnaugh maps (K-maps).

AFM.AQR.6 AFM.AQR.6

AFM.AQR.6.1Use the addition rule to count the number of outcomes in a disjoint set of sample spaces. Use the principle of inclusion-exclusion to count the number of outcomes in the union of sample spaces.

AFM.AQR.6.10Use the pigeonhole principle to prove statements about counting.

AFM.AQR.6.2Apply the axioms of probability to determine the probability of dependent and independent events, including use of the multiplication rule for independent events.

AFM.AQR.6.3Find expected value.

AFM.AQR.6.4Apply Bayes’ Theorem to determine conditional probability.

AFM.AQR.6.5Calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.

AFM.AQR.6.6Calculate the number of subsets of size r that can be chosen from a set of n elements.

AFM.AQR.6.7Calculate the number of combinations with repetitions of r elements from a set of n elements

AFM.AQR.6.8Prove combinatorial identities.

AFM.AQR.6.9Apply a combinatorial argument to prove the binomial theorem.

AFM.AQR.7 AFM.AQR.7

AFM.AQR.7.1Identify simple graphs, complete graphs, complete bipartite graphs, and trees. Identify graphs that have Euler and Hamiltonian cycles.

AFM.AQR.7.2Construct the complement and the line graph of a graph.

AFM.AQR.7.3Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.

AFM.AQR.7.4Prove statements about graph properties.

AFM.AQR.7.5Prove that every connected graph has a minimal spanning tree.

AFM.AQR.7.6Use Kruskal’s algorithm and Prim’s algorithm to determine the minimal spanning tree of a weighted graph.

AFM.AQRa Abstract & Quantitative Reasoning – Set Theory

AFM.AQRa.5Use set theory to describe relationships and equivalence when solving contextual, mathematical problems used to explain real-life phenomena.

AFM.AQRb Abstract & Quantitative Reasoning – Probabilities and Combinatorics

AFM.AQRb.6Calculate and solve combinatorics problems to make sense of a real-life, contextual problem.

AFM.AQRc Abstract & Quantitative Reasoning – Graph Theory

AFM.AQRc.7Apply graph theory to solve contextual, mathematical problems and to explain real-life phenomena.

AFM.LR.2 AFM.LR.2

AFM.LR.2.1Use a counterexample to disprove a statement.

AFM.LR.2.2Prove statements directly from definitions and previously proved statements.

AFM.LR.2.3Prove statements indirectly by proving the contrapositive of the statement.

AFM.LR.2.4Apply the method of reductio ad absurdum (proof by contradiction) to prove statements.

AFM.LR.2.5Use the method of mathematical induction to prove statements involving the positive integers.

AFM.LR.3 AFM.LR.3

AFM.LR.3.1Construct truth tables that represent conditional, biconditional, and quantified statements; use truth tables to determine whether the statement is true or false and use Venn diagrams to illustrate the relationship represented by these truth tables.

AFM.LR.3.2Represent logic operations such as AND, OR, NOT, NOR, and XOR (exclusive OR) using logical symbolism, determine whether statements involving these operations are true or false, and interpret such symbols into English.

AFM.LR.3.3Apply modus ponens and modus tollens to analyze logical arguments to determine whether it is valid, invalid, a tautology, or a contradiction.

AFM.LR.3.4Write the negation, converse, contrapositive, and inverse of a conditional statement and find the truth of each.

AFM.LR.3.5Represent the dichotomy between “true” and “false” with 1s and 0s. Use 1s and 0s to calculate whether a statement is true or false by constructing Boolean logic circuits.

AFM.LR.3.6Convert binary and hexadecimal numbers into decimal, and convert from binary to hexadecimal, and vice versa. Add binary integers and use 2’s complement to subtract binary integers.

AFM.LRa Logical Reasoning – Methods Of Proof

AFM.LRa.2Apply methods of proof to prove or disprove mathematical statements; explain reasoning and justify thinking through mathematical induction when formulating mathematical arguments.

AFM.LRb Logical Reasoning – Logical Symbolism and Binary

AFM.LRb.3Interpret, represent, and communicate logical arguments to explain reasoning and justify thinking when solving problems and to explain real-life phenomena.

AFM.MM Mathematical Modeling

AFM.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

AFM.MM.1.1Explain contextual, mathematical problems using a mathematical model.

AFM.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

AFM.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

AFM.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

AFM.NR Numerical Reasoning – Number Theory

AFM.NR.4 Apply number theory and number-theoretic operations to solve contextual, mathematical problems and to explain real-life phenomena.

AFM.NR.4.1Apply the divides relation to positive integers and calculate one integer modulo another integer.

AFM.NR.4.2Find the inverse of an integer for a certain modulus.

AFM.NR.4.3Calculate the floor and the ceiling of a real number.

AFM.NR.4.4Prove statements involving properties of numbers.

AFM.NR.4.5Prove statements involving the floor and ceiling functions.

AFM.NR.4.6Prove the Fundamental Theorem of Arithmetic, the Euclidean algorithm, and Fermat’s Little Theorem.

Advanced Mathematical Decision Making

AMDM.DSR Data & Statistical Reasoning – Investigative Research

AMDM.DSR.7 Conduct investigative research to solve real-life problems and answer statistical investigative questions involved in business and financial decision-making.

AMDM.DSR.7.1Apply statistical methods to design, conduct, and analyze statistical studies. Identify a contextual, real-life problem that can be answered using investigative research.

AMDM.DSR.7.2Build the skills and vocabulary necessary to analyze and critique reported statistical information, summaries, and graphical displays. Develop statistical investigative questions that can help solve a real-life problem involved in business and financial decision-making.

AMDM.DSR.7.3Create a statistical study using sound methodology to answer statistical investigative questions and to solve the real-life problem.

AMDM.DSR.7.4Explain how the sample size impacts the precision with which estimates of the population parameters can be made (i.e., the larger the sample size the more precision).

AMDM.DSR.7.5Recognize that random selection from a population plays a different role than random assignment in an experiment.

AMDM.DSR.7.6Incorporate random designs in data collection.

AMDM.DSR.7.7Describe ways in which big data can be used to make decisions in various business enterprises and in the context of business and financial decision-making.

AMDM.DSR.7.8Use distributions to identify the key features of the data collected.

AMDM.DSR.7.9Interpret results and make connections to the original research question.

AMDM.FGR Functional & Graphical Reasoning – Modeling With Functions

AMDM.FGR.9 Use functions to model problem situations in both discrete and continuous relationships.

AMDM.FGR.9.1Determine whether a problem situation involving two quantities is best modeled by a discrete or continuous relationship.

AMDM.FGR.9.2Use linear, exponential, logistic, and piecewise functions to construct a model.

AMDM.GSR Geometric & Spatial Reasoning – Deterministic Models

AMDM.GSR.10 Use functions to model problem situations in both discrete and continuous relationships.

AMDM.GSR.10.1Create and use two-dimensional and three-dimensional representations to model authentic situations.

AMDM.GSR.10.2Solve problems involving inaccessible distances using basic trigonometric principles including extensions of right triangle trigonometry.

AMDM.MM Mathematical Modeling

AMDM.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

AMDM.MM.1.1Explain contextual, mathematical problems using a mathematical model.

AMDM.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

AMDM.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

AMDM.MM.1.4Use relevant information to create various mathematical representations and structures to solve real-life problems.

AMDM.PAR AMDM.PAR

AMDM.PAR.11 AMDM.PAR.11

AMDM.PAR.11.1Represent situations and solve problems using vectors. in areas such as transportation, computer graphics, and the physics of force and motion.

AMDM.PAR.11.2Represent geometric transformations and solve problems using matrices.

AMDM.PAR.12 AMDM.PAR.12

AMDM.PAR.12.1Solve problems represented by a vertex-edge graphs.

AMDM.PAR.12.2Construct, analyze, and interpret flow charts to develop an algorithm to describe processes such as quality control procedures.

AMDM.PAR.12.3Investigate the scheduling of projects using Program Evaluation Review Technique (PERT).

AMDM.PAR.12.4Consider problems that can be resolved by coloring graphs.

AMDM.PAR.4 AMDM.PAR.4

AMDM.PAR.4.1Create and verify identification numbers.

AMDM.PAR.4.2Analyze and evaluate the mathematics behind various methods of voting and selection.

AMDM.PAR.4.3Evaluate various voting and selection processes to determine an appropriate method for a given situation.

AMDM.PAR.4.4Apply various ranking algorithms to determine an appropriate method for a given situation.

AMDM.PAR.8 AMDM.PAR.8

AMDM.PAR.8.1Use exponential functions to model change in a variety of financial situations.

AMDM.PAR.8.2Determine, represent, and analyze mathematical models for income, expenditures, and various types of loans and investments.

AMDM.PARa Patterning & Algebraic Reasoning – Identification Numbers, Voting, & Algorithms

AMDM.PARa.4Develop methods or algorithms to analyze discrete situations.

AMDM.PARb Patterning & Algebraic Reasoning – Financial Models

AMDM.PARb.8Create and analyze mathematical models to make decisions related to earning, investing, spending, and borrowing money.

AMDM.PARc Patterning & Algebraic Reasoning – Vectors & Matrices

AMDM.PARc.11Use functions to model problem situations in both discrete and continuous relationships.

AMDM.PARd Patterning & Algebraic Reasoning – Networks

AMDM.PARd.12Make informed decisions and solve problems with a variety of network models in quantitative situations.

AMDM.PR AMDM.PR

AMDM.PR.5 AMDM.PR.5

AMDM.PR.5.1Determine conditional probabilities and probabilities of compound events to make decisions in problem situations.

AMDM.PR.5.2Use probabilities to make and justify decisions about risks in everyday life.

AMDM.PR.6 AMDM.PR.6

AMDM.PR.6.1Calculate expected value to analyze mathematical fairness, payoff, and risk.

AMDM.PR.6.2Analyze real-life situations involving strategic interactions using the mathematics of zero-sum games.

AMDM.PR.6.3Construct a mathematical model of probabilistic situations to make mathematical assumptions.

AMDM.PRa Probabilistic Reasoning – Conditional Probabilities & Compound Events

AMDM.PRa.5Analyze the chances for success or failure in order to make decisions.

AMDM.PRb Probabilistic Reasoning – Mathematical Assumptions & Expected Value

AMDM.PRb.6Model strategic interaction among rational decision-makers.

AMDM.QPR AMDM.QPR

AMDM.QPR.2 AMDM.QPR.2

AMDM.QPR.2.1Apply proportions, ratios, rates, and percentages to various settings, including business, media, and consumerism.

AMDM.QPR.2.2Solve problems involving ratios in mechanical and agricultural contexts.

AMDM.QPR.2.3Use proportions to solve problems involving large quantities that are not easily measured.

AMDM.QPR.3 AMDM.QPR.3

AMDM.QPR.3.1Use averages and weighted averages to make decisions.

AMDM.QPR.3.2Calculate and interpret indices.

AMDM.QPRa Quantitative & Proportional Reasoning – Ratios, Rates, & Percents

AMDM.QPRa.2Make decisions and solve problems using ratios, rates, and percents in a variety of real-world applications.

AMDM.QPRb Quantitative & Proportional Reasoning – Averages & Indices

AMDM.QPRb.3Make predictions by analyzing averages and indices of large data sets through investigations of real-world contexts.

Calculus

C.FGR.2 C.FGR.2

C.FGR.2.1Estimate limits from graphs and tables of values.

C.FGR.2.2Find limits of sums, differences, products, and quotients using substitution.

C.FGR.2.3Represent asymptotic behavior using limits.

C.FGR.2.4Find limits of rational functions using algebraic techniques.

C.FGR.2.5Demonstrate continuity at a point using the definition and limit notation.

C.FGR.2.6Apply the Intermediate Value Theorem to a function over a closed interval.

C.FGR.3 C.FGR.3

C.FGR.3.1Interpret the derivative as an instantaneous rate of change that is a two-sided limit of an average rate of change.

C.FGR.3.2Demonstrate and apply the relationship between differentiability and continuity.

C.FGR.3.3Apply the concept of derivative geometrically, numerically, and analytically.

C.FGR.3.4Find the derivatives of sums, products, quotients, and composite functions.

C.FGR.3.5Find the derivatives of a variety of relations.

C.FGR.3.6Calculate higher order derivatives.

C.FGR.4 C.FGR.4

C.FGR.4.1Calculate the slope of a curve at a point.

C.FGR.4.2Write the equation of the tangent line to a curve at a point and use it to obtain a local linear approximation of a value near the point of tangency.

C.FGR.4.3Identify intervals where functions are increasing, decreasing, and constant by using the relationship between the function and the sign of its first derivative.

C.FGR.4.4Identify points of inflection and intervals of concavity of a function by using the second derivative of a function.

C.FGR.4.5Compare characteristics of f, f’, and f” graphically, numerically, analytically, and with technology.

C.FGR.4.6Apply Mean Value Theorem.

C.FGR.4.7Apply Extreme Value Theorem.

C.FGR.4.8Apply the derivative to real-world problems to find both local and absolute extrema, with and without technology.

C.FGR.4.9Model rates of change in applied situations.

C.FGRa Functional & Graphical Reasoning – Relationship Between Limits and Continuity

C.FGRa.2Apply limit notation and characteristics of continuity to analyze behaviors of functions.

C.FGRb Functional & Graphical Reasoning – Differentiation

C.FGRb.3Relate limits and continuity to the derivative as a rate of change and apply it to a variety of situations including modeling contexts.

C.FGRc Functional & Graphical Reasoning – Applications Of Differentiation

C.FGRc.4Apply derivatives to situations in order to draw conclusions including curve analysis and modeling rates of change in applications.

C.GSR Geometric & Spatial Reasoning – Indefinite and Definite Integrals

C.GSR.5 Analyze the relationship between the derivative and the integral using the Fundamental Theorem of Calculus.

C.GSR.5.1Use Riemann sums to approximate values of definite integrals.

C.GSR.5.2Interpret a definite integral as a limit of Riemann sums.

C.GSR.5.3Find the exact value of a definite integral using geometric formulas on a coordinate plane.

C.GSR.5.4Demonstrate the use of properties of definite integrals.

C.GSR.5.5Apply the Fundamental Theorem of Calculus as an interpretation of the accumulation in the rate of change of a function as equivalent to the change in the antiderivative over the interval.

C.GSR.5.6Apply Fundamental Theorem of Calculus to indefinite integrals to represent the family of antiderivatives.

C.GSR.5.7Apply integration by substitution to definite and indefinite integrals.

C.MM Mathematical Modeling

C.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

C.MM.1.1Explain contextual, mathematical problems using a mathematical model.

C.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

C.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

C.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

C.PAR Patterning & Algebraic Reasoning – Applications Of Integrals

C.PAR.6 Apply the definite integral and indefinite integral to contextual situations.

C.PAR.6.1Find a particular curve in a family of antiderivatives using an initial condition.

C.PAR.6.2Solve separable differential equations and use them to model real-world problems.

C.PAR.6.3Apply definite integrals to find the area between two curves.

C.PAR.6.4Apply definite integrals to find the average value of a function over a closed interval.

College Readiness Mathematics

CRM.DSR Data & Statistical Reasoning – Interpreting Data & Calculating Probabilities Of Compound Events

CRM.DSR.6 Make sense of and reason about variation in data using graphs, tables and probability models to solve problems and draw appropriate conclusions from solutions.

CRM.DSR.6.1Represent univariate data on the real number line.

CRM.DSR.6.10Calculate the conditional probability of A given B.

CRM.DSR.6.2Calculate, compare, and interpret shape, center, and spread of two or more univariate data sets, accounting for possible effects of extreme data points.

CRM.DSR.6.3Summarize categorical data for two categories in two-way frequency tables using relative frequencies in the context of the data.

CRM.DSR.6.4Represent bivariate data on a scatter plot and describe how the variables are related in terms of strength and direction.

CRM.DSR.6.5Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

CRM.DSR.6.6Compute using technology and interpret the correlation coefficient “r” of a linear fit.

CRM.DSR.6.7Distinguish between correlation and causation, and interpolation and extrapolation.

CRM.DSR.6.8Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events.

CRM.DSR.6.9Use the two-way frequency table to calculate conditional probabilities.

CRM.FGR Functional & Graphical Reasoning – Building & Interpreting Functions

CRM.FGR.4 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.

CRM.FGR.4.1Define a function through maps, sets, equations and graphs using function notation.

CRM.FGR.4.2Identify and sketch by hand the parent graph of functions expressed algebraically and show key characteristics of the graph using technology.

CRM.FGR.4.3Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function.

CRM.FGR.4.4Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

CRM.FGR.4.5Compare characteristics of two functions each represented in a different way.

CRM.FGR.4.6Construct linear and exponential functions, given a graph, a description of a relationship, or two input-output pairs.

CRM.FGR.4.7Construct arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect linear functions to arithmetic sequences and exponential functions to geometric sequences.

CRM.FGR.4.8Identify the effect on the parent graph of replacing f(x) by f(x) + k, kf(x), 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.

CRM.GSR Geometric & Spatial Reasoning – Measurement, Surface Area, Volume, & Right Triangle Trigonometry

CRM.GSR.5 Reason deductively and inductively about figures and their properties and make sense of geometric situations using measurements in real-world contexts.

CRM.GSR.5.1Use the distance formula, midpoint formula or slope to verify simple geometric properties.

CRM.GSR.5.2Use coordinates to compute perimeters of polygons, circumference of circles and areas of triangles, rectangles and circles.

CRM.GSR.5.3Informally derive the formulas for the volume and surface area of a cylinder, sphere, prism, pyramid, and cone.

CRM.GSR.5.4Use formulas for finding the volume and surface area of spheres, right and oblique prisms, cylinders, pyramids, and cones.

CRM.GSR.5.5Apply the Pythagorean Theorem and trigonometric ratios to solve problems involving right triangles.

CRM.MM Mathematical Modeling

CRM.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

CRM.MM.1.1Explain contextual, mathematical problems using a mathematical model.

CRM.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

CRM.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

CRM.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

CRM.NR Numerical & Quantitative Reasoning – Real Number System

CRM.NR.2 Utilize exact and approximate calculations to quantify real-world phenomena and solve problems.

CRM.NR.2.1Through multi-step/multi-operational problems, perform mathematical operations on real numbers demonstrating fluency using the order of operations.

CRM.NR.2.2Represent and solve problems using proportional reasoning with ratios, rates, proportions, and scaling.

CRM.NR.2.3Apply the rules of exponents to simplify numerical expressions, extending the properties of exponents to rational exponents.

CRM.NR.2.4Perform mathematical operations on real numbers to include numerical radical expressions and complex fractions.

CRM.NR.2.5Estimate solutions to problems with real numbers and use the estimates to assess the reasonableness of results in the context of the problem.

CRM.PAR Patterning & Algebraic Reasoning – Expressions, Equations, & Inequalities

CRM.PAR.3 Construct expressions, equations, and inequalities, and use them to represent and solve problems by choosing appropriate procedures and interpreting solutions in context.

CRM.PAR.3.1Create equations in one variable and use them to solve problems.

CRM.PAR.3.2Create inequalities in one variable and use them to solve problems.

CRM.PAR.3.3Using multiple representations, solve equations and inequalities and use the solutions to draw reasonable conclusions about a situation being modeled, including possible constraints.

CRM.PAR.3.4Solve quadratic equations using a variety of methods.

CRM.PAR.3.5Rearrange literal equations to highlight a specified variable using the same reasoning as in solving equations.

CRM.PAR.3.6Solve inequalities in one variable graphically and algebraically.

CRM.PAR.3.7Using multiple methods, create and solve systems of linear equations and inequalities.

CRM.PAR.3.8Solve a simple system of equations consisting of a linear and a quadratic equation in two variables. algebraically and graphically.

Differential Equations

DE.AR.2 DE.AR.2

DE.AR.2.1Classify differential equations by order and linearity.

DE.AR.2.2Solve separable differential equations for general solutions and initial value problems.

DE.AR.2.3Solve first-order linear differential equations and initial value problems using integrating factors.

DE.AR.2.4Use modeling or numerical methods to approximate solutions of first-order differential equations in context.

DE.AR.2.5Draw direction fields containing solutions curves for first-order differential equations by hand and using modeling.

DE.AR.2.6Solve first-order linear differential equations that apply to various real-world models including falling bodies, mixtures, population and the logistic equation, continuously compounded interest, and other physics applications.

DE.AR.3 DE.AR.3

DE.AR.3.1Determine whether a first- or second-order differential equation has a unique solution over a given interval by applying the Existence and Uniqueness Theorem.

DE.AR.3.10Determine ordinary points, recurrence relations, and change the index as they relate to series solutions to ordinary differential equations.

DE.AR.3.11Find series solutions to first and second-order non-linear initial value problems.

DE.AR.3.2Solve second-order linear homogeneous and non-homogeneous differential equations by finding characteristic equations, using the method of undetermined coefficients and variation of parameters.

DE.AR.3.3Solve second-order differential equations that apply to various real-world models.

DE.AR.3.4Use vector function notation when discussing the structure of solution sets for homogeneous systems as it pertains to the Wronskian.

DE.AR.3.5Determine the existence and uniqueness of solutions for second-order linear differential equations, determine a fundamental set of solutions, and verify that two solutions form a fundamental set by taking the Wronskian.

DE.AR.3.6Determine the structure of solution set to higher-order differential equations, apply the basic Existence and Uniqueness Theorem to higher-order differential equations, and use the generalizations of the Wronksian for higher order differential equations.

DE.AR.3.7Solve higher-order constant coefficient homogeneous differential equations.

DE.AR.3.8Solve special case non-homogeneous second order ordinary differential equations including Cauchy-Euler Equations.

DE.AR.3.9Find a second linearly dependent solution using reduction of order when given a solution to a non-homogeneous second-order differential equations.

DE.AR.4 DE.AR.4

DE.AR.4.1Determine whether a contextual situation results in a system of differential equations and apply the basic existence and uniqueness results for the corresponding initial value problems.

DE.AR.4.2Solve constant coefficient homogeneous systems using eigenvalues and eigenvectors. Solve systems with real, distinct eigenvalues, as well as those with repeated and imaginary eigenvalues.

DE.AR.4.3Draw phase portraits for solutions of homogeneous systems with constant coefficients.

DE.AR.4.4Solve non-homogeneous systems of ordinary differential equations using the method of undetermined coefficients and variation of parameters.

DE.AR.4.5Determine which non-linear systems are locally linear and identify the behavior of the system about each critical point.

DE.AR.4.6Plot locally linear systems.

DE.AR.4.7Use population models derived from locally linear systems.

DE.AR.5 DE.AR.5

DE.AR.5.1Use the integral definition to perform Laplace transforms for functions.

DE.AR.5.2Use a Laplace table to accurately and efficiently identify Laplace transforms.

DE.AR.5.3Perform inverse Laplace transforms using a variety of techniques.

DE.AR.5.4Solve first- and second-order differential equations using Laplace transforms that apply to fields such as electrical and mechanical engineering.

DE.AR.5.5Write piecewise functions as compositions of step (Heaviside) functions.

DE.AR.5.6Find the general uniqueness and existence of solutions for step functions, and use Laplace transforms to find solutions to step functions.

DE.AR.5.7Find the Laplace transform of the Dirac delta function.

DE.AR.5.8Solve linear systems of differential equations using Laplace transforms.

DE.ARa Abstract Reasoning – First Order Differential Equations

DE.ARa.2Solve contextual, mathematical problems involving first-order differential equations to explain real-life phenomena.

DE.ARb Abstract Reasoning – Higher Order Differential Equations

DE.ARb.3Solve contextual, mathematical problems involving second and higher order differential equations to explain real-life phenomena.

DE.ARc Abstract Reasoning – Systems Of Differential Of Equations

DE.ARc.4Solve contextual, mathematical problems involving systems of differential equations to explain real-life phenomena.

DE.ARd Abstract Reasoning – Laplace Transforms

DE.ARd.5Solve contextual, mathematical problems using Laplace transforms to explain real-life phenomena.

DE.MM Mathematical Modeling

DE.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

DE.MM.1.1Explain contextual, mathematical problems using a mathematical model.

DE.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

DE.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

DE.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

Engineering Calculus

EC.AR.2 EC.AR.2

EC.AR.2.1Build new mathematical knowledge through problem solving that involves the engineering design process.

EC.AR.2.10Present a technical design, using computer-generated model, for an assigned design project utilizing the appropriate scientific units (US standards and SI units).

EC.AR.2.11Use connections among mathematics, technology, and engineering in contextual situations.

EC.AR.2.12Develop vocabulary and communication skills by reading materials associated with engineering and technology education.

EC.AR.2.13Describe the issues of necessity that have influenced innovation and technological development.

EC.AR.2.14Explain the impact of key persons and historical events and their impact on engineering and society.

EC.AR.2.15Investigate the educational requirements and professional expectations associated with engineering career paths.

EC.AR.2.2Solve problems that arise in mathematics and in engineering contexts.

EC.AR.2.3Apply and adapt a variety of appropriate strategies to solve problems.

EC.AR.2.4Use visual and written communication to organize, record, and articulate coherent, mathematical thinking and to express basic design elements.

EC.AR.2.5Monitor and reflect on the process of mathematical problem solving and interpret solutions that arise in engineering contexts.

EC.AR.2.6Produce multiple representations for mathematics presented in engineering contexts.

EC.AR.2.7Select, apply, and translate among mathematical representations to solve problems that arise in engineering contexts.

EC.AR.2.8Use mathematical representations to model and interpret physical and engineering phenomena.

EC.AR.2.9Demonstrate fundamentals of technical sketching using computer-generated visuals in appropriate mathematical scaling.

EC.AR.3 EC.AR.3

EC.AR.3.1Determine the equations of lines and surfaces using vectors and 3D graphing.

EC.AR.3.2Apply dot and cross products of vectors to express equations of planes, parallelism, perpendicularity, angles.

EC.AR.3.3Describe the role of vectors in engineering applications, such as modeling the velocity of moving objects or static forces on structures and objects.

EC.AR.3.4Evaluate matrices and apply their properties to solve problems expressed as matrix equations.

EC.AR.3.5Compute limits of scalar and vector-valued functions.

EC.AR.3.6Identify and graph level curves of multivariate functions.

EC.AR.3.7Find the regions of continuity of multivariate functions.

EC.AR.4 EC.AR.4

EC.AR.4.1Compute the first and second partial derivatives of a function.

EC.AR.4.2Use the general chain rule to determine the partial derivatives of composite functions.

EC.AR.4.3Compute and apply the gradient of multivariable functions.

EC.AR.4.4Solve engineering optimization problems by applying partial differentiation or Lagrange multipliers.

EC.AR.4.5Utilize partial derivatives in developing the appropriate system balances in engineering problems.

EC.ARa Abstract Reasoning – Impact Of Engineering In Mathematics

EC.ARa.2Using the engineering design process, apply mathematical concepts and procedures to solve problems in engineering contexts and research the impact of engineering and technological advancement on mathematics and society.

EC.ARb Abstract Reasoning – Algebra In Engineering

EC.ARb.3Using the engineering design process, express spatial and functional relationships with vectors, functions, and analytic geometry in three dimensions, and use these relationships to solve real-life, mathematical problems and to explain engineering-based phenomena.

EC.ARc Abstract Reasoning – Partial Differentiation In Engineering

EC.ARc.4Define, describe, and represent the differentiation of functions of two independent variables and differential vectors to solve contextual, mathematical problems and to explain engineering-based phenomena.

EC.ARd Abstract Reasoning – Integration In Engineering

EC.ARd.5Interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain engineering-based phenomena.

EC.MM Mathematical Modeling

EC.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

EC.MM.1.1Explain contextual, mathematical problems using a mathematical model.

EC.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

EC.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

EC.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

EC.PR.5 EC.PR.5

EC.PR.5.1Manipulate integrals by changing the order of integration, introducing variable substitutions, or changing to curvilinear coordinates.

EC.PR.5.2Evaluate and apply line integrals that are independent of path.

EC.PR.5.3Apply properties of integrals to calculate and represent area, volume, or mass.

EC.PR.5.4Use integrals of vectors to define and apply the gradient, divergence, and the curl.

EC.PR.5.5Interpret the theorems of Green, Stokes, and Gauss and apply them to the study of real-world phenomena.

Geometry: Concepts & Connections

G.DSR Data & Statistical Reasoning; Probablistic Reasoning – Categorical Data In Two-Way Frequency Tables; Conditional Probability

G.DSR.11 Examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena.

G.DSR.11.1Construct and summarize categorical data for two categories in two-way frequency tables.

G.DSR.11.2Use categorical data in two-way frequency tables to calculate and interpret probabilities based on the investigation.

G.GSR.3 G.GSR.3

G.GSR.3.1Use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations.

G.GSR.3.2Verify experimentally the congruence properties of rotations, reflections, and translations: 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.

G.GSR.3.3Use geometric descriptions of rigid motions to draw the transformed figures and to predict the effect on a given figure. Describe a sequence of transformations from one figure to another and use transformation properties to determine congruence.

G.GSR.3.4Explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. Use congruency criteria for triangles to solve problems and to prove relationships in geometric figures.

G.GSR.4 G.GSR.4

G.GSR.4.1Use the undefined notions of point, line, line segment, plane, distance along a line segment, and distance around a circular arc to develop and use precise definitions and symbolic notations to prove theorems and solve geometric problems.

G.GSR.4.2Classify quadrilaterals in the coordinate plane by proving simple geometric theorems algebraically.

G.GSR.4.3Make formal geometric constructions with a variety of tools and methods.

G.GSR.4.4Prove and apply theorems about lines and angles to solve problems.

G.GSR.4.5Use geometric reasoning 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.

G.GSR.5 G.GSR.5

G.GSR.5.1Verify experimentally the properties of dilations.

G.GSR.5.2Given two figures, use and apply the definition of similarity in terms of similarity transformations.

G.GSR.5.3Use the properties of similarity transformations to establish criterion for two triangles to be similar. Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

G.GSR.5.4Construct formal proofs to justify and apply theorems about triangles.

G.GSR.6 G.GSR.6

G.GSR.6.1Explain 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.

G.GSR.6.2Explain and use the relationship between the sine and cosine of complementary angles.

G.GSR.6.3Use trigonometric ratios and the Pythagorean Theorem to solve for sides and angles of right triangles in applied problems.

G.GSR.7 G.GSR.7

G.GSR.7.1Explore and interpret a radian as the ratio of the arc length to the radius of a circle.

G.GSR.7.2Explore and explain the relationship between radian measures and degree measures and convert fluently between degree and radian measures.

G.GSR.7.3Use special right triangles on the unit circle to determine the values of sine, cosine, and tangent for 30°(π/6), 45°(π/4) and 60°(π/3) angle measures. Use reflections of triangles to determine reference angles and identify coordinate values in all four quadrants of the coordinate plane.

G.GSR.8 G.GSR.8

G.GSR.8.1Identify and apply angle relationships formed by chords, tangents, secants and radii with circles.

G.GSR.8.2Using similarity, derive the fact that the length of the arc (arc length) intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. Solve mathematically applicable problems involving applications of arc length and area of sector.

G.GSR.8.3Write and graph the equation of circles in standard form.

G.GSR.9 G.GSR.9

G.GSR.9.1Use volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems including right and oblique solids.

G.GSR.9.2Use geometric shapes, their measures, and their properties to describe objects and approximate volumes.

G.GSR.9.3Apply concepts of density based on area and volume in modeling situations

G.GSRa Geometric & Spatial Reasoning – Congruence

G.GSRa.3Experiment 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.

G.GSRb Geometric & Spatial Reasoning – Geometric Foundations, Constructions, and Proof

G.GSRb.4Establish facts between angle relations and generate valid arguments to defend facts established. Prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena.

G.GSRc Geometric & Spatial Reasoning – Similarity

G.GSRc.5Describe dilations in terms of center and scale factor and use these terms 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.

G.GSRd Geometric & Spatial Reasoning – Right Triangle Trigonometry

G.GSRd.6Examine 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.

G.GSRe Geometric & Spatial Reasoning – Trigonometry and The Unit Circle

G.GSRe.7Explore the concept of a radian measure and special right triangles.

G.GSRf Geometric & Spatial Reasoning – Circles

G.GSRf.8Examine and apply theorems involving circles; describe and derive arc length and area of a sector; and model and explain real-life frameworks involving circles.

G.GSRg Geometric & Spatial Reasoning – Equations and Measurement

G.GSRg.9Develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri’s principle; solve mathematically applicable problems involving volume; explore and visualize relationships between two-dimensional and three-dimensional objects to model and explain real-life phenomena.

G.MM Mathematical Modeling

G.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

G.MM.1.1Explain mathematically applicable problems using a mathematical model.

G.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

G.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a mathematically applicable situation.

G.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

G.PAR Patterning & Algebraic Reasoning – Polynomial Expressions

G.PAR.2 Interpret the structure of and perform operations with polynomials within a geometric framework.

G.PAR.2.1Interpret polynomial expressions of varying degrees that represent a quantity in terms of its given geometric framework.

G.PAR.2.2Perform operations with polynomials and prove that polynomials form a system analogous to the integers in that they are closed under these operations.

G.PAR.2.3Using algebraic reasoning, add, subtract, and multiply single variable polynomials.

G.PR Probabilistic Reasoning – Compound Events and Expected Values

G.PR.10 Solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions.

G.PR.10.1Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events. Apply the Addition Rule conceptually, P(A or B)= P(A) + P(B)-P(A and B), and interpret the answers in context.

G.PR.10.2Apply and interpret the general Multiplication Rule conceptually to independent events of a sample space, P(A and B) = [P(A)]x[P(B|A)] =[P(B)]x[P(A|B)] using contingency tables or tree diagrams.

G.PR.10.3Use conditional probability to interpret risk in terms of decision-making and investigate questions such as those involving false positives or false negatives from screening tests.

G.PR.10.4Define permutations and combinations and apply this understanding to compute probabilities of compound events and solve meaningful problems.

G.PR.10.5Interpret the probability distribution for a given random variable and interpret the expected value.

G.PR.10.6Develop a probability distribution for variables of interest using theoretical and empirical (observed) probabilities and calculate and interpret the expected value.

G.PR.10.7Calculate the expected value of a random variable and interpret it as the mean of a given probability distribution.

G.PR.10.8Compare the payoff values associated with the probability distribution for a random variable and make informed decisions based on expected value and measures of variability.

History Of Mathematics

HM.LMIR HM.LMIR

HM.LMIR.3 HM.LMIR.3

HM.LMIR.3.1Prove statements in a deductive system by using its definitions, postulates, and axioms

HM.LMIR.3.10Describe the cultural aspects of Greek society that influenced the way mathematics developed in ancient Greece.

HM.LMIR.3.11Describe the theories for the rise of intellectual thought in ancient Greece and the factors involved in its collapse.

HM.LMIR.3.12Analyze factors involved in the rise and fall of ancient Greek society.

HM.LMIR.3.2Prove the first five propositions in Book I of Euclid’s Elements.

HM.LMIR.3.3Construct a regular pentagon with a straight-edge and compass.

HM.LMIR.3.4Compute the areas of regular polygons by Heron’s formulas.

HM.LMIR.3.5Translate Greek geometric algebra into modern algebraic notation.

HM.LMIR.3.6Find the first four perfect numbers using Euclid's formula.

HM.LMIR.3.7Justify statements concerning figurate numbers using both graphical (as in the manner of the Greeks) and algebraic methods.

HM.LMIR.3.8Solve systems of linear and nonlinear equations using Diophantus' method.

HM.LMIR.3.9Explain the distinction made between number and magnitude, commensurable and incommensurable, and arithmetic and logistic, the cultural factors inherent in this distinction, and the logical crisis that occurred concerning incommensurable (irrational) magnitudes.

HM.LMIR.4 HM.LMIR.4

HM.LMIR.4.1Translate medieval mathematical problems that involve linear, quadratic, or cubic equations into modern notation and solve them in a variety of ways.

HM.LMIR.4.10Use historical multiplication and division algorithms.

HM.LMIR.4.2Use Khayyam’s geometric construction to find a solution to a cubic equation.

HM.LMIR.4.3Identify cyclic quadrilaterals and find associated lengths by Ptolemy’s Theorem.

HM.LMIR.4.4Investigate the relationships among the sides and angles of a spherical triangle.

HM.LMIR.4.5Describe the algebraic and geometric contributions of Islamic mathematicians in the Middle Ages.

HM.LMIR.4.6Describe the algebraic and geometric contributions of Chinese mathematicians in the Middle Ages.

HM.LMIR.4.7Describe the transition of Hindu-Arabic numerals from regional use in the 10th century to wide-spread use in the 15th century.

HM.LMIR.4.8Describe the transmission of ideas from the Greeks, through the Islamic peoples, to medieval Europe.

HM.LMIR.4.9Describe the influence of the Catholic Church and Charlemagne on the establishment of mathematics as one of the central pillars of education.

HM.LMIR.5 HM.LMIR.5

HM.LMIR.5.1Use historical multiplication and division algorithms.

HM.LMIR.5.2Use Cardano’s cubic formula to find a solution to a cubic equation.

HM.LMIR.5.3Explain the cultural factors that encouraged the development of algebra in 15th century Italy, and how this development influenced mathematical thought throughout Europe.

HM.LMIR.5.4Identify the works of Galileo, Copernicus, and Kepler as a landmark in scientific thought, describe the conflict between their explanation of the workings of the solar system and then-current perspectives, and contrast their works to those of Aristotle.

HM.LMIR.5.5Describe the mathematical contributions of Fermat, Pascal, and Descartes.

HM.LMIR.6 HM.LMIR.6

HM.LMIR.6.1Determine tangents to quadratic curves using the algebraic techniques of Fermat, Barrow, and Newton.

HM.LMIR.6.10Explain how the ancient Greek pattern of material axiomatics evolved into abstract axiomatics.

HM.LMIR.6.11Solve simple linear congruences of the form ax = b mod m.

HM.LMIR.6.12Use Fermat’s Little Theorem and Euler’s Theorem to simplify expressions of the form a^k mod m.

HM.LMIR.6.13Use Gauss’ Law of Quadratic Reciprocity to determine quadratic residues of two odd primes; i.e., solve quadratic congruences of the form x^2 = p mod q.

HM.LMIR.6.14Verify that the real primes which can be expressed as the sum of two squares are no longer prime in the field of Gaussian integers.

HM.LMIR.6.15Describe the mathematical contributions of Newton, Euler, and Gauss.

HM.LMIR.6.16Explore the history of African American mathematicians in the 17th , 18th, and 19th centuries and describe their contributions to mathematics.

HM.LMIR.6.17Explore the history of female mathematicians in the 17th, 18th, and 19th centuries and describe their contributions to mathematics.

HM.LMIR.6.2Describe the influence the French Revolution had on mathematics education.

HM.LMIR.6.3Prove that the summit angles of an isosceles birectangle are congruent, but that it is impossible to prove they are right without referring to the parallel postulate or one of its consequences.

HM.LMIR.6.4Compare and contrast the hypotheses of the acute angle (Hyperbolic), the right angle (Euclidean), and the obtuse angle (Spherical).

HM.LMIR.6.5Prove that under the hypothesis of the acute angle, similarity implies congruence.

HM.LMIR.6.6Describe the societal factors that inhibited the development of non-Euclidean geometry.

HM.LMIR.6.7Add, subtract, and multiply two quaternions.

HM.LMIR.6.8Investigate abstract algebra and group-theoretic concepts.

HM.LMIR.6.9Identify whether a given set with a binary operation is a group.

HM.LMIR.7 HM.LMIR.7

HM.LMIR.7.1Investigate the implications of infinite sets of real numbers.

HM.LMIR.7.2Compare and contrast denumerable and nondenumerable sets.

HM.LMIR.7.3Identify algebraic and transcendental numbers.

HM.LMIR.7.4Describe the mathematical contributions of Cantor.

HM.LMIR.7.5Describe the implications of Klein’s Erlangen Programme and Gödel's Incompleteness Theorem on the nature of mathematical discovery and proof.

HM.LMIR.7.6Explore the history of 20th century African American mathematicians and describe their contributions to mathematics.

HM.LMIR.7.7Explore the history of 20th century female mathematicians and describe their contributions to mathematics.

HM.LMIR.7.8Explore the history of 20th century Indian, Asian, Hispanic, Latin American mathematicians and describe their contributions to mathematics.

HM.LMIRa Logical, Mathematical & Investigative Reasoning – Ancient Greek Mathematics

HM.LMIRa.3Engage in the mathematical and cultural accomplishments of the ancient Greeks in order to grasp the foundational aspects of modern mathematics.

HM.LMIRb Logical, Mathematical & Investigative Reasoning – Mathematics In The Middles Ages

HM.LMIRb.4Engage in the mathematical and cultural accomplishments of the world’s societies in the 5th century through the 15th century in order to grasp the foundational aspects of modern mathematics.

HM.LMIRc Logical, Mathematical & Investigative Reasoning – Mathematics Of The Classical Era

HM.LMIRc.5Engage in the mathematical accomplishments of Europe in the 15th century through the early 17th century in order to grasp the foundational aspects of modern mathematics.

HM.LMIRd Logical, Mathematical & Investigative Reasoning – Modern Mathematics

HM.LMIRd.6Engage in the mathematical and cultural accomplishments of the world’s societies in the late 17th century through the early 20th century in order to grasp the foundational aspects of modern mathematics.

HM.LMIRe Logical, Mathematical & Investigative Reasoning – 20Th Century Mathematics

HM.LMIRe.7Investigate and describe modern mathematicians and their contributions to mathematics.

HM.MM Mathematical Modeling

HM.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

HM.MM.1.1Explain contextual, mathematical problems using a mathematical model.

HM.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

HM.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

HM.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

HM.NR Numerical Reasoning – Origins Of Mathematics

HM.NR.2 Explore and use historical number systems and computational methods.

HM.NR.2.1Use historical number systems to represent quantities.

HM.NR.2.2Use historical multiplication and division algorithms.

HM.NR.2.3Decompose fractions of the form 2/(pq) using the Egyptian method as recorded by Ahmes (Ahmose) in the Rhind Papyrus.

HM.NR.2.4Compute lengths, areas, and volumes according to historical formulas

HM.NR.2.5Describe the limitations of the Babylonian, Roman, Egyptian (hieratic and hieroglyphic), Chinese, and Greek number systems as compared to Hindu-Arabic numerals

HM.NR.2.6Identify the number system and notation used by a society as an influence on the types of mathematics developed by that society.

HM.NR.2.7Solve linear equations using the method of false position.

HM.NR.2.8Translate ancient mathematical problems that involve linear, quadratic, or cubic equations into modern notation and solve them in a variety of ways.

Linear Algebra With Computer Science Applications

LACS.ADR Abstract & Digital Reasoning – Programming

LACS.ADR.2 Investigate and describe real-life problems in linear algebra using an object-oriented programming language.

LACS.ADR.2.1Utilize sets, lists, dictionaries, indexing, and tuples in programming languages.

LACS.ADR.2.2Show and explain how to program and apply modules and control statements in programming languages.

LACS.ADR.2.3Program input and output features to read from and write to files in a programming assignment.

LACS.GSR LACS.GSR

LACS.GSR.3 LACS.GSR.3

LACS.GSR.3.1Use coordinates to represent points in n dimensions and define and use arithmetic operations on n-dimensional points.

LACS.GSR.3.10Use vector operations to program simple authentication schemes.

LACS.GSR.3.2Use vectors to find and interpret geometrical relationships between points in two and three dimensions, such as distance, and generalize these relationships to higher dimensions using n-dimensional vectors.

LACS.GSR.3.3Interpret adding, scaling, and linear combinations of vectors geometrically and algebraically.

LACS.GSR.3.4Find and use the dot product of two n-dimensional vectors.

LACS.GSR.3.5Use properties of the dot product to prove statements about vectors and to solve problems in context.

LACS.GSR.3.6Use the triangle inequality in n-dimensions.

LACS.GSR.3.7Find and use the cross product of two 3-dimensional vectors.

LACS.GSR.3.8Represent and perform vector operations using programming language classes that define the use of vectors.

LACS.GSR.3.9Apply perfect secrecy, all-or-nothing secret sharing, and solving lights out games to vectors over GF(2).

LACS.GSR.5 LACS.GSR.5

LACS.GSR.5.1Given a 2-by-2 or 3-by-3 linear transformation matrix, describe the transformation a geometric figure undergoes.

LACS.GSR.5.2Find matrices that represent scalings, reflections, and rotations of geometric figures.

LACS.GSR.5.3Find a matrix that represents a combination of transformations.

LACS.GSR.5.4Find the image of a point under a transformation.

LACS.GSR.5.5Find the area of a polygon given its coordinates using matrices; find the area of the image of a polygon after a transformation.

LACS.GSR.5.6Write code to perform transformations in two-dimensional geometry using matrix operations.

LACS.GSR.5.7Define functions from n dimensions to m dimensions as vectors and/or matrices.

LACS.GSR.5.8Find the image and preimage of a linear map using matrices; determine whether the linear map is one-to-one.

LACS.GSR.5.9Find and interpret geometrically the set of preimages of a vector under a given matrix.

LACS.GSRa Geometric & Spatial Reasoning – Vectors

LACS.GSRa.3Solve contextual, mathematical problems involving vectors to explain real-life phenomena.

LACS.GSRb Geometric & Spatial Reasoning – Matrices

LACS.GSRb.5Solve contextual, mathematical problems involving matrices as geometric transformations and to explain real-life phenomena.

LACS.MM Mathematical Modeling

LACS.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

LACS.MM.1.1Explain contextual, mathematical problems using a mathematical model.

LACS.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

LACS.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

LACS.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

LACS.PAR LACS.PAR

LACS.PAR.4 LACS.PAR.4

LACS.PAR.4.1Represent a linear system of three equations in three variables as an augmented matrix and reduce the matrix to row-echelon form.

LACS.PAR.4.10Solve a matrix equation using inverses; find all solutions to a matrix equation given one solution and the kernel.

LACS.PAR.4.11Improve the simple authentication scheme over GF(2).

LACS.PAR.4.12Show and explain how threshold secret sharing works in conjunction with Gaussian elimination through programming.

LACS.PAR.4.13Write code utilizing error-correcting concepts.

LACS.PAR.4.2Interpret the nature of the solution of a system from its row-echelon form, and if there are infinitely many solutions, express them as a vector equation.

LACS.PAR.4.3Determine whether a vector is a linear combination of other given vectors; find the linear combination of vectors that results in a given vector.

LACS.PAR.4.4Interpret linear dependence of vectors geometrically.

LACS.PAR.4.5Find the kernel of a matrix and explore the relationship between the kernel, the orthogonality of the vectors in the kernel, and the linear dependence of the rows/columns.

LACS.PAR.4.6Add two matrices, multiply a matrix by a scalar, find the transpose of a matrix.

LACS.PAR.4.7Determine when matrix multiplication is defined, and if defined, multiply two matrices by considering the matrix product as a dot product of a group of vectors.

LACS.PAR.4.8Determine when the inverse of a square matrix exists, and if it exists, find it by augmenting the identity matrix to the matrix and then use row operations.

LACS.PAR.4.9Decompose a matrix into its symmetric and skew-symmetric parts; decompose a matrix into its LU factorization.

LACS.PAR.7 LACS.PAR.7

LACS.PAR.7.1Determine whether a given set of vectors generates a vector space.

LACS.PAR.7.10Find an orthogonal basis for a given basis or subspace by applying the Gram-Schmidt orthonormalization process.

LACS.PAR.7.11Perform QR factorization of a matrix to solve matrix equations.

LACS.PAR.7.12Apply the method of least squares to find the line or parabola of best fit to approximate data in context.

LACS.PAR.7.13Apply the grow-and-shrink algorithm in the minimum spanning forest problem in GF(2).

LACS.PAR.7.14Apply the Exchange Lemma to image perspective rendering.

LACS.PAR.7.15Use bases to represent images and sounds as wavelets; perform wavelet transformation, implementation, and decomposition through programming.

LACS.PAR.7.16Program a Fast Fourier Transform to store a sequence of amplitude samples.

LACS.PAR.7.17Apply the Rank Theorem to demonstrate the simple authentication scheme.

LACS.PAR.7.2Justify whether a subset of a vector space is a subspace.

LACS.PAR.7.3Determine whether a given vector is in the linear span of a set of vectors.

LACS.PAR.7.4Determine whether two vector subspaces are orthogonal; find the orthogonal component of a given subspace.

LACS.PAR.7.5Determine whether a set of vectors is a basis for a vector space.

LACS.PAR.7.6Find the dimension of a vector space; find the dimensions of the row space, column space, and kernel for a given matrix; find the rank of a matrix.

LACS.PAR.7.7Find a matrix representing a linear map.

LACS.PAR.7.8Determine the change of representation for a linear transformation given two different bases on a vector space.

LACS.PAR.7.9Determine if two matrices are similar; determine if two matrices are orthogonal.

LACS.PAR.8 LACS.PAR.8

LACS.PAR.8.1Evaluate the determinant of a matrix along any row or column and use a recursive procedure for evaluating a determinant for matrices larger than 3-by-3.

LACS.PAR.8.10Find the dimension of the eigenspace corresponding to the eigenvalues of a symmetric matrix.

LACS.PAR.8.11Determine an orthogonal matrix that diagonalizes a given matrix.

LACS.PAR.8.12Apply eigenvalues and eigenvectors to problems in context.

LACS.PAR.8.2Justify properties of the determinant.

LACS.PAR.8.3Calculate the determinant of the product of two matrices; calculate the determinant of the transpose of a matrix.

LACS.PAR.8.4Determine if a matrix has a nonzero determinant and extend the nonzero determinant property to problems involving linear dependency, rank, and matrix inverses.

LACS.PAR.8.5Extend the definition and geometric interpretation of the cross product to n – 1 vectors in n dimensions.

LACS.PAR.8.6Use Cramer’s Rule to solve a system of linear equations.

LACS.PAR.8.7Find the characteristic polynomial of a matrix and interpret the characteristic polynomial geometrically.

LACS.PAR.8.8Find the eigenvalues and eigenvectors of a matrix and interpret them geometrically.

LACS.PAR.8.9Use a basis of eigenvectors to create a change of basis matrix.

LACS.PARa Patterning & Algebraic Reasoning – Matrices

LACS.PARa.4Solve contextual, mathematical problems involving matrices to explain real-life phenomena.

LACS.PARb Patterning & Algebraic Reasoning – Vector Spaces

LACS.PARb.7Solve contextual, mathematical problems using vector spaces to explain real-life phenomena.

LACS.PARc Patterning & Algebraic Reasoning – Eigenvalues and Eigenvectors

LACS.PARc.8Solve contextual, mathematical problems using eigenvalues and eigenvectors to explain real-life phenomena.

LACS.PR Probabilistic Reasoning – Markov Chains

LACS.PR.6 Using probabilistic and quantitative reasoning, solve contextual, mathematical problems using Markov chains to explain real-life phenomena.

LACS.PR.6.1Model a finite random process using transition matrices in a Markov chain.

LACS.PR.6.2Simulate the different stages of a Markov chain using random numbers.

LACS.PR.6.3Use matrix algebra to calculate the probability of future states of a Markov chain.

LACS.PR.6.4Determine the attractor for a regular Markov chain.

LACS.PR.6.5Use transition matrices to identify absorbing states of a Markov chain.

LACS.PR.6.6Apply Markov chains in context.

LACS.PR.6.7Write a program to model the probabilities of real-life phenomena using a Markov chain.

Mathematics Of Industry and Government

MIG.ARDDM MIG.ARDDM

MIG.ARDDM.2 MIG.ARDDM.2

MIG.ARDDM.2.1Use advanced linear programming to make decisions and interpret results in real-life contexts.

MIG.ARDDM.2.2Distinguish among continuous, integer, and binary contexts

MIG.ARDDM.2.3Model and interpret results of a contextual problem with three or more variables using linear programming.

MIG.ARDDM.2.4Solve problems with three or more variables using technology and principles of linear programming.

MIG.ARDDM.2.5Examine cause and effect of contextual changes.

MIG.ARDDM.3 MIG.ARDDM.3

MIG.ARDDM.3.1Find the optimal median location in a one-dimensional context.

MIG.ARDDM.3.2Find the optimal median location in a rectilinear context.

MIG.ARDDM.3.3Find the optimal location given three equally weighted, noncollinear points

MIG.ARDDM.3.4Find the optimal location in a set covering context.

MIG.ARDDM.4 MIG.ARDDM.4

MIG.ARDDM.4.1Relate context to a network representation.

MIG.ARDDM.4.2Apply appropriate recursive algorithms.

MIG.ARDDM.4.3Examine alternate decisions in response to contextual changes.

MIG.ARDDMa Abstract Reasoning & Deterministic Decision-Making – Linear Programming

MIG.ARDDMa.2Solve contextual, mathematical problems involving linear programming and use the mathematics as a model to make decisions about real life phenomena.

MIG.ARDDMb Abstract Reasoning & Deterministic Decision-Making – Optimal Locations

MIG.ARDDMb.3Solve contextual, mathematical problems involving optimal locations and use the mathematics as a model to make decisions about real life phenomena.

MIG.ARDDMc Abstract Reasoning & Deterministic Decision-Making – Optimal Paths

MIG.ARDDMc.4Solve contextual, mathematical problems involving optimal paths and use the mathematics as a model to make decisions about real life phenomena.

MIG.ARPDM MIG.ARPDM

MIG.ARPDM.5 MIG.ARPDM.5

MIG.ARPDM.5.1Use properties of normal distributions to make decisions about optimization and efficiency.

MIG.ARPDM.5.2Calculate, analyze and interpret theoretical and empirical probabilities using standardized and non-standardized data.

MIG.ARPDM.5.3Consider contextual factors and investigate issues within the decision-making process.

MIG.ARPDM.5.4Apply techniques to quality control settings.

MIG.ARPDM.6 MIG.ARPDM.6

MIG.ARPDM.6.1Calculate theoretical and empirical probabilities using standardized and non-standardized data.

MIG.ARPDM.6.2Analyze and interpret the probabilities in terms of context.

MIG.ARPDM.6.3Consider contextual factors and investigate issues within the decision-making process.

MIG.ARPDM.8 MIG.ARPDM.8

MIG.ARPDM.8.1Use technology to simulate a real-world situation.

MIG.ARPDM.8.2Analyze, evaluate, and interpret results of simulations.

MIG.ARPDM.8.3Examine alternate decisions in response to contextual changes of simulations.

MIG.ARPDM.9 MIG.ARPDM.9

MIG.ARPDM.9.1Develop and analyze fair methods for voting.

MIG.ARPDM.9.2Develop and analyze fair methods for apportioning representatives.

MIG.ARPDM.9.3Develop fair methods for setting voting district boundaries.

MIG.ARPDMa Abstract Reasoning & Probabilistic Decision-Making – Normal Distributions

MIG.ARPDMa.5Solve contextual, mathematical problems with normal distributions to make appropriate decisions.

MIG.ARPDMb Abstract Reasoning & Probabilistic Decision-Making – Binomial, Geometric, and Poisson Distributions

MIG.ARPDMb.6Solve contextual, mathematical problems using other distributions (e.g., binomial, geometric, and Poisson) as well as simulations to make appropriate decisions.

MIG.ARPDMc Abstract Reasoning & Probabilistic Decision-Making – Simulations

MIG.ARPDMc.8Use simulations to make appropriate decisions.

MIG.ARPDMd Abstract Reasoning & Probabilistic Decision-Making – Fair Representation

MIG.ARPDMd.9Using quantitative reasoning, determine fair methods to reflect the wishes of a larger population with representatives.

MIG.MM Mathematical Modeling

MIG.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

MIG.MM.1.1Explain contextual, mathematical problems using a mathematical model.

MIG.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

MIG.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

MIG.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

MIG.PR Probabilistic Reasoning – Probabilistic Models

MIG.PR.7 Use probabilistic models to make appropriate decisions.

MIG.PR.7.1Use program evaluation review technique (PERT) to investigate completion times of a project.

MIG.PR.7.2Develop and apply transition matrices to make predictions using Markov Chains.

MIG.PR.7.3Apply queuing theory

MIG.PR.7.4Consider contextual factors and investigate issues within the decision-making process.

Multivariable Calculus

MVC.AQR MVC.AQR

MVC.AQR.3 MVC.AQR.3

MVC.AQR.3.1Approximate the partial derivatives at a point of a function defined by a table of data.

MVC.AQR.3.2Find expressions for the first and second partial derivatives of a function.

MVC.AQR.3.3Use the total differential to approximate mathematical models.

MVC.AQR.3.4Represent the partial derivatives of a system of two functions in two variables using the Jacobian.

MVC.AQR.3.5Find the partial derivatives of the composition of functions using the general chain rule.

MVC.AQR.3.6Apply partial differentiation to problems of optimization, including problems requiring the use of the Lagrange multiplier.

MVC.AQR.3.7Find the family of solutions and the envelope of the family of solutions to differential equations, including Clairaut equations.

MVC.AQR.3.8Define and apply the gradient, the divergence, and the curl in terms of differential vector operations.

MVC.AQR.4 MVC.AQR.4

MVC.AQR.4.1Integrate functions of the form z = f(x, y) or w = f(x, y, z) through various techniques.

MVC.AQR.4.2Use, evaluate, and interpret double and triple integrals in terms of volume and mass.

MVC.AQR.4.3Represent and evaluate integrals of vector functions as double and triple integrals.

MVC.AQR.4.4Apply line and surface integral to functions representing real-world phenomena.

MVC.AQR.4.5Solve first-order exact differential equations.

MVC.AQR.4.6Use Green’s Theorem to evaluate line integrals in the plane; use Stokes’ Theorem to evaluate line integrals in space.

MVC.AQR.4.7Determine whether a line integral is independent of path and use line integrals in context.

MVC.AQR.4.8Use Gauss’ Divergence Theorem to evaluate surface integrals.

MVC.AQR.4.9Define and apply the gradient, the divergence, and the curl in terms of integrals of vector functions.

MVC.AQRa Abstract & Quantitative Reasoning – Partial Differentiation

MVC.AQRa.3Define, 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.

MVC.AQRb Abstract & Quantitative Reasoning – Integration

MVC.AQRb.4Interpret integrals of functions of two independent variables and of vector functions to solve contextual, mathematical problems and to explain real-life phenomena.

MVC.MM Mathematical Modeling

MVC.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

MVC.MM.1.1Explain contextual, mathematical problems using a mathematical model.

MVC.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

MVC.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

MVC.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

MVC.PAR Patterning & Algebraic Reasoning – Vectors, Functions, and Analytic Geometry In Three Dimensions

MVC.PAR.2 Express spatial and functional relationships with vectors, functions, and analytic geometry in three dimensions, and use these relationships to solve contextual, mathematical problems.

MVC.PAR.2.1Represent equations of lines in space using vectors.

MVC.PAR.2.2Express the analytic geometry of three dimensions in terms of the dot product and cross product of vectors.

MVC.PAR.2.3Use a linear system of equations to determine whether two planes intersect in a single point or a line, or whether they do not intersect at all.

MVC.PAR.2.4Evaluate functions of two independent variables at a point in the plane.

MVC.PAR.2.5Graph the level curves of functions of two independent variables.

MVC.PAR.2.6Investigate 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.

MVC.PAR.2.7Determine points or regions of discontinuity of functions of two independent variables.

Precalculus

PC.AGR.4 PC.AGR.4

PC.AGR.4.1Apply the fundamental trigonometric identities to simplify expressions and verify other identities.

PC.AGR.4.2Use sum, difference, double-angle, and half-angle formulas for sine, cosine, and tangent to establish other identities and apply them to solve problems.

PC.AGR.4.3Solve trigonometric equations arising in modeling contexts.

PC.AGR.4.4Prove and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

PC.AGR.4.5Determine the area of an oblique triangle.

PC.AGR.6 PC.AGR.6

PC.AGR.6.1Represent vector quantities as directed line segments; represent magnitude and direction of vectors in component form using appropriate mathematical notation.

PC.AGR.6.2Add and subtract vectors and multiply vectors by a scalar to find the resultant vector.

PC.AGR.6.3Add and subtract vectors on a coordinate plane using different methods.

PC.AGR.6.4Solve contextual vector problems, such as those involving velocity, force, and other quantities.

PC.AGR.6.5Sketch the graph of a curve represented parametrically, indicating the direction of motion.

PC.AGR.6.6Apply parametric equations to contextual problems.

PC.AGRa Algebraic & Geometric Reasoning – Trigonometric Identities and Equations

PC.AGRa.4Manipulate, prove, and apply trigonometric identities and equations to solve contextual mathematical problems.

PC.AGRb Algebraic & Graphical Reasoning – Vectors and Parametric Equations

PC.AGRb.6Represent and model vector quantities to solve problems in contextual situations.

PC.FGR.2 PC.FGR.2

PC.FGR.2.1Graph piecewise-defined functions, including step functions and absolute value functions.

PC.FGR.2.2Describe characteristics by interpreting the algebraic form and graph of a piecewise-defined function.

PC.FGR.2.3Represent the limit of a function using both the informal definition and the graphical interpretation in the context of piecewise-defined functions; interpret limits expressed in analytic notation.

PC.FGR.2.4Divide polynomials using various methods.

PC.FGR.2.5Graph rational functions and identify key characteristics.

PC.FGR.2.6Represent the behavior of a rational function using limit notation for vertical and horizontal asymptotes and end behavior.

PC.FGR.2.7Represent the limit of a function using both the informal definition and the graphical interpretation in the context of rational functions; interpret limits expressed in analytic notation.

PC.FGR.2.8Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise

PC.FGR.2.9Perform partial fraction decomposition of rational functions using non-repeated linear factors.

PC.FGR.3 PC.FGR.3

PC.FGR.3.1Use the concept of a radian as the ratio of the arc length to the radius of a circle to establish the existence of 2π radians in one revolution.

PC.FGR.3.2Utilize right triangles on the unit circle to determine the values of the six trigonometric ratios for π/6, π/4, and π/3. Use reflections of the triangles as reference angles to establish known values in all four quadrants of the coordinate plane.

PC.FGR.3.3Define the six trigonometric ratios in terms of x, y, and r using the unit circle centered at the origin of the coordinate plane. Interpret radian measures of angles as a rotation both counterclockwise and clockwise around the unit circle.

PC.FGR.3.4Derive the fundamental trigonometric identities.

PC.FGR.3.5Determine the value(s) of trigonometric functions for a set of given conditions.

PC.FGR.3.6Graph and write equations of trigonometric functions using period, phase shift, and amplitude in modeling contexts.

PC.FGR.3.7Classify the six trigonometric functions as even or odd and describe the symmetry.

PC.FGR.3.8Restrict the domain of a trigonometric function to create an invertible function and graph the inverse function. Evaluate inverse trigonometric expressions.

PC.FGRa Functional & Graphical Reasoning – Functions and Their Characteristics

PC.FGRa.2Analyze the behaviors of rational and piecewise functions to model contextual mathematical problems.

PC.FGRb Functional & Graphical Reasoning – Trigonometric Relationships and Functions

PC.FGRb.3Utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions.

PC.GSR Geometric & Spatial Reasoning – Conic Sections and Polar Equations

PC.GSR.5 Analyze the behaviors of conic sections and polar equations to model contextual mathematical problems.

PC.GSR.5.1Identify and graph different conic sections given the equations in standard form.

PC.GSR.5.2Identify different conic sections in general form and complete the square to convert the equation of a conic section into standard form.

PC.GSR.5.3Define polar coordinates and relate polar coordinates to Cartesian coordinates.

PC.GSR.5.4Classify special polar equations and apply to contextual situations.

PC.GSR.5.5Graph equations in the polar coordinate plane with and without the use of technology.

PC.MM Mathematical Modeling

PC.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

PC.MM.1.1Explain contextual, mathematical problems using a mathematical model.

PC.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

PC.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

PC.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

PC.PAR Patterning & Algebraic Reasoning – Sequences and Series

PC.PAR.7 Demonstrate how sequences and series apply to mathematical models in real-life situations.

PC.PAR.7.1Demonstrate that sequences are functions whose domain is the set of natural numbers.

PC.PAR.7.2Represent sequences graphically, numerically, and symbolically.

PC.PAR.7.3Determine the limit of a sequence if it exists.

PC.PAR.7.4Demonstrate that a series is the sum of the sequence and represent series graphically, numerically, and symbolically.

PC.PAR.7.5Describe the behavior of a series in terms of the limit of its partial sums.

PC.PAR.7.6Derive and use the sum formula of a finite geometric series to solve contextual problems to model real-life situations.

PC.PAR.7.7Derive and use the sum formula of an infinite geometric series to solve contextual problems to model real-life situations.

Statistical Reasoning

SR.DSR.2 SR.DSR.2

SR.DSR.2.1Formulate statistical investigative questions about a population using samples taken from the population.

SR.DSR.2.2Formulate comparative and associative statistical investigative questions for surveys, observational studies, and experiments to compare two or more groups or to investigate the association of two or more variables.

SR.DSR.2.3Formulate multivariable statistical investigative questions.

SR.DSR.2.4Formulate inferential statistical investigative questions regarding association and prediction.

SR.DSR.3 SR.DSR.3

SR.DSR.3.1Apply an appropriate data-collection plan when collecting primary or secondary data for the statistical investigative question of interest.

SR.DSR.3.2Distinguish between surveys, observational studies, and experiments.

SR.DSR.3.3Design sample surveys, experiments, and observational studies using accepted practices.

SR.DSR.3.4Distinguish between random selection and random assignment and identify their impact on conclusions.

SR.DSR.3.5Describe potential sources and effects of bias and confounding variables

SR.DSR.3.6Describe and adhere to the ethical use of data (e.g., sensitive information, privacy, and living subjects).

SR.DSR.3.7Identify when data can be generalized to a target population.

SR.DSR.4 SR.DSR.4

SR.DSR.4.1Summarize quantitative or categorical data using tables, graphical displays, and numerical summary statistics.

SR.DSR.4.2Summarize and describe relationships among multiple variables.

SR.DSR.4.3Use sampling distributions developed through simulation to describe the sample-to-sample variability of sample statistics.

SR.DSR.4.4Use sampling distributions to compute simulated p-values.

SR.DSR.4.5Describe the relationship between two quantitative variables by interpreting correlation (r) and a least-square regression line (using technology).

SR.DSR.4.6Use simulations to investigate associations between two categorical variables and to compare groups.

SR.DSR.5 SR.DSR.5

SR.DSR.5.1Use statistical evidence from analyses to answer the formulated statistical investigative questions.

SR.DSR.5.2Interpret the impact of outliers, missing values, or erroneous values on the results.

SR.DSR.5.3Use and interpret the p-value to determine whether the estimate for a population characteristic is plausible.

SR.DSR.5.4Interpret a given margin of error associated with an estimate of a population characteristic.

SR.DSR.5.5Explain the impact of multiple variables on one another.

SR.DSRa Data & Statistical Reasoning – Formulate Statistical Investigative Questions

SR.DSRa.2Formulate statistical investigative questions of interest to students that can be answered with data.

SR.DSRb Data & Statistical Reasoning – Collect Data

SR.DSRb.3Collect data by designing and implementing a plan to address the formulated statistical investigative question.

SR.DSRc Data & Statistical Reasoning – Analyze Data

SR.DSRc.4Analyze data by selecting and using appropriate graphical and numerical methods.

SR.DSRd Data & Statistical Reasoning – Interpret Results

SR.DSRd.5Interpret the results of the analysis, making connections to the formulated statistical investigative question.

SR.MM Mathematical Modeling

SR.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

SR.MM.1.1Explain contextual, mathematical problems using a mathematical model.

SR.MM.1.2Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or the humanities.

SR.MM.1.3Using abstract and quantitative reasoning, make decisions about information and data from a real-life situation.

SR.MM.1.4Use various mathematical representations and structures with this information to represent and solve real-life problems.

AP Calculus Learning Objectives

CB.Calc.LIM Limits

CB.Calc.LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

CB.Calc.LIM-1.ARepresent limits analytically using correct notation.

CB.Calc.LIM-1.BInterpret limits expressed in analytic notation.

CB.Calc.LIM-1.CEstimate limits of functions.

CB.Calc.LIM-1.DDetermine the limits of functions using limit theorems.

CB.Calc.LIM-1.EDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.

CB.Calc.LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

CB.Calc.LIM-2.AJustify conclusions about continuity at a point using the definition.

CB.Calc.LIM-2.BDetermine intervals over which a function is continuous.

CB.Calc.LIM-2.CDetermine values of x or solve for parameters that make discontinuous functions continuous, if possible.

CB.Calc.LIM-2.DInterpret the behavior of functions using limits involving infinity.

CB.Calc.LIM-3 The derivative of a function is defined as the limit of a difference quotient.

CB.Calc.LIM-3.AInterpret a limit as a definition of a derivative.

CB.Calc.LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.

CB.Calc.LIM-4.ADetermine limits of functions that result in indeterminate forms.

CB.Calc.LIM-5 Definite integrals can be approximated using geometric and numerical methods.

CB.Calc.LIM-5.AApproximate a definite integral using geometric and numerical methods.

CB.Calc.LIM-5.BInterpret the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-5.CRepresent the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-6 Improper integrals can be determined using limits of definite integrals.

CB.Calc.LIM-6.AEvaluate an improper integral or determine that the integral diverges.

CB.Calc.LIM-7 The behavior of a convergent or divergent series can be determined or estimated.

CB.Calc.LIM-7.ADetermine whether a series converges or diverges.

CB.Calc.LIM-7.BApproximate the sum of a series.

CB.Calc.LIM-8 Power series and Taylor series can represent functions.

CB.Calc.LIM-8.ARepresent a function at a point as a Taylor polynomial.

CB.Calc.LIM-8.BApproximate function values using a Taylor polynomial.

CB.Calc.LIM-8.CDetermine the error bound associated with a Taylor polynomial approximation.

CB.Calc.LIM-8.DDetermine the radius of convergence and interval of convergence for a power series.

CB.Calc.LIM-8.ERepresent a function as a Taylor series or a Maclaurin series.

CB.Calc.LIM-8.FInterpret Taylor series and Maclaurin series.

CB.Calc.LIM-8.GRepresent a given function as a power series.

CB.Calc.CHA Change

CB.Calc.CHA-1 The derivative of a function is the instantaneous rate of change of the function.

CB.Calc.CHA-1.AInterpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.

CB.Calc.CHA-2 Derivatives can be computed using definitions and rules.

CB.Calc.CHA-2.ADetermine average rates of change using difference quotients.

CB.Calc.CHA-2.BRepresent the derivative of a function as the limit of a difference quotient.

CB.Calc.CHA-2.CDetermine the equation of a line tangent to a curve at a given point.

CB.Calc.CHA-2.DEstimate derivatives.

CB.Calc.CHA-3 Derivatives can be used to model rates of change in applied contexts.

CB.Calc.CHA-3.AInterpret the meaning of a derivative in context.

CB.Calc.CHA-3.BCalculate rates of change in applied contexts.

CB.Calc.CHA-3.CInterpret rates of change in applied contexts.

CB.Calc.CHA-3.DCalculate related rates in applied contexts.

CB.Calc.CHA-3.EInterpret related rates in applied contexts.

CB.Calc.CHA-3.FApproximate a value on a curve using the equation of a tangent line.

CB.Calc.CHA-3.GCalculate derivatives of parametric functions.

CB.Calc.CHA-3.HCalculate derivatives of vector-valued functions.

CB.Calc.CHA-4 Integrals can be used to model accumulation and net change in applied contexts.

CB.Calc.CHA-4.AInterpret the meaning of areas associated with the graph of a rate of change in context.

CB.Calc.CHA-4.BDetermine the average value of a function using definite integrals.

CB.Calc.CHA-4.CDetermine values for positions and rates of change using definite integrals in problems involving rectilinear motion.

CB.Calc.CHA-4.DInterpret the meaning of a definite integral in accumulation problems.

CB.Calc.CHA-4.EDetermine net change using definite integrals in applied contexts.

CB.Calc.CHA-5 Definite integrals can be used to calculate areas, volumes, and lengths.

CB.Calc.CHA-5.ACalculate areas in the plane using the definite integral.

CB.Calc.CHA-5.BCalculate volumes of solids with known cross sections using definite integrals.

CB.Calc.CHA-5.CCalculate volumes of solids of revolution using definite integrals.

CB.Calc.CHA-5.DCalculate areas of regions defined by polar curves using definite integrals.

CB.Calc.CHA-6 Arc length can be calculated using definite integrals.

CB.Calc.CHA-6.ADetermine the length of a curve in the plane defined by a function, using a definite integral.

CB.Calc.CHA-6.BDetermine the length of a curve in the plane defined by parametric functions, using a definite integral.

CB.Calc.FUN Analysis of Functions

CB.Calc.FUN-1 Existence theorems can be used to draw conclusions about the behavior of functions.

CB.Calc.FUN-1.AExplain the behavior of a function on an interval using the Intermediate Value Theorem.

CB.Calc.FUN-1.BJustify conclusions about functions by applying the Mean Value Theorem over an interval.

CB.Calc.FUN-1.CJustify conclusions about functions by applying the Extreme Value Theorem.

CB.Calc.FUN-2 A function's derivative, when it exists, can be used to determine whether the function is increasing or decreasing.

CB.Calc.FUN-2.AExplain the relationship between differentiability and continuity.

CB.Calc.FUN-3 Derivatives can be calculated for various types of functions.

CB.Calc.FUN-3.ACalculate derivatives of familiar functions.

CB.Calc.FUN-3.BCalculate derivatives of products and quotients of differentiable functions.

CB.Calc.FUN-3.CCalculate derivatives of compositions of differentiable functions.

CB.Calc.FUN-3.DCalculate derivatives of implicitly defined functions.

CB.Calc.FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.

CB.Calc.FUN-3.FDetermine higher order derivatives of a function.

CB.Calc.FUN-3.GCalculate derivatives of functions written in polar coordinates.

CB.Calc.FUN-4 A function's derivative can be used to understand function behavior.

CB.Calc.FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.

CB.Calc.FUN-4.BCalculate minimum and maximum values in applied contexts or analysis of functions.

CB.Calc.FUN-4.CInterpret minimum and maximum values calculated in applied contexts.

CB.Calc.FUN-4.DDetermine critical points of implicit relations.

CB.Calc.FUN-4.EJustify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.

CB.Calc.FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.

CB.Calc.FUN-5.ARepresent accumulation functions using definite integrals.

CB.Calc.FUN-6 Recognizing and applying techniques for antidifferentiation.

CB.Calc.FUN-6.ACalculate a definite integral using areas and properties of definite integrals.

CB.Calc.FUN-6.BEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.

CB.Calc.FUN-6.CDetermine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

CB.Calc.FUN-6.DUse substitution and rearrangements into equivalent forms, such as long division and completing the square, for integrands requiring these techniques.

CB.Calc.FUN-6.EUse integration by parts for integrands requiring this technique.

CB.Calc.FUN-6.FUse integration by linear partial fractions for integrands requiring this technique.

CB.Calc.FUN-7 Solving differential equations allows us to model rates of change.

CB.Calc.FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.

CB.Calc.FUN-7.BVerify solutions to differential equations.

CB.Calc.FUN-7.CEstimate solutions to differential equations.

CB.Calc.FUN-7.DDetermine general solutions to differential equations.

CB.Calc.FUN-7.EDetermine particular solutions to differential equations.

CB.Calc.FUN-7.FInterpret the meaning of a differential equation and its variables in context.

CB.Calc.FUN-7.GDetermine general and particular solutions for problems involving differential equations in context.

CB.Calc.FUN-7.HInterpret the meaning of the logistic growth model in context.

CB.Calc.FUN-8 Solutions to differential equations can model motion in the plane.

CB.Calc.FUN-8.ADetermine a particular solution given a rate vector and initial conditions.

CB.Calc.FUN-8.BDetermine values for positions and rates of change in problems involving planar motion.

AP Statistics Learning Objectives

CB.Stats.VAR Variation and Distribution

CB.Stats.VAR-1 Exploring data through asking and answering statistical questions.

CB.Stats.VAR-1.AIdentify questions to be answered, based on variation in one-variable data.

CB.Stats.VAR-1.BIdentify variables in a set of data.

CB.Stats.VAR-1.CClassify types of variables.

CB.Stats.VAR-1.DIdentify questions to be answered about possible relationships in data.

CB.Stats.VAR-1.EIdentify questions to be answered about data collection methods.

CB.Stats.VAR-1.FIdentify questions suggested by patterns in data.

CB.Stats.VAR-1.GIdentify questions suggested by variation in statistics for samples collected from the same population.

CB.Stats.VAR-1.HIdentify questions suggested by variation in the shapes of distributions of samples taken from the same population.

CB.Stats.VAR-1.IIdentify questions suggested by probabilities of errors in statistical inference.

CB.Stats.VAR-1.JIdentify questions suggested by variation between observed and expected counts in categorical data.

CB.Stats.VAR-1.KIdentify questions suggested by variation in scatter plots.

CB.Stats.VAR-2 Normal distributions and comparing data.

CB.Stats.VAR-2.ACompare a data distribution to the normal distribution model.

CB.Stats.VAR-2.BDetermine proportions and percentiles from a normal distribution.

CB.Stats.VAR-2.CCompare measures of relative position in data sets.

CB.Stats.VAR-3 Designing and analyzing experiments.

CB.Stats.VAR-3.AIdentify the components of an experiment.

CB.Stats.VAR-3.BDescribe elements of a well-designed experiment.

CB.Stats.VAR-3.CCompare experimental designs and methods.

CB.Stats.VAR-3.DExplain why a particular experimental design is appropriate.

CB.Stats.VAR-3.EInterpret the results of a well-designed experiment.

CB.Stats.VAR-4 Probability rules and calculations.

CB.Stats.VAR-4.ACalculate probabilities for events and their complements.

CB.Stats.VAR-4.BInterpret probabilities for events.

CB.Stats.VAR-4.CExplain why two events are (or are not) mutually exclusive.

CB.Stats.VAR-4.DCalculate conditional probabilities.

CB.Stats.VAR-4.ECalculate probabilities for independent events and for the union of two events.

CB.Stats.VAR-5 Discrete random variables and probability distributions.

CB.Stats.VAR-5.ARepresent the probability distribution for a discrete random variable.

CB.Stats.VAR-5.BInterpret a probability distribution.

CB.Stats.VAR-5.CCalculate parameters for a discrete random variable.

CB.Stats.VAR-5.DInterpret parameters for a discrete random variable.

CB.Stats.VAR-5.ECalculate parameters for linear combinations of random variables.

CB.Stats.VAR-5.FDescribe the effects of linear transformations of parameters of random variables.

CB.Stats.VAR-6 Normal distributions and hypothesis testing for proportions.

CB.Stats.VAR-6.ACalculate the probability that a particular value lies in a given interval of a normal distribution.

CB.Stats.VAR-6.BDetermine the interval associated with a given area in a normal distribution.

CB.Stats.VAR-6.CDetermine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.

CB.Stats.VAR-6.DIdentify the null and alternative hypotheses for a population proportion.

CB.Stats.VAR-6.EIdentify an appropriate testing method for a population proportion.

CB.Stats.VAR-6.FVerify the conditions for making statistical inferences when testing a population proportion.

CB.Stats.VAR-6.GCalculate an appropriate test statistic and p-value for a population proportion.

CB.Stats.VAR-6.HIdentify the null and alternative hypotheses for a difference of two population proportions.

CB.Stats.VAR-6.IIdentify an appropriate testing method for the difference of two population proportions.

CB.Stats.VAR-6.JVerify the conditions for making statistical inferences when testing a difference of two population proportions.

CB.Stats.VAR-6.KCalculate an appropriate test statistic for the difference of two population proportions.

CB.Stats.VAR-7 t-distributions and hypothesis testing for means and regression.

CB.Stats.VAR-7.ADescribe t-distributions.

CB.Stats.VAR-7.BIdentify an appropriate testing method for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.CIdentify the null and alternative hypotheses for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.DVerify the conditions for the test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.ECalculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.FIdentify an appropriate selection of a testing method for a difference of two population means.

CB.Stats.VAR-7.GIdentify the null and alternative hypotheses for a difference of two population means.

CB.Stats.VAR-7.HVerify the conditions for the significance test for the difference of two population means.

CB.Stats.VAR-7.ICalculate an appropriate test statistic for a difference of two means.

CB.Stats.VAR-7.JIdentify the appropriate selection of a testing method for a slope of a regression model.

CB.Stats.VAR-7.KIdentify appropriate null and alternative hypotheses for a slope of a regression model.

CB.Stats.VAR-7.LVerify the conditions for the significance test for the slope of a regression model.

CB.Stats.VAR-7.MCalculate an appropriate test statistic for the slope of a regression model.

CB.Stats.VAR-8 Chi-square distributions and tests.

CB.Stats.VAR-8.ADescribe chi-square distributions.

CB.Stats.VAR-8.BIdentify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.CIdentify an appropriate testing method for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.DCalculate expected counts for the chi-square test for goodness of fit.

CB.Stats.VAR-8.EVerify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.

CB.Stats.VAR-8.FCalculate the appropriate statistic for the chi-square test for goodness of fit.

CB.Stats.VAR-8.GDetermine the p-value for the chi-square test for goodness of fit.

CB.Stats.VAR-8.HCalculate expected counts for two-way tables of categorical data.

CB.Stats.VAR-8.IIdentify the null and alternative hypotheses for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.JIdentify an appropriate testing method for comparing distributions in two-way tables of categorical data.

CB.Stats.VAR-8.KVerify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.

CB.Stats.VAR-8.LCalculate the appropriate statistic for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.MDetermine the p-value for a chi-square test for independence or homogeneity.

CB.Stats.UNC Uncertainty and Inference

CB.Stats.UNC-1 Representing and describing categorical and quantitative data.

CB.Stats.UNC-1.ARepresent categorical data using frequency or relative frequency tables.

CB.Stats.UNC-1.BDescribe categorical data represented in frequency or relative tables.

CB.Stats.UNC-1.CRepresent categorical data graphically.

CB.Stats.UNC-1.DDescribe categorical data represented graphically.

CB.Stats.UNC-1.ECompare multiple sets of categorical data.

CB.Stats.UNC-1.FClassify types of quantitative variables.

CB.Stats.UNC-1.GRepresent quantitative data graphically.

CB.Stats.UNC-1.HDescribe the characteristics of quantitative data distributions.

CB.Stats.UNC-1.ICalculate measures of center and position for quantitative data.

CB.Stats.UNC-1.JCalculate measures of variability for quantitative data.

CB.Stats.UNC-1.KExplain the selection of a particular measure of center and/or variability for describing a set of quantitative data.

CB.Stats.UNC-1.LRepresent summary statistics for quantitative data graphically.

CB.Stats.UNC-1.MDescribe summary statistics of quantitative data represented graphically.

CB.Stats.UNC-1.NCompare graphical representations for multiple sets of quantitative data.

CB.Stats.UNC-1.OCompare summary statistics for multiple sets of quantitative data.

CB.Stats.UNC-1.PCompare numerical and graphical representations for two categorical variables.

CB.Stats.UNC-1.QCalculate statistics for two categorical variables.

CB.Stats.UNC-1.RCompare statistics for two categorical variables.

CB.Stats.UNC-1.SRepresent bivariate quantitative data using scatterplots.

CB.Stats.UNC-2 Probability and simulation.

CB.Stats.UNC-2.AEstimate probabilities using simulation.

CB.Stats.UNC-3 Random variables and sampling distributions.

CB.Stats.UNC-3.AEstimate probabilities of binomial random variables using data from a simulation.

CB.Stats.UNC-3.BCalculate probabilities for a binomial distribution.

CB.Stats.UNC-3.CCalculate parameters for a binomial distribution.

CB.Stats.UNC-3.DInterpret probabilities and parameters for a binomial distribution.

CB.Stats.UNC-3.ECalculate probabilities for geometric random variables.

CB.Stats.UNC-3.FCalculate parameters of a geometric distribution.

CB.Stats.UNC-3.GInterpret probabilities and parameters for a geometric distribution.

CB.Stats.UNC-3.HEstimate sampling distributions using simulation.

CB.Stats.UNC-3.IExplain why an estimator is or is not unbiased.

CB.Stats.UNC-3.JCalculate estimates for a population parameter.

CB.Stats.UNC-3.KDetermine parameters of a sampling distribution for sample proportions.

CB.Stats.UNC-3.LDetermine whether a sampling distribution for a sample proportion can be described as approximately normal.

CB.Stats.UNC-3.MInterpret probabilities and parameters for a sampling distribution for a sample proportion.

CB.Stats.UNC-3.NDetermine parameters of a sampling distribution for a difference in sample proportions.

CB.Stats.UNC-3.ODetermine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.

CB.Stats.UNC-3.PInterpret probabilities and parameters for a sampling distribution for a difference in proportions.

CB.Stats.UNC-3.QDetermine parameters of a sampling distribution for a sample mean.

CB.Stats.UNC-3.RDetermine whether a sampling distribution of a sample mean can be described as approximately normal.

CB.Stats.UNC-3.SInterpret probabilities and parameters for a sampling distribution for a sample mean.

CB.Stats.UNC-3.TDetermine parameters of a sampling distribution for a difference in sample means.

CB.Stats.UNC-3.UDetermine whether a sampling distribution of a difference in sample means can be described as approximately normal.

CB.Stats.UNC-3.VInterpret probabilities and parameters for a sampling distribution for a difference in sample means.

CB.Stats.UNC-4 Confidence intervals for proportions and means.

CB.Stats.UNC-4.AIdentify an appropriate confidence interval procedure for a population proportion.

CB.Stats.UNC-4.BVerify the conditions for calculating confidence intervals for a population proportion.

CB.Stats.UNC-4.CDetermine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.

CB.Stats.UNC-4.DCalculate an appropriate confidence interval for a population proportion.

CB.Stats.UNC-4.ECalculate an interval estimate based on a confidence interval for a population proportion.

CB.Stats.UNC-4.FInterpret a confidence interval for a population proportion.

CB.Stats.UNC-4.GJustify a claim based on a confidence interval for a population proportion.

CB.Stats.UNC-4.HIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.

CB.Stats.UNC-4.IIdentify an appropriate confidence interval procedure for a comparison of population proportions.

CB.Stats.UNC-4.JVerify the conditions for calculating confidence intervals for a difference between population proportions.

CB.Stats.UNC-4.KCalculate an appropriate confidence interval for a comparison of population proportions.

CB.Stats.UNC-4.LCalculate an interval estimate based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.MInterpret a confidence interval for a difference of proportions.

CB.Stats.UNC-4.NJustify a claim based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.OIdentify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.PVerify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.QDetermine the margin of error for a given sample size for a one-sample t-interval.

CB.Stats.UNC-4.RCalculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.SInterpret a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.TJustify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.UIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.

CB.Stats.UNC-4.VIdentify an appropriate confidence interval procedure for a difference of two population means.

CB.Stats.UNC-4.WVerify the conditions for calculating confidence intervals for the difference of two population means.

CB.Stats.UNC-4.XDetermine the margin of error for the difference of two population means.

CB.Stats.UNC-4.YCalculate an appropriate confidence interval for a difference of two population means.

CB.Stats.UNC-4.ZInterpret a confidence interval for a difference of population means.

CB.Stats.UNC-5 Errors in statistical inference.

CB.Stats.UNC-5.AIdentify Type I and Type II errors.

CB.Stats.UNC-5.BCalculate the probability of a Type I and Type II errors.

CB.Stats.UNC-5.CIdentify factors that affect the probability of errors in significance testing.

CB.Stats.UNC-5.DInterpret Type I and Type II errors.

CB.Stats.DAT Data Analysis

CB.Stats.DAT-1 Bivariate data analysis and linear regression.

CB.Stats.DAT-1.ADescribe the characteristics of a scatter plot.

CB.Stats.DAT-1.BDetermine the correlation for a linear relationship.

CB.Stats.DAT-1.CInterpret the correlation for a linear relationship.

CB.Stats.DAT-1.DCalculate a predicted response value using a linear regression model.

CB.Stats.DAT-1.ERepresent differences between measured and predicted responses using residual plots.

CB.Stats.DAT-1.FDescribe the form of association of bivariate data using residual plots.

CB.Stats.DAT-1.GEstimate parameters for the least-squares regression line model.

CB.Stats.DAT-1.HInterpret coefficients for the least-squares regression line model.

CB.Stats.DAT-1.IIdentify influential points in regression.

CB.Stats.DAT-1.JCalculate a predicted response using a least-squares regression line for a transformed data set.

CB.Stats.DAT-2 Studies, sampling, and sources of bias.

CB.Stats.DAT-2.AIdentify the type of a study.

CB.Stats.DAT-2.BIdentify appropriate generalizations and determinations based on observational studies.

CB.Stats.DAT-2.CIdentify a sampling method, given a description of a study.

CB.Stats.DAT-2.DExplain why a particular sampling method is or is not appropriate for a given situation.

CB.Stats.DAT-2.EIdentify potential sources of bias in sampling methods.

CB.Stats.DAT-3 Interpreting and justifying conclusions from significance tests.

CB.Stats.DAT-3.AInterpret the p-value of a significance test for a population proportion.

CB.Stats.DAT-3.BJustify a claim about the population based on the results of a significance test for a population proportion.

CB.Stats.DAT-3.CInterpret the p-value of a significance test for a difference of population proportions.

CB.Stats.DAT-3.DJustify a claim about the population based on the results of a significance test for a difference of population proportions.

CB.Stats.DAT-3.EInterpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.DAT-3.FJustify a claim about the population based on the results of a significance test for a population mean.

CB.Stats.DAT-3.GInterpret the p-value of a significance test for a difference of population means.

CB.Stats.DAT-3.HJustify a claim about the population based on the results of a significance test for a difference of two population means in context.

CB.Stats.DAT-3.IInterpret the p-value for the chi-square test for goodness of fit.

CB.Stats.DAT-3.JJustify a claim about the population based on the results of a chi-square test for goodness of fit.

CB.Stats.DAT-3.KInterpret the p-value for the chi-square test for homogeneity or independence.

CB.Stats.DAT-3.LJustify a claim about the population based on the results of a chi-square test for homogeneity or independence.

CB.Stats.DAT-3.MInterpret the p-value of a significance test for the slope of a regression model.

CB.Stats.DAT-3.NJustify a claim about the population based on the results of a significance test for the slope of a regression model.

Standards on this page are sourced from published state frameworks. Goblins aligns tutoring to these standards.