Tennessee Math Standards - Tennessee Academic Standards for Mathematics

1031 standards, Kindergarten through AP.

Tennessee recently revised its Academic Standards for Mathematics to include dual high school pathways: traditional sequences (Algebra I and II, Geometry) or integrated pathways (Math I, II, and III), plus Statistics, Precalculus, and Math Ready. K-8 standards feed into these options for diverse student trajectories.

Grade K

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K.CC Counting and Cardinality (CC)
K.CC.A Know number names and the counting sequence.
K.CC.A.1Count to 100 by ones, fives, and tens. Count backward from 10.
K.CC.A.2Count forward by ones beginning from any given number within the known sequence (instead of having to begin at 1).
K.CC.A.3Write numbers from 0 to 20. Represent a quantity of objects with a written number 0-20.
K.CC.A.4Recognize, describe, extend, and create patterns and explain a simple rule for a pattern using concrete materials. Analyze the structure of the repeating pattern by identifying the unit (core) of the pattern.
K.CC.B Count to tell the number of objects.
K.CC.B.5 Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.B.5.aWhen counting objects 1-20, say the number names in the standard order, using one-to-one correspondence.
K.CC.B.5.bRecognize that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
K.CC.B.5.cRecognize that each successive number name refers to a quantity that is one greater and each previous number is one less.
K.CC.B.6Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, a circle, or as many as 10 things in a scattered configuration. Given a number from 1-20, count out that many objects.
K.CC.C Compare numbers.
K.CC.C.7Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group.
K.CC.C.8Compare two given numbers up to 10, when written as numerals, using the terms greater than, less than, or equal to. (Students need not use comparison symbols here.)
K.G Geometry (G)
K.G.A Identify and describe shapes and solids.
K.G.A.1Describe objects in the environment using names of shapes and solids (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). Describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, between, and next to.
K.G.A.2Correctly name shapes and solids (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres) regardless of their orientations or overall size.
K.G.A.3Identify shapes (squares, circles, triangles, rectangles, and hexagons) as two-dimensional and solids (cubes, cones, cylinders, and spheres) as three-dimensional.
K.G.B Analyze, compare, create, and compose shapes.
K.G.B.4Describe similarities and differences between two- and three-dimensional shapes/solids, in different sizes and orientations.
K.G.B.5Model shapes/solids in the world by building or drawing them.
K.G.B.6Compose a figure using simple shapes/solids and identify smaller shapes/solids within the figure.
K.MD Measurement and Data (MD)
K.MD.A Describe and compare measurable attributes.
K.MD.A.1Describe the measurable attributes of an object, such as length (long/short), height (tall/short), or weight (heavy/light).
K.MD.A.2Directly compare two objects with a measurable attribute in common, to describe which object has more of/less of the attribute. For example, directly compare the heights of two children and describe one child as taller/shorter.
K.MD.B Work with money.
K.MD.B.3Identify the penny, nickel, dime, and quarter based on their attributes (size and color) and recognize the value of each.
K.MD.C Classify objects and count the number of objects in each category.
K.MD.C.4Sort a collection of objects into a given category, with 10 or fewer in each category. Compare the categories by group size.
K.NBT Number and Operations in Base Ten (NBT)
K.NBT.A Work with numbers 11–19 to gain foundations for place value.
K.NBT.A.1Compose and decompose numbers from 11 to 19 into a group of ten ones and some more ones by using objects or drawings (e.g., 18 equals 10 + 8). Record the composition or decomposition using a drawing or by writing an equation.
K.OA Operations and Algebraic Thinking (OA)
K.OA.A Represent and solve problems involving addition and subtraction.
K.OA.A.1Represent addition and subtraction with objects, fingers, drawings, acting out situations, verbal explanations, expressions, or equations.
K.OA.A.2Add and subtract within 10 to solve contextual problems with result/total unknown involving situations of add to, take from, and put together/take apart. Use objects, drawings, or equations to represent the problem.
K.OA.A.3Decompose numbers less than or equal to 10 into addend pairs in more than one way (e.g., 5 = 2 + 3 and 5 = 4 + 1) by using objects or drawings. Record each decomposition using a drawing or writing an equation.
K.OA.A.4Find the number that makes 10, when added to any given number, from 1 to 9 using objects or drawings. Record the answer using a drawing or writing an equation.
K.OA.A.5Use mental strategies flexibly to develop fluency in addition and subtraction within 10.

Grade 1

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1.G Geometry (G)
1.G.A Reason about shapes/solids and their attributes.
1.G.A.1Distinguish between attributes that define a shape (e.g., number of sides and vertices) versus attributes that do not define the shape (e.g., color, orientation, overall size); build and draw two-dimensional shapes to possess defining attributes.
1.G.A.2Create a composite figure and use the composite figure to make new figures by using two-dimensional shapes (rectangles, squares, hexagons, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional solids (cubes, spheres, rectangular prisms, cones, and cylinders).
1.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of, the shares. Understand for these examples that partitioning into more equal shares creates smaller shares.
1.MD Measurement and Data (MD)
1.MD.A Measure lengths indirectly and by iterating length units.
1.MD.A.1Order three objects by length. Compare the lengths of two objects indirectly by using a third object. For example, to compare indirectly the heights of Bill and Susan: if Bill is taller than mother and mother is taller than Susan, then Bill is taller than Susan.
1.MD.A.2Measure the length of an object using non-standard units (paper clips, cubes, etc.) and express this length as a whole number of units.
1.MD.B Work with time and money.
1.MD.B.3Recognize a clock as a measurement tool. Tell and write time in hours and half-hours using analog and digital clocks.
1.MD.B.4Count the value of a set of like coins less than one dollar using the ¢ symbol only.
1.MD.C Represent and interpret data.
1.MD.C.5Organize, represent, and interpret data with up to three categories using pictographs, bar graphs, and tally charts. Ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
1.NBT Number and Operations in Base Ten (NBT)
1.NBT.A Extend the counting sequence.
1.NBT.A.1Count to 120, by ones, twos, and fives starting at any multiple of that number. Count backward from 20. Read and write numbers to 120 and represent a quantity of objects with a written number.
1.NBT.A.2Recognize, describe, extend, and create patterns when counting by ones, twos, fives, and tens and use those patterns to predict the next number in the counting sequence up to 120 through counting or building with concrete materials. For example: 1, 3, 5, ...; 2, 4, 6, ...; 5, 10, 15, ...; etc.
1.NBT.B Understand place value.
1.NBT.B.3Know that the digits of a two-digit number represent groups of tens and ones (e.g., 39 can be represented as 39 ones, 2 tens and 19 ones, or 3 tens and 9 ones).
1.NBT.B.4Compare two two-digit numbers based on the meanings of the digits in each place and use the symbols >, =, and < to show the relationship.
1.NBT.C Use place value understanding and properties of operations to add and subtract.
1.NBT.C.5Add a two-digit number to a one-digit number and a two-digit number to a multiple of ten (within 100). Use concrete models, drawings, strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to explain the reasoning used.
1.NBT.C.6Mentally find 10 more or 10 less than a given two-digit number without having to count by ones and explain the reasoning used.
1.NBT.C.7Subtract multiples of 10 from any number in the range of 10-99 using concrete models, drawings, strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
1.OA Operations and Algebraic Thinking (OA)
1.OA.A Represent and solve problems involving addition and subtraction.
1.OA.A.1Add and subtract within 20 to solve contextual problems, with unknowns in all positions, involving situations of add to, take from, put together/take apart, and compare. Use objects, drawings, and equations with a symbol for the unknown number to represent the problem. NOTE: While start unknown situations may be introduced in first grade, they are not expected to be mastered until second grade.
1.OA.A.2Add three whole numbers whose sum is within 20 to solve contextual problems using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
1.OA.B Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.B.3Apply properties of operations (additive identity, commutative, and associative) as strategies to add and subtract. (Students need not use formal terms for these properties.)
1.OA.B.4Understand the relationship between addition and subtraction by representing subtraction as an unknown-addend problem. For example, to solve 10 – 8 = __, a student can use 8 + __ = 10.
1.OA.C Add and subtract within 20.
1.OA.C.5Add and subtract within 20 using strategies such as counting on, counting back, making 10, related known facts, and composing/decomposing numbers with an emphasis on making ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9 or adding 6 + 7 by creating the known equivalent 6 + 4 + 3 = 10 + 3 = 13 OR 6 + 6 + 1 = 12 + 1).
1.OA.C.6Use mental strategies flexibly and efficiently to develop fluency in addition and subtraction within 20. By the end of grade 1, know all sums and differences up to 10.
1.OA.D Work with addition and subtraction equations.
1.OA.D.7Understand the meaning of the equal sign (e.g., 6 = 6; 5 + 2 = 4 + 3; 7 = 8 – 1). Determine if equations involving addition and subtraction are true or false.
1.OA.D.8Determine the unknown whole number in an addition or subtraction equation with sums/differences within 20, with the unknown in any position (e.g., 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?).

Grade 2

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2.G Geometry (G)
2.G.A Reason about shapes and their attributes.
2.G.A.1Identify triangles, quadrilaterals, pentagons, and hexagons. Draw two-dimensional shapes having specified attributes (as determined directly or visually, not by measuring), such as a given number of angles/vertices or a given number of sides of equal length.
2.G.A.2Partition a rectangle into rows and columns of same-sized squares and find the total number of squares.
2.G.A.3Partition circles and rectangles into two, three, and four equal shares. Describe the shares using the words halves, thirds, fourths, half of, a third of, and a fourth of, and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
2.MD Measurement and Data (MD)
2.MD.A Measure and estimate lengths in standard units.
2.MD.A.1Measure the length of an object in whole number units by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2.MD.A.2Measure the length of an object using two different whole number units of measure and describe how the two measurements relate to the size of the unit chosen.
2.MD.A.3Estimate lengths using whole number units of inches, feet, yards, centimeters, and meters.
2.MD.A.4Measure, using whole number lengths, to determine how much longer one object is than another and express the difference in terms of a standard unit of length.
2.MD.B Relate addition and subtraction to length.
2.MD.B.5Add and subtract within 100 to solve contextual problems, with the unknown in any position, involving lengths that are given in the same units by using drawings and equations with a symbol for the unknown to represent the problem.
2.MD.B.6Represent whole numbers as lengths from 0 on a number line and know that the points corresponding to the numbers on the number line are equally spaced. Use a number line to represent whole number sums and differences of lengths within 100.
2.MD.C Work with time and money.
2.MD.C.7Tell and write time in quarter hours and to the nearest five minutes (in a.m. and p.m.) using analog and digital clocks.
2.MD.C.8Solve contextual problems involving amounts less than one dollar including quarters, dimes, nickels, and pennies using the ¢ symbol appropriately. Solve contextual problems involving whole number dollar amounts up to $100 using the $ symbol appropriately.
2.MD.D Represent and interpret data.
2.MD.D.10Draw a pictograph (with a key of values of 1, 2, 5, or 10) and a bar graph (with intervals of one) to represent a data set with up to four categories. Solve addition and subtraction problems related to the data in a graph.
2.MD.D.9Given a set of data, create a line plot, where the horizontal scale is marked off in whole-number units.
2.NBT Number and Operations in Base Ten (NBT)
2.NBT.A Understand place value.
2.NBT.A.1Know that the three digits of a three-digit number represent amounts of hundreds, tens, and ones (e.g., 706 can be represented in multiple ways as 7 hundreds, 0 tens, and 6 ones; 706 ones; or 70 tens and 6 ones).
2.NBT.A.2Recognize, describe, extend, and create patterns when counting by ones, twos, fives, tens, and hundreds and use those patterns to predict the next number in the counting sequence up to 1000 through counting. For example: 111, 113, 115, ...; 82, 84, 86, ...; 370, 380, 390....; 100, 200, 300,…; etc.
2.NBT.A.3Read and write numbers to 1000 using standard form, word form, and expanded form. For example, write 234 as 200 + 30 + 4.
2.NBT.A.4Compare two three-digit numbers based on the meanings of the digits in each place and use the symbols >, =, and < to show the relationship.
2.NBT.B Use place value understanding and properties of operations to add and subtract.
2.NBT.B.5Fluently add and subtract within 100 using properties of operations, strategies based on place value, and/or the relationship between addition and subtraction.
2.NBT.B.6Add up to four two-digit numbers using properties of operations and strategies based on place value.
2.NBT.C Use place value understanding and properties of operations to add and subtract.
2.NBT.C.7Add and subtract within 1000 using concrete models, drawings, strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to explain the reasoning used. (Explanations may include words, drawing, or objects.)
2.NBT.C.8Mentally add or subtract 10 or 100 to/from any given number within 1000.
2.OA Operations and Algebraic Thinking (OA)
2.OA.A Represent and solve problems involving addition and subtraction.
2.OA.A.1Add and subtract within 100 to solve one- and two-step contextual problems, with unknowns in all positions, involving situations of add to, take from, put together/take apart, and compare. Use objects, drawings, and equations with a symbol for the unknown number to represent the problem.
2.OA.B Add and subtract within 30.
2.OA.B.2Fluently add and subtract within 30 using mental strategies. By the end of 2nd grade, know all sums of two one-digit numbers and related subtraction facts.
2.OA.C Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.3Determine whether a group of objects (up to 20) has an odd or even number of members by pairing objects or counting them by 2s. Write an equation to express an even number as a sum of two equal addends.
2.OA.C.4Use repeated 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. For example, a 3 by 4 array can be expressed as 3 + 3 + 3 + 3 = 12 or 4 + 4 + 4 = 12.
2.OA.D Solve problems involving addition and subtraction and identify and explain patterns in arithmetic.
2.OA.D.1Identify arithmetic patterns in an addition or hundreds chart and explain them using properties of operations. For example, analyze patterns in the addition chart and observe an alternating pattern of even and odd numbers (because each time we move to the right one box or down one box, we are adding one more to our sum: (2 + 3) + 1 = 2 + (3 + 1)= 2 + 4 which uses the associative property of addition).

Grade 3

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3.G Geometry (G)
3.G.A Reason about shapes and their attributes.
3.G.A.1Understand that shapes in different categories may share attributes and that the shared attributes can define a larger category. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals and recognize examples of quadrilaterals that do not belong to any of these subcategories.
3.G.A.2Partition shapes into parts with equal areas. Recognize that equal shares of identical wholes need not have the same shape. Express the area of each part as a unit fraction of the whole.
3.G.A.3Determine if a figure is a polygon.
3.MD Measurement and Data (MD)
3.MD.A Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.A.1 Solve contextual problems in time and money.
3.MD.A.1.aTell and write time to the nearest minute and measure time intervals in minutes. Solve contextual problems involving addition and subtraction of time intervals in minutes.
3.MD.A.1.bSolve one-step contextual problems involving amounts less than one dollar including quarters, dimes, nickels, and pennies using the ¢ symbol appropriately. Solve contextual problems involving whole number dollar amounts up to $1000 using the $ symbol appropriately.
3.MD.A.2Measure the mass of objects and liquid volume using standard units of grams (g), kilograms (kg), milliliters (ml), and liters (l). Estimate the mass of objects and liquid volume using benchmarks. For example, a large paper clip is about one gram, so a box of about 100 large clips is about 100 grams.
3.MD.B Represent and interpret data.
3.MD.B.3Draw a pictograph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in graphs.
3.MD.B.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units: whole numbers, halves, or quarters.
3.MD.C Geometric measurement: understand and apply concepts of area and relate area to multiplication and to addition.
3.MD.C.5 Recognize that plane figures have an area and understand concepts of area measurement.
3.MD.C.5.aUnderstand that a square with side length 1 unit, called "a unit square," is said to have "one square unit" of area and can be used to measure area.
3.MD.C.5.bUnderstand that a plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.C.6Measure areas by counting unit squares (square centimeters, square meters, square inches, square feet, and improvised units).
3.MD.C.7 Relate area of rectangles to the operations of multiplication and addition.
3.MD.C.7.aFind the area of a rectangle with whole-number side lengths by tiling it and show that the area is the same as would be found by multiplying the side lengths.
3.MD.C.7.bMultiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real-world and mathematical problems and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.C.7.cUse tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and (b + c) is the sum of (a x b) and (a x c). Use area models to represent the distributive property in mathematical reasoning. For example, in a rectangle with dimensions 4 by 6, students can decompose the rectangle into 4 x 3 and 4 x 3 to find the total area of 4 x 6.
3.MD.C.7.dRecognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems.
3.MD.D Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.D.8Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exploring rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.NBT Number and Operations in Base Ten (NBT)
3.NBT.A Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.A.1Round whole numbers to the nearest 10 or 100 using understanding of place value and use a number line to explain how the number was rounded.
3.NBT.A.2Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3.NBT.A.3Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.
3.NBT.A.4Read and write multi-digit whole numbers (less than or equal to 100,000) using standard form, word form, and expanded form (e.g., 23,456 can be written as 20,000 + 3,000 + 400 + 50 + 6).
3.NF Number and Operations – Fractions (NF)
3.NF.A Develop understanding of fractions as numbers.
3.NF.A.1Understand a unit fraction, 1/b, as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a non-unit fraction, n/b, as the quantity formed by n parts of size 1/b. For example, 3/4 represents a quantity formed by 3 parts of size 1/4.
3.NF.A.2 Understand a fraction as a number on the number line. Represent fractions on a number line.
3.NF.A.2.aRepresent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint locates the number 1/b on the number line. For example, on a number line from 0 to 1, students can partition it into 4 equal parts and recognize that each part represents a length of 1/4 and the first part has an endpoint at 1/4 on the number line.
3.NF.A.2.bRepresent a fraction n/b on a number line diagram by marking off n lengths 1/b from 0. Recognize that the resulting interval has size n/b and that its endpoint locates the number n/b on the number line. For example, 5/3 is the distance from 0 when there are 5 iterations of 1/3.
3.NF.A.3 Explain equivalence of fractions and compare fractions by reasoning about their size.
3.NF.A.3.aUnderstand two fractions as equivalent (equal) if they are the same size or the same point on a number line.
3.NF.A.3.bRecognize and generate simple equivalent fractions (e.g., 1/2 = 2/4, 4/6 = 2/3) and explain why the fractions are equivalent using a visual fraction model.
3.NF.A.3.cExpress whole numbers as fractions and recognize fractions that are equivalent to whole numbers. For example, express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point on a number line diagram.
3.NF.A.3.dCompare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Use the symbols >, =, or < to show the relationship and justify the conclusions.
3.OA Operations and Algebraic Thinking (OA)
3.OA.A Represent and solve problems involving multiplication and division.
3.OA.A.1Interpret the factors and products in whole number multiplication equations (e.g., 4 x 7 is 4 groups of 7 objects with a total of 28 objects or 4 strings measuring 7 inches each with a total length of 28 inches).
3.OA.A.2Interpret the dividend, divisor, and quotient in whole number division equations (e.g., 28 ÷ 7 can be interpreted as 28 objects divided into 7 equal groups with 4 objects in each group or 28 objects divided so there are 7 objects in each of the 4 equal groups).
3.OA.A.3Multiply and divide within 100 to solve contextual problems, with the unknown in any positions, in situations involving equal groups, arrays/area, and measurement quantities using strategies based on place value, the properties of operations, and the relationship between multiplication and division (e.g., contexts including computations such as 3 x ? = 24, 6 x 16 = ?, ? ÷ 8 = 3, or 96 ÷ 6 = ?).
3.OA.A.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers within 100. For example, determine the unknown number that makes the equation true in each of the equations: 8 x ? = 48, 5 = ? ÷ 3, 6 x 6 = ?.
3.OA.B Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known (commutative property of multiplication). 3 x 5 x 2 can be solved by (3 x 5) x 2 or 3 x (5 x 2) (associative property of multiplication). One way to find 8 x 7 is by using 8 x (5 + 2) = (8 x 5) + (8 x 2). By knowing that 8 x 5 = 40 and 8 x 2 = 16, then 8 x 7 = 40 + 16 = 56 (distributive property of multiplication over addition).
3.OA.B.6Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
3.OA.C Multiply and divide within 100.
3.OA.C.7Fluently multiply and divide within 100, using strategies such as the properties of operations or the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8). By the end of 3rd grade, know all products of two one-digit numbers and related division facts.
3.OA.D Solve problems involving the four operations and identify and explain patterns in arithmetic.
3.OA.D.8Solve two-step contextual problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
3.OA.D.9Identify patterns in a multiplication chart and explain them using properties of operations. For example, in the multiplication chart, observe that 4 times a number is always even (because 4 x 6 = (2 x 2) x 6 = 2 x (2 x 6), which uses the associative property of multiplication) or, for example, observe that 6 times 7 is one more group of 7 than 5 times 7 (because 6 x 7 = (5 + 1) x 7 = (5 x 7) + (1 x 7), which uses the distributive property of multiplication over addition).

Grade 4

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4.G Geometry (G)
4.G.A Draw and identify lines and angles and classify shapes by properties of their lines and angles.
4.G.A.1Draw points, lines, line segments, rays, angles (right, acute, obtuse, straight, reflex), and perpendicular and parallel lines. Identify these in two-dimensional figures.
4.G.A.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size. Classify triangles based on the measure of the angles as right, acute, or obtuse.
4.G.A.3Recognize and draw lines of symmetry for two-dimensional figures.
4.MD Measurement and Data (MD)
4.MD.A Estimate and solve problems involving measurement.
4.MD.A.1Measure and estimate to determine relative sizes of measurement units within a single system of measurement involving length, liquid volume, and mass/weight of objects using customary and metric units.
4.MD.A.2Solve one- or two-step real-world problems involving whole number measurements (including length, liquid volume, mass/weight, time, and money) with all four operations within a single system of measurement. (Contexts need not include conversions.)
4.MD.A.3Know and apply the area and perimeter formulas for rectangles in real-world and mathematical contexts. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
4.MD.B Represent and interpret data.
4.MD.B.4Make a line plot to display a data set of measurements in fractions of the same unit (1/2 or 1/4 or 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
4.MD.C Geometric measurement: understand concepts of angle and measure angles.
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint; and understand concepts of angle measurement.
4.MD.C.5.aUnderstand that an angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle.
4.MD.C.5.bUnderstand that an angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles. An angle that turns through n one-degree angles is said to have an angle measure of n degrees and represents a fractional portion of the circle.
4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems. (e.g., by using an equation with a symbol for the unknown angle measure).
4.NBT Number and Operations in Base Ten (NBT)
4.NBT.A Generalize place value understanding for multi-digit whole numbers.
4.NBT.A.1Recognize that in a multi-digit whole number (less than or equal to 1,000,000), a digit in one place represents 10 times as much as it represents in the place to its right. For example, recognize that 7 in 700 is 10 times bigger than the 7 in 70 because 700 ÷ 70 = 10 and 70 x 10 = 700.
4.NBT.A.2Read and write multi-digit whole numbers (less than or equal to 1,000,000) using standard form, word form, and expanded notation (e.g. the expanded notation of 4256 is written as (4 x 1000) + (2 x 100) + (5 x 10) + (6 x 1)). Compare two multi-digit numbers based on meanings of the digits in each place and use the symbols >, =, and < to show the relationship.
4.NBT.A.3Round multi-digit whole numbers to any place (up to and including the hundred-thousand place) using understanding of place value and use a number line to explain how the number was rounded.
4.NBT.B Use place value understanding and properties of operations to perform multi-digit arithmetic.
4.NBT.B.4Fluently add and subtract within 1,000,000 using efficient strategies and algorithms.
4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NF Number and Operations – Fractions (NF)
4.NF.A Extend understanding of fraction equivalence and comparison.
4.NF.A.1Explain why a fraction a/b is equivalent to a fraction (a x n)/(b x n) or (a ÷ n)/(b ÷ n) using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. For example, 3/4 = (3 x 2)/(4 x 2) = 6/8.
4.NF.A.2Compare two fractions with different numerators and different denominators by creating common denominators or common numerators or by comparing to a benchmark such as 0 or 1/2 or 1. Recognize that comparisons are valid only when the two fractions refer to the same whole. Use the symbols >, =, or < to show the relationship and justify the conclusions.
4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. For example, 4/5 = 1/5 + 1/5 + 1/5 + 1/5.
4.NF.B.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.
4.NF.B.3.bDecompose a fraction into a sum of fractions with the same denominator in more than one way (e.g., 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8) recording each decomposition by an equation. Justify decompositions using a visual fraction model.
4.NF.B.3.cAdd and subtract mixed numbers with like denominators by replacing each mixed number with an equivalent fraction and/or by using properties of operations and the relationship between addition and subtraction.
4.NF.B.3.dSolve contextual problems involving addition and subtraction of fractions referring to the same whole and having like denominators
4.NF.B.4 Apply and extend understanding of multiplication as repeated addition to multiply a whole number by a fraction.
4.NF.B.4.aUnderstand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × 1/4, recording the conclusion by the equation 5/4 = 5 x 1/4.
4.NF.B.4.bUnderstand a multiple of a/b as a multiple of 1/b and use this understanding to multiply a whole number by a fraction. For example, use a visual fraction model to express 3 × 2/5 as 6 × 1/5, recognizing this product as 6/5. (In general, n x a/b = (n x a)/b = (n x a) x 1/b.)
4.NF.B.4.cSolve contextual problems involving multiplication of a whole number by a fraction (e.g., by using visual fraction models and equations to represent the problem). For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 4 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
4.NF.C Understand decimal notation for fractions and compare decimal fractions.
4.NF.C.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.
4.NF.C.6Read and write decimal notation for fractions with denominators 10 or 100. Locate these decimals on a number line.
4.NF.C.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Use the symbols >, =, or < to show the relationship and justify the conclusions.
4.OA Operations and Algebraic Thinking (OA)
4.OA.A Use the four operations with whole numbers to solve problems.
4.OA.A.1Interpret a multiplication equation as a comparison (e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as much as 5). Represent verbal/written statements of multiplicative comparisons as multiplication equations.
4.OA.A.2Multiply or divide to solve contextual problems involving multiplicative comparison, and distinguish multiplicative comparison from additive comparison. For example, school A has 300 students and school B has 600 students: to say that school B has two times as many students is an example of multiplicative comparison; to say that school B has 300 more students is an example of additive comparison.
4.OA.A.3Solve multi-step contextual problems (posed with whole numbers and having whole-number answers using the four operations) including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity.
4.OA.B Gain familiarity with factors and multiples.
4.OA.B.4Find factor pairs for whole numbers in the range 1-100 using models. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number is prime or composite and whether the given number is a multiple of a given one-digit number.
4.OA.C Generate and analyze patterns.
4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Grade 5

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5.G Geometry (G)
5.G.A Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1Graph ordered pairs and label points using the first quadrant of the coordinate plane. Understand in the ordered pair that the first number indicates the horizontal distance traveled along the x-axis from the origin and the second number indicates the vertical distance traveled along the y-axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
5.G.A.2Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane and interpret coordinate values of points in the context of the situation.
5.G.B Classify two-dimensional figures into categories based on their properties.
5.G.B.3Classify two-dimensional figures in a hierarchy based on properties. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.MD Measurement and Data (MD)
5.MD.A Convert like measurement units within a given measurement system from a larger unit to a smaller unit.
5.MD.A.1Convert customary and metric measurement units within a single system by expressing measurements of a larger unit in terms of a smaller unit. Use these conversions to solve multi-step real-world problems involving distances, intervals of time, liquid volumes, masses of objects, and money (including problems involving simple fractions or decimals). For example, 3.6 liters and 4.1 liters can be combined as 7.7 liters or 7700 milliliters.
5.MD.B Represent and interpret data.
5.MD.B.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
5.MD.C Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3.aUnderstand that a cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume and can be used to measure volume.
5.MD.C.3.bUnderstand that a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.C.4Measure volume by counting unit cubes, using cubic centimeters, cubic inches, cubic feet, and improvised units.
5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume of right rectangular prisms.
5.MD.C.5.aFind the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent whole-number products of three factors as volumes (e.g., to represent the associative property of multiplication).
5.MD.C.5.bKnow and apply the formulas V = l x w x h and V = B x h (where B represents the area of the base) for rectangular prisms with whole number edge lengths in the context of solving real-world and mathematical problems.
5.MD.C.5.cRecognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real-world problems.
5.NBT Number and Operations in Base Ten (NBT)
5.NBT.A Understand the place value system.
5.NBT.A.1Recognize 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.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.A.3Read and write decimals to thousandths using standard form, word form, and expanded notation (e.g., the expanded notation of 347.392 is written as (3 x 100) + (4 x 10) + (7 x 1) + (3 x (1/10)) + (9 x (1/100)) + (2 x (1/1000))). Compare two decimals to thousandths based on meanings of the digits in each place and use the symbols >, =, and < to show the relationship.
5.NBT.A.4Round decimals to the nearest hundredth, tenth, or whole number using understanding of place value, and use a number line to explain how the number was rounded.
5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.B.5Fluently multiply multi-digit whole numbers (up to three-digit by four-digit factors) using efficient strategies and algorithms.
5.NBT.B.6Find whole-number quotients and remainders of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations. Assess the reasonableness of answers using estimation strategies. (Limit multiplication problems so that the product does not exceed thousandths. Limit division problems so that either the dividend or the divisor is a whole number.)
5.NF Number and Operations – Fractions (NF)
5.NF.A Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.A.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12 or 3/5 + 7/10 = 6/10 + 7/10 = 13/10.
5.NF.A.2Solve contextual problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
5.NF.B Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.3Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). For example, 3/4 = 3 ÷ 4 so when 3 wholes are shared equally among 4 people, each person has a share of size 3/4. Solve contextual problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers by using visual fraction models or equations to represent the problem. For example, if 8 people want to share 49 sheets of construction paper equally, how many sheets will each person receive? Between what two whole numbers does your answer lie?
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number or a fraction by a fraction.
5.NF.B.4.aInterpret the product a/b x q as a x (q ÷ b) (partition the quantity q into b equal parts and then multiply by a). Interpret the product a/b x q as (a x q) ÷ b (multiply a times the quantity q and then partition the product into b equal parts). For example, use a visual fraction model or write a story context to show that 2/3 x 6 can be interpreted as 2 x (6 ÷ 3) or (2 x 6) ÷ 3. Do the same with 2/3 x 4/5 = 8/15. (In general, a/b x c/d = ac/bd.)
5.NF.B.4.bFind the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles and represent fraction products as rectangular areas.
5.NF.B.5 Interpret multiplication as scaling (resizing).
5.NF.B.5.aCompare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. For example, know if the product will be greater than, less than, or equal to the factors.
5.NF.B.5.bExplain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explain why multiplying a given number by a fraction between 0 and 1 results in a product less than the given number; and relate the principle of fraction equivalence a/b = (a x n)/(b x n) to the effect of multiplying a/b by 1.
5.NF.B.6Solve real-world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem.
5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.NF.B.7.aInterpret division of a unit fraction by a non-zero whole number and compute such quotients. For example, use visual models and the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3. In other words, when thirds are partitioned into 4 equal groups, they become twelfths.
5.NF.B.7.bInterpret division of a whole number by a unit fraction and compute such quotients. For example, use visual models and the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4 (i.e., there are 20 groups of 1/5 inside 4 wholes and connect this to ? x (1/5) = 4).
5.NF.B.7.cSolve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions and non-unit fractions by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3 cup servings are in 2 cups of raisins?
5.OA Operations and Algebraic Thinking (OA)
5.OA.A Write and interpret numerical expressions.
5.OA.A.1Use parentheses and/or brackets in numerical expressions involving whole numbers and evaluate expressions having these symbols using the conventional order by applying the Order of Operations. (When applying the order of operations, the evaluation of exponents need not be included.)
5.OA.A.2Write numerical expressions that record calculations with numbers and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product.
5.OA.B Analyze patterns and relationships.
5.OA.B.3 Generate two numerical patterns using two given rules. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences.
5.OA.B.3.aIdentify relationships between corresponding terms in two numerical patterns.
5.OA.B.3.bForm ordered pairs (limited to first quadrant) consisting of corresponding terms from two numerical patterns and graph the ordered pairs on a coordinate plane.

Grade 6

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6.EE Expressions and Equations(EE)
6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 Write, read, and evaluate expressions in which variables stand for numbers.
6.EE.A.2.aWrite expressions that record operations with numbers and with variables. For example, express the calculation "Subtract y from 5" as 5 – y.
6.EE.A.2.bIdentify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
6.EE.A.2.cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
6.EE.A.3Apply the properties of operations (including, but not limited to, commutative, associative, and distributive properties) to generate equivalent expressions. (The distributive property of multiplication over addition is prominent here. Negative coefficients are not an expectation at this grade level.) For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4Identify when expressions are equivalent (i.e., when the expressions name the same number regardless of which value is substituted into them). For example, the expression 5b + 3b is equivalent to (5 +3)b, which is equivalent to 8b.
6.EE.B Reason about and solve one-variable equations and inequalities.
6.EE.B.5Understand that a solution to an equation or inequality is the value(s) that makes that statement true. Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.6Use variables to represent numbers and write expressions when solving real-world and mathematical problems; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.7Solve real-world and mathematical problems by writing and solving one-step equations of the form x + p = q, px = q, x – p = q, and x/p = q for cases in which p, q, and x are all nonnegative rational numbers and p ≠ 0. (Complex fractions are not an expectation at this grade level.)
6.EE.B.8Interpret and write an inequality of the form x > c, x < c, x ≤ c, or x ≥ c which represents a condition or constraint in a real-world or mathematical problem. Recognize that inequalities have infinitely many solutions; represent solutions of inequalities on number line diagrams.
6.EE.C Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another. For example, Susan is putting money in her savings account by depositing a set amount each week ($50). Represent her savings account balance with respect to the number of weekly deposits (s = 50w, illustrating the relationship between balance amount s and number of weeks w).
6.EE.C.9.aWrite an equation in the form of y = px where y, p, and x are all non-negative and p ≠ 0, to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.
6.EE.C.9.bAnalyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
6.G Geometry (G)
6.G.A Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; know and apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.2Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh where B is the area of the base to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
6.G.A.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side that joins two vertices (vertical or horizontal segments only). Apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
6.NS The Number System (NS)
6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.A.1Interpret and compute quotients of fractions, and solve real-world and mathematical problems involving division of fractions by fractions (e.g., connecting visual fraction models and equations to represent the problem is suggested). For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 times 8/9 is 2/3 ((a/b) ÷ (c/d) = ad/bc). Further example: How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.B Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.2Fluently divide multi-digit numbers using a standard algorithm.
6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm and making connections to previous conceptual work with each operation.
6.NS.B.4Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation as well as describing situations in which opposite quantities can combine to make 0.
6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself. For example, –(–3) = 3, and that 0 is its own opposite.
6.NS.C.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6.cFind and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7 Understand ordering and absolute value of rational numbers.
6.NS.C.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
6.NS.C.7.bWrite, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3º C > –7º C to express the fact that –3º C is warmer than –7º C.
6.NS.C.7.cUnderstand the absolute value of a rational number as its distance from 0 on the number line and distinguish comparisons of absolute value from statements about order in a real-world context. For example, an account balance of -24 dollars represents a greater debt than an account balance -14 dollars because -24 is located to the left of -14 on the number line.
6.NS.C.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.RP Ratios and Proportional Relationships (RP)
6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Make a distinction between ratios and fractions. For example, the ratio of wings to beaks in a bird house at the zoo was 2:1, because for every 2 wings there was 1 beak. Another example could be for every vote candidate A received, candidate C received nearly three votes.
6.RP.A.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0. Use rate language in the context of a ratio relationship. For example, this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. Also, we paid $75 for 15 hamburgers, which is a rate of $5 per hamburger. (Expectations for unit rates in 6th grade are limited to non-complex fractions).
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
6.RP.A.3.aMake 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.RP.A.3.bSolve unit rate problems including those involving unit pricing and constant speed. For example, if a runner ran 10 miles in 90 minutes, running at that speed, how long will it take him to run 6 miles? How fast is he running in miles per hour?
6.RP.A.3.cFind a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6.RP.A.3.dUse ratio reasoning to convert customary and metric measurement units (within the same system); manipulate and transform units appropriately when multiplying or dividing quantities.
6.SP Statistics and Probability (SP)
6.SP.A Develop understanding of statistical variability.
6.SP.A.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
6.SP.A.2Understand that a set of data collected to answer a statistical question has a distribution which can be described by its measures of center (mean, median, mode), measures of variation (range only), and overall shape.
6.SP.A.3Recognize that a measure of center (mean, median, mode) for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.B Summarize and describe distributions.
6.SP.B.4Display a single set of numerical data using dot plots (line plots), box plots, pie charts and stem plots.
6.SP.B.5 Summarize numerical data sets in relation to their context.
6.SP.B.5.aReport the number of observations.
6.SP.B.5.bDescribe the nature of the attribute under investigation, including how it was measured and its units of measurement.
6.SP.B.5.cGive quantitative measures of center (median and/or mean) and variability (range) as well as describing any overall pattern with reference to the context in which the data were gathered.
6.SP.B.5.dRelate the choice of measures of center to the shape of the data distribution and the context in which the data were gathered.

Grade 7

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7.EE Expressions and Equations(EE)
7.EE.A Use properties of operations to generate equivalent expressions.
7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.2Rewrite and connect equivalent expressions in different forms in a contextual problem to provide multiple ways of interpreting the problem and investigating how the quantities in it are related. For example, shoes are on sale at a 25% discount. How is the discounted price P related to the original cost C of the shoes? C – 0.25C = P. In other words, P is 75% of the original cost since C – 0.25C can be written as 0.75C.
7.EE.B Solve real-world and mathematical problems using numerical and algebraic expressions and equations and inequalities.
7.EE.B.3 Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers presented in any form (whole numbers, fractions, and decimals).
7.EE.B.3.aApply properties of operations to calculate with numbers in any form; convert between forms as appropriate.
7.EE.B.3.bAssess the reasonableness of answers using mental computation and estimation strategies.
7.EE.B.4 Use variables to represent quantities in a real-world and mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.aSolve real-world and mathematical problems leading to equations of the form px + q = r and p(x + q) = r where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4.bSolve real-world and mathematical problems leading to inequalities of the form px + q > r, px + q < r, px + q ≥ r, and px + q ≤ r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality on a number line and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
7.G Geometry (G)
7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.A.1Solve problems involving scale drawings of congruent and similar geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2Draw triangles with given conditions: three angle measures or three side measures. Notice when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.B Solve real-world and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.3Know the formulas for the area and circumference of a circle and use them to solve problems. Explore the relationships between the radius, the circumference, and the area of a circle, and the number π.
7.G.B.4Know and use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.B.5Solve real-world and mathematical problems involving area of two-dimensional figures composed of triangles, quadrilaterals, and polygons, and volume and surface area of three-dimensional objects composed of cubes and right prisms.
7.NS The Number System (NS)
7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.1.aUnderstand 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. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
7.NS.A.1.bUnderstand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7.NS.A.1.cApply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7.NS.A.2.aUnderstand that multiplication is extended from fractions to all rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.A.2.bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.A.2.cApply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.dConvert a rational number to a decimal using long division; know that the decimal form of a rational number terminates or eventually repeats.
7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
7.RP Ratios and Proportional Relationships (RP)
7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.A.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 15 minutes, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.
7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.2.aDecide whether two quantities are in a proportional relationship (e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin).
7.RP.A.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.cUse the concept of equality to represent proportional relationships with equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.A.2.dExplain 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.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems. Examples: batting averages, recipes, simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error, etc.
7.SP Statistics and Probability (SP)
7.SP.A Use random sampling to draw inferences about a population.
7.SP.A.1Explore how statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.A.2Collect and use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
7.SP.B Draw informal comparative inferences about two populations.
7.SP.B.3Informally compare the measures of center (mean, median, mode) of two numerical data distributions with similar variabilities. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team; on a dot plot or box plot, the separation between the two distributions of heights is noticeable.
7.SP.B.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a 7th grade science book are generally longer than the words in a chapter of a 4th grade science book.
7.SP.C Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5Recognize that the probability of a chance event is a number between 0 and 1 and interpret the likelihood of the event occurring.
7.SP.C.6 Calculate theoretical and experimental probability of simple events.
7.SP.C.6.aApproximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.
7.SP.C.6.bCalculate the theoretical probability of a simple event.
7.SP.C.6.cCompare theoretical probabilities to experimental probabilities; explain any possible sources of discrepancy. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.C.7 Develop a probability model and use it to find experimental or theoretical probabilities of events.
7.SP.C.7.aUse a uniform probability model, with equal probability assigned to all outcomes, to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7.bDevelop a probability model, including non-uniform models, by observing frequencies in data generated from a chance process. Use the model to estimate the probabilities of events. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.D Summarize and describe numerical data sets.
7.SP.D.8 Summarize a numerical data set in relation to its context.
7.SP.D.8.aGive quantitative measures of center (median and/or mean) and variability (range and/or interquartile range), as well as describe any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
7.SP.D.8.bRelate and understand the choice of measures of center (median and/or mean) and variability (range and/or interquartile range) to the shape of the data distribution and the context in which the data were gathered.

Grade 8

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8.EE Expressions and Equations(EE)
8.EE.A Work with radicals and integer exponents.
8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 x 3^-5 = 3^-3 = 1/3^3 = 1/27.
8.EE.A.2Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.
8.EE.A.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 x 10^8 and the population of the world as 7 x 10^9, and determine that the world population is more than 20 times larger.
8.EE.A.4Using technology, solve real-world problems with numbers expressed in decimal and scientific notation. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading).
8.EE.B Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6Use 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; know and apply the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.EE.C Analyze and solve linear equations, linear inequalities, and systems of two linear equations.
8.EE.C.7 Solve linear equations in one variable.
8.EE.C.7.aGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.C.7.bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms.
8.EE.C.8 Analyze and solve systems of two linear equations graphically.
8.EE.C.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8.bEstimate solutions by graphing a system of two linear equations in two variables. Identify solutions by inspecting graphs of a system of linear equations in two variables.
8.EE.C.9By graphing on the coordinate plane or by analyzing a given graph, determine the solution set of a linear inequality in one or two variables.
8.F Functions (F)
8.F.A Define, evaluate, and compare functions.
8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in 8th grade.)
8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and another linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3Know and interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
8.F.B Use functions to model relationships between quantities.
8.F.B.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.
8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
8.G Geometry (G)
8.G.A Understand and describe the effects of transformations on two-dimensional figures and use informal arguments to establish facts about angles.
8.G.A.1 Describe the effect of translations, rotations, reflections, and dilations on two-dimensional figures using coordinates.
8.G.A.1.aVerify informally that lines are taken to lines, and determine when line segments are taken to line segments of the same length.
8.G.A.1.bVerify informally that angles are taken to angles of the same measure.
8.G.A.1.cVerify informally that parallel lines are taken to parallel lines.
8.G.A.1.dMake connections between dilations and scale factors.
8.G.A.2Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
8.G.B Understand and apply the Pythagorean Theorem.
8.G.B.3Explain a model of the Pythagorean Theorem and its converse.
8.G.B.4Know and apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.5Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.C.6Apply the formulas for the volumes of cones, cylinders, and spheres to solve real-world and mathematical problems.
8.NS The Number System (NS)
8.NS.A Know that there are numbers that are not rational, and approximate them by rational numbers.
8.NS.A.1Know that real numbers that are not rational are called irrational (e.g., π, √2, etc.). Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually or terminates, and convert a decimal expansion which repeats eventually or terminates into a rational number.
8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers by locating them approximately on a number line diagram. Estimate the value of irrational expressions (such as π^2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.SP Statistics and Probability (SP)
8.SP.A Investigate patterns of association in bivariate data.
8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2Know that straight lines are widely used to model linear relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.A.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercepts. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
8.SP.B Investigate chance processes and develop, use, and evaluate probability models
8.SP.B.4 Find probabilities of and represent sample spaces for compound events using organized lists, tables, tree diagrams, and simulation.
8.SP.B.4.aUnderstand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
8.SP.B.4.bRepresent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

Algebra I | A1

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A1.A A1.A
A1.A.APR Algebra – Arithmetic with Polynomials and Rational Expressions (A.APR)
A1.A.APR.A Perform arithmetic operations on polynomials.
A1.A.APR.A.1Add, subtract, and multiply polynomials. Use these operations to demonstrate that polynomials form a closed system that adhere to the same properties of operations as the integers.
A1.A.CED Algebra – Creating Equations (A.CED)
A1.A.CED.A Create equations that describe numbers or relationships.
A1.A.CED.A.1Create equations and inequalities in one variable and use them to solve problems in a real-world context.
A1.A.CED.A.2Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.
A1.A.CED.A.3Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.
A1.A.CED.A.4Rearrange formulas to isolate a quantity of interest using algebraic reasoning.
A1.A.REI Algebra – Reasoning with Equations and Inequalities (A.REI)
A1.A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
A1.A.REI.A.1Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.
A1.A.REI.B Solve equations and inequalities in one variable.
A1.A.REI.B.2 Solve linear and absolute value equations and inequalities in one variable.
A1.A.REI.B.2.aSolve linear equations and inequalities, including compound inequalities, in one variable. Represent solutions algebraically and graphically.
A1.A.REI.B.2.bSolve absolute value equations and inequalities in one variable. Represent solutions algebraically and graphically.
A1.A.REI.B.3 Solve quadratic equations and inequalities in one variable.
A1.A.REI.B.3.aSolve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when a quadratic equation has solutions that are not real numbers.
A1.A.REI.B.3.bSolve quadratic inequalities using the graph of the related quadratic equation.
A1.A.REI.C Solve systems of equations.
A1.A.REI.C.4Write and solve a system of linear equations in real-world context.
A1.A.REI.D Represent and solve equations and inequalities graphically.
A1.A.REI.D.5Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
A1.A.REI.D.6Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.
A1.A.REI.D.7Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
A1.A.SSE Algebra – Seeing Structure in Expressions (A.SSE)
A1.A.SSE.A Interpret the structure of expressions.
A1.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
A1.A.SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
A1.A.SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity.
A1.F A1.F
A1.F.BF Functions – Building Functions (F.BF)
A1.F.BF.A Build a function that models a relationship between two quantities.
A1.F.BF.A.1 Build a function that describes a relationship between two quantities.
A1.F.BF.A.1.aDetermine steps for calculation, a recursive process, or an explicit expression from a context.
A1.F.BF.B Build new functions from existing functions.
A1.F.BF.B.2Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given graphs.
A1.F.IF Functions – Interpreting Functions (F.IF)
A1.F.IF.A Understand the concept of function and use function notation.
A1.F.IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
A1.F.IF.A.2 Use function notation.
A1.F.IF.A.2.aUse function notation to evaluate functions for inputs in their domains, including functions of two variables.
A1.F.IF.A.2.bInterpret statements that use function notation in terms of a context.
A1.F.IF.A.3Understand geometric formulas as functions.
A1.F.IF.B Interpret functions that arise in applications in terms of the context.
A1.F.IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
A1.F.IF.B.5Relate the domain of a function to its graph and, where applicable, to the context of the function it models.
A1.F.IF.B.6Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.
A1.F.IF.C Analyze functions using different representations.
A1.F.IF.C.7Graph functions expressed algebraically and show key features of the graph by hand and using technology.
A1.F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
A1.F.IF.C.8.aRewrite quadratic functions to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a real-world context.
A1.F.IF.C.9 Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.
A1.F.IF.C.9.aCompare properties of two different functions. Functions may be of different types and/or represented in different ways.
A1.F.IF.C.9.bCompare properties of the same function on two different intervals or represented in two different ways.
A1.F.LE Functions – Linear and Exponential Models (F.LE)
A1.F.LE.A Construct and compare linear and exponential models and solve problems.
A1.F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
A1.F.LE.A.1.aKnow that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.
A1.F.LE.A.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
A1.F.LE.A.1.cRecognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.
A1.F.LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.
A1.F.LE.B Interpret expressions for functions in terms of the situation they model.
A1.F.LE.B.3Interpret the parameters in a linear or exponential function in terms of a context.
A1.N A1.N
A1.N.Q Number and Quantity – Quantities (N.Q)
A1.N.Q.A Reason quantitatively and use units to understand problems.
A1.N.Q.A.1 Use units as a way to understand real-world problems.
A1.N.Q.A.1.aChoose and interpret the scale and the origin in graphs and data displays.
A1.N.Q.A.1.bUse appropriate quantities in formulas, converting units as necessary.
A1.N.Q.A.1.cDefine and justify appropriate quantities within a context for the purpose of modeling.
A1.N.Q.A.1.dChoose an appropriate level of accuracy when reporting quantities.
A1.S A1.S
A1.S.ID Statistics and Probability – Interpreting Categorical and Quantitative Data (S.ID)
A1.S.ID.A Summarize, represent, and interpret data on a single count or measurement variable.
A1.S.ID.A.1Use measures of center to solve real-world and mathematical problems.
A1.S.ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (mean, median, and/or mode) and spread (range, interquartile range) of two or more different data sets.
A1.S.ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.
A1.S.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
A1.S.ID.B.4Represent data from two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
A1.S.ID.C Interpret linear models.
A1.S.ID.C.5Interpret the rate of change and the constant term of a linear model in the context of data.
A1.S.ID.C.6Use technology to compute the correlation coefficient of a linear model; interpret the correlation coefficient in the context of the data.
A1.S.ID.C.7Explain the differences between correlation and causation. Recognize situations where an additional factor may be affecting correlated data.

Algebra II | A2

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A2.A A2.A
A2.A.APR Algebra – Arithmetic with Polynomials and Rational Expressions (A.APR)
A2.A.APR.A Understand the relationship between zeros and factors of polynomials.
A2.A.APR.A.1Know and apply the Factor Theorem: For a polynomial p(x) and a number a, p(a) = 0 if and only if (x – a) is a factor of p(x).
A2.A.APR.A.2Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
A2.A.CED Algebra – Creating Equations (A.CED)
A2.A.CED.A Create equations that describe numbers or relationships.
A2.A.CED.A.1Create equations and inequalities in one variable and use them to solve problems in a real-world context.
A2.A.CED.A.2Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.
A2.A.CED.A.3Rearrange formulas to isolate a quantity of interest using algebraic reasoning.
A2.A.REI Algebra – Reasoning with Equations and Inequalities (A.REI)
A2.A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
A2.A.REI.A.1Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.
A2.A.REI.A.2Solve radical equations in one variable, and identify extraneous solutions when they exist.
A2.A.REI.B Solve systems of equations.
A2.A.REI.B.3Write and solve a system of linear equations in a real-world context.
A2.A.REI.B.4Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically, graphically, and using technology.
A2.A.SSE Algebra – Seeing Structure in Expressions (A.SSE)
A2.A.SSE.A Interpret the structure of expressions.
A2.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
A2.A.SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
A2.A.SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity.
A2.F A2.F
A2.F.BF Functions – Building Functions (F.BF)
A2.F.BF.A Build a function that models a relationship between two quantities.
A2.F.BF.A.1 Build a function that describes a relationship between two quantities.
A2.F.BF.A.1.aCombine standard function types using arithmetic operations.
A2.F.BF.A.1.bCombine standard function types using composition.
A2.F.BF.A.2Define sequences as functions, including recursive definitions, whose domain is a subset of the integers. Write explicit and recursive formulas for arithmetic and geometric sequences in context and connect them to linear and exponential functions.
A2.F.BF.B Build new functions from existing functions.
A2.F.BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.
A2.F.BF.B.4 Find the inverse of a function.
A2.F.BF.B.4.aDetermine whether a function is one-to-one.
A2.F.BF.B.4.bFind the inverse of a function on an appropriate domain.
A2.F.BF.B.4.cGiven an invertible function on an appropriate domain, identify the domain of the inverse function.
A2.F.IF Functions – Interpreting Functions (F.IF)
A2.F.IF.A Interpret functions that arise in applications in terms of the context.
A2.F.IF.A.1For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
A2.F.IF.A.2Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.
A2.F.IF.A.3Understand geometric formulas as functions.
A2.F.IF.B Analyze functions using different representations.
A2.F.IF.B.4Graph functions expressed algebraically and show key features of the graph by hand and using technology.
A2.F.IF.B.5 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
A2.F.IF.B.5.aRewrite quadratic functions to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a real-world context.
A2.F.IF.B.5.bKnow and use the properties of exponents to interpret expressions for exponential functions in terms of a real-world context.
A2.F.IF.B.6 Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.
A2.F.IF.B.6.aCompare properties of two different functions. Functions may be of different types and/or represented in different ways.
A2.F.IF.B.6.bCompare properties of the same function on two different intervals or represented in two different ways.
A2.F.LE Functions – Linear, Quadratic, and Exponential Models (F.LE)
A2.F.LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
A2.F.LE.A.1 Know the relationship between exponential functions and logarithmic functions.
A2.F.LE.A.1.aSolve exponential equations using a variety of strategies, including logarithms.
A2.F.LE.A.1.bUnderstand that a logarithm is the solution to ab^ct = d, where a, b, c, and d are numbers.
A2.F.LE.A.1.cEvaluate logarithms using technology.
A2.F.LE.A.2Know that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or cubically.
A2.N A2.N
A2.N.M Number and Quantity – Matrices (N.M)
A2.N.M.A Perform operations on matrices and use matrices in applications.
A2.N.M.A.1Use matrices to represent data in a real-world context. Interpret rows, columns, and dimensions of matrices in terms of the context.
A2.N.M.A.2 Perform operations on matrices in a real-world context.
A2.N.M.A.2.aMultiply a matrix by a scalar to produce a new matrix.
A2.N.M.A.2.bAdd and/or subtract matrices by hand and using technology.
A2.N.M.A.2.cMultiply matrices of appropriate dimensions, by hand in simple cases and using technology for more complicated cases.
A2.N.M.A.2.dDescribe the roles that zero matrices and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real number system.
A2.N.M.A.3Create and use augmented matrices to solve systems of linear equations in real-world contexts, by hand and using technology.
A2.N.Q Number and Quantity – Quantities (N.Q)
A2.N.Q.A Reason quantitatively and use units to understand problems.
A2.N.Q.A.1 Use units as a way to understand real-world problems.
A2.N.Q.A.1.aChoose and interpret the scale and the origin in graphs and data displays.
A2.N.Q.A.1.bUse appropriate quantities in formulas, converting units as necessary.
A2.N.Q.A.1.cDefine and justify appropriate quantities within a context for the purpose of modeling.
A2.N.Q.A.1.dChoose an appropriate level of accuracy when reporting quantities.
A2.N.RN Number and Quantity – The Real Number System (N.RN)
A2.N.RN.A Extend the properties of exponents to rational exponents.
A2.N.RN.A.1 Extend the properties of integer exponents to rational exponents.
A2.N.RN.A.1.aDevelop the meaning of rational exponents by applying the properties of integer exponents.
A2.N.RN.A.1.bExplain why x^1/n can be written as the nth root of x.
A2.N.RN.A.1.cRewrite expressions involving radicals and rational exponents using the properties of exponents.
A2.S A2.S
A2.S.CP Statistics and Probability – Conditional Probability and the Rules of Probability (S.CP)
A2.S.CP.A Understand independence and conditional probability and use them to create visual representations of data.
A2.S.CP.A.1Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Categorize events as independent or dependent.
A2.S.CP.B Understand and apply basic concepts of probability.
A2.S.CP.B.2 Apply statistical counting techniques.
A2.S.CP.B.2.aUse the Fundamental Counting Principle to compute probabilities of compound events and solve problems.
A2.S.CP.B.2.bUse permutations and combinations to compute probabilities of compound events and solve problems.
A2.S.CP.B.3Use the Law of Large Numbers to assess the validity of a statistical claim.
A2.S.CP.C Use the rules of probability to compute probabilities of compound events in a uniform probability model.
A2.S.CP.C.4Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the given context.
A2.S.IC Statistics and Probability – Making Inferences and Justifying Conclusions (S.IC)
A2.S.IC.A Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
A2.S.IC.A.1Recognize the purposes of and differences among sample surveys, experiments, and observational studies.
A2.S.IC.A.2Identify potential sources of bias in statistical studies.
A2.S.IC.A.3Distinguish between a statistic and a parameter; Evaluate reports based on data and recognize when poor conclusions are drawn from well-collected data.
A2.S.ID Statistics and Probability – Interpreting Categorical and Quantitative Data (S.ID)
A2.S.ID.A Summarize, represent, and interpret data on a single count or measurement variable.
A2.S.ID.A.1Use statistics appropriate to the shape of the data distribution to compare center (mean, median, and/or mode) and spread (range, standard deviation) of two or more different data sets.
A2.S.ID.A.2Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages using the Empirical Rule.
A2.S.ID.A.3Compute, interpret, and compare z-scores for normally distributed data in a real-world context.
A2.S.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
A2.S.ID.B.4Represent data from two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Geometry | G

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G.C Geometry – Circles (G.C)
G.C.A Find areas of sectors of circles.
G.C.A.1Use proportional relationships between the area of a circle and the area of a sector within the circle to solve problems in a real-world context.
G.CO Geometry – Congruence (G.CO)
G.CO.A Experiment with transformations in the plane.
G.CO.A.1Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.
G.CO.A.2Given a rectangle, parallelogram, trapezoid, or regular polygon, determine the transformations that carry the shape onto itself and describe them in terms of the symmetry of the figure.
G.CO.A.3Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.A.4Given a geometric figure, draw the image of the figure after a sequence of one or more rigid motions, by hand and using technology. Identify a sequence of rigid motions that will carry a given figure onto another.
G.CO.B Understand congruence in terms of rigid motions.
G.CO.B.5Given two figures, use the definition of congruence in terms of rigid motions to determine informally if they are congruent.
G.CO.B.6Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.B.7Explain how the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) follow from the definition of congruence in terms of rigid motions.
G.CO.C Use geometric theorems to justify relationships.
G.CO.C.10Use definitions and theorems about parallelograms to solve problems and to justify relationships in geometric figures.
G.CO.C.8Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.
G.CO.C.9Use definitions and theorems about triangles to solve problems and to justify relationships in geometric figures.
G.CO.D Perform geometric constructions.
G.CO.D.11Perform formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
G.CO.D.12Use geometric constructions to solve geometric problems in context, by hand and using technology.
G.GMD Geometry – Geometric Measurement and Dimension (G.GMD)
G.GMD.A Explain volume and surface area formulas and use them to solve problems.
G.GMD.A.1Understand and explain the formulas for the volume and surface area of a cylinder, cone, prism, and pyramid.
G.GMD.A.2Use volume and surface area formulas for cylinders, cones, prisms, pyramids, and spheres to solve problems in a real-world context.
G.GPE Geometry – Expressing Geometric Properties with Equations (G.GPE)
G.GPE.A Use coordinates to solve problems and justify simple geometric theorems algebraically.
G.GPE.A.1Use coordinates to justify geometric relationships algebraically and to solve problems.
G.GPE.A.2Use the slope criteria for parallel and perpendicular lines to solve problems and to justify relationships in geometric figures.
G.GPE.A.3Understand the relationship between the Pythagorean Theorem and the distance formula and use an efficient method to solve problems on the coordinate plane.
G.MG Geometry – Modeling with Geometry (G.MG)
G.MG.A Apply geometric concepts in modeling situations.
G.MG.A.1Use geometric shapes, their measures, and their properties to model objects found in a real-world context for the purpose of approximating solutions to problems.
G.N G.N
G.N.Q Number and Quantity – Quantities (N.Q)
G.N.Q.A Reason quantitatively and use units to solve problems.
G.N.Q.A.1 Use units as a way to understand real-world problems.
G.N.Q.A.1.aUse appropriate quantities in formulas, converting units as necessary.
G.N.Q.A.1.bDefine and justify appropriate quantities within a context for the purpose of modeling.
G.N.Q.A.1.cChoose an appropriate level of accuracy when reporting quantities.
G.S G.S
G.S.CP Statistics and Probability – Conditional Probability and the Rules of Probability (S.CP)
G.S.CP.A Understand independence and conditional probability and use them to create visual representations of data.
G.S.CP.A.1 Use set notation to represent contextual situations.
G.S.CP.A.1.aDescribe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or", "and", "not").
G.S.CP.A.1.bFlexibly move between visual models (Venn diagrams, frequency tables, etc.) and set notation.
G.S.CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
G.S.CP.B.2Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the given context.
G.S.CP.B.3 Understand and apply the Addition Rule.
G.S.CP.B.3.aExplain the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B) in terms of visual models (Venn diagrams, frequency tables, etc.).
G.S.CP.B.3.bApply the Addition Rule to solve problems and interpret the answer in terms of the given context.
G.S.CP.C Apply geometric concepts to situations involving probability.
G.S.CP.C.4Calculate probabilities using geometric figures.
G.SRT Geometry – Similarity, Right Triangles, and Trigonometry (G.SRT)
G.SRT.A Understand similarity in terms of similarity transformations.
G.SRT.A.1Use properties of dilations given by a center and a scale factor to solve problems and to justify relationships in geometric figures.
G.SRT.A.2Define similarity in terms of transformations. Use transformations to determine whether two figures are similar.
G.SRT.B Use similarity to solve problems and justify relationships.
G.SRT.B.3Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.
G.SRT.C Define trigonometric ratios and solve problems involving triangles.
G.SRT.C.4 Use side ratios in right triangles to define trigonometric ratios.
G.SRT.C.4.aUnderstand 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.SRT.C.4.bExplain and use the relationship between the sine and cosine of complementary angles.
G.SRT.C.5 Solve triangles.
G.SRT.C.5.aKnow and use the Pythagorean Theorem and trigonometric ratios (sine, cosine, tangent, and their inverses) to solve right triangles in a real-world context.
G.SRT.C.5.bKnow and use relationships within special right triangles to solve problems in a real-world context.
G.SRT.C.5.cUse the Law of Sines and Law of Cosines to solve non-right triangles in a real-world context.

Integrated Math I | M1

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M1.A M1.A
M1.A.CED Algebra – Creating Equations (A.CED)
M1.A.CED.A Create equations that describe numbers or relationships
M1.A.CED.A.1Create equations and inequalities in one variable and use them to solve problems in a real-world context.
M1.A.CED.A.2Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.
M1.A.CED.A.3Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.
M1.A.CED.A.4Rearrange formulas to isolate a quantity of interest using algebraic reasoning.
M1.A.REI Algebra – Reasoning with Equations and Inequalities (A.REI)
M1.A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
M1.A.REI.A.1Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.
M1.A.REI.B Solve equations and inequalities in one variable.
M1.A.REI.B.2 Solve linear and absolute value equations and inequalities in one variable.
M1.A.REI.B.2.aSolve linear equations and inequalities, including compound inequalities, in one variable. Represent solutions algebraically and graphically.
M1.A.REI.B.2.bSolve absolute value equations and inequalities in one variable. Represent solutions algebraically and graphically.
M1.A.REI.C Solve systems of equations.
M1.A.REI.C.3Write and solve a system of linear equations in a real-world context.
M1.A.REI.D Represent and solve equations and inequalities graphically.
M1.A.REI.D.4Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
M1.A.REI.D.5Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.
M1.A.REI.D.6Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
M1.A.SSE Algebra – Seeing Structure in Expressions (A.SSE)
M1.A.SSE.A Interpret the structure of expressions.
M1.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
M1.A.SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
M1.A.SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity.
M1.F M1.F
M1.F.BF Functions – Building Functions (F.BF)
M1.F.BF.A Build a function that models a relationship between two quantities.
M1.F.BF.A.1 Build a function that describes a relationship between two quantities.
M1.F.BF.A.1.aDetermine steps for calculation, a recursive process, or an explicit expression from a context.
M1.F.BF.A.2Define sequences as functions, including recursive definitions, whose domain is a subset of the integers. Write explicit and recursive formulas for arithmetic and geometric sequences in context and connect them to linear and exponential functions.
M1.F.IF Functions – Interpreting Functions (F.IF)
M1.F.IF.A Understand the concept of a function and use function notation.
M1.F.IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
M1.F.IF.A.2 Use function notation.
M1.F.IF.A.2.aUse function notation to evaluate functions for inputs in their domains, including functions of two variables.
M1.F.IF.A.2.bInterpret statements that use function notation in terms of a context.
M1.F.IF.A.3Understand geometric formulas as functions.
M1.F.IF.B Interpret functions that arise in applications in terms of the context.
M1.F.IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
M1.F.IF.B.5Relate the domain of a function to its graph and, where applicable, to the context of the function it models.
M1.F.IF.C Analyze functions using different representations.
M1.F.IF.C.6 Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.
M1.F.IF.C.6.aCompare properties of two different functions. Functions may be of different types and/or represented in different ways.
M1.F.IF.C.6.bCompare properties of the same function on two different intervals or represented in two different ways.
M1.F.LE Functions – Linear and Exponential Models (F.LE)
M1.F.LE.A Construct and compare linear and exponential models and solve problems.
M1.F.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
M1.F.LE.A.1.aKnow that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.
M1.F.LE.A.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
M1.F.LE.A.1.cRecognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.
M1.F.LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.
M1.F.LE.B Interpret expressions for functions in terms of the situation they model.
M1.F.LE.B.3Interpret the parameters in a linear or exponential function in terms of a context.
M1.G M1.G
M1.G.CO Geometry – Congruence (G.CO)
M1.G.CO.A Experiment with transformations in the plane.
M1.G.CO.A.1Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.
M1.G.CO.A.2Given a rectangle, parallelogram, trapezoid, or regular polygon, determine the transformations that carry the shape onto itself and describe them in terms of the symmetry of the figure.
M1.G.CO.B Use geometric theorems to justify relationships.
M1.G.CO.B.3Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.
M1.G.CO.B.4Use definitions and theorems about triangles to solve problems and to justify relationships in geometric figures.
M1.G.CO.C Perform geometric constructions.
M1.G.CO.C.5Perform formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
M1.G.CO.C.6Use geometric constructions to solve geometric problems in context, by hand and using technology.
M1.G.GPE Geometry – Geometric Properties with Equations (G.GPE)
M1.G.GPE.A Use coordinates to solve problems and justify simple geometric theorems algebraically.
M1.G.GPE.A.1Use coordinates to solve problems and justify geometric relationships algebraically.
M1.G.GPE.A.2Use the slope criteria for parallel and perpendicular lines to solve problems and to justify relationships in geometric figures.
M1.G.GPE.A.3Understand the relationship between the Pythagorean Theorem and the distance formula and use an efficient method to solve problems on the coordinate plane.
M1.N M1.N
M1.N.M Number and Quantity – Matrices (N.M)
M1.N.M.A Perform operations on matrices and use matrices in applications.
M1.N.M.A.1Use matrices to represent data in a real-world context. Interpret rows, columns, and dimensions of matrices in terms of the context.
M1.N.M.A.2 Perform operations on matrices in a real-world context.
M1.N.M.A.2.aMultiply a matrix by a scalar to produce a new matrix.
M1.N.M.A.2.bAdd and/or subtract matrices by hand and using technology.
M1.N.M.A.2.cMultiply matrices of appropriate dimensions, by hand in simple cases and using technology for more complicated cases.
M1.N.M.A.2.dDescribe the roles that zero matrices and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real number system.
M1.N.M.A.3Create and use augmented matrices to solve systems of linear equations in real-world contexts, by hand and using technology.
M1.N.Q Number and Quantity – Quantities (N.Q)
M1.N.Q.A Reason quantitatively and use units to understand problems.
M1.N.Q.A.1 Use units as a way to understand real-world problems.
M1.N.Q.A.1.aChoose and interpret the scale and the origin in graphs and data displays.
M1.N.Q.A.1.bUse appropriate quantities in formulas, converting units as necessary.
M1.N.Q.A.1.cDefine and justify appropriate quantities within a context for the purpose of modeling.
M1.N.Q.A.1.dChoose an appropriate level of accuracy when reporting quantities.
M1.S M1.S
M1.S.ID Statistics and Probability – Interpreting Categorical and Quantitative Data (S.ID)
M1.S.ID.A Summarize, represent, and interpret data on two categorical and quantitative variables.
M1.S.ID.A.1Represent data from two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
M1.S.ID.B Interpret linear models.
M1.S.ID.B.2Interpret the rate of change and the constant term of a linear model in the context of the data.
M1.S.ID.B.3Use technology to compute the correlation coefficient of a linear model; interpret the correlation coefficient in the context of the data.
M1.S.ID.B.4Explain the differences between correlation and causation. Recognize situations where an additional factor may be affecting correlated data.

Integrated Math II | M2

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M2.A M2.A
M2.A.APR Algebra – Arithmetic with Polynomials and Rational Expressions (A.APR)
M2.A.APR.A Perform arithmetic operations on polynomials.
M2.A.APR.A.1Add, subtract, and multiply polynomials. Use these operations to demonstrate that polynomials form a closed system that adhere to the same properties of operations as the integers.
M2.A.APR.B Understand the relationship between zeros and factors of polynomials.
M2.A.APR.B.2Know and apply the Factor Theorem: For a polynomial p(x) and a number a, p(a) = 0 if and only if (x – a) is a factor of p(x).
M2.A.CED Algebra – Creating Equations (A.CED)
M2.A.CED.A Create equations that describe numbers or relationships.
M2.A.CED.A.1Create equations and inequalities in one variable and use them to solve problems in a real-world context.
M2.A.CED.A.2Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.
M2.A.CED.A.3Rearrange formulas to isolate a quantity of interest using algebraic reasoning.
M2.A.REI Algebra – Reasoning with Equations and Inequalities (A.REI)
M2.A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
M2.A.REI.A.1Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.
M2.A.REI.B Solve equations and inequalities in one variable.
M2.A.REI.B.2 Solve quadratic equations and inequalities in one variable.
M2.A.REI.B.2.aSolve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when a quadratic equation has non-real solutions.
M2.A.REI.B.2.bSolve quadratic inequalities using the graph of the related quadratic equation.
M2.A.REI.B.3Solve radical equations in one variable and identify extraneous solutions when they exist.
M2.A.REI.C Solve systems of equations.
M2.A.REI.C.4Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically, graphically, and using technology.
M2.A.REI.D Represent and solve equations and inequalities graphically.
M2.A.REI.D.5Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.
M2.A.SSE Algebra – Seeing Structure in Expressions (A.SSE)
M2.A.SSE.A Interpret the structure of expressions.
M2.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
M2.A.SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficient.
M2.A.SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity.
M2.F M2.F
M2.F.BF Functions – Building Functions (F.BF)
M2.F.BF.A Build a function that models a relationship between two quantities.
M2.F.BF.A.1 Build a function that describes a relationship between two quantities.
M2.F.BF.A.1.aCombine standard function types using arithmetic operations.
M2.F.BF.B Build new functions from existing functions.
M2.F.BF.B.2Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given graphs.
M2.F.IF Functions – Interpreting Functions (F.IF)
M2.F.IF.A Understand the concept of function and use function notation.
M2.F.IF.A.1 Use function notation.
M2.F.IF.A.1.aUse function notation to evaluate functions for inputs in their domains, including functions of two variables.
M2.F.IF.A.1.bInterpret statements that use function notation in terms of a context.
M2.F.IF.A.2Understand geometric formulas as functions.
M2.F.IF.B Interpret functions that arise in applications in terms of the context.
M2.F.IF.B.3For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
M2.F.IF.B.4Relate the domain of a function to its graph and, where applicable, to the context of the function it models.
M2.F.IF.B.5Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.
M2.F.IF.C Analyze functions using different representation.
M2.F.IF.C.6Graph functions expressed algebraically and show key features of the graph by hand and using technology.
M2.F.IF.C.7 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
M2.F.IF.C.7.aRewrite quadratic functions to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a real-world context.
M2.F.IF.C.7.bKnow and use the properties of exponents to interpret expressions for exponential functions in terms of a real-world context.
M2.F.IF.C.8 Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.
M2.F.IF.C.8.aCompare properties of two different functions. Functions may be of different types and/or represented in different ways.
M2.F.IF.C.8.bCompare properties of the same function on two different intervals or represented in two different ways.
M2.G M2.G
M2.G.CO Geometry – Congruence (G.CO)
M2.G.CO.A Experiment with transformations in the plane.
M2.G.CO.A.1Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.
M2.G.CO.A.2Given a rectangle, parallelogram, trapezoid, or regular polygon, determine the transformations that carry the shape onto itself and describe them in terms of the symmetry of the figure.
M2.G.CO.A.3Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
M2.G.CO.A.4Given a geometric figure, draw the image of the figure after a sequence of one or more rigid motions, by hand and using technology. Identify a sequence of rigid motions that will carry a given figure onto another.
M2.G.CO.B Understand congruence in terms of rigid motions.
M2.G.CO.B.5Given two figures, use the definition of congruence in terms of rigid motions to determine informally if they are congruent.
M2.G.CO.B.6Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
M2.G.CO.B.7Explain how the criteria for triangle congruence (ASA, SAS, AAS, SSS, and HL) follow from the definition of congruence in terms of rigid motions.
M2.G.CO.C Use geometric theorems to justify relationships.
M2.G.CO.C.8Use definitions and theorems about triangles to solve problems and to justify relationships in geometric figures.
M2.G.CO.C.9Use definitions and theorems about parallelograms to solve problems and to justify relationships in geometric figures.
M2.G.SRT Geometry – Similarity, Right Triangles, and Trigonometry (G.SRT)
M2.G.SRT.A Understand similarity in terms of similarity transformations.
M2.G.SRT.A.1Use properties of dilations given by a center and a scale factor to solve problems and to justify relationships in geometric figures.
M2.G.SRT.A.2Define similarity in terms of transformations. Use transformations to determine whether two figures are similar.
M2.G.SRT.B Use similarity to solve problems and justify relationships.
M2.G.SRT.B.3Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures.
M2.N M2.N
M2.N.Q Number and Quantity – Quantities (N.Q)
M2.N.Q.A Reason quantitatively and use units to understand problems.
M2.N.Q.A.1 Use units as a way to understand real-world problems.
M2.N.Q.A.1.aChoose and interpret the scale and the origin in graphs and data displays.
M2.N.Q.A.1.bUse appropriate quantities in formulas, converting units as necessary.
M2.N.Q.A.1.cDefine and justify appropriate quantities within a context for the purpose of modeling.
M2.N.Q.A.1.dChoose an appropriate level of accuracy when reporting quantities.
M2.N.RN Number and Quantity – The Real Number System (N.RN)
M2.N.RN.A Extend the properties of exponents to rational exponents.
M2.N.RN.A.1 Extend the properties of integer exponents to rational exponents.
M2.N.RN.A.1.aDevelop the meaning of rational exponents by applying the properties of integer exponents.
M2.N.RN.A.1.bExplain why x^1/n can be written as the nth root of x.
M2.N.RN.A.1.cRewrite expressions involving radicals and rational exponents using the properties of exponents.
M2.S M2.S
M2.S.ID Statistics and Probability – Interpreting Categorical and Quantitative Data (S.ID)
M2.S.ID.A Summarize, represent, and interpret data on two categorical and quantitative variables.
M2.S.ID.A.1Represent data from two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Integrated Math III | M3

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M3.A M3.A
M3.A.APR Algebra – Arithmetic with Polynomials and Rational Expressions (A.APR)
M3.A.APR.A Understand the relationship between zeros and factors of polynomials.
M3.A.APR.A.1Know and apply the Factor Theorem: For a polynomial p(x) and a number a, p(a) = 0 if and only if (x – a) is a factor of p(x).
M3.A.APR.A.2Identify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial.
M3.A.CED Algebra – Creating Equations (A.CED)
M3.A.CED.A Create equations that describe numbers or relationships.
M3.A.CED.A.1Create equations and inequalities in one variable and use them to solve problems in a real-world context.
M3.A.CED.A.2Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.
M3.A.CED.A.3Rearrange formulas to isolate a quantity of interest using algebraic reasoning.
M3.A.REI Algebra – Reasoning with Equations and Inequalities (A.REI)
M3.A.REI.A Understand solving equations as a process of reasoning and explain the reasoning.
M3.A.REI.A.1Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.
M3.A.REI.A.2Solve radical equations in one variable and identify extraneous solutions when they exist.
M3.A.SSE Algebra – Seeing Structure in Expressions (A.SSE)
M3.A.SSE.A Interpret the structure of expressions.
M3.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
M3.A.SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
M3.A.SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity.
M3.F M3.F
M3.F.BF Functions – Building Functions (F.BF)
M3.F.BF.A Build new functions from existing functions.
M3.F.BF.A.1 Build a function that describes a relationship between two quantities.
M3.F.BF.A.1.aCombine standard function types using composition.
M3.F.BF.A.2Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given graphs.
M3.F.BF.A.3 Find the inverse of a function.
M3.F.BF.A.3.aDetermine whether a function is one-to-one.
M3.F.BF.A.3.bFind the inverse of a function on an appropriate domain.
M3.F.BF.A.3.cGiven an invertible function on an appropriate domain, identify the domain of the inverse function.
M3.F.IF Functions – Interpreting Functions (F.IF)
M3.F.IF.A Understand the concept of a function and use function notation.
M3.F.IF.A.1 Use function notation.
M3.F.IF.A.1.aUse function notation to evaluate functions for inputs in their domains, including functions of two variables.
M3.F.IF.A.1.bInterpret statements that use function notation in terms of a context.
M3.F.IF.A.2Understand geometric formulas as functions.
M3.F.IF.B Interpret functions that arise in applications in terms of the context.
M3.F.IF.B.3For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
M3.F.IF.B.4Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.
M3.F.IF.C Analyze functions using different representations.
M3.F.IF.C.5Graph functions expressed algebraically and show key features of the graph by hand and using technology.
M3.F.IF.C.6 Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.
M3.F.IF.C.6.aCompare properties of two different functions. Functions may be of different types and/or represented in different ways.
M3.F.IF.C.6.bCompare properties of the same function on two different intervals or represented in two different ways.
M3.F.LE Functions – Linear, Quadratic, and Exponential Models (F.LE)
M3.F.LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
M3.F.LE.A.1Know that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or cubically.
M3.F.LE.A.2 Know the relationship between exponential functions and logarithmic functions.
M3.F.LE.A.2.aSolve exponential equations using a variety of strategies, including logarithms.
M3.F.LE.A.2.bUnderstand that a logarithm is the solution to ab^ct = d, where a, b, c, and d are numbers.
M3.F.LE.A.2.cEvaluate logarithms using technology.
M3.G M3.G
M3.G.C Geometry – Circles (G.C)
M3.G.C.A Find areas of sectors of circles.
M3.G.C.A.1Use proportional relationships between the area of a circle and the area of a sector within the circle to solve problems and represent solutions in a real-world context.
M3.G.GMD Geometry – Geometric Measurement and Dimension (G.GMD)
M3.G.GMD.A Explain volume and surface area formulas and use them to solve problems.
M3.G.GMD.A.1Understand and explain the formulas for the volume and surface area of a cylinder, cone, prism, and pyramid.
M3.G.GMD.A.2Use volume and surface area formulas for cylinders, cones, prisms, pyramids, and spheres to solve problems in a real-world context.
M3.G.MG Geometry – Modeling with Geometry (G.MG)
M3.G.MG.A Apply geometric concepts in modeling situations.
M3.G.MG.A.1Use geometric shapes, their measures, and their properties to model objects found in a real-world context for the purpose of approximating solutions to problems.
M3.G.SRT Geometry – Similarity, Right Triangles, and Trigonometry (G.SRT)
M3.G.SRT.A Define trigonometric ratios and solve problems involving triangles.
M3.G.SRT.A.1 Use side ratios in right triangles to define trigonometric ratios.
M3.G.SRT.A.1.aUnderstand 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.
M3.G.SRT.A.1.bExplain and use the relationship between the sine and cosine of complementary angles.
M3.G.SRT.A.2 Solve triangles.
M3.G.SRT.A.2.aKnow and use the Pythagorean Theorem and trigonometric ratios (sine, cosine, tangent, and their inverses) to solve right triangles in a real-world context.
M3.G.SRT.A.2.bKnow and use relationships within special right triangles to solve problems in a real-world context.
M3.G.SRT.A.2.cUse the Law of Sines and Law of Cosines to solve non-right triangles in a real-world context.
M3.N M3.N
M3.N.Q Number and Quantity – Quantities (N.Q)
M3.N.Q.A Reason quantitatively and use units to understand problems.
M3.N.Q.A.1 Use units as a way to understand real-world problems.
M3.N.Q.A.1.aChoose and interpret the scale and the origin in graphs and data displays.
M3.N.Q.A.1.bUse appropriate quantities in formulas, converting units as necessary.
M3.N.Q.A.1.cDefine and justify appropriate quantities within a context for the purpose of modeling.
M3.N.Q.A.1.dChoose an appropriate level of accuracy when reporting quantities.
M3.S M3.S
M3.S.CP Statistics and Probability – Conditional Probability and the Rules of Probability (S.CP)
M3.S.CP.A Understand independence and conditional probability and use them to create visual representations of data.
M3.S.CP.A.1 Use set notation to represent contextual situations.
M3.S.CP.A.1.aDescribe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or", "and", "not").
M3.S.CP.A.1.bFlexibly move between visual models (Venn diagrams, frequency tables, etc.) and set notation.
M3.S.CP.A.2Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Categorize events as independent or dependent.
M3.S.CP.B Understand and apply basic concepts of probability.
M3.S.CP.B.3 Apply statistical counting techniques.
M3.S.CP.B.3.aUse the Fundamental Counting Principle to compute probabilities of compound events and solve problems.
M3.S.CP.B.3.bUse permutations and combinations to compute probabilities of compound events and solve problems.
M3.S.CP.B.4Use the Law of Large Numbers to assess the validity of a statistical claim.
M3.S.CP.C Use the rules of probability to compute probabilities of compound events in a uniform probability model.
M3.S.CP.C.5Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A and interpret the answer in terms of the given context.
M3.S.CP.C.6 Understand and apply the Addition Rule.
M3.S.CP.C.6.aExplain the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B) in terms of visual models (Venn diagrams, frequency tables, etc.).
M3.S.CP.C.6.bApply the Addition Rule to solve problems and interpret the answer in terms of the given context.
M3.S.CP.D Apply geometric concepts to situations involving probability.
M3.S.CP.D.7Calculate probabilities using geometric figures.
M3.S.IC Statistics and Probability – Making Inferences and Justifying Conclusions (S.IC)
M3.S.IC.A Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
M3.S.IC.A.1Recognize the purposes of and differences among sample surveys, experiments, and observational studies.
M3.S.IC.A.2Identify potential sources of bias in statistical studies.
M3.S.IC.A.3Distinguish between a statistic and a parameter. Evaluate reports based on data and recognize when poor conclusions are drawn from well-collected data.
M3.S.ID Statistics and Probability – Interpreting Categorical and Quantitative Data (S.ID)
M3.S.ID.A Summarize, represent, and interpret data on a single count or measurement variable.
M3.S.ID.A.1Use measures of center to solve real-world and mathematical problems.
M3.S.ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (mean, median, and/or mode) and spread (range, interquartile range, and standard deviation) of two or more different data sets.
M3.S.ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points.
M3.S.ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages using the Empirical Rule.
M3.S.ID.A.5Compute, interpret, and compare z-scores for normally distributed data in a real-world context.
M3.S.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
M3.S.ID.B.6Represent data from two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Mathematical Reasoning for Decision Making | MR

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MR.A MR.A
MR.A.LP Algebra – Linear Programming (A.LP)
MR.A.LP.A Use linear programming techniques to solve real-world problems.
MR.A.LP.A.1Read, interpret, and solve linear programming problems graphically and by computational methods.
MR.A.LP.B Solve real-world optimization problems.
MR.A.LP.B.2Use linear programming to solve optimization problems (for example, optimizing profit for a small business).
MR.A.LP.B.3Interpret the meaning of the maximum or minimum value in terms of the objective function.
MR.D MR.D
MR.D.ID Data Analysis, Statistics, and Probability – Organize and Interpret Data (D.ID)
MR.D.ID.A Analyze data from multiple viewpoints and perspectives.
MR.D.ID.A.1Organize, analyze, and interpret data for problem solving (for example, compare data related to costs of living; analyze survey data; provide a circle graph that demonstrates the percentages of income that support various expenses).
MR.D.ID.A.2Determine whether a set of data supports a given assertion (for example, whether a data set collected on Tennessee residents can be generalized to support an assertion about all Americans; whether a data set supports, or is large enough to support, the validity of a claim).
MR.D.ID.A.3Develop facility with representations of a data set and explain why some representations are more accurate or relevant than others in a given context.
MR.D.ID.A.4Interpret and use measures of central tendency and spread to solve problems and make informed decisions.
MR.D.ID.A.5Calculate expected value in real-world situations (such as lottery return on investment, expected value of each possession in sports, and expected payoff in a game of chance).
MR.D.ID.A.6Evaluate and compare two investments or strategies where one investment or strategy is safer but has lower expected value. Include large and small investments and situations with serious consequences.
MR.D.ID.A.7Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Evaluate strategies and make decisions based on expected values (for example, whether a team should pursue a higher-scoring option with a smaller probability of success or a lower-scoring option with a higher probability of success; whether a homeowner should file a small insurance claim given the probability that the monthly cost of insurance will rise as a result).
MR.D.ND Data Analysis, Statistics, and Probability – Normal Probability Distribution (D.ND)
MR.D.ND.A Work with the normal distribution in real-world situations.
MR.D.ND.A.1Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
MR.D.ND.B Work with the confidence intervals in real-world situations.
MR.D.ND.B.2Understand and interpret confidence levels and confidence intervals (for example, use the weights of randomly sampled boxes of cereal compared to the expected tolerances to determine whether the machinery is operating properly).
MR.G MR.G
MR.G.GMD Geometry and Measurement – Geometric Measurement (G.GMD)
MR.G.GMD.A Understand the role of precision in measurement.
MR.G.GMD.A.1Use standard units (metric and non-metric) to accurately measure objects to within 0.1 of the unit used.
MR.G.GMD.A.2Use precise measurements (within 0.1 of the unit used) to calculate area, surface area, and volume/capacity (emphasize common two-and three-dimensional shapes).
MR.G.GMD.A.3Understand and explain the effects that an error in measurement will have on a calculation that uses the erroneous measurement (for example, whether an error of 0.1 unit in length affects the calculated vs. actual measurement of the volume of an object, and whether that error is compounded by errors in other measurements used in the calculation).
MR.G.GMD.B Accurately use standard and non-standard units in measurement.
MR.G.GMD.B.10Discuss the various examples and consequences of innumeracy; consider poor estimation, improper experimental design, inappropriate comparisons, and scientific notation comparisons.
MR.G.GMD.B.4Use standard units of measure to develop accurately estimated measurements of commonly available non-standard instruments of measurement (for example, establish the length of hand span in inches or centimeters; length of arm span or stride length in feet or yards; the area of a floor tile in square inches or square feet; the volume of a gallon of milk or a water bottle or a soda can in cubic inches or cubic centimeters, etc.).
MR.G.GMD.B.5Understand and explain the consequences of relying on non-standard units of measure (for example, explain why paper clip length or pencil length are not standard units of measure and how failing to use mutually agreed-upon units can lead to erroneous assumptions, calculations, or conclusions).
MR.G.GMD.B.6Use the established dimensions of common non-standard measuring instruments to estimate other measurements using standard units to a given tolerance (for example, use stride length to estimate the length of a hallway to within 10% of the actual length in feet; use the estimated volume of a finger in cubic centimeters to estimate the amount of liquid in a glass).
MR.G.GMD.B.7Estimate the area, surface area, volume, or capacity of an object using the established dimensions of common non-standard measuring instruments to determine measurements in standard units with and without using technology (for example, use the number of floor tiles along a wall to estimate the area of the floor of a room and use the height of a person to estimate the height of the room, then find the volume of the room based on those estimations; use the size of a milk jug to estimate the number of gallons in a tank of water).
MR.G.GMD.B.8Estimate the amount of error in a calculation that is based on using established dimensions of common non-standard measuring instruments (for example, if a person's stride length is 30 inches plus/minus 2 inches, and the person uses stride length to measure the length and width of a plot of land, determine the estimated error in calculating the area of the plot of land).
MR.G.GMD.B.9Understand and use unit conversions in estimations involving both standard and non-standard units (for example, determine how many boxes of flooring will be needed to cover a floor of given dimensions if 10% waste is assumed; how many gallons of paint will be needed to paint a room of a given size; how many bags of fertilizer will be needed to fertilize a yard of a given size).
MR.N MR.N
MR.N.NQ Financial Mathematics (N.NQ)
MR.N.NQ.A Use financial mathematics to make personal financial decisions.
MR.N.NQ.A.1Define common terms associated with finance (such as interest, compound interest, annuities, retirement funds, amortizations, future value, and present value) and know how each term is related to personal finance.
MR.N.NQ.A.2Calculate compound interest within the context of personal finance (such as credit card debt, home/car purchase, personal loans, and amortization schedules) and use the results to make decisions (for example, determine which home financing option is best).
MR.N.NQ.A.3Calculate net pay using gross pay (weekly, biweekly, monthly, or annual) and both fixed and variable deductions (such as withholding tax, Social Security tax, insurance costs, retirement investments and other contributory benefits).
MR.N.NQ.A.4Access and use published data (such as cost of city or state utilities, housing, city or state taxes, meals, and other costs of living) to estimate and compare monthly living expenses based on location, identified needs, and personal preferences or desired lifestyles.
MR.N.NQ.A.5Access and use published data (such as average life expectancy based on location and/or health issues, investment data, retirement funds, and annuity data) to calculate and compare retirement investments (such as total savings and monthly payouts) based on projected income.
MR.N.NQ.A.6Access and use published data to create depreciation schedules and analyze the depreciation of various assets (such as cars, business equipment, and store fixtures).
MR.N.NQ.A.7Access and use published data to calculate income tax based on projected gross annual income, returns on investments, tax deductions and tax credits, and other factors that affect calculations.
MR.N.NQ.A.8Develop a personal mid-term (three to five years) financial plan based on anticipated income, projected living expenses, projected retirement or other savings, and other factors that affect personal finances.
MR.N.NQ.B Use financial mathematics to make personal financial decisions.
MR.N.NQ.B.10Define common terms associated with business finance (such as assets, liabilities, revenue, expenses, net profit, net loss, profit margin, and return on investment) and know how each term is related to business finance.
MR.N.NQ.B.11Access and use published data to develop a three-year financial plan for starting and running a small business (including projected income and projected fixed and variable costs such as licenses, rent and utilities, city and state taxes, cost of goods sold, etc.).
MR.N.NQ.B.9Compare the components of a small business plan to the components of a personal financial plan (i.e., identify components that are common to both plans and components that are unique to a small business plan).

Precalculus | P

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P.A P.A
P.A.C Algebra – Conic Sections (A.C)
P.A.C.A Understand the properties of conic sections and model real-world phenomena.
P.A.C.A.1Display all of the conic sections as portions of a cone.
P.A.C.A.2Know and write the equation of a circle of given center and radius using the Pythagorean Theorem.
P.A.C.A.3Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
P.A.C.A.4From an equation in standard form, graph the appropriate conic section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an understanding of the relationship between their standard algebraic form and the graphical characteristics.
P.A.C.A.5Transform equations of conic sections to convert between general and standard form.
P.A.PE Algebra – Parametric Equations (A.PE)
P.A.PE.A Describe and use parametric equations.
P.A.PE.A.1Graph curves parametrically (by hand and with appropriate technology).
P.A.PE.A.2Eliminate parameters by rewriting parametric equations as a single equation.
P.A.REI Algebra – Reasoning with Equations and Inequalities (A.REI)
P.A.REI.A Solve systems of equations and nonlinear inequalities.
P.A.REI.A.1Represent a system of linear equations as a single matrix equation in a vector variable.
P.A.REI.A.2Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
P.A.REI.A.3Solve rational and radical equations in one variable, and identify extraneous solutions when they exist.
P.A.REI.A.4Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if one-variable), by hand and with appropriate technology.
P.A.REI.A.5Solve systems of nonlinear inequalities by graphing.
P.A.S Algebra – Sequences and Series (A.S)
P.A.S.A Understand and use sequences and series.
P.A.S.A.1Demonstrate an understanding of sequences by representing them recursively and explicitly.
P.A.S.A.2Use sigma notation to represent a series; expand and collect expressions in both finite and infinite settings.
P.A.S.A.3 Derive and use the formulas for the general term and summation of finite or infinite arithmetic and geometric series, if they exist.
P.A.S.A.3.aDetermine whether a given arithmetic or geometric series converges or diverges.
P.A.S.A.3.bFind the sum of a given geometric series (both infinite and finite).
P.A.S.A.3.cFind the sum of a finite arithmetic series.
P.A.S.A.4Understand that series represent the approximation of a number when truncated; estimate truncation error in specific examples.
P.A.S.A.5Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal’s Triangle.
P.F P.F
P.F.BF Functions – Building Functions (F.BF)
P.F.BF.A Build new functions from existing functions.
P.F.BF.A.1Understand how the algebraic properties of an equation transform the geometric properties of its graph (for example, given a function, describe the transformation of the graph resulting from the manipulation of the algebraic properties of the equation such as translations, stretches, reflections, and changes in periodicity and amplitude).
P.F.BF.A.2Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and composition of functions.
P.F.BF.A.3Compose functions (for example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time).
P.F.BF.A.4Construct the difference quotient for a given function and simplify the resulting expression.
P.F.BF.A.5 Find inverse functions (including exponential, logarithmic, and trigonometric).
P.F.BF.A.5.aCalculate the inverse of a function, f (x), with respect to each of the functional operations; in other words, the additive inverse, − f(x), the multiplicative inverse, 1/f(x), and the inverse with respect to composition, f^−1(x). Understand the algebraic and graphical implications of each type.
P.F.BF.A.5.bVerify by composition that one function is the inverse of another.
P.F.BF.A.5.cRead values of an inverse function from a graph or a table, given that the function has an inverse.
P.F.BF.A.5.dRecognize a function is invertible if and only if it is one-to-one. Produce an invertible function from a non-invertible function by restricting the domain.
P.F.BF.A.6Explain why the graph of a function and its inverse are reflections of one another over the line y = x.
P.F.GT Functions – Graphing Trigonometric Functions (F.GT)
P.F.GT.A Model periodic phenomena with trigonometric functions.
P.F.GT.A.1Interpret transformations of trigonometric functions.
P.F.GT.A.2Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.
P.F.GT.A.3Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.
P.F.GT.A.4Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.
P.F.GT.A.5Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
P.F.GT.A.6Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.
P.F.GT.A.7Graph the inverse trigonometric functions and identify their key characteristics.
P.F.GT.A.8Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology and interpret them in terms of the context.
P.F.IF Functions – Interpreting Functions (F.IF)
P.F.IF.A Analyze functions using different representations.
P.F.IF.A.1Determine whether a function is even, odd, or neither.
P.F.IF.A.2Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real-world problems that can be modeled with these functions (by hand and with appropriate technology).
P.F.IF.A.3Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential, polynomial, logarithmic, trigonometric, and rational).
P.F.IF.A.4Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax^2 + c.
P.F.IF.A.5Visually locate critical points on the graphs of functions and determine if each critical point is a minimum, a maximum, or point of inflection. Describe intervals where the function is increasing or decreasing and where different types of concavity occur.
P.F.IF.A.6Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.
P.F.IF.A.7Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers (for example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1).
P.F.TA P.F.TA
P.F.TA.A P.F.TA.A
P.F.TA.A.3Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and explain how to use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
P.F.TA.A.4Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
P.F.TF Functions – Trigonometric Functions (F.TF)
P.F.TF.A Extend the domain of trigonometric functions using the unit circle.
P.F.TF.A.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
P.F.TF.A.2Convert from radians to degrees and from degrees to radians.
P.F.TF.A.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
P.G P.G
P.G.AT Geometry – Applied Trigonometry (G.AT)
P.G.AT.A Use trigonometry to solve problems.
P.G.AT.A.1Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.
P.G.AT.A.2Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
P.G.AT.A.3Derive and apply the formulas for the area of sector of a circle.
P.G.AT.A.4Calculate the arc length of a circle subtended by a central angle.
P.G.AT.A.5Prove the Laws of Sines and Cosines and use them to solve problems.
P.G.AT.A.6Understand and apply the Law of Sines (including the ambiguous case) and the Law of Cosines to find unknown measurements in right and non-right triangles (such as surveying problems and resultant forces).
P.G.PC Geometry – Polar Coordinates (G.PC)
P.G.PC.A Use polar coordinates.
P.G.PC.A.1Graph functions in polar coordinates.
P.G.PC.A.2Convert between rectangular and polar coordinates.
P.G.PC.A.3Represent situations and solve problems involving polar coordinates.
P.G.TI Geometry – Trigonometric Identities (G.TI)
P.G.TI.A Apply trigonometric identities to rewrite expressions and solve equations.
P.G.TI.A.1Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, reciprocal, quotient, sum/difference, double-angle, and half-angle.
P.G.TI.A.2Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
P.N P.N
P.N.CN The Complex Number System (N.CN)
P.N.CN.A Perform complex number arithmetic and understand the representation on the complex plane.
P.N.CN.A.1Know there is a complex number i such that i^2 = –1, and every complex number has the form a + bi with a and b real.
P.N.CN.A.2Perform arithmetic operations with complex numbers expressing answers in the form a + bi.
P.N.CN.A.3Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
P.N.CN.A.4Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
P.N.CN.A.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation (for example, (–1 + 3i)^3 = 8 because (–1 + 3i) has modulus 2 and argument 120°).
P.N.CN.A.6Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
P.N.CN.B Use complex numbers in polynomial identities and equations.
P.N.CN.B.7Extend polynomial identities to the complex numbers (for example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
P.N.CN.B.8Solve quadratic equations with real coefficients that have complex solutions.
P.N.CN.B.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
P.N.NE Number Expressions (N.NE)
P.N.NE.A Represent, interpret, compare, and simplify number expressions.
P.N.NE.A.1Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or compare them.
P.N.NE.A.2Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
P.N.NE.A.3Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of π and e.
P.N.NE.A.4Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers are not.
P.N.NE.A.5Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
P.N.VM Vector and Matrix Quantities (N.VM)
P.N.VM.A Represent and model with vector quantities.
P.N.VM.A.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, V→).
P.N.VM.A.2Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
P.N.VM.A.3Solve problems involving velocity and other quantities that can be represented by vectors.
P.N.VM.B Understand the graphic representation of vectors and vector arithmetic.
P.N.VM.B.4 Add and subtract vectors.
P.N.VM.B.4.aAdd vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
P.N.VM.B.4.bGiven two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
P.N.VM.B.4.cUnderstand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
P.N.VM.B.5 Multiply a vector by a scalar.
P.N.VM.B.5.aRepresent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise (e.g., as c(v_x , v_y) = (cv_x , cv_y).
P.N.VM.B.5.bCompute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
P.N.VM.B.6Calculate and interpret the dot product of two vectors.
P.N.VM.C Perform operations on matrices and use matrices in applications.
P.N.VM.C.10Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
P.N.VM.C.7Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
P.N.VM.C.8Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
P.N.VM.C.9Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
P.S P.S
P.S.MD Statistics and Probability – Model with Data (S.MD)
P.S.MD.A Model data using regressions equations.
P.S.MD.A.1Create scatter plots, analyze patterns, and describe relationships for bivariate data (linear, polynomial, trigonometric, or exponential)to model real-world phenomena and to make predictions.
P.S.MD.A.2Determine a regression equation to model a set of bivariate data. Justify why this equation best fits the data.
P.S.MD.A.3Use a regression equation, modeling bivariate data, to make predictions. Identify possible considerations regarding the accuracy of predictions when interpolating or extrapolating.

Statistics | S

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S.1 Topic 1: Sampling and Data
S.1.aUnderstand the investigative process of statistics and differentiate between descriptive and inferential statistics.
S.1.bDifferentiate between a population and a sample.
S.1.cConstruct a simple random sample.
S.1.dUnderstand the differences between stratified sampling, cluster sampling, systematic sampling, and convenience sampling.
S.1.eDetermine when samples of convenience are acceptable and how sampling bias and error can occur.
S.1.fIdentify and classify data as either qualitative or quantitative and classify quantitative data as either discrete or continuous data.
S.1.gDisplay and interpret qualitative data with graphs: pie graphs, bar graphs, and pareto charts.
S.1.hDifferentiate between levels of measurement: nominal, ordinal, interval, and ratio.
S.1.iCreate a frequency distribution from a list of quantitative and/or qualitative data.
S.1.jCalculate relative frequencies and cumulative frequencies using a frequency distribution table.
S.1.kUnderstand differences between a designed experiment and an observational study.
S.1.lDifferentiate between the types of variables used in a designed experiment.
S.1.mUnderstand different methods used in an experiment to isolate effects of the explanatory variable.
S.2 Topic 2: Descriptive Statistics
S.2.aDisplay and interpret graphs using quantitative data including stem-and-leaf plots, line graphs, and box plots.
S.2.bConstruct a histogram from a frequency distribution table.
S.2.cInterpret data using histograms and time series graphs.
S.2.dAnalyze a frequency distribution table and determine the sample size, class width and class midpoints.
S.2.eRecognize, describe, and calculate the measures of locations of data: quartiles, median, five number summary, interquartile range outliers, upper and lower fences, and percentiles.
S.2.fDistinguish between a parameter and a statistic.
S.2.gCalculate and differentiate between different measures of center: mean, median, and mode.
S.2.hCalculate the mean of a frequency distribution: GPA and weighted grade.
S.2.iInterpret the shape of the distribution from a graph: normal/symmetric, skewed, or uniform.
S.2.jCalculate and differentiate between different measures of spread: range, variance, and standard deviation.
S.2.kDetermine if a data value is unusual based on standard deviations.
S.3 Topic 3: Probability
S.3.aUnderstand and use terminology and symbols of probability.
S.3.bList the elements of events and the sample space from an experiment.
S.3.cUnderstand the concept of randomness: flipping a coin, rolling a die, and drawing a card from a standard 52 card deck.
S.3.dDifferentiate between and calculate different types of probabilities: empirical and theoretical.
S.3.eExplain the Law of Large Numbers.
S.3.fCalculate and interpret probabilities using the complement rule, addition rule, and multiplication rule.
S.3.gDifferentiate between and calculate probabilities for different types of events: independent, dependent, with or without replacement, conditional, and mutually exclusive.
S.3.hUse Venn diagrams and lists to solve probability problems when appropriate.
S.4 Topic 4: Discrete Random Variables
S.4.aIdentify the random variable in a probability experiment.
S.4.bRecognize and understand discrete probability distribution functions.
S.4.cCreate a probability distribution for the values of a discrete random variable.
S.4.dUse a probability function to determine probabilities associated with a discrete random variable.
S.4.eCalculate and interpret the mean (expected value), variance, and standard deviation for discrete random variables and binomial probability distributions.
S.4.fDetermine when a probability distribution should be classified as a discrete binomial probability distribution, and calculate probabilities associated with such a distribution.
S.5 Topic 5: Continuous Random Variables and the Normal Distribution
S.5.aRecognize and understand continuous probability density functions.
S.5.bUse a probability density curve to describe a population, including a normal population.
S.5.cCalculate and interpret the area under a probability density curve.
S.5.dCalculate and interpret a z-score, understanding the concept of standardizing data.
S.5.eCalculate and interpret z-scores using the Empirical Rule, understanding the general properties of the normal distribution.
S.5.fUse technology to calculate the area under the curve for any normal distribution model: left, right, and between.
S.5.gUse technology to calculate percentiles, quartiles, and other numerical values of X for a specified area under a normal curve.
S.6 Topic 6: Central Limit Theorem
S.6.aRecognize the characteristics of the mean of sample means taken from different types of populations: normal and non-normal.
S.6.bCalculate the mean of sample means taken from different types of populations: normal and non-normal.
S.6.cDescribe how the means of samples calculated from a non-normal population might be distributed.
S.6.dApply the Central Limit Theorem to normal and non-normal populations and compute probabilities of a sample mean.
S.6.eDetermine whether the Central Limit Theorem can be used for a given situation.
S.6.fAssess the impact of sample size on sampling variability.
S.7 Topic 7: Confidence Intervals
S.7.aRead and write confidence intervals using two different forms: point estimate plus/minus margin of error (error bound) and interval notation.
S.7.bCalculate and interpret confidence intervals for estimating a population mean and a population proportion.
S.7.cCalculate the margin of error (error bound) using sample statistics.
S.7.dPredict if a confidence interval will become wider or narrower given larger or smaller sample sizes as well as higher or lower confidence levels.
S.7.eFind the point estimate and margin of error (error bound) when given a confidence interval.
S.7.fEstimate the sample size necessary to estimate a population mean.
S.7.gRecognize the difference between the sample mean and the population mean, as well as the difference between the sample standard deviation and standard error of the mean.
S.7.hFind critical values for z and t given a value of alpha and degrees of freedom.
S.7.iEstimate the sample size necessary to estimate a population proportion.
S.8 Topic 8: Hypothesis Testing
S.8.aDetermine the appropriate null and alternative hypotheses when presented with a problem.
S.8.bDifferentiate between Type I and Type II errors.
S.8.cUnderstand and list the assumptions needed to conduct z-tests and t-tests.
S.8.dDetermine whether to reject or fail to reject the null hypothesis using the p-value method.
S.8.eDetermine if a test is left-tailed, right-tailed, or two-tailed.
S.8.fDifferentiate between independent group and matched pair sampling.
S.8.gCalculate test statistics and p-values for hypotheses tests: single proportion, single mean, and difference between two means.
S.8.hConduct hypotheses tests for a single proportion and a single mean.
S.8.iTest hypotheses regarding the difference of two independent means (assume the variances are not pooled).
S.8.jDraw conclusions and make inferences about claims based on hypotheses tests.
S.9 Topic 9: Regression and Correlation
S.9.aDifferentiate between the independent (explanatory variable, x) and the dependent (response variable, y) in a bivariate data set.
S.9.bCreate a scatter plot and determine the type of relationship that exists between two variables: positive or negative correlation and weak or strong correlation.
S.9.cCalculate and interpret the correlation coefficient using technology.
S.9.dCalculate the line of best fit and interpret the coefficient of determination.
S.9.eUse the line of best fit to make conclusions about the relationship between two variables, understanding correlation does not imply causation.
S.9.fCalculate a residual using the line of best fit.
S.9.gUse the p-value to determine if a line of best fit is statistically significant.
S.9.hFor a given value of x, find the appropriate estimated value of y.
S.9.iDistinguish between interpolated and extrapolated values and explain why interpolated values are more reliable.
S.9.jPerform a residual analysis to check assumptions of regression.

AP Calculus Learning Objectives

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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

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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.