Kentucky Math Standards - Kentucky Academic Standards for Mathematics

900 standards, Kindergarten through AP.

Kentucky was an early adopter of Common Core, adding state specific modifications across K-8 and high school conceptual categories. The result blends rigorous national content with local customization.

Kindergarten

K.CC Counting and Cardinality

K.CC.1 Count. (MP.7, MP.8)

K.CC.1.aCount to 100 by ones and by tens.

K.CC.1.bCount backwards from 30 by ones.

K.CC.2Count forward beginning from a given number within the known sequence within 100 (instead of having to begin at 1). (MP.7)

K.CC.3 Represent numbers. (MP.2, MP.7, MP.8)

K.CC.3.aWrite numbers from 0 to 20.

K.CC.3.bRepresent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

K.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality. (MP.2, MP.8)

K.CC.4.aWhen counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

K.CC.4.bUnderstand 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.4.cUnderstand that each successive number name refers to a quantity that is one larger.

K.CC.5 Given a number from 1-20, count out that many objects. (MP.2, MP.3)

K.CC.5.aCount to answer “how many?” questions with as many as 20 things arranged in a line, a rectangular array, or a circle.

K.CC.5.bCount to answer “how many?” questions with as many as 10 things in a scattered configuration.

K.CC.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group. (MP.1, MP.3, MP.6)

K.CC.7Compare two numbers between 1 and 10 presented as written numerals. (MP.2)

K.G.1 Name and describe shapes in the environment. (MP.6)

K.G.1.aDescribe objects in the environment using names of shapes.

K.G.1.bDescribe the relative positions of these objects using terms above, below, in front of, behind and next to.

K.G.2Correctly name shapes regardless of orientations or overall size. (MP.7)

K.G.3Identify shapes as two-dimensional or three-dimensional. (MP.3, MP.6)

K.G.4Describe the similarities, differences and attributes of two and three dimensional shapes using different sizes and orientations. (MP.3, MP.7)

K.G.5Model shapes in the world by building figures from components and drawing shapes. (MP.1, MP.5)

K.G.6Compose simple shapes to form larger shapes. (MP.3, MP.5)

K.MD Measurement and Data

K.MD.1Describe measurable attributes (length, height, weight, width, depth) of an object or a set of objects using appropriate vocabulary. (MP.3, MP.6)

K.MD.2Directly compare two objects with a measurable attribute in common, to see which object has “more of”/ “less of” the attribute and describe the difference. (MP.2, MP.6)

K.MD.3Classify and sort objects or people by attributes. Limit objects or people in each category to be less than or equal to 10. (MP.3, MP.6)

K.MD.4Recognize and identify coins by name (penny, nickel, dime, quarter). (MP.6)

K.NBT Numbers and Operations in Base Ten

K.NBT.1Compose and decompose numbers from 11 to 19 using quantities (numbers with units) of ten ones and some further ones. Understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. (MP.3, MP.4, MP.7)

K.OA Operations and Algebraic Thinking

K.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, expressions, or equations. (MP.2, MP.4)

K.OA.2Solve addition and subtraction word problems and add and subtract within 10 by using objects or drawings to represent the problem. (MP.5)

K.OA.3 Decompose numbers less than or equal to 10. (MP.2, MP.4)

K.OA.3.aDecompose numbers into two groups in more than one way by using objects or drawings and record each decomposition by a drawing or equation.

K.OA.3.bUse objects or drawings to demonstrate equality as the balancing of quantities.

K.OA.4For any number from 1 to 9, find the number that makes 10 when added to the given number by using objects or drawings and record the answer with a drawing or equation. (MP.7, MP.8)

K.OA.5Fluently add and subtract within 5. (MP.2, MP.7)

Grade 1

1.G.1Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes. (MP.7)

1.G.2 Compose shapes. (MP.1, MP.4)

1.G.2.aCompose two-dimensional shapes to create rectangles, squares, trapezoids, triangles, half-circles and quarter-circles composite shape and compose new shapes from the composite shapes.

1.G.2.bUse three-dimensional shapes (cubes, right rectangular prisms, right circular cones and right circular cylinders) to create a composite shape and compose new shapes from the composite shapes.

1.G.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 decomposing into more equal shares creates smaller shares. (MP.3, MP.6)

1.MD Measurement and Data

1.MD.1Order three objects by length; compare the lengths of two objects indirectly by using a third object. (MP.6)

1.MD.2Express the length of an object as a whole number of same-size length units, by laying multiple copies of a shorter object (the length unit) end to end with no gaps or overlaps. (MP.2, MP.5)

1.MD.3 Assign values to time and money. (MP.6, MP.8)

1.MD.3.aTell and write time in hours and half-hours using analog and digital clocks.

1.MD.3.bIdentify the coins by values (penny, nickel, dime, quarter).

1.MD.4 Investigate questions involving categorical data. (MP.1, MP.3, MP.4, MP.6)

1.MD.4.aPose a question that can be answered by gathering data.

1.MD.4.bDetermine strategy for gathering data from peers.

1.MD.4.cOrganize and represent data in a table/chart with up to three categories.

1.MD.4.dInterpret data to answer questions about the table/chart that connects to the question posed, including 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 Numbers and Operations in Base Ten

1.NBT.1 Count and represent numbers. (MP.2, MP.5, MP.8)

1.NBT.1.aCount forward to and backward from 120, starting at any number less than 120.

1.NBT.1.bIn this range, read and write numerals and represent a number of objects with a written numeral.

1.NBT.2 Understand the two-digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: (MP.5, MP.7)

1.NBT.2.a10 can be thought of as a bundle of ten ones—called a “ten.”

1.NBT.2.bThe numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight or nine ones.

1.NBT.2.cThe numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight or nine tens (and 0 ones).

1.NBT.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. (MP.2)

1.NBT.4 Add within 100 including adding a two-digit number and a one-digit number. Add a two-digit number and a multiple of 10. (MP.7, MP.2, MP.3)

1.NBT.4.a Add within 100 using…

1.NBT.4.a.1concrete models or drawings

1.NBT.4.a.2strategies based on place value

1.NBT.4.a.3properties of operations

1.NBT.4.a.4the relationship between addition and subtraction

1.NBT.4.bRelate the addition strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

1.NBT.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. (MP.2, MP.8)

1.NBT.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences). (MP.3, MP.5)

1.NBT.6.a Subtract using:

1.NBT.6.a.1concrete models or drawings

1.NBT.6.a.2strategies based on place value

1.NBT.6.a.3properties of operations

1.NBT.6.a.4the relationship between addition and subtraction

1.NBT.6.bRelate the subtraction strategy to a written method and explain the reasoning used.

1.OA Operations and Algebraic Thinking

1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions. (MP.1, MP.2)

1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, by using objects, drawings and equations with a symbol for one unknown number to represent the problem. (MP.1, MP.4, MP.5)

1.OA.3Apply properties of operations as strategies to add and subtract. (MP.2, MP.7)

1.OA.4Understand subtraction as an unknown-addend problem. (MP.2, MP.7)

1.OA.5Relate counting to addition and subtraction. (MP.5, MP.8)

1.OA.6 Add and subtract within 20. (MP.2, MP.7, MP.8)

1.OA.6.aFluently add and subtract within 10.

1.OA.6.bAdd and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making 10; decomposing a number leading to a 10; using the relationship between addition and subtraction; creating equivalent but easier or known sums.

1.OA.7Understand the meaning of the equal sign and determine if equations involving addition and subtraction are true or false. (MP.2, MP.3)

1.OA.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. (MP.1, MP.2)

Grade 2

2.G.1Recognize and draw shapes having specified attributes, such as a given number of angles or sides. Identify triangles, quadrilaterals, pentagons, hexagons and cubes (identify number of faces). (MP.4, MP.7)

2.G.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. (MP.6, MP.8)

2.G.3Partition circles and rectangles into two, three, or four equal shares; describe the shares using the words halves, thirds, half of, a third of, etc.; and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. (MP.2, MP.3)

2.MD Measurement and Data

2.MD.1Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks and measuring tapes. (MP.5, MP.6)

2.MD.10Create a pictograph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put together, take-apart and compare problems using information presented in a bar graph. (MP.2, MP.6)

2.MD.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. (MP.3, MP.5)

2.MD.3Estimate lengths using units of inches, feet, yards, centimeters and meters. (MP.2, MP.6)

2.MD.4Measure to determine how much longer one object is than another, expressing the length difference in terms of either a customary or metric standard length unit. (MP.5, MP.6)

2.MD.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units by using drawings and equations with a symbol for the unknown number to represent the problem. (MP.1, MP.4)

2.MD.6Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2, ... and represent whole-number sums and differences within 100 on a number line. (MP.3, MP.4)

2.MD.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. (MP.5, MP.6)

2.MD.8Solve word problems with adding and subtracting within 100, (not using dollars and cents simultaneously) using the ＄and ￠ symbols appropriately (not including decimal notation). (MP.1, MP.5)

2.MD.9 Investigate questions involving measurements. (MP.1, MP.6)

2.MD.9.aIdentify a statistical question focused on measurements.

2.MD.9.bGenerate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object.

2.MD.9.cShow the measurements by making a dot plot, where the horizontal scale is marked off in whole-number units.

2.NBT Numbers and Operations in Base Ten

2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens and ones. Understand the following as special cases: (MP.2, MP.7)

2.NBT.1.a100 can be thought of as a bundle of ten tens—called a “hundred.”

2.NBT.1.bThe numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2.NBT.2Count forwards and backwards within 1000; skip-count by 5s, 10s and 100s. (MP.8, MP.1, MP.6)

2.NBT.3Read and write numbers to 1000 using base-ten numerals, number names and expanded form. (MP.7)

2.NBT.4Compare two three-digit numbers based on meanings of the hundreds, tens and ones digits, using >, =, and < symbols to record the results of comparisons. (MP.1, MP.6)

2.NBT.5Fluently add and subtract within 100 using strategies based on place value, properties of operations and/or the relationship between addition and subtraction. (MP.2, MP.8)

2.NBT.6Add up to four two-digit numbers using strategies based on place value and properties of operations. (MP.2, MP.7)

2.NBT.7 Add and subtract within 1000. (MP.1, MP.5)

2.NBT.7.a Represent and solve addition and subtraction problems using…

2.NBT.7.a.1concrete models or drawings

2.NBT.7.a.2strategies based on place value

2.NBT.7.a.3properties of operations

2.NBT.7.a.4the relationship between addition and subtraction

2.NBT.7.a.5relate drawings and strategies to expressions or equations

2.NBT.7.bUnderstand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

2.NBT.8Mentally add 10 or 100 to a given number 100–900 and mentally subtract 10 or 100 from a given number 100–900. (MP.7, MP.8)

2.NBT.9Explain why addition and subtraction strategies work, using place value and the properties of operations. (MP.3, MP.7)

2.OA Operations and Algebraic Thinking

2.OA.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions, by using drawings and equations with a symbol for the unknown number to represent the problem. (MP.1, MP.2 and MP.4)

2.OA.2Fluently add and subtract within 20 using mental strategies. (MP.2, MP.7, MP.8)

2.OA.3Determine whether a group of objects (up to 20) has an odd or even number of members; write an equation to express an even number as a sum of two equal addends. (MP.2, MP.7)

2.OA.4Use 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. (MP.2, MP.4)

Grade 3

3.G.1 Classify polygons by attributes. (MP.6, MP.7)

3.G.1.aRecognize and classify polygons based on the number of sides and vertices (triangles, quadrilaterals, pentagons and hexagons).

3.G.1.bRecognize and classify quadrilaterals (rectangles, squares, parallelograms, rhombuses, trapezoids) by side lengths and understanding shapes in different categories may share attributes and the shared attributes can define a larger category.

3.G.1.cIdentify shapes that do not belong to a given category or subcategory.

3.G.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. (MP.2, MP.5)

3.MD Measurement and Data

3.MD.1Tell and write time to the nearest minute and measure elapsed time intervals in minutes. Solve word problems involving addition and subtraction of time intervals within and across the hour in minutes. (MP.4, MP.6, MP.1, MP.4)

3.MD.2 Measure and solve problems involving mass and liquid volume. (MP.1, MP.6)

3.MD.2.aMeasure and estimate masses and liquid volumes of objects using standard units of grams (g), kilograms (kg) and liters (L).

3.MD.2.bAdd, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.

3.MD.3 Investigate questions involving categorical data. (MP.3, MP.5, MP.6)

3.MD.3.aIdentify a statistical question focused on categorical data and gather data.

3.MD.3.bCreate a scaled pictograph and a scaled bar graph to represent a data set (using technology or by hand).

3.MD.3.cMake observations from the graph about the question posed, including “how many more” and “how many less” questions.

3.MD.4 Investigate questions involving numerical data. (MP.1, MP.3, MP.6)

3.MD.4.aIdentify a statistical question focused on numerical data.

3.MD.4.bGenerate measurement data by measuring lengths using rulers marked with halves and fourths of an inch.

3.MD.4.cShow the data by making a dot plot where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

3.MD.4.dMake observations from the graph about the question posed, including questions about the shape of the data and compare responses.

3.MD.5Recognize area as an attribute of plane figures and understand concepts of area measurement. (MP.5)

3.MD.6Measure areas by counting unit squares (square cm, square m, square in, square ft. and improvised units). (MP.5, MP.6)

3.MD.7 Relate area to the operations of multiplication and addition. (MP.1, MP.8)

3.MD.7.aFind the area of a rectangle with whole-number side lengths by tiling it and show the area is the same as would be found by multiplying the side lengths.

3.MD.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.7.cUse tiling to show in a concrete case the area of a rectangle with whole-number side lengths a and b+c is the sum of a×b and a×c. Use area models to represent the distributive property in mathematical reasoning.

3.MD.7.dRecognize area as additive. Find areas of figures that can be decomposed into non-overlapping rectangles by adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

3.MD.8 Solve real world and mathematical problems involving perimeters of polygons. (MP.1, MP.4)

3.MD.8.aFind the perimeter given the side lengths of a polygon.

3.MD.8.bFind an unknown side length, given the perimeter and some lengths.

3.MD.8.cDraw rectangles with the same perimeter and different areas or with the same area and different perimeters.

3.NBT Numbers and Operations in Base Ten

3.NBT.1Use place value understanding to round whole numbers to the nearest 10 or 100. (MP.7)

3.NBT.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. (MP.2, MP.3)

3.NBT.3Multiply one-digit whole numbers by multiples of 10 in the range of 10–90 using strategies based on place value and properties of operations. (MP.7, MP.8)

3.NF Numbers and Operations—Fractions

3.NF.1Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. (MP.2, MP.7)

3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line. (MP.4)

3.NF.2.a Represent a fraction 1/b (unit fraction) on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.

3.NF.2.a.1Recognize each part has size 1/b

3.NF.2.a.2a unit fraction, 1/b is located 1/b of a whole unit from 0 on the number line

3.NF.2.bRepresent a non-unit fraction a/b on a number line by marking off lengths of 1/b (unit fractions) from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the non-unit fraction a/b on the number line.

3.NF.3 Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. (MP.2, MP.3)

3.NF.3.aUnderstand two fractions as equivalent (equal) if they are the same size, or same point on a number line.

3.NF.3.bRecognize and generate simple equivalent fractions. Explain why the fractions are equivalent through writing or drawing.

3.NF.3.cExpress whole numbers as fractions and recognize fractions that are equivalent to whole numbers.

3.NF.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. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.

3.OA Operations and Algebraic Thinking

3.OA.1Interpret and demonstrate products of whole numbers. (MP.1, MP.5)

3.OA.2Interpret and demonstrate whole-number quotients of whole numbers, where objects are partitioned into equal shares. (MP.2, MP.5)

3.OA.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities, by using drawings and equations with a symbol for the unknown number to represent the problem. (MP.1, MP.4)

3.OA.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers. (MP.6, MP.7)

3.OA.5Apply properties of operations as strategies to multiply and divide. (MP.3, MP.4)

3.OA.6Understand division as an unknown-factor problem. (MP.2)

3.OA.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. (MP.2, MP.8)

3.OA.8Use various strategies to solve two-step word problems using the four operations (involving only whole numbers with whole number answers). 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. (MP.1, MP.4)

3.OA.9Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations. (MP.3, MP.8)

Grade 4

4.G.1Draw points, lines, line segments, rays, angles (right, acute, obtuse) and perpendicular and parallel lines. Identify these in two-dimensional figures. (MP.5, MP.6)

4.G.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence of absence of angles of a specified size. Recognize right triangles as a category and identify right triangles. (MP.7)

4.G.3 Identify lines of symmetry. (MP.5, MP.7)

4.G.3.aRecognize a line of symmetry for a two-dimensional figure.

4.G.3.bIdentify line-symmetric figures and draw lines of symmetry.

4.MD Measurement and Data

4.MD.1 Know relative size of measurement units (mass, weight, liquid volume, length, time) within one system of units (metric system, U.S. standard system and time). (MP.5, MP.6)

4.MD.1.aUnderstand the relationship of measurement units within any given measurement system.

4.MD.1.bWithin any given measurement system, express measurements in a larger unit in terms of a smaller unit.

4.MD.1.cRecord measurement equivalents in a two-column table.

4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects and money. (MP.1, MP.4)

4.MD.2.aSolve measurement problems involving whole number, simple fractions or decimals.

4.MD.2.bSolve problems that require converting a given measurement from a larger unit to a smaller unit within a common measurement system, such as 2 km=2,000 m.

4.MD.2.cVisually display measurement quantities using representations such as number lines that feature a measurement scale.

4.MD.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems. (MP.1, MP.3)

4.MD.4 Use dot plots to analyze data to a statistical question. (MP.1, MP.6)

4.MD.4.aIdentify a statistical question focused on numerical data.

4.MD.4.bMake a dot plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).

4.MD.4.cSolve problems involving addition and subtraction of fractions by using information presented in dot plots.

4.MD.5Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint and understand concepts of angle measurement. (MP.7)

4.MD.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. (MP.5, MP.6)

4.MD.7Recognize angle measure as additive. When an angle is 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. (MP.1, MP.4)

4.NBT Numbers and Operations in Base Ten

4.NBT.1Recognize in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. (MP.7)

4.NBT.2 Represent and compare multi-digit whole numbers. (MP.2, MP.7)

4.NBT.2.aRead and write multi-digit whole numbers using base-ten numerals, number names and expanded form.

4.NBT.2.bCompare two multi-digit numbers based on meanings of the digit in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.3Use place value understanding to round multi-digit whole numbers to any place. (MP.2, MP.6)

4.NBT.4Fluently add and subtract multi-digit whole numbers using an algorithm. (MP.2, MP.8)

4.NBT.5 Multiply whole numbers:(MP.3, MP.4, MP.8)

4.NBT.5.aUp to four digit number by a one-digit number – Multiply 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.5.bTwo-digit number by two-digit number – Multiply 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.6 Divide up to four-digit dividends by one-digit divisors. Find whole number quotients and remainders using: (MP.3, MP.7, MP.8)

4.NBT.6.astrategies based on place value – Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

4.NBT.6.bthe properties of operations – Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

4.NBT.6.cthe relationship between multiplication and division – Illustrate and explain the calculation by using equations, rectangular arrays and/or area models.

4.NF Numbers and Operations—Fractions

4.NF.1 Understand and generate equivalent fractions. (MP.4, MP.7, MP.8)

4.NF.1.aUse visual fraction models to recognize and generate equivalent fractions that have different numerators/denominators even though they are the same size.

4.NF.1.bExplain why a fraction a/b is equivalent to a fraction (n×a)/(n×b).

4.NF.2Compare two fractions with different numerators and different denominators using the symbols <, =, or >. Recognize comparisons are valid only when the two fractions refer to the same whole. Justify the conclusions. (MP.2, MP.3)

4.NF.3 Understand a fraction a/b with a>1 as a sum of fractions 1/b. (MP.1, MP.5, MP.7)

4.NF.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.

4.NF.3.bDecomposing a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions.

4.NF.3.cAdd and subtract mixed numbers with like denominators.

4.NF.3.dSolve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.

4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (MP.5, MP.8)

4.NF.4.aUnderstand a fraction a/b as a multiple of 1/b.

4.NF.4.bUnderstand a multiple of a/b as a multiple of 1/b and use this understanding to multiply a fraction by a whole number.

4.NF.4.cSolve word problems involving multiplication of a fraction by a whole number.

4.NF.5 Convert and add fractions with denominators of 10 and 100. (MP.5, MP.7)

4.NF.5.aConvert a fraction with a denominator of 10 to an equivalent fraction with a denominator of 100.

4.NF.5.bAdd two fractions with respective denominators 10 and 100.

4.NF.6Use decimal notation for fractions with denominators 10 or 100. (MP.4, MP.7)

4.NF.7 Compare two decimals to hundredths. (MP.2, MP.3, MP.5)

4.NF.7.aCompare two decimals to hundredths by reasoning about their size.

4.NF.7.bRecognize that comparisons are valid only when the two decimals refer to the same whole.

4.NF.7.cRecord the results of comparisons with the symbols >, =, or < and justify the conclusions.

4.OA Operations and Algebraic Thinking

4.OA.1Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations. (MP.2, MP.4)

4.OA.2Multiply or divide to solve word problems involving multiplicative comparisons by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (MP.1, MP.2, MP.3)

4.OA.3 Solve multistep problems. (MP.1, MP.4)

4.OA.3.aPerform operations in the conventional order when there are no parentheses to specify a particular order.

4.OA.3.bSolve multistep word 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. Assess the reasonableness of answers using mental computations and estimation strategies including rounding.

4.OA.4 Find factors and multiples of numbers in the range 1-100. (MP.5, MP.7)

4.OA.4.aFind all factor pairs for a given whole number.

4.OA.4.bRecognize that a whole number is a multiple of each of its factors.

4.OA.4.cDetermine whether a given whole number is a multiple of a given one-digit number.

4.OA.4.dDetermine whether a given whole number is prime or composite.

4.OA.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern not explicit in the rule itself. (MP.2, MP.3)

Grade 5

5.G.1Use a pair perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis and the second number indicates how far to travel in the direction of the second. (MP.4, MP.7)

5.G.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. (MP.1, MP.6)

5.G.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. (MP.3, MP.6)

5.G.4Classify two-dimensional figures in a hierarchy based on properties. (MP.1, MP.7)

5.MD Measurement and Data

5.MD.1Convert among different size measurement units (mass, weight, liquid volume, length, time) within one system of units (metric system, U.S. standard system and time). (MP.3, MP.8)

5.MD.2Identify and gather data for statistical questions focused on both categorical and numerical data. Select an appropriate data display (bar graph, pictograph, dot plot). Make observations from the graph about the questions posed. (MP.4, MP.5, MP.6)

5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. (MP.6)

5.MD.3.aA 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.3.bA 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.4Measure volumes by counting unit cubic cm, cubic in, cubic ft. and improvised units. (MP.5, MP.6)

5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. (MP.1, MP.4, MP.8)

5.MD.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 threefold whole-number products as volumes.

5.MD.5.bApply the formulas V=l×w×h and V=B×h for rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

5.MD.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 Numbers and Operations in Base Ten

5.NBT.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. (MP.2, MP.7)

5.NBT.2 Multiply and divide by powers of 10. (MP.3, MP.8)

5.NBT.2.aExplain patterns in the number of zeros of the product when multiplying a number by powers of 10.

5.NBT.2.bExplain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

5.NBT.2.cUse whole-number exponents to denote powers of 10.

5.NBT.3 Read, write and compare decimals to thousandths. (MP.2, MP.5, MP.7)

5.NBT.3.aRead and write decimals to thousandths using base-ten numerals, number names and expanded form.

5.NBT.3.bCompare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

5.NBT.4Use place value understanding to round decimals to any place. (MP.5, MP.7)

5.NBT.5Fluently multiply multi-digit whole numbers (not to exceed four-digit by two-digit multiplication) using an algorithm. (MP.7, MP.8)

5.NBT.6 Divide up to four-digit dividends by two-digit divisors. (MP.2, MP.3, MP.4)

5.NBT.6.a Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors using…

5.NBT.6.a.1strategies based on place value

5.NBT.6.a.2the properties of operations

5.NBT.6.a.3the relationship between multiplication and division

5.NBT.6.bIllustrate and explain the calculation by using equations, rectangular arrays and/or area models.

5.NBT.7 Operations with decimals to hundredths. (MP.2, MP.3, MP.5)

5.NBT.7.a Add, subtract, multiply and divide decimals to hundredths using…

5.NBT.7.a.1concrete models or drawings

5.NBT.7.a.2strategies based on place value

5.NBT.7.a.3properties of operations

5.NBT.7.a.4the relationship between addition and subtraction

5.NBT.7.bRelate the strategy to a written method and explain the reasoning used.

5.NF Numbers and Operations—Fractions

5.NF.1 Efficiently add and subtract fractions with unlike denominators (including mixed numbers) by… (MP.2, MP.3)

5.NF.1.ausing reasoning strategies, such as counting up on a number line or creating visual fraction models

5.NF.1.bfinding common denominators

5.NF.2 Solve word problems involving addition and subtraction of fractions. (MP.1, MP.4)

5.NF.2.aSolve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators.

5.NF.2.bUse benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

5.NF.3Interpret a fraction as division of the numerator by the denominator (a/b=a÷b). Solve words 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. (MP.4, MP.8)

5.NF.4 Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction. (MP.1)

5.NF.4.aInterpret the product (a/b)×q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a×q÷b.

5.NF.4.bFind the area of a rectangle with fractional side lengths by tiling it with 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.5 Interpret multiplication as scaling (resizing), by: (MP.2, MP.6)

5.NF.5.aComparing 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.

5.NF.5.bExplaining 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); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b=(n×a)/(n×b) to the effect of multiplying a/b by 1.

5.NF.6Solve real world problems involving multiplication of fractions and mixed numbers. (MP.4, MP.5)

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (MP.1, MP.4, MP.8)

5.NF.7.aInterpret division of a unit fraction by a non-zero whole number and compute such quotients.

5.NF.7.bInterpret division of a whole number by a unit fraction and compute such quotients.

5.NF.7.cSolve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.

5.OA Operations and Algebraic Thinking

5.OA.1Use parentheses, brackets or braces in numerical expressions and evaluate expressions that include symbols. (MP.1, MP.3)

5.OA.2Write simple expressions with numbers and interpret numerical expressions without evaluating them. (MP.2, MP.7)

5.OA.3 Generate numerical patterns for situations. (MP.2, MP.4)

5.OA.3.aGenerate a rule for growing patterns, identifying the relationship between corresponding terms (x, y).

5.OA.3.bGenerate patterns using one or two given rules (x, y).

5.OA.3.cUse tables, ordered pairs and graphs to represent the relationship between the quantities.

Grade 6

6.EE Expressions and Equations

6.EE.1Write and evaluate numerical expressions involving whole-number exponents. (MP.2, MP.6)

6.EE.2 Write, read and evaluate expressions in which letters stand for numbers. (MP.1, MP.3, MP.4)

6.EE.2.aWrite expressions that record operations with numbers and with letters standing for numbers.

6.EE.2.bIdentify parts of an expression using mathematical terms (sums, term, product, factor, quotient, coefficient); view one or more parts of an expression in a single entity.

6.EE.2.cEvaluate expressions for specific values of their variables, including values that are non-negative rational numbers. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

6.EE.3Apply the properties of operations to generate equivalent expressions. (MP.7, MP.8)

6.EE.4Identify when two expressions are equivalent when the two expressions name the same number regardless of which value is substituted into them. (MP.2, MP.3, MP.7)

6.EE.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (MP.1, MP.2, MP.7)

6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or depending on the purpose at hand, any number in a specified set. (MP.2, MP.6)

6.EE.7Solve real-world and mathematical problems by writing and solving equations of the form x+p=q and px=q for cases in which p, q and x are all nonnegative rational numbers. (MP.1, MP.2, MP.3, MP.4)

6.EE.8Write an inequality of the form x>c, x<cm x≥c, or x≤c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of these forms have infinitely many solutions; represent solutions of such inequalities on vertical and horizontal number lines. (MP.3, MP.7)

6.EE.9 Use variables to represent two quantities in a real-world problem that changes in relationship to one another. (MP.3, MP.4, MP.7)

6.EE.9.aAppropriately recognize one quantity as the dependent variable and the other as the independent variable.

6.EE.9.bWrite an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable.

6.EE.9.cAnalyze the relationship between the dependent and independent variables using graphs and tables and relate these to the question.

6.G.1Find the area of right triangles, other triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and quadrilaterals; apply these techniques in the context of solving real-world and mathematical problems. (MP.1, MP.6, MP.8)

6.G.2Find the volume of a right rectangular prism with rational number edge lengths. Apply the formulas V=lwh and V=Bh to find volumes of right rectangular prisms with rational number edge lengths in the context of solving real-world and mathematical problems. (MP.2, MP.5, MP.6)

6.G.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. (MP.4, MP.5, MP.6)

6.G.4Classify three-dimensional figures including cubes, prisms, pyramids, cones and spheres. (MP.2, MP.3)

6.NS The Number System

6.NS.1Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions. (MP.1, MP.2, MP.3)

6.NS.2 Fluently divide multi-digit numbers using an algorithm. (MP.7, MP.8)

6.NS.2.aConvert a rational number to a decimal using long division.

6.NS.2.bKnow that the decimal form of a rational number terminates in 0s or eventually repeats.

6.NS.3Fluently add, subtract, multiply and divide multi-digit decimals using an algorithm for each operation. (MP.2, MP.6)

6.NS.4Use 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. (MP.8)

6.NS.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values; use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. (MP.1, MP.2, MP.4)

6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes, using appropriate range and intervals, to represent points on the line and in the plane, that include negative numbers and coordinates. (MP.2, MP.4)

6.NS.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize 0 is its own opposite and the opposite of a negative number is a positive, and the opposite of a negative number is a positive, such as−(−3)=3.

6.NS.6.bFind 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.6.cUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize the similarity between whole numbers, their negative opposites and their positions on a number line, ordered pairs differ only by signs and their locations on one or both axes.

6.NS.7 Understand ordering and absolute value of rational numbers. (MP.1, MP.2, MP.4)

6.NS.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

6.NS.7.bWrite, interpret and explain statements of order for rational numbers in real-world contexts.

6.NS.7.cUnderstand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

6.NS.7.dDistinguish comparisons of absolute value from statements about order.

6.NS.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. (MP.5, MP.7)

6.RP Ratios and Proportional Relationships

6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (MP.2, MP.6)

6.RP.2Understand the concept of a unit rate a/b associated with a ratio a:b with B≠0 and use rate language in the context of a ratio relationship. (MP.2, MP.6)

6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems. (MP.1, MP.4, MP.7)

6.RP.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.3.bSolve rate problems including those involving unit pricing and constant speed.

6.RP.3.cUse ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

6.SP Statistics and Probability

6.SP.0 Apply the four-step investigative process for statistical reasoning. (MP.1, MP.4)

6.SP.0.aFormulate Questions: Formulate a statistical question as one that anticipates variability and can be answered with data.

6.SP.0.bCollect Data: Design and use a plan to collect appropriate data to answer a statistical question.

6.SP.0.cAnalyze Data: Select appropriate graphical methods and numerical measures to analyze data by displaying variability within a group, comparing individual to individual and comparing individual to group.

6.SP.0.dInterpret Results: Draw logical conclusions and make generalizations from the data based on the original question.

6.SP.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. (MP.1, MP.3, MP.6)

6.SP.2Understand that a set of numerical data collected to answer a statistical question has a distribution which can be described by its center, spread and overall shape. (MP.2, MP.6, MP.7)

6.SP.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number to describe a typical value, while a measure of variation describes how the values in the distribution vary. (MP.2., MP.5, MP.6)

6.SP.4Display the distribution of numerical data in plots on a number line, including dot plots, histograms and box plots. (MP.6, MP.7)

6.SP.5 Summarize numerical data sets in relation to their context, such as by: (MP.3, MP.7)

6.SP.5.aReporting the number of observations.

6.SP.5.bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.

6.SP.5.cDetermining quantitative measures of center (median and/or mean) to describe distribution of numerical data.

6.SP.5.dDescribing distributions of numerical data qualitatively relating to shape (using terms such as cluster, mode(s), gap, symmetric, uniform, skewed-left, skewed-right and the presence of outliers) and quantitatively relating to spread/variability (using terms such as range and interquartile range).

6.SP.5.eRelating the choice of measures of center and variability to the shape of the data distribution.

Grade 7

7.EE Expressions and Equations

7.EE.1Apply properties of operations as strategies to add, subtract, factor and expand linear expressions with rational coefficients. (MP.2, MP.3)

7.EE.2Understand that rewriting an expression in different forms in a problem context can clarify the problem and how the quantities in it are related. (MP.7, MP.8)

7.EE.3Solve real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. (MP.1, MP.4, MP.6)

7.EE.4 Use variables to represent quantities in a real-world or mathematical problem and construct equations and inequalities to solve problems by reasoning about the quantities. (MP.2, MP.4)

7.EE.4.aSolve word problems leading to equations of the form px+q=r and p(x+q)=r, where p, q and r are specific rational numbers. Solve equations of these forms. Graph the solution set of the equality and interpret it in context of the problem.

7.EE.4.bSolve word problems leading to inequalities of the form px+q>r, px+q<r, px+q≥r, px+q≤r; where p, q and r are specific rational numbers. Graph the solution set of the inequality and interpret it in context of the problem.

7.G.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. (MP.1, MP.2, MP.5)

7.G.2Draw (freehand, with ruler and protractor and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. (MP.6, MP.7)

7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. (MP.5, MP.6)

7.G.4 Use formulas for area and circumference of circles and their relationships. (MP.1, MP.2, MP.8)

7.G.4.aApply the formulas for the area and circumference of a circle to solve real-world and mathematical problems.

7.G.4.bExplore and understand the relationship between the radius, diameter, circumference and area of a circle.

7.G.5Apply properties of supplementary, complementary, vertical and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (MP.3, MP.6, MP.7)

7.G.6 Solve problems involving area of two-dimensional objects and surface area and volume of three-dimensional objects. (MP.3, MP.4, MP.5)

7.G.6.aSolve real-world and mathematical problems involving area of two-dimensional objects composed of triangles, quadrilaterals and other polygons.

7.G.6.bSolve real-world and mathematical problems involving volume and surface area, using nets as needed, of three-dimensional objects including cubes, pyramids and right prisms.

7.NS The Number System

7.NS.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. (MP.2, MP.4, MP.7)

7.NS.1.aDescribe situations in which opposite quantities combine to make 0.

7.NS.1.bUnderstand 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.1.cUnderstand 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.1.dApply properties of operations as strategies to add and subtract rational numbers.

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. (MP.2, MP.7, MP.8)

7.NS.2.aUnderstand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1)=1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

7.NS.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.2.cApply properties of operations as strategies to multiply and divide rational numbers.

7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers. (MP.1, MP.2, MP.5)

7.RP Ratios and Proportional Relationships

7.RP.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. (MP.2, MP.6)

7.RP.2 Recognize and represent proportional relationships between quantities. (MP.1, MP.2, MP.3)

7.RP.2.aDecide whether two quantities represent a proportional relationship.

7.RP.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships.

7.RP.2.cRepresent proportional relationships by equations.

7.RP.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.3 Use percents to solve mathematical and real-world problems. (MP.5, MP.6)

7.RP.3.aFind a percent of a quantity as a rate per 100; solve problems involving finding the whole, a part and a percent, given two of these.

7.RP.3.bUse proportional relationships to solve multistep ratio and percent problems.

7.SP Statistics and Probability

7.SP.0Create displays, including circle graphs (pie charts), scaled pictographs and bar graphs, to compare and analyze distributions of categorical data from both matching and different-sized samples. (MP.2, MP.3, MP.6)

7.SP.1Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. (MP.3, MP.6)

7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. (MP.2, MP.3, MP.7)

7.SP.2.aGenerate multiple samples of categorical data of the same size to gauge the variation in estimates or predictions.

7.SP.2.bGenerate multiple samples (or simulated samples) of numerical data to gauge the variation in estimates or predictions.

7.SP.2.cGauge how far off an estimate or prediction might be related to a population character of interest.

7.SP.3Describe the degree of visual overlap (and separation) from the graphical representations of two numerical data distributions (box plots, dot plots) with similar variabilities with similar contexts (same variable), measuring the difference between the centers (medians or means) by expressing this difference as a multiple of a measure of variability (interquartile range when comparing medians or the mean absolute deviation when comparing means). (MP.1, MP.5, MP.7)

7.SP.4Calculate and use measures of center (mean and median) and measures of variability (interquartile range when comparing medians and mean absolute deviation when comparing means) for numerical data from random samples to draw informal comparative inferences about two populations. (MP.2, MP.5, MP.7)

7.SP.5Describe the probability of a chance event is a number between 0 and 1, which tells how likely the event is, from impossible (0) to certain (1). A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely and a probability near 1 indicates a likely event. (MP.5, MP.6, MP.7)

7.SP.6Approximate 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. (MP.1, MP.2)

7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. (MP.4, MP.7, MP.8)

7.SP.7.aDevelop a uniform probability model by assigning equal probability to all outcomes and use the model to determine probabilities of events.

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

7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams and simulation. (MP.2, MP.4, MP.7)

7.SP.8.aExplain 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.

7.SP.8.bRepresent sample spaces for compound events described in everyday language using methods such as organized lists, tables and tree diagrams.

7.SP.8.cDesign and use a simulation to generate frequencies for compound events.

Grade 8

8.EE Expressions and Equations

8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions. (MP.3, MP.7, MP.8)

8.EE.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. Know that perfect squares and perfect cubes are rational. (MP.5, MP.6)

8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 (Scientific Notation) to estimate very large or very small quantities and express how many times larger or smaller one is than the other. (MP.3, MP.5, MP.6)

8.EE.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology. (MP.2, MP.5, MP.6)

8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. (MP.2, MP.3, MP.4)

8.EE.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 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. (MP.3, MP.4, MP.7)

8.EE.7 Solve linear equations in one variable. (MP.2, MP.3, MP.7)

8.EE.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.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.8 Analyze and solve a system of two linear equations. (MP.1, MP.3, MP.4)

8.EE.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; understand that a system of two linear equations may have one solution, no solution, or infinitely many solutions.

8.EE.8.bSolve systems of two linear equations in two variables algebraically by using substitution where at least one equation contains at least one variable whose coefficient is 1 and by inspection for simple cases.

8.EE.8.cSolve real-world and mathematical problems leading to two linear equations in two variables.

8.F.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. (MP.7, MP.8)

8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (MP.1, MP.2, MP.4)

8.F.3 Understand properties of linear functions. (MP.7)

8.F.3.aInterpret the equation y=mx+b as defining a linear function, whose graph is a straight line.

8.F.3.bIdentify and give examples of functions that are not linear.

8.F.4 Construct a function to model a linear relationship between two quantities. (MP.4, MP.5, MP.8)

8.F.4.aDetermine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph.

8.F.4.bInterpret 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.5 Use graphs to represent functions. (MP.3, MP.7)

8.F.5.aDescribe qualitatively the functional relationship between two quantities by analyzing a graph.

8.F.5.bSketch a graph that exhibits the qualitative features of a function that has been described verbally.

8.G.1 Verify experimentally the properties of rotations, reflections and translations: (MP.5, MP.6)

8.G.1.aLines are congruent to lines.

8.G.1.bLine segments are congruent to line segments of the same length.

8.G.1.cAngles are congruent to angles of the same measure.

8.G.1.dParallel lines are congruent to parallel lines.

8.G.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections and translations. Given two congruent figures, describe a sequence that exhibits the congruence between them. (MP.2, MP.7)

8.G.3Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. (MP.3, MP.5, MP.6)

8.G.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations and dilations. Given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. (MP.2, MP.5, MP.7)

8.G.5Use 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. (MP.3)

8.G.6Explain a proof of the Pythagorean Theorem and its converse. (MP.3, MP.7)

8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. (MP.1, MP.2, MP.4)

8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (MP.5, MP.6)

8.G.9Apply the formulas for the volumes and surface areas of cones, cylinders and spheres and use them to solve real-world and mathematical problems. (MP.1, MP.7, MP.8)

8.NS The Number System

8.NS.1Understand informally that every number has a decimal expansion; the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat. Know that other numbers are called irrational. (MP.2, MP.6, MP.7)

8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram and estimate the value of expressions. (MP.2, MP.7, MP.8)

8.SP Statistics and Probability

8.SP.1Construct and interpret scatter plots for bivariate numerical data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association and nonlinear association. (MP.2, MP.7)

8.SP.2Know that lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a line and informally assess the model fit by judging the closeness of the data points to the line. (MP.2)

8.SP.3Use the equation of a linear model to solve problems in the context of bivariate numerical data, interpreting the slope and intercept. (MP.2, MP.4)

HS

HS.A Conceptual Category Algebra

HS.A.1 Interpret expressions that represent a quantity in terms of its context. (MP.2, MP.6)

HS.A.1.aInterpret parts of an expression, such as terms, factors and coefficients.

HS.A.1.bInterpret complicated expressions, given a context, by viewing one or more of their parts as a single entity.

HS.A.10(+)Rewrite simple rational expressions in different forms. (MP.7, MP.8)

HS.A.11(+)Add, subtract, multiply and divide rational algebraic expressions. (MP.2, MP.3)

HS.A.12Create equations and inequalities in one variable and use them to solve problems. (MP.1, MP.4)

HS.A.13Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (MP.2, MP.5)

HS.A.14Create a system of equations or inequalities to represent constraints within a modeling context. Interpret the solution(s) to the corresponding system as viable or nonviable options within the context. (MP.4, MP.5)

HS.A.15Rearrange formulas to solve a literal equation, highlighting a quantity of interest, using the same reasoning as in solving equations. (MP.2, MP.7)

HS.A.16Understand each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. (MP.1, MP.3)

HS.A.17 Solve and justify equations in one variable. Justify the solutions and give examples showing how extraneous solutions may arise. (MP.3, MP.5, MP.7)

HS.A.17.aSolve rational equations written as proportions in one variable.

HS.A.17.bSolve radical equations in one variable.

HS.A.18Solve linear equations and inequalities in one variable, including literal equations with coefficients represented by letters. (MP.2, MP.7)

HS.A.19 Solve quadratic equations in one variable. (MP.1, MP.8)

HS.A.19.aSolve quadratic equations by taking square roots, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a±bi for real numbers a and b.

HS.A.19.b(+) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2=q that has the same solutions. Derive the quadratic formula from this form.

HS.A.19.c(+) Solve quadratic equations by completing the square.

HS.A.2Use the structure of an expression to identify ways to rewrite it and consistently look for opportunities to rewrite expressions in equivalent forms. (MP.7, MP.8)

HS.A.20 Solve systems of linear equations in two variables. (MP.3, MP.6)

HS.A.20.aUnderstand a system of two equations in two variables has the same solution as a new system formed by replacing one of the original equations with an equivalent equation.

HS.A.20.bSolve systems of linear equations with graphs, substitution and elimination, focusing on pairs of linear equations in two variables.

HS.A.21Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (MP.3, MP.6)

HS.A.22(+) Use matrices to solve a system of equations. (MP.4, MP.7)

HS.A.22(+).aRepresent a system of linear equations as a single matrix equation in a vector variable.

HS.A.22(+).bFind the inverse of a matrix if it exists.

HS.A.22(+).cUse matrices to solve systems of linear equations (using technology for matrices of dimension 3×3 or greater).

HS.A.23Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. (MP.1, MP.4)

HS.A.24Justify that the solutions of the equations f(x)=g(x) are the x-coordinates of the points where the graphs of y=f(x) and y=g(x) intersect. Find the approximate solutions graphically, using technology or tables. (MP.3, MP.5)

HS.A.25 Graph linear inequalities in two variables. (MP.5, MP.6)

HS.A.25.aGraph the solutions to a linear inequality as a half-plane (excluding the boundary in the case of a strict inequality).

HS.A.25.bGraph the solution set to a system of linear inequalities as the intersection of the corresponding half-planes.

HS.A.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (MP.5, MP.7)

HS.A.3.aWrite the standard form of a given polynomial and identify the terms, coefficients, degree, leading coefficient and constant term.

HS.A.3.bFactor a quadratic expression to reveal the zeros of the function it defines.

HS.A.3.cUse the properties of exponents to rewrite exponential expressions.

HS.A.3.d(+) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

HS.A.4(+)Derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and use the formula to solve problems. (MP.1, MP.4)

HS.A.5Add, subtract and multiply polynomials. (MP.7, MP.8)

HS.A.6(+)Know and apply the Remainder Theorem. (MP.1, MP.8)

HS.A.7Identify roots of polynomials when suitable factorizations are available. Know these roots become the zeros (x-intercepts) for the corresponding polynomial function. (MP.2, MP.5, MP.7)

HS.A.8(+)Prove polynomial identities and use them to describe numerical relationships. (MP.2, MP.3, MP.6)

HS.A.9(+)Know 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. (MP.7, MP.8)

HS.C Calculus(+)

HS.C.1(+) Understand limits. (MP.2, MP.3)

HS.C.1(+).aApply limits to a variety of functions, including piecewise functions.

HS.C.1(+).b(+)(+) Prove that the limit of a function exists, based upon the definition of a limit.

HS.C.10(+)Communicate an understanding of continuity using precise mathematical symbols and language. (MP.2, MP.6)

HS.C.11(+) Define derivatives. (MP.5, MP.8)

HS.C.11(+).aDefine the derivative of a function as the limit of the difference quotient.

HS.C.11(+).bUnderstand this limit of the difference quotient can be interpreted as an instantaneous rate of change or the slope of a tangent line.

HS.C.12(+)Use average rate of change to estimate the derivative from a table of values or a graph. (MP.2, MP.8)

HS.C.13(+)Understand the derivative as a function. (MP.2, MP.5)

HS.C.14(+)Apply the definition of derivative to find derivative values and derivative functions. (MP.2, MP.3)

HS.C.15(+)Explain why differentiability implies continuity yet continuity does not imply differentiability. (MP.3, MP.6)

HS.C.16(+)Understand and apply the Mean Value Theorem, including numerical, graphical and algebraic representations. (MP.2, MP.5)

HS.C.17(+)Understand the relationship between the concavity of a function and the sign of the second derivative. (MP.2, MP.3)

HS.C.18(++)Understand Rolle’s Theorem as a special case of the Mean Value Theorem. (MP.2, MP.3)

HS.C.19(+)Efficiently find derivatives of functions with and without technology. (MP.2, MP.5)

HS.C.2(+)Demonstrate an understanding of limits by estimating and finding the limit of a function at a point graphically, numerically and algebraically. (MP.5, MP.8)

HS.C.20(+)Understand and use derivative rules for sums, differences, products and quotients of two functions and calculate the derivative of a composite function using the chain rule. (MP.2, MP.3)

HS.C.21(+)Use implicit differentiation to find a derivative in an equation of two variables. (MP.1, MP.2)

HS.C.22(+)Use implicit differentiation to find the derivative of the inverse of a function. (MP.2, MP.3)

HS.C.23(+)Understand the relationship between the increasing and decreasing behavior of a function and the sign of the first derivative of the function. (MP.1, MP.2)

HS.C.24(+)Use the first derivative to analyze curves and identify relative extrema. (MP.2, MP.3)

HS.C.25(+)Understand the relationship of concavity to the second derivative. (MP.2, MP.5)

HS.C.26(+)Use the second derivative to find points of inflection. (MP.2, MP.3)

HS.C.27(+)Use the second derivative to analytically locate intervals on which a function is concave up, concave down or neither. (MP.2, MP.3)

HS.C.28(+)Describe how graphical characteristics of a given function, the first derivative of that function and the second derivative of that function interrelate. (MP.2, MP.5)

HS.C.29(+)Use derivatives to express rate of change in a variety of contexts. (MP.2, MP.4)

HS.C.3(+)Apply properties and theorems of limits, including limits of indeterminate forms. (MP.2, MP.3)

HS.C.30(+)Use derivatives to solve a variety of problems including related rates, optimization, tangent line approximations and growth and decay models. (MP.1, MP.4)

HS.C.31(+)Use differentiation to solve problems involving velocity, speed and acceleration. (MP.1, MP.2)

HS.C.32(+) Understand and apply differential equations. (MP.1, MP.4)

HS.C.32(+).aVerify solutions to differential equations and use them to model real-world problems with and without technology.

HS.C.32(+).bSolve separable differential equations and use them in modeling real-world problems with and without technology.

HS.C.33(+)Understand the definite integral of a function over an interval. Interpret a definite integral as a limit of Riemann Sums and as net accumulation of change. (MP.2, MP.5)

HS.C.34(+)Write a Riemann sum that represents the definition of a definite integral. (MP.2, MP.3)

HS.C.35(+)Calculate the values of Riemann Sums over equal subdivisions to approximate definite integrals of functions represented graphically and numerically (using tables). Use left-hand sums, right-hand sums, midpoint sums and trapezoidal sums. (MP.2, MP.3)

HS.C.36(+)Recognize differentiation and integration as inverse operations. (MP.2, MP.8)

HS.C.37(+)Understand how the Fundamental Theorem of Calculus connects differentiation and integration and use it to evaluate definite and indefinite integrals and to represent particular antiderivatives. (MP.2, MP.3)

HS.C.38(+)Perform analytical and graphical analysis of functions using the Fundamental Theorem of Calculus. (MP.2, MP.5)

HS.C.39(+)Understand and use the definite integral of a function over an interval and understand its use as a mathematical tool. (MP.1, MP.2)

HS.C.4(+)Communicate understanding of limits using precise mathematical symbols and language. (MP.3, MP.6)

HS.C.40(+)Find antiderivatives of a variety of basic functions including power, exponential, logarithmic and trigonometric and apply basic properties of definite integrals. (MP.2, MP.7)

HS.C.41(+)Use substitution techniques and change of limits of integration to find antiderivatives. (MP.2, MP.3)

HS.C.42(+)Find particular antiderivatives given initial conditions. (MP.1, MP.2)

HS.C.43(+)Model, solve and interpret applications of antiderivatives including finding area, velocity, acceleration and volume of a solid. (MP.1, MP.4)

HS.C.44(+)Apply integration to solve problems including particle motion and exponential growth and decay. (MP.1, MP.4)

HS.C.45(+)Describe the application of integration to a variety of problems using precise mathematical language and notation. (MP.4, MP.6)

HS.C.5(+)Describe asymptotic behavior (analytically and graphically) in terms of infinite limits and limits at infinity. (MP.2, MP.5)

HS.C.6(+)Discuss the end behavior of functions; identify representative functions for each type of end behavior using precise mathematical symbols and language. (MP.2, MP.6)

HS.C.7(+)Understand and use the limit definition of continuity to determine whether a given function is continuous at a specific point. (MP.2, MP.3)

HS.C.8(+)Define and identify different types of discontinuity—removable (hole) or non-removable (jump, asymptote)—in terms of limits. (MP.3, MP.6)

HS.C.9(+) Understand and apply continuous function theorems. (MP.2, MP.3)

HS.C.9(+).aApply the Intermediate Value Theorem to continuous functions.

HS.C.9(+).bApply the Extreme Value Theorem to continuous functions.

HS.CaCalculus—Limits

HS.CbCalculus—Function Behavior

HS.CcCalculus—Continuity

HS.CdCalculus—Understanding the Derivative

HS.CeCalculus—Applications of Derivatives

HS.CfCalculus—Understanding Integration

HS.CgCalculus—Applications of Integration

HS.F Conceptual Category Functions

HS.F.1 Understand properties and key features of functions and the different ways functions can be represented. (MP.2, MP.4, MP.7)

HS.F.1.aUnderstand 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.

HS.F.1.bUsing appropriate function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.

HS.F.1.cFor 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.

HS.F.1.dRelate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

HS.F.1.eCompare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

HS.F.10Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents with the use of technology. (MP.1, MP.7)

HS.F.11 Distinguish between situations that can be modeled with linear functions and with exponential functions. (MP.3, MP.8)

HS.F.11.aRecognize and justify that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

HS.F.11.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.

HS.F.11.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

HS.F.12Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (MP.7, MP.8)

HS.F.13Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (MP.7, MP.8)

HS.F.14Interpret the parameters in a linear or exponential function in terms of a context. (MP.1, MP.2)

HS.F.15(+)Understand the relationship of radian measure of an angle to its arc length. (MP.1, MP.6)

HS.F.16(+) Understand and use the unit circle. (MP.7, MP.8)

HS.F.16(+).aExplain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

HS.F.16(+).bUse special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6 and 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.

HS.F.16(+).cUse the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

HS.F.17(+)Choose trigonometric functions to model periodic phenomena with specified period, midline and amplitude. (MP.4, MP.5, MP.6)

HS.F.18(+)Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (MP.2, MP.3)

HS.F.19(+)Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology and interpret them in terms of the context. (MP.4, MP.5)

HS.F.2Recognize that arithmetic and geometric sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (MP.7, MP.8)

HS.F.20(+) Proving identities and formulas within the context of trigonometry. (MP.3, MP.7)

HS.F.20(+).aProve the Pythagorean identity and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

HS.F.20(+).bProve the addition and subtraction formulas for sine, cosine and tangent and use them to solve problems.

HS.F.3 Understand average rate of change of a function over an interval. (MP.2, MP.4)

HS.F.3.aCalculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval.

HS.F.3.bEstimate the rate of change from a graph.

HS.F.4 Graph functions expressed symbolically and show key features of the graph, with and without using technology (computer, graphing calculator). (MP.4, MP.5)

HS.F.4.aGraph linear and quadratic functions and show intercepts, maxima and minima.

HS.F.4.bGraph square root, cube root and absolute value functions.

HS.F.4.cGraph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior.

HS.F.4.dGraph exponential and logarithmic functions, showing intercepts and end behavior.

HS.F.4.e(+) Graph trigonometric functions, showing period, midline and amplitude.

HS.F.4.f(+) Graph piecewise functions, including step functions.

HS.F.4.g(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.

HS.F.5 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (MP.3, MP.6)

HS.F.5.aIdentify zeros, extreme values and symmetry of the graph within the context of a quadratic function.

HS.F.5.bUse the properties of exponents to interpret expressions for exponential functions and classify the exponential function as representing growth or decay.

HS.F.6 Write a function that describes a relationship between two quantities. (MP.4, MP.7)

HS.F.6.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.

HS.F.6.bCombine standard function types using arithmetic operations.

HS.F.6.c(+) Compose functions.

HS.F.7 Use arithmetic and geometric sequences to model situations and scenarios. (MP.4, MP.8)

HS.F.7.aUse formulas (explicit and recursive) to generate terms for arithmetic and geometric sequences.

HS.F.7.bWrite formulas to model arithmetic and geometric sequences and apply those formulas in realistic situations.

HS.F.7.c(+) Translate between recursive and explicit formulas.

HS.F.8 Understand the effects of transformations on the graph of a function. (MP.3, MP.5)

HS.F.8.aIdentify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx) and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs.

HS.F.8.bExperiment with cases and illustrate an explanation of the effects on the graph using technology.

HS.F.9 Find inverse functions. (MP.2, MP.6)

HS.F.9.aGiven the equation of an invertible function, find the inverse.

HS.F.9.b(+) Verify by composition that one function is the inverse of another.

HS.F.9.c(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

HS.F.9.d(+) Produce an invertible function from a non-invertible function by restricting the domain.

HS.FaFunctions—Interpreting Functions

HS.FbFunctions—Building Functions

HS.FcFunctions—Linear, Quadratic and Exponential Functions

HS.FdFunctions—Trigonometric Functions

HS.G Conceptual Category Geometry

HS.G.1 Know and apply precise definitions of the language of Geometry: (MP.3, MP.6)

HS.G.1.aUnderstand properties of line segments, angles and circle.

HS.G.1.bUnderstand properties of and differences between perpendicular and parallel lines.

HS.G.10Apply the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (MP.3, MP.6)

HS.G.11 Understand theorems about triangles. (MP.1, MP.3)

HS.G.11.aApply theorems about triangles.

HS.G.11.b(+) Prove theorems about triangles.

HS.G.11.cUse similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

HS.G.12 Understand properties of right triangles. (MP.3, MP.4)

HS.G.12.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 (sine, cosine and tangent).

HS.G.12.bExplain and use the relationship between the sine and cosine of complementary angles.

HS.G.12.cUse trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

HS.G.13(+)Derive the formula A=1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (MP.6, MP.7)

HS.G.14(+) Understand and apply the Law of Sines and the Law of Cosines. (MP.1, MP.3)

HS.G.14(+).aUse the Law of Sines and Cosines to find unknown measurements in right and non-right triangles.

HS.G.14(+).bProve the Laws of Sines and Cosines and use them to solve problems.

HS.G.15Verify using dilations that all circles are similar. (MP.5, MP.8)

HS.G.16 Identify and describe relationships among angles and segments within the context of circles involving: (MP.3, MP.5, MP.7)

HS.G.16.aRecognize differences between and properties of inscribed, central and circumscribed angles.

HS.G.16.bUnderstand relationships between inscribed angles and the diameter of a circle.

HS.G.16.cUnderstand the relationship between the radius of a circle and the line drawn through the point of tangency on that radius.

HS.G.17(+) Apply basic construction procedures within the context of a circle. (MP.5, MP.6)

HS.G.17(+).aConstruct the inscribed and circumscribed circles of a triangle.

HS.G.17(+).bConstruct a tangent line from a point outside a given circle to the circle.

HS.G.18(+) Understand the relationship between an intercepted arc length within a circle and the radius of the circle. (MP.2, MP.3)

HS.G.18(+).aDerive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius. Derive the formula for the area of a sector.

HS.G.18(+).bDefine the radian measure of the angle as the measure of a central angle that intercepts an arc equal in length to the radius of the circle.

HS.G.19 Understand the relationship between the algebraic form and the geometric representation of a circle. (MP.6, MP.8)

HS.G.19.aWrite the equation of a circle of given center and radius using the Pythagorean Theorem.

HS.G.19.b(+) Derive and write the equation of a circle of given center and radius using the Pythagorean Theorem.

HS.G.19.c(+) Complete the square to find the center and radius of a circle given by an equation.

HS.G.2 Representing transformations in the plane. (MP.5, MP.7)

HS.G.2.aDescribe transformations as functions that take points in the plane as inputs and give other points as outputs.

HS.G.2.bCompare transformations that preserve distance and angle measures to those that do not.

HS.G.2.cGiven a rectangle, parallelogram, trapezoid, or regular polygon, formally describe the rotations and reflections that carry it onto itself, using properties of these figures.

HS.G.20(+) Derive the equations of conic sections. (MP.2, MP.7)

HS.G.20(+).aDerive the equation of a parabola given a focus and directrix.

HS.G.20(+).bDerive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

HS.G.21Use coordinates to justify and prove simple geometric theorems algebraically. (MP.2, MP.6)

HS.G.22Justify and apply the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. (MP.3, MP.7)

HS.G.23 Find measurements among points within the coordinate plane. (MP.2, MP.8)

HS.G.23.aUse points from the coordinate plane to find the coordinates of a midpoint of a line segment and the distance between the endpoints of a line segment.

HS.G.23.bFind the point on a directed line segment between two given points that partitions the segment in a given ratio.

HS.G.24 Use coordinates within the coordinate plane to calculate measurements of two dimensional figures. (MP.2, MP.4)

HS.G.24.aCompute the perimeters of various polygons.

HS.G.24.bCompute the areas of triangles, rectangles and other quadrilaterals.

HS.G.25 Analyze and determine the validity of arguments for the formulas for the various figures and shapes. (MP.3, MP.7)

HS.G.25.aFinding the circumference and area of a circle.

HS.G.25.bFinding the volume of a sphere, prism, cylinder, pyramid and cone.

HS.G.26(+)Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. (MP.2, MP.5)

HS.G.27Use volume formulas to solve problems for cylinders, pyramids, cones, spheres, prisms. (MP.4, MP.6)

HS.G.28Identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects. (MP.5, MP.7)

HS.G.29Use geometric shapes, their measures and their properties to describe objects in real world settings. (MP.1, MP.4)

HS.G.3(+)Develop formal definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments. (MP.6, MP.7)

HS.G.30Apply concepts of density based on area and volume in modeling situations, using appropriate units of measurement. (MP.4, MP.6)

HS.G.31Apply geometric methods to solve design problems. (MP.1, MP.4)

HS.G.4 Understand the effects of transformations of geometric figures. (MP.2, MP.8)

HS.G.4.aGiven a geometric figure and a rotation, reflection, or translation, draw the transformed figure.

HS.G.4.bSpecify a sequence of transformations that will carry a given figure onto another.

HS.G.4.cUse geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

HS.G.5 Know and apply the concepts of triangle congruence: (MP.3, MP.6)

HS.G.5.aUse 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.

HS.G.5.bExplain how the criteria for triangle congruence (ASA, SAS and SSS) follow from the definition of congruence in terms of rigid motions.

HS.G.6Apply theorems for lines, angles, triangles, parallelograms. (MP.2, MP.3)

HS.G.7 Prove theorems about geometric figures. (MP.6, MP.7)

HS.G.7.aConstruct formal proofs to justify theorems for lines, angles and triangles.

HS.G.7.b(+) Construct formal proofs to justify theorems for parallelograms.

HS.G.8 Create and apply geometric constructions. (MP.5, MP.6)

HS.G.8.aMake formal geometric constructions with a variety of tools and methods.

HS.G.8.bApply basic construction procedures to construct more complex figures.

HS.G.9 Understand properties of dilations. (MP.5, MP.7)

HS.G.9.aVerify the properties that result from that dilations given by a center and a scale factor.

HS.G.9.bVerify that a dilation produces an image that is similar to the pre-image.

HS.GaGeometry—Congruence

HS.GbGeometry—Similarity, Right Triangles and Trigonometry

HS.GcGeometry—Circles

HS.GdGeometry—Expressing Geometric Properties with Equations

HS.GeGeometry—Geometric Measurement and Dimensions

HS.GfGeometry—Modeling with Geometry

HS.N Conceptual Category Number and Quantity

HS.N.1Extend the properties of integer exponents to rational exponents, allowing for the expression of radicals in terms of rational exponents. (MP.2, MP.7)

HS.N.10(+)Extend polynomial identities to the complex numbers. (MP.7, MP.8)

HS.N.11(+)Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (MP.1, MP.3)

HS.N.12(+) Understand and apply properties of vectors. (MP.1, MP.6)

HS.N.12(+).aRecognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes.

HS.N.12(+).bFind the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

HS.N.12(+).cSolve problems involving velocity and other quantities that can be represented by vectors.

HS.N.13(+) Perform operations with vectors (addition, subtraction and multiplication by a scalar). (MP.3, MP.7)

HS.N.13(+).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.

HS.N.13(+).bGiven two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

HS.N.13(+).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.

HS.N.13(+).dRepresent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise.

HS.N.13(+).eCompute 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).

HS.N.14Use matrices to represent and manipulate data. (MP.4, MP.5)

HS.N.15 Perform operations with matrices. (MP.7, MP.8)

HS.N.15.aAdd, subtract and multiply matrices of appropriate dimensions.

HS.N.15.bMultiply matrices by scalars to produce new matrices.

HS.N.16(+) Understand properties of square and identity matrices. (MP.3, MP.7)

HS.N.16(+).aUnderstand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

HS.N.16(+).bUnderstand 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.

HS.N.16(+).cWork with 2×2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

HS.N.17(+)Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. (MP.2, MP.7)

HS.N.2Rewrite expressions involving radicals and rational exponents using the properties of exponents. (MP.7)

HS.N.3(+)Justify why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. (MP.3, MP.6)

HS.N.4 Use units in context as a way to understand problems and to guide the solution of multi-step problems; (MP.5, MP.6)

HS.N.4.aChoose and interpret units consistently in formulas

HS.N.4.bChoose and interpret the scale and the origin in graphs and data displays

HS.N.5Define appropriate units in context for the purpose of descriptive modeling. (MP.1, MP.6)

HS.N.6Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (MP.2, MP.6)

HS.N.7 Understanding properties of complex numbers. (MP.7, MP.8)

HS.N.7.aKnow 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.

HS.N.7.bUse the relation i^2=-1 and the commutative, associative and distributive properties to add, subtract and multiply complex numbers.

HS.N.7.c(+) Find the conjugate of a complex number and use it to find the quotient of complex numbers.

HS.N.8(+) Understanding representations of complex numbers using the complex plane. (MP.2, MP.5)

HS.N.8(+).aRepresent 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.

HS.N.8(+).bRepresent addition, subtraction, multiplication, modulus and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

HS.N.8(+).cCalculate 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.

HS.N.9Solve quadratic equations with real coefficients that have complex solutions. (MP.1, MP.2)

HS.NaNumber and Quantity—The Real Number System

HS.NbNumber and Quantity—Quantities

HS.NcNumber and Quantity—The Complex Number System

HS.NdNumber and Quantity—Vector and Matrix Quantities

HS.SP Conceptual Category Statistics and Probability

HS.SP.1Represent the distribution of data with plots on the real number line (stem plots, dot plots, histograms and box plots). (MP.4, MP.5)

HS.SP.10Decide if a specified model is consistent with the results from a simulation. (MP.3, MP.6)

HS.SP.11Recognize the purposes of and differences among sample surveys, experiments and observational studies; explain how randomization relates to each. (MP.3, MP.8)

HS.SP.12Use data from a sample survey to estimate a population mean or proportion and explain how bias may be involved in the process. (MP.4, MP.7)

HS.SP.13Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between estimates or statistics are significant. (MP.3, MP.8)

HS.SP.14 Describe events as subsets of a sample space. Use characteristics (or categories) of the outcomes, such as, (MP.1, MP.2)

HS.SP.14.aas unions, “A or B,” that are mutually exclusive events to calculate basic probabilities.

HS.SP.14.bas unions, “A or B,” that are non-mutually exclusive events to calculate basic probabilities.

HS.SP.14.cas intersections, “A and B” to calculate basic probabilities.

HS.SP.14.das complements of other events, “not A” to calculate basic probabilities.

HS.SP.15 Understand the concept of independence. (MP.1, MP.6)

HS.SP.15.aUnderstand that two events A and B are independent if the probability of A and B occurring together is the product of their individual probabilities, P(A)×P(B).

HS.SP.15.b(+) Determine whether two events are independent and provide a justification to support the decision.

HS.SP.15.cRecognize and explain the concept of independence in everyday language and everyday situations.

HS.SP.16 Understand the concept of conditional probability. (MP.1, MP.3)

HS.SP.16.aUnderstand the conditional probability of A given B as P(A and B)/P(B).

HS.SP.16.b(+) Interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A and the conditional probability of B given A is the same as the probability of B.

HS.SP.16.cRecognize and explain the concept of conditional probability in everyday language and everyday situations.

HS.SP.16.dFind 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 model.

HS.SP.17(+)Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide whether events are independent and to approximate conditional probabilities. (MP.2, MP.4)

HS.SP.18(+)Apply the General Multiplication Rule, P(A and B)=P(A)P(B|A)=P(B)P(A|B), in a uniform probability model and interpret the answer in terms of the model. (MP.1, MP.2)

HS.SP.19 Use permutations and combinations to compute probabilities. (MP.1, MP.8)

HS.SP.19.aDistinguish between situations that can be modeled using counting techniques, including Fundamental Counting Principle, permutations and combinations.

HS.SP.19.bPerform calculations using the appropriate counting technique, including simple probabilities.

HS.SP.19.c(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

HS.SP.2Use statistics appropriate to the shape of the numerical data distribution to compare center (median, mean) and spread (interquartile range when comparing medians and standard deviation when comparing means) of different data distributions. (MP.2, MP.6)

HS.SP.20(+)Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same appropriate graphical displays as for data distributions. (MP.3, MP.6)

HS.SP.21(+)Calculate the expected value of a random variable; interpret it as the mean of the probability distribution and use the value in analyzing decisions. (MP.1, MP.8)

HS.SP.22(+) Develop a probability distribution for a random variable. (MP.2, MP.8)

HS.SP.22(+).aFind an expected value based on a sample space in which theoretical probabilities can be calculated.

HS.SP.22(+).bFind an expected value based on a sample space in which empirical probabilities can be calculated.

HS.SP.23(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. (MP.6, MP.8)

HS.SP.23(+).aFind the expected payoff for a game of chance.

HS.SP.23(+).bEvaluate and compare strategies on the basis of expected values.

HS.SP.23(+).cUse calculated expected values to make fair decisions and formulate strategies.

HS.SP.3Interpret differences in shape, center and spread in the context of the distributions of the numerical data, accounting for the presence and possible effects of extreme data points (outliers). (MP.1, MP.7)

HS.SP.4(+)When appropriate, fit a normal distribution to a numerical data set for given mean and standard deviation and then estimate population percentages using the Empirical Rule and recognize that there are data sets for which such a procedure is not appropriate. (MP.1, MP.3)

HS.SP.5Summarize categorical data for two or more categories in frequency tables. Calculate and interpret joint, marginal and conditional relative frequencies (probabilities) in the context of the data, recognizing possible associations and trends in the data. (MP.2, MP.7)

HS.SP.6 Represent data on two quantitative variables on a scatter plot and describe how the explanatory and response variables are related. (MP.3, MP.4, MP.5)

HS.SP.6.aCalculate an appropriate mathematical model, or use a given mathematical model, for data to solve problems in context.

HS.SP.6.bInformally assess the fit of a model (through calculating correlation for linear data, plotting, calculating and/or analyzing residuals).

HS.SP.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. (MP.1, MP.2)

HS.SP.8 Understand the role and purpose of correlation in linear regression. (MP.5, MP.6)

HS.SP.8.aUse technology to compute correlation coefficient of a linear fit.

HS.SP.8.bInterpret the meaning of the correlation within the context of the data.

HS.SP.8.cDescribe the limitations of correlation when establishing causation.

HS.SP.9Understand statistics as a process for making inferences and justifying conclusions about population parameters based on a random sample from that population. (MP.1, MP.3)

HS.SPaStatistics and Probability—Interpreting Categorical and Quantitative Data

HS.SPbStatistics and Probability—Making Inferences and Justifying Conclusions

HS.SPcStatistics and Probability—Conditional Probability and the Rules of Probability

HS.SPdStatistics and Probability—Using Probability to Make Decisions

AP Calculus Learning Objectives

CB.Calc.LIM Limits

CB.Calc.LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

CB.Calc.LIM-1.ARepresent limits analytically using correct notation.

CB.Calc.LIM-1.BInterpret limits expressed in analytic notation.

CB.Calc.LIM-1.CEstimate limits of functions.

CB.Calc.LIM-1.DDetermine the limits of functions using limit theorems.

CB.Calc.LIM-1.EDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.

CB.Calc.LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

CB.Calc.LIM-2.AJustify conclusions about continuity at a point using the definition.

CB.Calc.LIM-2.BDetermine intervals over which a function is continuous.

CB.Calc.LIM-2.CDetermine values of x or solve for parameters that make discontinuous functions continuous, if possible.

CB.Calc.LIM-2.DInterpret the behavior of functions using limits involving infinity.

CB.Calc.LIM-3 The derivative of a function is defined as the limit of a difference quotient.

CB.Calc.LIM-3.AInterpret a limit as a definition of a derivative.

CB.Calc.LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.

CB.Calc.LIM-4.ADetermine limits of functions that result in indeterminate forms.

CB.Calc.LIM-5 Definite integrals can be approximated using geometric and numerical methods.

CB.Calc.LIM-5.AApproximate a definite integral using geometric and numerical methods.

CB.Calc.LIM-5.BInterpret the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-5.CRepresent the limiting case of the Riemann sum as a definite integral.

CB.Calc.LIM-6 Improper integrals can be determined using limits of definite integrals.

CB.Calc.LIM-6.AEvaluate an improper integral or determine that the integral diverges.

CB.Calc.LIM-7 The behavior of a convergent or divergent series can be determined or estimated.

CB.Calc.LIM-7.ADetermine whether a series converges or diverges.

CB.Calc.LIM-7.BApproximate the sum of a series.

CB.Calc.LIM-8 Power series and Taylor series can represent functions.

CB.Calc.LIM-8.ARepresent a function at a point as a Taylor polynomial.

CB.Calc.LIM-8.BApproximate function values using a Taylor polynomial.

CB.Calc.LIM-8.CDetermine the error bound associated with a Taylor polynomial approximation.

CB.Calc.LIM-8.DDetermine the radius of convergence and interval of convergence for a power series.

CB.Calc.LIM-8.ERepresent a function as a Taylor series or a Maclaurin series.

CB.Calc.LIM-8.FInterpret Taylor series and Maclaurin series.

CB.Calc.LIM-8.GRepresent a given function as a power series.

CB.Calc.CHA Change

CB.Calc.CHA-1 The derivative of a function is the instantaneous rate of change of the function.

CB.Calc.CHA-1.AInterpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.

CB.Calc.CHA-2 Derivatives can be computed using definitions and rules.

CB.Calc.CHA-2.ADetermine average rates of change using difference quotients.

CB.Calc.CHA-2.BRepresent the derivative of a function as the limit of a difference quotient.

CB.Calc.CHA-2.CDetermine the equation of a line tangent to a curve at a given point.

CB.Calc.CHA-2.DEstimate derivatives.

CB.Calc.CHA-3 Derivatives can be used to model rates of change in applied contexts.

CB.Calc.CHA-3.AInterpret the meaning of a derivative in context.

CB.Calc.CHA-3.BCalculate rates of change in applied contexts.

CB.Calc.CHA-3.CInterpret rates of change in applied contexts.

CB.Calc.CHA-3.DCalculate related rates in applied contexts.

CB.Calc.CHA-3.EInterpret related rates in applied contexts.

CB.Calc.CHA-3.FApproximate a value on a curve using the equation of a tangent line.

CB.Calc.CHA-3.GCalculate derivatives of parametric functions.

CB.Calc.CHA-3.HCalculate derivatives of vector-valued functions.

CB.Calc.CHA-4 Integrals can be used to model accumulation and net change in applied contexts.

CB.Calc.CHA-4.AInterpret the meaning of areas associated with the graph of a rate of change in context.

CB.Calc.CHA-4.BDetermine the average value of a function using definite integrals.

CB.Calc.CHA-4.CDetermine values for positions and rates of change using definite integrals in problems involving rectilinear motion.

CB.Calc.CHA-4.DInterpret the meaning of a definite integral in accumulation problems.

CB.Calc.CHA-4.EDetermine net change using definite integrals in applied contexts.

CB.Calc.CHA-5 Definite integrals can be used to calculate areas, volumes, and lengths.

CB.Calc.CHA-5.ACalculate areas in the plane using the definite integral.

CB.Calc.CHA-5.BCalculate volumes of solids with known cross sections using definite integrals.

CB.Calc.CHA-5.CCalculate volumes of solids of revolution using definite integrals.

CB.Calc.CHA-5.DCalculate areas of regions defined by polar curves using definite integrals.

CB.Calc.CHA-6 Arc length can be calculated using definite integrals.

CB.Calc.CHA-6.ADetermine the length of a curve in the plane defined by a function, using a definite integral.

CB.Calc.CHA-6.BDetermine the length of a curve in the plane defined by parametric functions, using a definite integral.

CB.Calc.FUN Analysis of Functions

CB.Calc.FUN-1 Existence theorems can be used to draw conclusions about the behavior of functions.

CB.Calc.FUN-1.AExplain the behavior of a function on an interval using the Intermediate Value Theorem.

CB.Calc.FUN-1.BJustify conclusions about functions by applying the Mean Value Theorem over an interval.

CB.Calc.FUN-1.CJustify conclusions about functions by applying the Extreme Value Theorem.

CB.Calc.FUN-2 A function's derivative, when it exists, can be used to determine whether the function is increasing or decreasing.

CB.Calc.FUN-2.AExplain the relationship between differentiability and continuity.

CB.Calc.FUN-3 Derivatives can be calculated for various types of functions.

CB.Calc.FUN-3.ACalculate derivatives of familiar functions.

CB.Calc.FUN-3.BCalculate derivatives of products and quotients of differentiable functions.

CB.Calc.FUN-3.CCalculate derivatives of compositions of differentiable functions.

CB.Calc.FUN-3.DCalculate derivatives of implicitly defined functions.

CB.Calc.FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.

CB.Calc.FUN-3.FDetermine higher order derivatives of a function.

CB.Calc.FUN-3.GCalculate derivatives of functions written in polar coordinates.

CB.Calc.FUN-4 A function's derivative can be used to understand function behavior.

CB.Calc.FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.

CB.Calc.FUN-4.BCalculate minimum and maximum values in applied contexts or analysis of functions.

CB.Calc.FUN-4.CInterpret minimum and maximum values calculated in applied contexts.

CB.Calc.FUN-4.DDetermine critical points of implicit relations.

CB.Calc.FUN-4.EJustify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.

CB.Calc.FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.

CB.Calc.FUN-5.ARepresent accumulation functions using definite integrals.

CB.Calc.FUN-6 Recognizing and applying techniques for antidifferentiation.

CB.Calc.FUN-6.ACalculate a definite integral using areas and properties of definite integrals.

CB.Calc.FUN-6.BEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.

CB.Calc.FUN-6.CDetermine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

CB.Calc.FUN-6.DUse substitution and rearrangements into equivalent forms, such as long division and completing the square, for integrands requiring these techniques.

CB.Calc.FUN-6.EUse integration by parts for integrands requiring this technique.

CB.Calc.FUN-6.FUse integration by linear partial fractions for integrands requiring this technique.

CB.Calc.FUN-7 Solving differential equations allows us to model rates of change.

CB.Calc.FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.

CB.Calc.FUN-7.BVerify solutions to differential equations.

CB.Calc.FUN-7.CEstimate solutions to differential equations.

CB.Calc.FUN-7.DDetermine general solutions to differential equations.

CB.Calc.FUN-7.EDetermine particular solutions to differential equations.

CB.Calc.FUN-7.FInterpret the meaning of a differential equation and its variables in context.

CB.Calc.FUN-7.GDetermine general and particular solutions for problems involving differential equations in context.

CB.Calc.FUN-7.HInterpret the meaning of the logistic growth model in context.

CB.Calc.FUN-8 Solutions to differential equations can model motion in the plane.

CB.Calc.FUN-8.ADetermine a particular solution given a rate vector and initial conditions.

CB.Calc.FUN-8.BDetermine values for positions and rates of change in problems involving planar motion.

AP Statistics Learning Objectives

CB.Stats.VAR Variation and Distribution

CB.Stats.VAR-1 Exploring data through asking and answering statistical questions.

CB.Stats.VAR-1.AIdentify questions to be answered, based on variation in one-variable data.

CB.Stats.VAR-1.BIdentify variables in a set of data.

CB.Stats.VAR-1.CClassify types of variables.

CB.Stats.VAR-1.DIdentify questions to be answered about possible relationships in data.

CB.Stats.VAR-1.EIdentify questions to be answered about data collection methods.

CB.Stats.VAR-1.FIdentify questions suggested by patterns in data.

CB.Stats.VAR-1.GIdentify questions suggested by variation in statistics for samples collected from the same population.

CB.Stats.VAR-1.HIdentify questions suggested by variation in the shapes of distributions of samples taken from the same population.

CB.Stats.VAR-1.IIdentify questions suggested by probabilities of errors in statistical inference.

CB.Stats.VAR-1.JIdentify questions suggested by variation between observed and expected counts in categorical data.

CB.Stats.VAR-1.KIdentify questions suggested by variation in scatter plots.

CB.Stats.VAR-2 Normal distributions and comparing data.

CB.Stats.VAR-2.ACompare a data distribution to the normal distribution model.

CB.Stats.VAR-2.BDetermine proportions and percentiles from a normal distribution.

CB.Stats.VAR-2.CCompare measures of relative position in data sets.

CB.Stats.VAR-3 Designing and analyzing experiments.

CB.Stats.VAR-3.AIdentify the components of an experiment.

CB.Stats.VAR-3.BDescribe elements of a well-designed experiment.

CB.Stats.VAR-3.CCompare experimental designs and methods.

CB.Stats.VAR-3.DExplain why a particular experimental design is appropriate.

CB.Stats.VAR-3.EInterpret the results of a well-designed experiment.

CB.Stats.VAR-4 Probability rules and calculations.

CB.Stats.VAR-4.ACalculate probabilities for events and their complements.

CB.Stats.VAR-4.BInterpret probabilities for events.

CB.Stats.VAR-4.CExplain why two events are (or are not) mutually exclusive.

CB.Stats.VAR-4.DCalculate conditional probabilities.

CB.Stats.VAR-4.ECalculate probabilities for independent events and for the union of two events.

CB.Stats.VAR-5 Discrete random variables and probability distributions.

CB.Stats.VAR-5.ARepresent the probability distribution for a discrete random variable.

CB.Stats.VAR-5.BInterpret a probability distribution.

CB.Stats.VAR-5.CCalculate parameters for a discrete random variable.

CB.Stats.VAR-5.DInterpret parameters for a discrete random variable.

CB.Stats.VAR-5.ECalculate parameters for linear combinations of random variables.

CB.Stats.VAR-5.FDescribe the effects of linear transformations of parameters of random variables.

CB.Stats.VAR-6 Normal distributions and hypothesis testing for proportions.

CB.Stats.VAR-6.ACalculate the probability that a particular value lies in a given interval of a normal distribution.

CB.Stats.VAR-6.BDetermine the interval associated with a given area in a normal distribution.

CB.Stats.VAR-6.CDetermine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.

CB.Stats.VAR-6.DIdentify the null and alternative hypotheses for a population proportion.

CB.Stats.VAR-6.EIdentify an appropriate testing method for a population proportion.

CB.Stats.VAR-6.FVerify the conditions for making statistical inferences when testing a population proportion.

CB.Stats.VAR-6.GCalculate an appropriate test statistic and p-value for a population proportion.

CB.Stats.VAR-6.HIdentify the null and alternative hypotheses for a difference of two population proportions.

CB.Stats.VAR-6.IIdentify an appropriate testing method for the difference of two population proportions.

CB.Stats.VAR-6.JVerify the conditions for making statistical inferences when testing a difference of two population proportions.

CB.Stats.VAR-6.KCalculate an appropriate test statistic for the difference of two population proportions.

CB.Stats.VAR-7 t-distributions and hypothesis testing for means and regression.

CB.Stats.VAR-7.ADescribe t-distributions.

CB.Stats.VAR-7.BIdentify an appropriate testing method for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.CIdentify the null and alternative hypotheses for a population mean with unknown sigma, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.DVerify the conditions for the test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.ECalculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.

CB.Stats.VAR-7.FIdentify an appropriate selection of a testing method for a difference of two population means.

CB.Stats.VAR-7.GIdentify the null and alternative hypotheses for a difference of two population means.

CB.Stats.VAR-7.HVerify the conditions for the significance test for the difference of two population means.

CB.Stats.VAR-7.ICalculate an appropriate test statistic for a difference of two means.

CB.Stats.VAR-7.JIdentify the appropriate selection of a testing method for a slope of a regression model.

CB.Stats.VAR-7.KIdentify appropriate null and alternative hypotheses for a slope of a regression model.

CB.Stats.VAR-7.LVerify the conditions for the significance test for the slope of a regression model.

CB.Stats.VAR-7.MCalculate an appropriate test statistic for the slope of a regression model.

CB.Stats.VAR-8 Chi-square distributions and tests.

CB.Stats.VAR-8.ADescribe chi-square distributions.

CB.Stats.VAR-8.BIdentify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.CIdentify an appropriate testing method for a distribution of proportions in a set of categorical data.

CB.Stats.VAR-8.DCalculate expected counts for the chi-square test for goodness of fit.

CB.Stats.VAR-8.EVerify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.

CB.Stats.VAR-8.FCalculate the appropriate statistic for the chi-square test for goodness of fit.

CB.Stats.VAR-8.GDetermine the p-value for the chi-square test for goodness of fit.

CB.Stats.VAR-8.HCalculate expected counts for two-way tables of categorical data.

CB.Stats.VAR-8.IIdentify the null and alternative hypotheses for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.JIdentify an appropriate testing method for comparing distributions in two-way tables of categorical data.

CB.Stats.VAR-8.KVerify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.

CB.Stats.VAR-8.LCalculate the appropriate statistic for a chi-square test for homogeneity or independence.

CB.Stats.VAR-8.MDetermine the p-value for a chi-square test for independence or homogeneity.

CB.Stats.UNC Uncertainty and Inference

CB.Stats.UNC-1 Representing and describing categorical and quantitative data.

CB.Stats.UNC-1.ARepresent categorical data using frequency or relative frequency tables.

CB.Stats.UNC-1.BDescribe categorical data represented in frequency or relative tables.

CB.Stats.UNC-1.CRepresent categorical data graphically.

CB.Stats.UNC-1.DDescribe categorical data represented graphically.

CB.Stats.UNC-1.ECompare multiple sets of categorical data.

CB.Stats.UNC-1.FClassify types of quantitative variables.

CB.Stats.UNC-1.GRepresent quantitative data graphically.

CB.Stats.UNC-1.HDescribe the characteristics of quantitative data distributions.

CB.Stats.UNC-1.ICalculate measures of center and position for quantitative data.

CB.Stats.UNC-1.JCalculate measures of variability for quantitative data.

CB.Stats.UNC-1.KExplain the selection of a particular measure of center and/or variability for describing a set of quantitative data.

CB.Stats.UNC-1.LRepresent summary statistics for quantitative data graphically.

CB.Stats.UNC-1.MDescribe summary statistics of quantitative data represented graphically.

CB.Stats.UNC-1.NCompare graphical representations for multiple sets of quantitative data.

CB.Stats.UNC-1.OCompare summary statistics for multiple sets of quantitative data.

CB.Stats.UNC-1.PCompare numerical and graphical representations for two categorical variables.

CB.Stats.UNC-1.QCalculate statistics for two categorical variables.

CB.Stats.UNC-1.RCompare statistics for two categorical variables.

CB.Stats.UNC-1.SRepresent bivariate quantitative data using scatterplots.

CB.Stats.UNC-2 Probability and simulation.

CB.Stats.UNC-2.AEstimate probabilities using simulation.

CB.Stats.UNC-3 Random variables and sampling distributions.

CB.Stats.UNC-3.AEstimate probabilities of binomial random variables using data from a simulation.

CB.Stats.UNC-3.BCalculate probabilities for a binomial distribution.

CB.Stats.UNC-3.CCalculate parameters for a binomial distribution.

CB.Stats.UNC-3.DInterpret probabilities and parameters for a binomial distribution.

CB.Stats.UNC-3.ECalculate probabilities for geometric random variables.

CB.Stats.UNC-3.FCalculate parameters of a geometric distribution.

CB.Stats.UNC-3.GInterpret probabilities and parameters for a geometric distribution.

CB.Stats.UNC-3.HEstimate sampling distributions using simulation.

CB.Stats.UNC-3.IExplain why an estimator is or is not unbiased.

CB.Stats.UNC-3.JCalculate estimates for a population parameter.

CB.Stats.UNC-3.KDetermine parameters of a sampling distribution for sample proportions.

CB.Stats.UNC-3.LDetermine whether a sampling distribution for a sample proportion can be described as approximately normal.

CB.Stats.UNC-3.MInterpret probabilities and parameters for a sampling distribution for a sample proportion.

CB.Stats.UNC-3.NDetermine parameters of a sampling distribution for a difference in sample proportions.

CB.Stats.UNC-3.ODetermine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.

CB.Stats.UNC-3.PInterpret probabilities and parameters for a sampling distribution for a difference in proportions.

CB.Stats.UNC-3.QDetermine parameters of a sampling distribution for a sample mean.

CB.Stats.UNC-3.RDetermine whether a sampling distribution of a sample mean can be described as approximately normal.

CB.Stats.UNC-3.SInterpret probabilities and parameters for a sampling distribution for a sample mean.

CB.Stats.UNC-3.TDetermine parameters of a sampling distribution for a difference in sample means.

CB.Stats.UNC-3.UDetermine whether a sampling distribution of a difference in sample means can be described as approximately normal.

CB.Stats.UNC-3.VInterpret probabilities and parameters for a sampling distribution for a difference in sample means.

CB.Stats.UNC-4 Confidence intervals for proportions and means.

CB.Stats.UNC-4.AIdentify an appropriate confidence interval procedure for a population proportion.

CB.Stats.UNC-4.BVerify the conditions for calculating confidence intervals for a population proportion.

CB.Stats.UNC-4.CDetermine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.

CB.Stats.UNC-4.DCalculate an appropriate confidence interval for a population proportion.

CB.Stats.UNC-4.ECalculate an interval estimate based on a confidence interval for a population proportion.

CB.Stats.UNC-4.FInterpret a confidence interval for a population proportion.

CB.Stats.UNC-4.GJustify a claim based on a confidence interval for a population proportion.

CB.Stats.UNC-4.HIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.

CB.Stats.UNC-4.IIdentify an appropriate confidence interval procedure for a comparison of population proportions.

CB.Stats.UNC-4.JVerify the conditions for calculating confidence intervals for a difference between population proportions.

CB.Stats.UNC-4.KCalculate an appropriate confidence interval for a comparison of population proportions.

CB.Stats.UNC-4.LCalculate an interval estimate based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.MInterpret a confidence interval for a difference of proportions.

CB.Stats.UNC-4.NJustify a claim based on a confidence interval for a difference of proportions.

CB.Stats.UNC-4.OIdentify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.PVerify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.QDetermine the margin of error for a given sample size for a one-sample t-interval.

CB.Stats.UNC-4.RCalculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.SInterpret a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.TJustify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.

CB.Stats.UNC-4.UIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.

CB.Stats.UNC-4.VIdentify an appropriate confidence interval procedure for a difference of two population means.

CB.Stats.UNC-4.WVerify the conditions for calculating confidence intervals for the difference of two population means.

CB.Stats.UNC-4.XDetermine the margin of error for the difference of two population means.

CB.Stats.UNC-4.YCalculate an appropriate confidence interval for a difference of two population means.

CB.Stats.UNC-4.ZInterpret a confidence interval for a difference of population means.

CB.Stats.UNC-5 Errors in statistical inference.

CB.Stats.UNC-5.AIdentify Type I and Type II errors.

CB.Stats.UNC-5.BCalculate the probability of a Type I and Type II errors.

CB.Stats.UNC-5.CIdentify factors that affect the probability of errors in significance testing.

CB.Stats.UNC-5.DInterpret Type I and Type II errors.

CB.Stats.DAT Data Analysis

CB.Stats.DAT-1 Bivariate data analysis and linear regression.

CB.Stats.DAT-1.ADescribe the characteristics of a scatter plot.

CB.Stats.DAT-1.BDetermine the correlation for a linear relationship.

CB.Stats.DAT-1.CInterpret the correlation for a linear relationship.

CB.Stats.DAT-1.DCalculate a predicted response value using a linear regression model.

CB.Stats.DAT-1.ERepresent differences between measured and predicted responses using residual plots.

CB.Stats.DAT-1.FDescribe the form of association of bivariate data using residual plots.

CB.Stats.DAT-1.GEstimate parameters for the least-squares regression line model.

CB.Stats.DAT-1.HInterpret coefficients for the least-squares regression line model.

CB.Stats.DAT-1.IIdentify influential points in regression.

CB.Stats.DAT-1.JCalculate a predicted response using a least-squares regression line for a transformed data set.

CB.Stats.DAT-2 Studies, sampling, and sources of bias.

CB.Stats.DAT-2.AIdentify the type of a study.

CB.Stats.DAT-2.BIdentify appropriate generalizations and determinations based on observational studies.

CB.Stats.DAT-2.CIdentify a sampling method, given a description of a study.

CB.Stats.DAT-2.DExplain why a particular sampling method is or is not appropriate for a given situation.

CB.Stats.DAT-2.EIdentify potential sources of bias in sampling methods.

CB.Stats.DAT-3 Interpreting and justifying conclusions from significance tests.

CB.Stats.DAT-3.AInterpret the p-value of a significance test for a population proportion.

CB.Stats.DAT-3.BJustify a claim about the population based on the results of a significance test for a population proportion.

CB.Stats.DAT-3.CInterpret the p-value of a significance test for a difference of population proportions.

CB.Stats.DAT-3.DJustify a claim about the population based on the results of a significance test for a difference of population proportions.

CB.Stats.DAT-3.EInterpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.

CB.Stats.DAT-3.FJustify a claim about the population based on the results of a significance test for a population mean.

CB.Stats.DAT-3.GInterpret the p-value of a significance test for a difference of population means.

CB.Stats.DAT-3.HJustify a claim about the population based on the results of a significance test for a difference of two population means in context.

CB.Stats.DAT-3.IInterpret the p-value for the chi-square test for goodness of fit.

CB.Stats.DAT-3.JJustify a claim about the population based on the results of a chi-square test for goodness of fit.

CB.Stats.DAT-3.KInterpret the p-value for the chi-square test for homogeneity or independence.

CB.Stats.DAT-3.LJustify a claim about the population based on the results of a chi-square test for homogeneity or independence.

CB.Stats.DAT-3.MInterpret the p-value of a significance test for the slope of a regression model.

CB.Stats.DAT-3.NJustify a claim about the population based on the results of a significance test for the slope of a regression model.

Standards on this page are sourced from published state frameworks. Goblins aligns tutoring to these standards.