Arizona Math Standards

814 standards, Kindergarten through AP.

Kindergarten

K.CC Counting and Cardinality (CC)
K.CC.A Know number names and the count sequence.
K.CC.A.1Count to 100 by ones and by tens.
K.CC.A.2Count forward from a given number other than one, within the known sequence (e.g., "Starting at the number 5, count up to 11.").
K.CC.A.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0 to 20 (with 0 representing a count of no objects).
K.CC.B Count to tell the number of objects.
K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.B.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 (one to one correspondence).
K.CC.B.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 (cardinality).
K.CC.B.4.cUnderstand that each successive number name refers to a quantity that is one larger (hierarchical inclusion).
K.CC.B.5Count to answer questions about “How many?” when 20 or fewer objects are arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1 to 20, count out that many objects.
K.CC.C Compare numbers.
K.CC.C.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group. (Include groups with up to ten objects.)
K.CC.C.7Compare two numbers between 0 and 10 presented as written numerals.
K.G Geometry (G)
K.G.A Identify and describe shapes.
K.G.A.1Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
K.G.A.2Correctly name shapes regardless of their orientation or overall size (e.g., circle, triangle, square, rectangle, rhombus, trapezoid, hexagon, cube, cone, cylinder, sphere).
K.G.A.3Identify shapes as two-dimensional (lying in a plane, flat) or three-dimensional (solid).
K.G.B Analyze, compare, create, and compose shapes.
K.G.B.4Analyze and compare two-dimensional and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/corners), and other attributes (e.g., having sides of equal length).
K.G.B.5Model shapes in the world by building shapes from components (e.g., use sticks and clay balls) and drawing shapes.
K.G.B.6Use simple shapes to form composite shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
K.MD Measurement and Data (MD)
K.MD.A Describe and compare measurable attributes.
K.MD.A.1Describe measurable attributes of a single object (e.g., length and weight).
K.MD.A.2Directly compare two objects with a measurable attribute in common to see which object has “more of” or “less of” the attribute, and describe the difference (e.g., directly compare the length of 10 cubes to a pencil and describe one as longer or shorter).
K.MD.B Classify objects and count the number of objects in each category.
K.MD.B.3Classify objects into given categories; count the number in each category and sort the categories by count. (Note: limit category counts to be less than or equal to 10.)
K.NBT Number and Operations in Base Ten (NBT)
K.NBT.A Work with numbers 11 to 19 to gain foundations for place value.
K.NBT.A.1Compose and decompose numbers from 11 to 19 into ten ones and additional ones by using objects, drawings and/or equations. Understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones (e.g., 18 = 10 + 8).
K.NBT.B Use place value understanding and properties of operations to add and subtract.
K.NBT.B.2Demonstrate understanding of addition and subtraction within 10 using place value.
K.OA Operations and Algebraic Thinking (OA)
K.OA.A Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.A.1Represent addition and subtraction concretely.
K.OA.A.2Solve addition and subtraction word problems and add and subtract within 10. See Table 1.
K.OA.A.3Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., using fingers, objects, symbols, tally marks, drawings, expressions).
K.OA.A.4For any number from 1 to 9, find the number that makes 10 when added to the given number (e.g., using fingers, objects, symbols, tally marks, drawings, or equation).
K.OA.A.5Fluently add and subtract within 5.

Grade 1

1.G Geometry (G)
1.G.A Reason with shapes and their attributes.
1.G.A.1Distinguish between defining attributes (triangles are closed and 3 sided) versus non-defining attributes (color, orientation, overall size) for two-dimensional shapes; build and draw shapes that possess defining attributes.
1.G.A.2Compose two-dimensional shapes or three-dimensional shapes to create a composite shape.
1.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters. Describe the whole as two of, or four of the shares. Understand that decomposing into more equal shares creates smaller shares.
1.MD Measurement and Data (MD)
1.MD.A Measure lengths indirectly and by iterating length units.
1.MD.A.1Order three objects by length. Compare the lengths of two objects indirectly by using a third object.
1.MD.A.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. (Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.)
1.MD.B Work with time and money.
1.MD.B.3aTell and write time in hours and half-hours using analog and digital clocks.
1.MD.B.3bIdentify coins by name and value (pennies, nickels, dimes and quarters).
1.MD.C Represent and interpret data.
1.MD.C.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
1.NBT Number and Operations in Base Ten (NBT)
1.NBT.A Extend the counting sequence.
1.NBT.A.1Count to 120 by 1's, 2's, and 10's starting at any number less than 100. In this range, read and write numerals and represent a number of objects with a written numeral.
1.NBT.B Understand place value.
1.NBT.B.2 Understand that the two digits of a two-digit number represent groups of tens and ones. Understand the following as special cases:
1.NBT.B.2.a10 can be thought of as a group of ten ones — called a “ten”.
1.NBT.B.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.B.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.B.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NBT.C Use place value understanding and properties of operations to add and subtract.
1.NBT.C.4Demonstrate understanding of addition within 100, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form.
1.NBT.C.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.
1.NBT.C.6Subtract multiples of 10 in the range of 10 to 90 (positive or zero differences), using objects or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form.
1.OA Operations and Algebraic Thinking (OA)
1.OA.A Represent and solve problems involving addition and subtraction.
1.OA.A.1Use addition and subtraction within 20 to solve word problems with unknowns in all positions (e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem).
1.OA.A.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 (e.g., by using objects, drawings, and/or equations with a symbol for the unknown number to represent the problem).
1.OA.B Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.B.3Apply properties of operations (commutative and associative properties of addition) as strategies to add and subtract through 20. (Students need not use formal terms for these properties.)
1.OA.B.4Understand subtraction as an unknown-addend problem within 20 (e.g., subtract 10 - 8 by finding the number that makes 10 when added to 8).
1.OA.C Add and subtract within 10.
1.OA.C.5Relate counting to addition and subtraction (e.g., by using counting on 2 to add 2).
1.OA.C.6Fluently add and subtract within 10.
1.OA.D Work with addition and subtraction equations.
1.OA.D.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false (e.g., Which of the following equations are true and which are false? 6 + 1 = 6 - 1, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2).
1.OA.D.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers (e.g., determine the unknown number that makes the equation true in each of the equations 8 + □ = 11, 5 = □ - 3, 6 + 6 = □).

Grade 2

2.G Geometry (G)
2.G.A Reason with shapes and their attributes.
2.G.A.1Identify and describe specified attributes of two-dimensional and three-dimensional shapes, according to the number and shape of faces, number of angles, and the number of sides and/or vertices. Draw two-dimensional shapes based on the specified attributes (e.g., triangles, quadrilaterals, pentagons, and hexagons).
2.G.A.2Partition a rectangle into rows and columns of same-size rectangles and count to find the total number of rectangles.
2.G.A.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, fourths, half of, third of, fourth of, and describe the whole as two halves, three thirds, or four fourths. Recognize that equal shares of identical wholes need not have the same shape.
2.MD Measurement and Data (MD)
2.MD.A Measure and estimate lengths in standard units.
2.MD.A.1Measure the length of an object by selecting and using appropriate tools (e.g., ruler, meter stick, yardstick, measuring tape).
2.MD.A.2Measure the length of an object twice, using different standard length units for the two measurements; describe how the two measurements relate to the size of the unit chosen. Understand that depending on the size of the unit, the number of units for the same length varies.
2.MD.A.3Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.A.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
2.MD.B Relate addition and subtraction to length.
2.MD.B.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same unit.
2.MD.B.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.
2.MD.C Work with time and money.
2.MD.C.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.C.8Solve word problems involving collections of money, including dollar bills, quarters, dimes, nickels, and pennies. Record the total using $ and appropriately.
2.MD.D Represent and interpret data.
2.MD.D.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in the graph.
2.MD.D.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
2.NBT Number and Operations in Base Ten (NBT)
2.NBT.A Understand place value.
2.NBT.A.1 Understand that the three digits of a three-digit number represent groups of hundreds, tens, and ones (e.g., 706 equals 7 hundreds, 0 tens, and 6 ones and also equals 70 tens and 6 ones). Understand the following as special cases:
2.NBT.A.1.a100 can be thought of as a group of ten tens—called a “hundred.”
2.NBT.A.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.A.2Count within 1000; skip count by 5's, 10's and 100's.
2.NBT.A.3Read and write numbers up to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.A.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
2.NBT.B Use place value understanding and properties of operations to add and subtract.
2.NBT.B.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.NBT.B.6Add up to three two-digit numbers using strategies based on place value and properties of operations.
2.NBT.B.7Demonstrate understanding of addition and subtraction within 1000, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written form.
2.NBT.B.8Mentally add 10 or 100 to a given number in the range of 100 and 900, and mentally subtract 10 or 100 from a given number in the range of 100 and 900.
2.NBT.B.9Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)
2.OA Operations and Algebraic Thinking (OA)
2.OA.A Represent and solve problems involving addition and subtraction.
2.OA.A.1Use addition and subtraction within 100 to solve one- and two-step word problems. Represent a word problem as an equation with a symbol for the unknown.
2.OA.B Add and subtract within 20.
2.OA.B.2Fluently add and subtract within 20. By the end of Grade 2, know from memory all sums of two one-digit numbers.
2.OA.C Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.3Determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by pairing objects or counting them by 2's).
2.OA.C.4Use addition to find the total number of objects arranged in rectangular arrays (with up to 5 rows and 5 columns). Write an equation to express the total as a sum of equal addends.

Grade 3

3.G Geometry (G)
3.G.A Reason with shapes and their attributes.
3.G.A.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others)may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples quadrilaterals that do not belong to any of these subcategories.
3.G.A.2Partition shapes into b parts with equal areas. Express the area of each part as a unit fraction 1/b of the whole. (Grade 3 expectations are limited to fractions with denominators b = 2,3,4,6,8.)
3.MD Measurement and Data (MD)
3.MD.A Solve problems involving measurement.
3.MD.A.1aTell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes (e.g., representing the problem on a number line diagram).
3.MD.A.1bSolve word problems involving money through $20.00, using symbols $, ".", ₵.
3.MD.A.2Measure and estimate liquid volumes and masses of objects using metric units. (Excludes compound units such as cm3 and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units. Excludes multiplicative comparison problems (problems involving notions of “times as much”).
3.MD.B Represent and interpret data.
3.MD.B.3Create a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs.
3.MD.B.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch to the nearest quarter-inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
3.MD.C Geometric measurement: Understand concepts of area and perimeter.
3.MD.C.5 Understand area as an attribute of plane figures and understand concepts of area measurement.
3.MD.C.5.aA square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
3.MD.C.5.bA plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.C.6Measure areas by counting unit squares (e.g., square cm, square m, square in, square ft, and improvised units).
3.MD.C.7 Relate area to the operations of multiplication and addition.
3.MD.C.7.aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.C.7.bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.C.7.cUse tiling to show that the area of a rectangle with whole-number side lengths a and b + c is the sum of a b and a ח c. Use area models to represent the distributive property in mathematical reasoning.
3.MD.C.7.dUnderstand that rectilinear figures can be decomposed into non-overlapping rectangles and that the sum of the areas of these rectangles is identical to the area of the original rectilinear figure. Apply this technique to solve problems in real-world contexts.
3.MD.C.8Solve real-world and mathematical problems involving perimeters of plane figures and areas of rectangles, including finding the perimeter given the side lengths, finding an unknown side length. Represent rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.NBT Number and Operations in Base Ten (NBT)
3.NBT.A Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.A.1Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.A.2Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3.NBT.A.3Multiply one-digit whole numbers by multiples of 10 in the range 10 to 90 using strategies based on place value and the properties of operations (e.g., 9 x 80, 5 x 60).
3.NF Number and Operations – Fractions (NF)
3.NF.A Understand fractions as numbers.
3.NF.A.1Understand a fraction (1/b) as the quantity formed by one 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.
3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.A.2.aRepresent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Understand that each part has size 1/b and that the end point of the part based at 0 locates the number 1/b on the number line.
3.NF.A.2.bRepresent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Understand that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line including values greater than 1.
3.NF.A.2.cUnderstand a fraction 1/b as a special type of fraction that can be referred to as a unit fraction (e.g. 1/2, 1/4).
3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.A.3.aUnderstand two fractions as equivalent if they have the same relative size compared to 1 whole.
3.NF.A.3.bRecognize and generate simple equivalent fractions. Explain why the fractions are equivalent.
3.NF.A.3.cExpress whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
3.NF.A.3.dCompare two fractions with the same numerator or the same denominator by reasoning about their size. Understand that comparisons are valid only when the two fractions refer to the same whole. Record results of comparisons with the symbols >, =, or <, and justify conclusions.
3.OA Operations and Algebraic Thinking (OA)
3.OA.A Represent and solve problems involving whole number multiplication and division.
3.OA.A.1Interpret products of whole numbers as the total number of objects in equal groups (e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each).
3.OA.A.2Interpret whole number quotients of whole numbers (e.g., interpret 56 8 as the number of objects in each group when 56 objects are partitioned equally into 8 groups, or as a number of groups when 56 objects are partitioned into equal groups of 8 objects each). See Table 2.
3.OA.A.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.
3.OA.A.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers For example, determine the unknown number that makes the equation true in each of the equations 8 x □ = 48, 5 = □ ÷ 3, 6 x 6 = □.
3.OA.B Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5Apply properties of operations as strategies to multiply and divide. Properties include commutative and associative properties of multiplication and the distributive property. (Students do not need to use the formal terms for these properties.)
3.OA.B.6Understand division as an unknown-factor problem (e.g., find 32 8 by finding the number that makes 32 when multiplied by 8).
3.OA.C Multiply and divide within 100.
3.OA.C.7Fluently multiply and divide within 100. By the end of Grade 3, know from memory all multiplication products through 10 x 10 and division quotients when both the quotient and divisor are less than or equal to 10.
3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.10When solving problems, assess the reasonableness of answers using mental computation and estimation strategies including rounding.
3.OA.D.8Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Utilize understanding of the Order of Operations when there are no parentheses.
3.OA.D.9Identify patterns in the addition table and the multiplication table and explain them using properties of operations (e.g. observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends).

Grade 4

4.G Geometry (G)
4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.A.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
4.G.A.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size (e.g., understand right triangles as a category, and identify right triangles).
4.G.A.3Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
4.MD Measurement and Data (MD)
4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
4.MD.A.1Know relative sizes of measurement units within one system of units which could include km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit and in a smaller unit in terms of a larger unit. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1,12), 2,24), (3,36).
4.MD.A.2Use the four operations to solve word problems and problems in real-world context involving distances, intervals of time (hr, min, sec), liquid volumes, masses of objects, and money, including decimals and problems involving fractions with like denominators, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using a variety of representations, including number lines that feature a measurement scale.
4.MD.A.3Apply the area and perimeter formulas for rectangles in mathematical problems and problems in real-world contexts including problems with unknown side lengths.
4.MD.B Represent and interpret data.
4.MD.B.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.
4.MD.C Geometric measurement: Understand concepts of angle and measure angles.
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
4.MD.C.5.aAn angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles. b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7Understand angle measures as additive. (When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.) Solve addition and subtraction problems to find unknown angles on a diagram within mathematical problems as well as problems in real-world contexts.
4.NBT Number and Operations in Base Ten (NBT)
4.NBT.A Generalize place value understanding for multi-digit whole numbers.
4.NBT.A.1Apply concepts of place value, multiplication, and division to understand that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
4.NBT.A.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.A.3Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.B Use place value understanding and properties of operations to perform multi-digit arithmetic.
4.NBT.B.4Fluently add and subtract multi-digit whole numbers using a standard algorithm.
4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.B.6Demonstrate understanding of division by finding whole-number quotients and remainders with up to four-digit dividends and one-digit divisors.
4.NF Number and Operations – Fractions (NF)
4.NF.A Extend understanding of fraction equivalence and ordering.
4.NF.A.1Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to understand and generate equivalent fractions.
4.NF.A.2 Compare two fractions with different numerators and different denominators (e.g., by creating common denominators or numerators and by comparing to a benchmark fraction).
4.NF.A.2.aUnderstand that comparisons are valid only when the two fractions refer to the same size whole.
4.NF.A.2.bRecord the results of comparisons with symbols >, =, or <, and justify the conclusions.
4.NF.B Build fractions from unit fractions by applying and extending previous understanding of operations on whole numbers.
4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of unit fractions (1/b).
4.NF.B.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.
4.NF.B.3.bDecompose a fraction into a sum of fractions with the same denominator in more than one way (e.g., 3/8 = 1/8 + 1/8+1/8; 3/8 = 2/8 + 1/8; 2 1/8 = 1 + 1 + 1/8 + or 2 1/8 = 8/8 + 8/8 + 1/8).
4.NF.B.3.cAdd and subtract mixed numbers with like denominators (e.g., by using properties of operations and the relationship between addition and subtraction and/or by replacing each mixed number with an equivalent fraction).
4.NF.B.3.dSolve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
4.NF.B.4 Build fractions from unit fractions.
4.NF.B.4.aUnderstand a fraction a/b as a multiple of a unit fraction 1/b. In general, a/b = a x 1/b.
4.NF.B.4.bUnderstand a multiple of a/b as a multiple of a unit fraction 1/b, and use this understanding to multiply a whole number by a fraction. In general, n x a/b = (n x a)/b
4.NF.B.4.cSolve word problems involving multiplication of a whole number by a fraction. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
4.NF.C Understand decimal notation for fractions, and compare decimal fractions.
4.NF.C.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 (tenths) and 100 (hundredths). For example, express 3/10 as 30/100, and ad 3/10 + 4/100 = 34/100. (Note: Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators, in general, is not a requirement at this grade.)
4.NF.C.6Use decimal notation for fractions with denominators 10 (tenths) or 100 (hundredths), and locate these decimals on a number line.
4.NF.C.7Compare two decimals to hundredths by reasoning about their size. Understand that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <.
4.OA Operations and Algebraic Thinking (OA)
4.OA.A Use the four operations with whole numbers to solve problems.
4.OA.A.1Represent verbal statements of multiplicative comparisons as multiplication equations. Interpret a multiplication equation as a comparison (e.g., 35 is the number of objects in 5 groups, each containing 7 objects, and is also the number of objects in 7 groups, each containing 5 objects).
4.OA.A.2Multiply or divide within 1000 to solve word problems involving multiplicative comparison (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison). See Table 2.
4.OA.A.3Solve multistep word problems using the four operations, including problems in which remainders must be interpreted. Understand how the remainder is a fraction of the divisor. Represent these problems using equations with a letter standing for the unknown quantity.
4.OA.B Gain familiarity with factors and multiples.
4.OA.B.4Find all factor pairs for a whole number in the range 1 to 100 and understand that a whole number is a multiple of each of its factors.
4.OA.C Generate and analyze patterns.
4.OA.C.5Generate a number pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself and explain the pattern informally (e.g., given the rule “add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers).
4.OA.C.6When solving problems, assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 5

5.G Geometry (G)
5.G.A Graph points on the coordinate plane to solve mathematical problems as well as problems in real-world context.
5.G.A.1Understand and describe a coordinate system as perpendicular number lines, called axes, that intersect at the origin (0 , 0). Identify a given point in the first quadrant of the coordinate plane using an ordered pair of numbers, called coordinates. Understand that the first number (x) indicates the distance traveled on the horizontal axis, and the second number (y) indicates the distance traveled on the vertical axis.
5.G.A.2Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.G.B Classify two-dimensional figures into categories based on their properties.
5.G.B.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
5.G.B.4Classify two-dimensional figures in a hierarchy based on properties.
5.MD Measurement and Data (MD)
5.MD.A Convert like measurement units within a given measurement system.
5.MD.A.1Convert among different-sized standard measurement units within a given measurement system, and use these conversions in solving multi-step, real-world problems.
5.MD.B Represent and interpret data.
5.MD.B.2Make a line plot to display a data set of measurements in fractions of a unit (1/8, 1/2, 3/4). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
5.MD.C Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition.
5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3.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.C.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.C.4Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.MD.C.5 Relate volume to the operations of multiplication and addition and solve mathematical problems and problems in real-world contexts involving volume.
5.MD.C.5.aFind the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes (e.g., to represent the associative property of multiplication).
5.MD.C.5.bUnderstand and use the formulas V = l x w x h and V = B x h, where in this case B is the area of the base (B = l x w), for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths to solve mathematical problems and problems in real-world contexts.
5.MD.C.5.cUnderstand volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms, applying this technique to solve mathematical problems and problems in real-world contexts.
5.NBT Number and Operations in Base Ten (NBT)
5.NBT.A Understand the place value system.
5.NBT.A.1Apply concepts of place value, multiplication, and division to understand that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
5.NBT.A.3 Read, write, and compare decimals to thousandths.
5.NBT.A.3.aRead and write decimals to thousandths using base-ten numerals, number names, and expanded form.
5.NBT.A.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.A.4Use place value understanding to round decimals to any place.
5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.B.5Fluently multiply multi-digit whole numbers using a standard algorithm.
5.NBT.B.6Apply and extend understanding of division to find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors.
5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths, connecting objects or drawings to strategies based on place value, properties of operations, and/or the relationship between operations. Relate the strategy to a written form.
5.NF Number and Operations – Fractions (NF)
5.NF.A Use equivalent fractions to add and subtract fractions.
5.NF.A.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12).
5.NF.A.2Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by using a variety of representations, equations, and visual models to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (e.g. recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2).
5.NF.B Use previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.3Interpret a fraction as the number that results from dividing the whole number numerator by the whole number denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people, each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number and a fraction by a fraction.
5.NF.B.4.aInterpret the product (a/b) x q as a parts of a partition of q into b equal parts. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation.
5.NF.B.4.bInterpret the product of a fraction multiplied by a fraction (a/b) x (c/d). Use a visual fraction model and create a story context for this equation. For example, use a visual fraction model to show (2/3) x (4/5) = 8/15, and create a story context for this equation. In general, (a/b) x (c/d) = ac/bd.
5.NF.B.4.cFind the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.B.5 Interpret multiplication as scaling (resizing), by:
5.NF.B.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.B.5.bExplaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number; 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 x a)/(n x b) to the effect of multiplying a/b by 1.
5.NF.B.6Solve problems in real-world contexts involving multiplication of fractions, including mixed numbers, by using a variety of representations including equations and models.
5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.NF.B.7.aInterpret division of a unit fraction by a non-zero whole number, and compute such quotients. Use the relationship between multiplication and division to justify conclusions.
5.NF.B.7.bInterpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to justify conclusions (e.g., 4 (1/5) = 20 because 20 x (1/5) = 4).
5.NF.B.7.cSolve problems in real-world context involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, using a variety of representations.
5.OA Operations and Algebraic Thinking (OA)
5.OA.A Write and interpret numerical expressions.
5.OA.A.1Use parentheses and brackets in numerical expressions, and evaluate expressions with these symbols (Order of Operations).
5.OA.A.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them (e.g., express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognize that 3 x (18,932 + 921) is three times as large as 18,932 + 921, without having to calculate the indicated sum or product).
5.OA.B Analyze patterns and relationships.
5.OA.B.3Generate two numerical patterns using two given rules (e.g., generate terms in the resulting sequences). Identify and explain the apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane (e.g., given the rule "add 3" and the starting number 0, and given the rule "add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence).
5.OA.B.4Understand primes have only two factors and decompose numbers into prime factors.

Grade 6

6.EE Expressions and Equations (EE)
6.EE.A Apply and extend previous understanding of arithmetic to algebraic expressions.
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 Write, read, and evaluate algebraic expressions.
6.EE.A.2.aWrite expressions that record operations with numbers and variables.
6.EE.A.2.bIdentify parts of an expression using mathematical terms (sum, term, product, factor, quotient, and coefficient); view one or more parts of an expression as a single entity.
6.EE.A.2.cEvaluate expressions given specific values of their variables. Include expressions that arise from formulas used to solve mathematical problems and problems in real-world context. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
6.EE.A.3Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x.
6.EE.A.4Identify when two expressions are equivalent. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
6.EE.B Reason about and solve one-variable equations and inequalities.
6.EE.B.5Understand solving an equation or inequality as a process of reasoning to find the value(s) of the variables that make that equation or inequality true. Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.6Use variables to represent numbers and write expressions when solving mathematical problems and problems in real-world context; understand that a variable can represent an unknown number or any number in a specified set.
6.EE.B.7Solve mathematical problems and problems in real-world context by writing and solving equations of the form x + p = q, x - p = q, px = q, and x/p = q for cases in which p, q and x are all non-negative rational numbers.
6.EE.B.8Write an inequality of the form x > c, x < c, x ≥ c, or x ≤ c to represent a constraint or condition to solve mathematical problems and problems in real-world context. Recognize that inequalities have infinitely many solutions; represent solutions of such inequalities on number lines.
6.EE.C Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.C.9Use variables to represent two quantities that change in relationship to one another to solve mathematical problems and problems in real-world context. Write an equation to express one quantity (the dependent variable) in terms of the other quantity (the independent variable). Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
6.G Geometry (G)
6.G.A Solve mathematical problems and problems in real-world context involving area, surface area, and volume.
6.G.A.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques to solve mathematical problems and problems in real-world context.
6.G.A.2Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Understand and use the formula V = B h, where in this case, B is the area of the base (B = l x w) to find volumes of right rectangular prisms with fractional edge lengths in mathematical problems and problems in real-world context.
6.G.A.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 to solve mathematical problems and problems in a real-world context.
6.G.A.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques to solve mathematical problems and problems in real-world context.
6.NS The Number System (NS)
6.NS.A Apply and extend previous understanding of multiplication and division to divide fractions by fractions.
6.NS.A.1Interpret and compute quotients of fractions to solve mathematical problems and problems in real-world context involving division of fractions by fractions using visual fraction models and equations to represent the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3. In general, a/b ÷ c/d = ad/bc.
6.NS.B Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.2Fluently divide multi-digit numbers using a standard algorithm.
6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.
6.NS.B.4 Use previous understanding of factors to find the greatest common factor and the least common multiple.
6.NS.B.4.aFind the greatest common factor of two whole numbers less than or equal to 100.
6.NS.B.4.bFind the least common multiple of two whole numbers less than or equal to 12.
6.NS.B.4.cUse the distributive property to express a sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9+2).
6.NS.C Apply and extend previous understanding of numbers to the system of rational numbers.
6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative numbers to represent quantities in real-world context, explaining the meaning of 0 in each situation.
6.NS.C.6 Understand a rational number can be represented as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself and that 0 is its own opposite.
6.NS.C.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6.cFind and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7 Understand ordering and absolute value of rational numbers.
6.NS.C.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line.
6.NS.C.7.bWrite, interpret, and explain statements of order for rational numbers in real-world context.
6.NS.C.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 real-world context.
6.NS.C.7.dDistinguish comparisons of absolute value from statements about order in mathematical problems and problems in real-world context.
6.NS.C.8Solve mathematical problems and problems in real-world context by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.RP Ratio and Proportion (RP)
6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1Understand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship between two quantities. For example, "There were 2/3 as many men as women at the concert.”
6.RP.A.2Understand the concept of a unit rate a/b associated with a ratio a : b with b ≠ 0, and use rate language (e.g., for every, for each, for each 1, per) in the context of a ratio relationship. (Complex fraction notation is not an expectation for unit rates in this grade level.)
6.RP.A.3 Use ratio and rate reasoning to solve mathematical problems and problems in real-world context (e.g., by reasoning about data collected from measurements, tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
6.RP.A.3.aMake tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.A.3.bSolve unit rate problems including those involving unit pricing and constant speed.
6.RP.A.3.cFind a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity). Solve percent problems with the unknown in all positions of the equation.
6.RP.A.3.dUse ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
6.SP Statistics and Probability (SP)
6.SP.A Develop understanding of statistical variability.
6.SP.A.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for variability in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.
6.SP.A.2Understand that a set of data collected to answer a statistical question has a distribution whose general characteristics can be described by its center, spread, and overall shape.
6.SP.A.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation uses a single number to describe the spread of the data set.
6.SP.B Summarize and describe distributions.
6.SP.B.4Display and interpret numerical data by creating plots on a number line including histograms, dot plots, and box plots.
6.SP.B.5 Summarize numerical data sets in relation to their context by:
6.SP.B.5.aReporting the number of observations.
6.SP.B.5.bDescribing the nature of the attribute under investigation including how it was measured and its units of measurement.
6.SP.B.5.cGiving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
6.SP.B.5.dRelating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Grade 7

7.EE Expressions and Equations (EE)
7.EE.A Use properties of operations to generate equivalent expressions.
7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.2Rewrite an expression in different forms, and understand the relationship between the different forms and their meanings in a problem context. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."
7.EE.B Solve mathematical problems and problems in real-world context using numerical and algebraic expressions and equations.
7.EE.B.3Solve multi-step mathematical problems and problems in real-world context posed with positive and negative rational numbers in any form. Convert between forms as appropriate and assess the reasonableness of answers. For example, If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50 per hour.
7.EE.B.4 Use variables to represent quantities in mathematical problems and problems in real-world context, and construct simple equations and inequalities to solve problems.
7.EE.B.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 fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
7.EE.B.4.bSolve word problems leading to inequalities of the form px+q > r or px+q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
7.G Geometry (G)
7.G.A Draw, construct, and describe geometrical figures, and describe the relationships between them.
7.G.A.1Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2Draw geometric shapes with given conditions using a variety of methods. 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.
7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures.
7.G.B Solve mathematical problems and problems in real-world context involving angle measure, area, surface area, and volume.
7.G.B.4Understand and use the formulas for the area and circumference of a circle to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure.
7.G.B.6Solve mathematical problems and problems in a real-world context involving area, of two-dimensional objects composed of triangles, quadrilaterals, and other polygons. Solve mathematical problems and problems in real-world context involving volume and surface area of three-dimensional objects composed of cubes and right prisms.
7.NS The Number System (NS)
7.NS.A Apply and extend previous understanding of operations with fractions to add, subtract, multiply, and divide rational numbers except division by zero.
7.NS.A.1 Add and subtract integers and other rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.1.aDescribe situations in which opposite quantities combine to make 0.
7.NS.A.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 context.
7.NS.A.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 context.
7.NS.A.1.dApply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2 Multiply and divide integers and other rational numbers.
7.NS.A.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 context.
7.NS.A.2.bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world context.
7.NS.A.2.cApply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.dConvert a rational number to decimal form using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats.
7.NS.A.3Solve mathematical problems and problems in real-world context involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions where a/b ÷ c/d when a,b,c,and d are all integers and b,c, and d ≠ 0.
7.RP Ratio and Proportion (RP)
7.RP.A Analyze proportional relationships and use them to solve mathematical problems and problems in real-world context.
7.RP.A.1Compute unit rates associated with ratios involving both simple and complex fractions, including ratios of quantities measured in like or different units.
7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.2.aDecide whether two quantities are in a proportional relationship (e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin).
7.RP.A.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.cRepresent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.A.2.dExplain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (e.g., simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error).
7.SP Statistics and Probability (SP)
7.SP.A Use random sampling to draw inferences about a population.
7.SP.A.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.
7.SP.A.2Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
7.SP.B Draw informal comparative inferences about two populations.
7.SP.B.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.B.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
7.SP.C Investigate chance processes and develop, use and evaluate probability models.
7.SP.C.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.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. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies. If the agreement is not good, explain possible sources of the discrepancy.
7.SP.C.7.aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7.bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

Grade 8

8.EE Expressions and Equations (EE)
8.EE.A Work with radicals and integer exponents.
8.EE.A.1Understand and apply the properties of integer exponents to generate equivalent numerical expressions.
8.EE.A.2 Use 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. Know that √2 is irrational.
8.EE.A.2.aEvaluate square roots of perfect squares less than or equal to 225.
8.EE.A.2.bEvaluate cube roots of perfect cubes less than or equal to 1000.
8.EE.A.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and express how many times larger or smaller one is than the other.
8.EE.A.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.
8.EE.B Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5Graph proportional relationships interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at (0, b).
8.EE.C Analyze and solve linear equations, inequalities, and pairs of simultaneous linear equations.
8.EE.C.7 Fluently solve linear equations and inequalities in one variable.
8.EE.C.7.aGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solution. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.C.7.bSolve linear equations and inequalities with rational number coefficients, including solutions that require expanding expressions using the distributive property and collecting like terms.
8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8.bSolve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations including cases of no solution and infinite number of solutions. Solve simple cases by inspection.
8.EE.C.8.cSolve mathematical problems and problems in real-world context leading to two linear equations in two variables.
8.F Functions (F)
8.F.A Define, evaluate, and compare functions.
8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)
8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length in not linear because its graph contains the points (1,1), (2,4), and (3,9) which are not on a straight line.
8.F.B Use functions to model relationships between quantities.
8.F.B.4Given a description of a situation, generate a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or a graph. Track how the values of the two quantities change together. Interpret the rate of change and initial value of a linear function in terms of the situation it models, its graph, or its table of values.
8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
8.G Geometry (G)
8.G.A Understand congruence and similarity.
8.G.A.1Verify experimentally the properties of rotations, reflections, and translations. Properties include: lines are taken to lines, line segments are taken to line segments of the same length, angles are taken to angles of the same measure, parallel lines are taken to parallel lines.
8.G.A.2Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that demonstrates congruence.
8.G.A.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4Understand that a two-dimensional figure is similar to another if, and only if, one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that demonstrates similarity.
8.G.A.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. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B Understand and apply the Pythagorean Theorem.
8.G.B.6Understand the Pythagorean Theorem and its converse.
8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world context and mathematical problems in two and three dimensions.
8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.C.9Understand and use formulas for volumes of cones, cylinders and spheres and use them to solve real-world context and mathematical problems.
8.NS The Number System (NS)
8.NS.A Understand that there are irrational numbers, and approximate them using rational numbers.
8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion. Know that numbers whose decimal expansions do not terminate in zeros or in a repeating sequence of fixed digits are called irrational.
8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers. Locate them approximately on a number line diagram, and estimate their values.
8.NS.A.3Understand that given any two distinct rational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b. Given any two distinct irrational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b.
8.SP Statistics and Probability (SP)
8.SP.A Investigate patterns of association in bivariate data.
8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate and describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.A.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
8.SP.B Investigate chance processes and develop, use, and evaluate probability models.
8.SP.B.5 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
8.SP.B.5.aUnderstand that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
8.SP.B.5.bRepresent sample spaces for compound events using organized lists, tables, tree diagrams and other methods. Identify the outcomes in the sample space which compose the event.
8.SP.B.5.cDesign and use a simulation to generate frequencies for compound events.

Algebra 1

A1.A-APR Algebra – Arithmetic with Polynomials and Rational Expressions (A-APR)
A1.A-APR.A Perform arithmetic operations on polynomials.
A1.A-APR.A.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A1.A-APR.B Understand the relationship between zeros and factors of polynomials.
A1.A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic and cubic polynomials in which linear and quadratic factors are available.
A1.A-CED Algebra – Creating Equations (A-CED)
A1.A-CED.A Create equations that describe numbers or relationships.
A1.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities that are linear, quadratic, or exponential.
A1.A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A1.A-CED.A.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A1.A-CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
A1.A-REI Algebra – Reasoning with Equations and Inequalities (A-REI)
A1.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
A1.A-REI.A.1Explain each step in solving linear and quadratic equations 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.
A1.A-REI.B Solve equations and inequalities in one variable.
A1.A-REI.B.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A1.A-REI.B.4 Solve quadratic equations in one variable.
A1.A-REI.B.4.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – k)^2 = q that has the same solutions. Derive the quadratic formula from this form.
A1.A-REI.B.4.bSolve quadratic equations by inspection (e.g., x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Focus on solutions for quadratic equations that have real roots. Include cases that recognize when a quadratic equation has no real solutions.
A1.A-REI.C Solve systems of equations.
A1.A-REI.C.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
A1.A-REI.C.6Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.
A1.A-REI.D Represent and solve equations and inequalities graphically.
A1.A-REI.D.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve, which could be a line.
A1.A-REI.D.11Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x) =g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Focus on cases where f(x) and/or g(x) are linear, absolute value, quadratic, and exponential functions.
A1.A-REI.D.12Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary in the case of a strict inequality, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
A1.A-SSE Algebra – Seeing Structure in Expressions (A-SSE)
A1.A-SSE.A Interpret the structure of expressions.
A1.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
A1.A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
A1.A-SSE.A.1.bInterpret expressions by viewing one or more of their parts as a single entity.
A1.A-SSE.A.2Use structure to identify ways to rewrite numerical and polynomial expressions. Focus on polynomial multiplication and factoring patterns.
A1.A-SSE.B Write expressions in equivalent forms to solve problems.
A1.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A1.A-SSE.B.3.aFactor a quadratic expression to reveal the zeros of the function it defines.
A1.A-SSE.B.3.bComplete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A1.F-BF Functions – Building Functions (F-BF)
A1.F-BF.A Build a function that models a relationship between two quantities.
A1.F-BF.A.1Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from real-world context. Focus on linear, absolute value, quadratic, exponential, and piecewise-defined functions (limited to the aforementioned functions).
A1.F-BF.B Build new functions from existing functions.
A1.F-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph. Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).
A1.F-IF Functions – Interpreting Functions (F-IF)
A1.F-IF.A Understand the concept of a function and use function notation.
A1.F-IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
A1.F-IF.A.2Evaluate a function for inputs in the domain, and interpret statements that use function notation in terms of a context.
A1.F-IF.A.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
A1.F-IF.B Interpret functions that arise in applications in terms of the context.
A1.F-IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Include problem-solving opportunities utilizing real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums. Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).
A1.F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
A1.F-IF.B.6Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Focus on linear, absolute value, quadratic, and exponential functions.
A1.F-IF.C Analyze functions using different representations.
A1.F-IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Functions include linear, exponential, quadratic, and piecewise-defined functions (limited to the aforementioned functions).
A1.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
A1.F-IF.C.8.aUse the process of factoring and completing the square of a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
A1.F-IF.C.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).
A1.F-LE Functions – Linear, Quadratic, and Exponential Models (F-LE)
A1.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
A1.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
A1.F-LE.A.1.aProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
A1.F-LE.A.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
A1.F-LE.A.1.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
A1.F-LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input/output pairs.
A1.F-LE.A.3Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.
A1.F-LE.B Interpret expressions for functions in terms of the situation they model.
A1.F-LE.B.5Interpret the parameters in a linear or exponential function with integer exponents utilizing real world context.
A1.N-Q Number and Quantity – Quantities (N-Q)
A1.N-Q.A Reason quantitatively and use units to solve problems.
A1.N-Q.A.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
A1.N-Q.A.2Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
A1.N-Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.
A1.N-RN Number and Quantity – The Real Number System (N-RN)
A1.N-RN.B Use properties of rational and irrational numbers.
A1.N-RN.B.3Explain 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.
A1.S-CP Statistics and Probability – Conditional Probability and the rules of Probability (S-CP)
A1.S-CP.A Understand independence and conditional probability and use them to interpret data.
A1.S-CP.A.1Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events.
A1.S-CP.A.2Use the Multiplication Rule for independent events to understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
A1.S-ID Statistics and Probability – Summarize, represent, and interpret data on a single count or measurement variable. (S-ID)
A1.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable.
A1.S-ID.A.1Represent real-value data with plots for the purpose of comparing two or more data sets.
A1.S-ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
A1.S-ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of outliers if present.
A1.S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
A1.S-ID.B.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data, including joint, marginal, and conditional relative frequencies. Recognize possible associations and trends in the data.
A1.S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the quantities are related.
A1.S-ID.B.6.aFit a function to the data; use functions fitted to data to solve problems in the context of the data. Focus on linear models.
A1.S-ID.B.6.bInformally assess the fit of a function by plotting and analyzing residuals.
A1.S-ID.C Interpret linear models.
A1.S-ID.C.7Interpret the slope as a rate of change and the constant term of a linear model in the context of the data.
A1.S-ID.C.8Compute and interpret the correlation coefficient of a linear relationship.
A1.S-ID.C.9Distinguish between correlation and causation.

Algebra 2

A2.A-APR Algebra – Arithmetic with Polynomials and Rational Expressions (A-APR)
A2.A-APR.B Understand the relationship between zeros and factors of polynomials.
A2.A-APR.B.2Know and apply the Remainder and Factor Theorem: For a polynomial p(x) and a number a, the remainder on division by (x – a) is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A2.A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic, cubic, and quartic polynomials including polynomials for which factors are not provided.
A2.A-APR.C Use polynomial identities to solve problems.
A2.A-APR.C.4Prove polynomial identities and use them to describe numerical relationships.
A2.A-APR.D Rewrite rational expressions.
A2.A-APR.D.6Rewrite rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or for the more complicated examples, a computer algebra system.
A2.A-CED Algebra – Creating Equations (A-CED)
A2.A-CED.A Create equations that describe numbers or relationships.
A2.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities arising from linear, quadratic, rational, and exponential functions.
A2.A-REI Algebra – Reasoning with Equations and Inequalities (A-REI)
A2.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
A2.A-REI.A.1Explain each step in solving an 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. Extend from quadratic equations to rational and radical equations.
A2.A-REI.A.2Solve rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A2.A-REI.B Solve equations and inequalities in one variable.
A2.A-REI.B.4Fluently solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b.
A2.A-REI.C Solve systems of equations.
A2.A-REI.C.7Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x^2 + y^2 = 3.
A2.A-REI.D Represent and solve equations and inequalities graphically.
A2.A-REI.D.11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Include problems in real-world context. Extend from linear, quadratic, and exponential functions to cases where f(x) and/or g(x) are polynomial, rational, exponential, and logarithmic functions.
A2.A-SSE Algebra – Seeing Structure in Expressions (A-SSE)
A2.A-SSE.A Interpret the structure of expressions.
A2.A-SSE.A.2Use structure to identify ways to rewrite polynomial and rational expressions. Focus on polynomial operations and factoring patterns.
A2.A-SSE.B Write expressions in equivalent forms to solve problems.
A2.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Include problem-solving opportunities utilizing real-world context and focus on expressions with rational exponents.
A2.A-SSE.B.3.cUse the properties of exponents to transform expressions for exponential functions.
A2.A-SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, use the quadratic formula to solve problems such as calculating mortgage payments on a fixed rate mortgage.
A2.F-BF Functions – Building Functions (F-BF)
A2.F-BF.A Build a function that models a relationship between two quantities.
A2.F-BF.A.1 Write a function that describes a relationship between two quantities. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, exponential, and piecewise-defined functions. Include problem-solving opportunities utilizing real-world context.
A2.F-BF.A.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.
A2.F-BF.A.1.bCombine function types using arithmetic operations and function composition.
A2.F-BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
A2.F-BF.B Build new functions from existing functions.
A2.F-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, and exponential functions, and piecewise-defined functions.
A2.F-BF.B.4 Find inverse functions.
A2.F-BF.B.4.aUnderstand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, recognizing that functions f and g are inverse functions if and only if f(x) = y and g(y) = x for all values of x in the domain of f and all values of y in the domain of g.
A2.F-BF.B.4.bUnderstand that if a function contains a point (a,b), then the graph of the inverse relation of the function contains the point (b,a).
A2.F-BF.B.4.cInterpret the meaning of and relationship between a function and its inverse utilizing real-world context.
A2.F-IF Functions – Interpreting Functions (F-IF)
A2.F-IF.B Interpret functions that arise in applications in terms of the context.
A2.F-IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Include problem-solving opportunities utilizing a real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Extend from linear, quadratic and exponential to include polynomial, radical, logarithmic, rational, sine, cosine, tangent, exponential, and piecewise-defined functions.
A2.F-IF.B.6Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, sine, cosine, tangent, exponential, and piecewise-defined functions.
A2.F-IF.C Analyze functions using different representations.
A2.F-IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend from linear, quadratic and exponential functions to include square root, cube root, polynomial, exponential, logarithmic, sine, cosine, tangent and piecewise-defined functions.
A2.F-IF.C.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions and classify those functions as exponential growth or decay.
A2.F-IF.C.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.). Extend from linear, quadratic and exponential functions to include polynomial, radical, logarithmic, rational, trigonometric, exponential, and piecewise-defined functions.
A2.F-LE Functions – Linear, Quadratic, and Exponential Models (F-LE)
A2.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
A2.F-LE.A.4For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithms that are not readily found by hand or observation using technology.
A2.F-LE.B Interpret expressions for functions in terms of the situation they model.
A2.F-LE.B.5Interpret the parameters in an exponential function with rational exponents utilizing real-world context.
A2.F-TF Functions – Trigonometric Functions (F-TF)
A2.F-TF.A Extend the domain of trigonometric functions using the unit circle.
A2.F-TF.A.1Understand radian measure of an angle as the length of the arc on any circle subtended by the angle, measured in units of the circle's radius.
A2.F-TF.A.2Explain how the unit circle in the coordinate plane enables the extension of sine and cosine functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
A2.F-TF.B Model periodic phenomena with trigonometric functions.
A2.F-TF.B.5Create and interpret trigonometric functions that model periodic phenomena with specified amplitude, frequency, and midline.
A2.F-TF.C Apply trigonometric identities.
A2.F-TF.C.8Use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 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.
A2.N-CN Number and Quantity – The Complex Number System (N–CN)
A2.N-CN.A Perform arithmetic operations with complex numbers.
A2.N-CN.A.1Apply the relation i^2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Write complex numbers in the form (a+bi) with a and b real.
A2.N-CN.C Use complex numbers in polynomial identities and equations.
A2.N-CN.C.7Solve quadratic equations with real coefficients that have complex solutions.
A2.N-Q Number and Quantity – Quantities (N-Q)
A2.N-Q.A Reason quantitatively and use units to solve problems.
A2.N-Q.A.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
A2.N-Q.A.2Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
A2.N-Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.
A2.N-RN Number and Quantity – The Real Number System (N-RN)
A2.N-RN.A Extend the properties of exponents to rational exponents.
A2.N-RN.A.1Explain how the definition of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
A2.N-RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
A2.S-CP Statistics and Probability – Conditional Probability and the Rules of Probability (S-CP)
A2.S-CP.A Understand independence and conditional probability and use them to interpret data.
A2.S-CP.A.3Understand the conditional probability of A given B as P(A and B)/P(B), and 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.
A2.S-CP.A.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
A2.S-CP.A.5Recognize and explain the concepts of conditional probability and independence utilizing real-world context.
A2.S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
A2.S-CP.B.6Use Bayes Rule to find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
A2.S-CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
A2.S-CP.B.8Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
A2.S-IC Statistics and Probability – Making Inferences and Justifying Conclusions (S-IC)
A2.S-IC.A Understand and evaluate random processes underlying statistical experiments.
A2.S-IC.A.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
A2.S-IC.A.2Explain whether a specified model is consistent with results from a given data-generating process.
A2.S-IC.B Make inferences and justify conclusions from experiments, and observational studies.
A2.S-IC.B.3Recognize the purposes of and differences between designed experiments, sample surveys and observational studies.
A2.S-IC.B.4Use data from a sample survey to estimate a population mean or proportion; recognize that estimates are unlikely to be correct and the estimates will be more precise with larger sample sizes.
A2.S-ID Statistics and Probability – Interpreting Categorical and Quantitative Data (S-ID)
A2.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable.
A2.S-ID.A.4Use the mean and standard deviation of a data set to fit it to a normal curve, and use properties of the normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, or tables to estimate areas under the normal curve.
A2.S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
A2.S-ID.B.6 Represent data of two quantitative variables on a scatter plot, and describe how the quantities are related. Extend to polynomial and exponential models.
A2.S-ID.B.6.aFit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.
A2.S-ID.C Interpret models.
A2.S-ID.C.10Interpret parameters of exponential models.

Geometry

G.G-C Geometry – Circles (G-C)
G.G-C.A Understand and apply theorems about circles.
G.G-C.A.1Prove that all circles are similar.
G.G-C.A.2Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G.G-C.A.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G.G-C.B Find arc lengths and areas of sectors of circles.
G.G-C.B.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Convert between degrees and radians.
G.G-CO Geometry – Congruence (G-CO)
G.G-CO.A Experiment with transformations in the plane.
G.G-CO.A.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.G-CO.A.2Represent and describe transformations in the plane as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.
G.G-CO.A.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G.G-CO.A.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.G-CO.A.5Given a geometric figure and a rotation, reflection, or translation draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another.
G.G-CO.B Understand congruence in terms of rigid motions.
G.G-CO.B.6Use geometric definitions 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.
G.G-CO.B.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.G-CO.B.8Explain how the criteria for triangle congruence (ASA, AAS, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.G-CO.C Prove geometric theorems.
G.G-CO.C.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180; base angles of isosceles triangle are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.G-CO.C.11Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and rectangles are parallelograms with congruent diagonals.
G.G-CO.C.9en a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
G.G-CO.D Make geometric constructions.
G.G-CO.D.12Make formal geometric constructions with a variety of tools and methods. Constructions include: copying segments; copying angles; bisecting segments; bisecting angles; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G.G-CO.D.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle; with a variety of tools and methods.
G.G-GMD Geometry – Geometric Measurement and Dimension (G-GMD)
G.G-GMD.A Explain volume formulas and use them to solve problems.
G.G-GMD.A.1Analyze and verify the formulas for the volume of a cylinder, pyramid, and cone.
G.G-GMD.A.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems utilizing real-world context.
G.G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects.
G.G-GMD.B.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
G.G-GPE Geometry – Expressing Geometric Properties with Equations (G-GPE)
G.G-GPE.A Translate between the geometric description and the equation for a conic section.
G.G-GPE.A.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
G.G-GPE.B Use coordinates to prove geometric theorems algebraically.
G.G-GPE.B.4Use coordinates to algebraically prove or disprove geometric relationships algebraically. Relationships include: proving or disproving geometric figures given specific points in the coordinate plane; and proving or disproving if a specific point lies on a given circle.
G.G-GPE.B.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems, including finding the equation of a line parallel or perpendicular to a given line that passes through a given point.
G.G-GPE.B.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G.G-GPE.B.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.
G.G-MG Geometry – Modeling with Geometry (G-MG)
G.G-MG.A Apply geometric concepts in modeling situations.
G.G-MG.A.1Use geometric shapes, their measures, and their properties to describe objects utilizing real-world context.
G.G-MG.A.2Apply concepts of density based on area and volume in modeling situations utilizing real-world context.
G.G-MG.A.3Apply geometric methods to solve design problems utilizing real-world context.
G.G-SRT Geometry – Similarity, Right Triangles, and Trigonometry (G-SRT)
G.G-SRT.A Understand similarity in terms of similarity transformations.
G.G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
G.G-SRT.A.1.aDilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G.G-SRT.A.1.bThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.G-SRT.A.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.G-SRT.A.3Use the properties of similarity transformations to establish the AA, SAS, and SSS criterion for two triangles to be similar.
G.G-SRT.B Prove theorems involving similarity.
G.G-SRT.B.4Prove theorems about triangles. Theorems include: an interior line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.G-SRT.B.5Use congruence and similarity criteria to prove relationships in geometric figures and solve problems utilizing real-world context.
G.G-SRT.C Define trigonometric ratios and solve problems involving right triangles.
G.G-SRT.C.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.G-SRT.C.7Explain and use the relationship between the sine and cosine of complementary angles.
G.G-SRT.C.8Use trigonometric ratios (including inverse trigonometric ratios) and the Pythagorean Theorem to find unknown measurements in right triangles utilizing real-world context.
G.N-Q Number and Quantity – Quantities (N-Q)
G.N-Q.A Reason quantitatively and use units to solve problems.
G.N-Q.A.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
G.N-Q.A.2Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
G.N-Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

Plus Standards for High School

P.A-APR Algebra – Arithmetic with Polynomials and Rational Expressions (A–APR)
P.A-APR.C Use polynomial identities to solve problems.
P.A-APR.C.5Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.
P.A-APR.D Rewrite rational expressions.
P.A-APR.D.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
P.A-REI Algebra – Reasoning with Equations and Inequalities (A-REI)
P.A-REI.C Solve systems of equations.
P.A-REI.C.8Represent a system of linear equations as a single matrix equation in a vector variable.
P.A-REI.C.9Find the inverse of a matrix if it exists, and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
P.CM-DM Contemporary Mathematics – Discrete Mathematics (CM-DM)
P.CM-DM.A Understand and apply vertex-edge graph topics
P.CM-DM.A.1Study the following topics related to vertex-edge graph: Euler circuits, Hamilton circuits, shortest path, vertex coloring, and adjacency matrices.
P.CM-DM.A.2Understand, analyze, and apply vertex-edge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in real-world and abstract settings.
P.CM-DM.A.3Devise, analyze, and apply algorithms for solving vertex-edge graph problems.
P.CM-DM.A.4Extend work with adjacency matrices for graphs, such as interpreting row sums and using the nth power of the adjacency matrix to count paths of length n in a graph.
P.F-BF Functions – Building Functions (F-BF)
P.F-BF.A Build a function that models a relationship between two quantities.
P.F-BF.A.1 Write a function that describes a relationship between two quantities.
P.F-BF.A.1.cCompose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
P.F-BF.B Build new functions from existing functions.
P.F-BF.B.4 Find inverse functions.
P.F-BF.B.4.bVerify by composition that one function is the inverse of another.
P.F-BF.B.4.cRead values of an inverse function from a graph or a table, given that the function has an inverse.
P.F-BF.B.4.dProduce an invertible function from a non-invertible function by restricting the domain.
P.F-BF.B.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
P.F-IF Functions – Interpreting Functions (F–IF)
P.F-IF.C Analyze functions using different representations.
P.F-IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
P.F-TF Functions – Trigonometric Functions (F–TF)
P.F-TF.A Extend the domain of trigonometric functions using the unit circle.
P.F-TF.A.3Use 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.
P.F-TF.A.4Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
P.F-TF.B Model periodic phenomena with trigonometric functions.
P.F-TF.B.6Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
P.F-TF.B.7Use inverse functions to solve trigonometric equations utilizing real world context; evaluate the solution and interpret them in terms of context.
P.F-TF.C Apply trigonometric identities.
P.F-TF.C.9Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
P.G-C Geometry – Circles (G-C)
P.G-C.A Understand and apply theorems about circles.
P.G-C.A.4Construct a tangent line from a point outside a given circle to the circle.
P.G-GMD Geometry – Geometric Measurement and Dimension (G-GMD)
P.G-GMD.A Explain volume formulas and use them to solve problems.
P.G-GMD.A.2Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
P.G-GPE Geometry – Expressing Geometric Properties with Equations (G-GPE)
P.G-GPE.A Translate between the geometric description and the equation for a conic section.
P.G-GPE.A.2Derive the equation of a parabola given a focus and directrix.
P.G-GPE.A.3Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
P.G-SRT Geometry – Similarity, Right Triangles, and Trigonometry (G-SRT)
P.G-SRT.D Apply trigonometry to general triangles.
P.G-SRT.D.10Prove the Laws of Sines and Cosines and use them to solve problems.
P.G-SRT.D.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
P.G-SRT.D.9Derive the formula A = ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
P.N-CN Number and Quantity – The Complex Number System (N–CN)
P.N-CN.A Perform arithmetic operations with complex numbers.
P.N-CN.A.3Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
P.N-CN.B Represent complex numbers and their operations on the complex plane.
P.N-CN.B.4Represent complex numbers on the complex plane in rectangular and polar form, including real and imaginary numbers, and explain why the rectangular and polar forms of a given complex number represent the same number.
P.N-CN.B.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)^3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
P.N-CN.B.6Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
P.N-CN.C Use complex numbers in polynomial identities and equations.
P.N-CN.C.8Extend polynomial identities to the complex numbers.
P.N-CN.C.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
P.N-VM Number and Quantity – Vector and Matrix Quantities (N–VM)
P.N-VM.A Represent and model with vector quantities.
P.N-VM.A.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
P.N-VM.A.2Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
P.N-VM.A.3Solve problems involving velocity and other quantities that can be represented by vectors.
P.N-VM.B P.N-VM.B
P.N-VM.B.4 Add and subtract vectors.
P.N-VM.B.4.aAdd vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
P.N-VM.B.4.bGiven two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
P.N-VM.B.4.cUnderstand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
P.N-VM.B.5 Multiply a vector by a scalar.
P.N-VM.B.5.aRepresent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise e.g., as c(v_x, v_y) = (cv_x, cv_y).
P.N-VM.B.5.bCompute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
P.N-VM.C Perform operations on matrices and use matrices in applications.
P.N-VM.C.10Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
P.N-VM.C.11Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
P.N-VM.C.12Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
P.N-VM.C.6Use matrices to represent and manipulate data.
P.N-VM.C.7Multiply matrices by scalars to produce new matrices.
P.N-VM.C.8Add, subtract, and multiply matrices of appropriate dimensions.
P.N-VM.C.9Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
P.N-VN P.N-VN
P.N-VN.BPerform operations on vectors.
P.S-CP Statistics and Probability – Conditional Probability and the Rules of Probability (S-CP)
P.S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
P.S-CP.B.9Use permutations and combinations to compute probabilities of compound events and solve problems.
P.S-IC Statistics and Probability – Making Inferences and Justifying Conclusions (S-IC)
P.S-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
P.S-IC.B.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
P.S-IC.B.4Use data from a random sample to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
P.S-IC.B.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
P.S-IC.B.6Evaluate reports based on data.
P.S-MD Statistics and Probability – Using Probability to Make Decisions (S-MD)
P.S-MD.A Calculate expected values and use them to solve problems.
P.S-MD.A.1Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
P.S-MD.A.2Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
P.S-MD.A.3Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated. Find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
P.S-MD.A.4Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically. Find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
P.S-MD.B Use probability to evaluate outcomes of decisions.
P.S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
P.S-MD.B.5.aFind the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
P.S-MD.B.5.bEvaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
P.S-MD.B.6Use randomization to make fair decisions based on probabilities.
P.S-MD.B.7Analyze decisions and strategies using probability concepts.

Precalculus

PC.RFR Reasoning with Functions and Relations (RFR)
PC.RFR.AF Analyze functions
PC.RFR.AF.1Interpret parameters of a function defined by an expression in the context of the situation.
PC.RFR.AF.2Sketch the graph of a function that models a relationship between two quantities, identifying key features.
PC.RFR.AF.3Interpret key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities.
PC.RFR.AF.4Use limits to describe long-range behavior, asymptotic behavior, and points of discontinuity.
PC.RFR.AF.5Sketch the graph of all six trigonometric functions, identifying key features.
PC.RFR.BF Building Functions
PC.RFR.BF.1Model relationships between quantities that require adding, subtracting, multiplying, and/or dividing functions
PC.RFR.BF.2Model relationships through composition and attend to the restrictions of the domain.
PC.RFR.BF.3Rewrite a function as a composition of functions.
PC.RFR.BF.4Determine if a function has an inverse. If so, find the inverse. If not, define a restriction on the domain that meets the requirement for invertibility and find the inverse on the restricted domain.
PC.RFR.BF.5Interpret the meanings of quantities involving functions and their inverses.
PC.RFR.BF.6Verify by analytical methods that one function is the inverse of another.
PC.RFR.ETT PC.RFR.ETT
PC.RFR.ETT.1Model real-world situations involving trigonometry.
PC.RFR.ETT.2Apply the Law of Sines and Law of Cosines to solve problems.
PC.RFR.ETT.3Use trigonometry to find the area of triangles.
PC.RFR.ETT.4Use 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.
PC.RFR.ETT.5Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
PC.RFR.ETT.6Use inverse functions to solve trigonometric equations utilizing real world context; evaluate the solution and interpret them in terms of context.
PC.RFR.IC Interpreting Conics
PC.RFR.IC.1Model real-world situations which involve conic sections.
PC.RFR.IC.2Identify key features of conic sections (foci, directrix, radii, axes, asymptotes, center) graphically and algebraically.
PC.RFR.IC.3Sketch a graph of a conic section using its key features.
PC.RFR.IC.4Use the key features of a conic section to write its equation.
PC.RFR.IC.5Given a quadratic equation of the form ax2+ by2 + cx + dy + e = 0, determine if the equation is a circle, ellipse, parabola, or hyperbola.
PC.RFR.ISS Interpreting Sequences and Series
PC.RFR.ISS.1Model real-world situations involving sequences or series using recursive and/or explicit definitions.
PC.RFR.ISS.2Use covariational reasoning to describe sequences and series.
PC.RFR.ISS.3Represent finite or infinite series using sigma notation.
PC.RFR.ISS.4Find the sums of finite or infinite series, if they exist.
PC.RM Reasoning with Matrices
PC.RM.UM Using Matrices
PC.RM.UM.1Use matrices to represent and manipulate data.
PC.RM.UM.2Use matrix operations to solve problems. Add, subtract, and multiply matrices of appropriate dimensions. Multiply matrices by scalars to produce new matrices.
PC.RM.UM.3Find the inverse and determinant of a matrix.
PC.RM.UM.4Use matrices to solve systems of linear equations.
PC.RT Reasoning with Trigonometry
PC.RT.EPE Exploring Polar Equations
PC.RT.EPE.1Graph polar equations.
PC.RT.EPE.2Analyze and interpret the graphs of polar equations.
PC.RT.EPE.3Use polar equations to solve problems.
PC.RT.ETTExtending Triangle Trigonometry
PC.RT.RTS Reasoning with Trigonometric Structure
PC.RT.RTS.1Use the structure of a trigonometric expression to identify ways to rewrite it.
PC.RT.RTS.2Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
PC.RT.RTS.3Solve trigonometric equations.
PC.RV Reasoning with Vectors
PC.RV.EV Exploring Vectors
PC.RV.EV.1Recognize vector quantities as having both magnitude and direction.
PC.RV.EV.2Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
PC.RV.EV.3Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
PC.RV.EV.4Solve problems involving velocity and other quantities that can be represented by vectors.
PC.RV.EV.5Add and subtract vectors, and multiply a vector by a scalar.
PC.RV.MP Modeling with Parametrics
PC.RV.MP.1Model real-world contexts with parametric equations.
PC.RV.MP.2Use parametric equations to solve problems.
PC.RV.MP.3Graph parametric equations and identify orientation.
PC.RV.MP.4Analyze and interpret the graphs of parametric equations.

Quantitative Reasoning

QR.CR Covariational Reasoning (CR)
QR.CR.1Analyze and compare growth and decay using absolute and relative change utilizing real-world contexts.
QR.CR.2Compare, reason and communicate about proportional and non-proportional models utilizing real-world contexts.
QR.CR.3Identify, create, and use appropriate models for bivariate data sets (i.e. linear, exponential) to estimate solutions for contextual questions, identify patterns and identify how changing parameters affect the models.
QR.DMR Discrete Mathematical Reasoning (DMR)
QR.DMR.1Understand, analyze, and apply vertex-edge graphs to model and make informed decisions related to paths, circuits, networks, and relationships in real-world settings. Encompasses P.CM-DM.A.1, P.CM-DM.A.2
QR.DMR.2Devise, analyze, and apply algorithms for solving vertex-edge graph problems. P.CM-DM.A.3
QR.DMR.3Extend work with adjacency matrices for graphs, such as interpreting row sums and using the nth power of the adjacency matrix to count paths of length n in a graph. P.CM-DM.A.4
QR.FR Financial Reasoning (FR)
QR.FR.1Identify and research a career goal. Develop a plan and time table for achieving it including educational/training requirements, costs, and other factors (e.g. cost versus savings, income and debt).
QR.FR.2Understand and apply strategies to monitor income and expenses, plan for spending, implement a diversified investment strategy, and save for future goals.
QR.FR.3Use models to solve and communicate about contextual financial questions such as credit card debt, installment savings, amortization schedules, mortgage and other loan scenarios.
QR.FR.4Identify and explain personal and societal consequences of financial decisions.
QR.NR Numerical Reasoning (NR)
QR.NR.1Represent quantities, using equivalent forms when appropriate, to investigate and describe quantitative and geometric relationships and solve problems in real-world contexts.
QR.NR.2Reason, model, and communicate with and about percentages (change, incorrect, deceptive, relative and absolute).
QR.NR.3Understand and compare magnitudes of numbers utilizing real-world context. Understand the importance and impact of unit selection.
QR.NR.4Use and justify estimation skills, and know why, how, and when to estimate results. Assess and justify the reasonableness of estimations using the context and comparisons to other known values.
QR.SPR Statistical and Probabilistic Reasoning (SPR)
QR.SPR.1Reason and communicate about the validity of claims based on empirical, theoretical, and subjective probabilities. Draw conclusions or make decisions related to risk, pay-off, expected value, and false negatives/positives in various probabilistic contexts. Encompasses P.S-CP.B.9, P.S-MD.A.2, P.S-MD.A.3, P.S-MD.A.4, P.S-MD.B.5, P.S-MD.B.7
QR.SPR.2Analyze statistical information and identify limitations, strengths, or lack of information in studies including data collection methods (e.g. sampling, experimental, observational) and possible sources of bias. Identify errors or misuses of statistics to justify particular conclusions. Encompasses P.S-IC.B.3
QR.SPR.3Represent numerical summaries and visual displays of real-world data to make informed decisions. Reason, communicate, and describe strengths, limitations, and fallacies of various displays. Encompasses P.S-IC.B.6
QR.SPR.4Represent center, shape, and spread of two or more data sets. Reason, communicate, and compare data sets in context.

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.