CA CCSS and IXL

List of all standards with links

to practice questions

K.K.CC Counting and Cardinality

K. Know number names and the count sequence.
- K.K.CC.1 Count to 100 by ones and by tens.
- K.K.CC.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
- K.K.CC.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
K. Count to tell the number of objects.
- K.K.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality.
  - K.K.CC.4.a When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
  - K.K.CC.4.b Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
  - K.K.CC.4.c Understand that each successive number name refers to a quantity that is one larger.
- K.K.CC.5 Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
K. Compare numbers.
- K.K.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
- K.K.CC.7 Compare two numbers between 1 and 10 presented as written numerals.
  - Compare two numbers - up to 10 (K-G.6)

K.K.OA Operations and Algebraic Thinking

K. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
- K.K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
- K.K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
- K.K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
- K.K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
  - Count to fill a ten frame (K-C.17)
  - Complete the addition sentence - make ten (K-J.5)
- K.K.OA.5 Fluently add and subtract within 5.

K.K.NBT Number and Operations in Base Ten

K. Work with numbers 11–19 to gain foundations for place value.
- K.K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
  - Count tens and ones - up to 20 (K-D.16)
  - Write tens and ones - up to 20 (K-D.17)

K.K.MD Measurement and Data

K. Describe and compare measurable attributes.
- K.K.MD.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
- K.K.MD.2 Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference.
K. Classify objects and count the number of objects in each category.
- K.K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

K.K.G Geometry

K. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
- K.K.G.1 Describe 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.K.G.2 Correctly name shapes regardless of their orientations or overall size.
- K.K.G.3 Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").
  - Two-dimensional and three-dimensional shapes (K-W.1)
K. Analyze, compare, create, and compose shapes.
- K.K.G.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length).
- K.K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
- K.K.G.6 Compose simple shapes to form larger shapes.

1.1.OA Operations and Algebraic Thinking

1 Represent and solve problems involving addition and subtraction.
- 1.1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
- 1.1.OA.2 Solve 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 equations with a symbol for the unknown number to represent the problem.
  - Add three numbers - word problems (1-E.12)
1 Understand and apply properties of operations and the relationship between addition and subtraction.
- 1.1.OA.3 Apply properties of operations as strategies to add and subtract.
- 1.1.OA.4 Understand subtraction as an unknown-addend problem.
  - Complete the addition sentence - sums up to 10 (1-D.3)
  - Complete the addition sentence - make ten (1-E.7)
1 Add and subtract within 20.
- 1.1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
- 1.1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
1 Work with addition and subtraction equations.
- 1.1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
- 1.1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

1.1.NBT Number and Operations in Base Ten

1 Extend the counting sequence.
- 1.1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
1 Understand place value.
- 1.1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
  - Counting tens and ones - up to 99 (1-A.8)
  - Hundred chart (1-A.14)
  - Place value models up to 100 (1-M.3)
  - Write numbers as tens and ones (1-M.5)
  - 1.1.NBT.2.a 10 can be thought of as a bundle of ten ones - called a "ten."
  - 1.1.NBT.2.b The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
  - 1.1.NBT.2.c The 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).
    - Convert between tens and ones (1-M.4)
- 1.1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
  - Comparing numbers up to 100 (1-K.3)
1 Use place value understanding and properties of operations to add and subtract.
- 1.1.NBT.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models 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 method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
- 1.1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
  - Ten more or less (1-J.8)
- 1.1.NBT.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models 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 method and explain the reasoning used.
  - Subtract multiples of 10 (1-I.3)

1.1.MD Measurement and Data

1 Measure lengths indirectly and by iterating length units.
- 1.1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object.
  - Compare objects: length and height (1-P.2)
- 1.1.MD.2 Express 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.
  - Measure using objects (1-P.3)
1 Tell and write time.
- 1.1.MD.3 Tell and write time in hours and half-hours using analog and digital clocks.
1 Represent and interpret data.
- 1.1.MD.4 Organize, 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.

1.1.G Geometry

1 Reason with shapes and their attributes.
- 1.1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
- 1.1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
- 1.1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
  - Equal parts - halves and fourths (1-X.1)
  - Halves and fourths (1-X.6)

2.2.OA Operations and Algebraic Thinking

2 Represent and solve problems involving addition and subtraction.
- 2.2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
2 Add and subtract within 20.
- 2.2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
2 Work with equal groups of objects to gain foundations for multiplication.
- 2.2.OA.3 Determine 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 2s; write an equation to express an even number as a sum of two equal addends.
  - Even or odd (2-A.9)
  - Add doubles - complete the sentence (2-E.12)
- 2.2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

2.2.NBT Number and Operations in Base Ten

2 Understand place value.
- 2.2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
  - Place value models - up to hundreds (2-M.2)
  - Identify a digit up to the hundreds place (2-M.4)
  - Place value - up to hundreds (2-M.6)
  - Convert to/from a number - up to hundreds (2-M.11)
  - 2.2.NBT.1.a 100 can be thought of as a bundle of ten tens - called a "hundred."
  - 2.2.NBT.1.b The 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).
- 2.2.NBT.2 Count within 1000; skip-count by 2s, 5s, 10s, and 100s.
- 2.2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
- 2.2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
  - Comparing numbers up to 1,000 (2-B.2)
2 Use place value understanding and properties of operations to add and subtract.
- 2.2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
- 2.2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.
  - Add three numbers up to two digits each (2-G.13)
  - Add four or more numbers up to two digits each (2-G.15)
- 2.2.NBT.7 Add and subtract within 1000, using concrete models 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 method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
- 2.2.NBT.7.1 Use estimation strategies to make reasonable estimates in problem solving.
  - Estimate sums (2-N.5)
- 2.2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
  - Input/output tables - add and subtract by 100 (2-L.13)
  - Input/output tables - add and subtract by 10 or 100 (2-L.14)
- 2.2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

2.2.MD Measurement and Data

2 Measure and estimate lengths in standard units.
- 2.2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
  - Measure using an inch ruler (2-S.2)
  - Measure using a centimeter ruler (2-S.8)
- 2.2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
  - Which customary unit of length is appropriate? (2-S.3)
  - Which metric unit of length is appropriate? (2-S.9)
- 2.2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.
  - Which metric unit of length is appropriate? (2-S.9)
- 2.2.MD.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
  - Customary units of length: word problems (2-S.4)
  - Metric units of length: word problems (2-S.10)
2 Relate addition and subtraction to length.
- 2.2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
  - Customary units of length: word problems (2-S.4)
  - Metric units of length: word problems (2-S.10)
- 2.2.MD.6 Represent 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 Work with time and money.
- 2.2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year).
- 2.2.MD.8 Solve word problems involving combinations of dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.
2 Represent and interpret data.
- 2.2.MD.9 Generate 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.
  - Create line plots (2-R.7)
- 2.2.MD.10 Draw 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 a bar graph.
  - Interpret bar graphs II (2-R.3)
  - Which bar graph is correct? (2-R.4)

2.2.G Geometry

2 Reason with shapes and their attributes.
- 2.2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
- 2.2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
  - Area (2-V.3)
- 2.2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
  - Equal parts (2-W.1)

3.3.OA Operations and Algebraic Thinking

3 Represent and solve problems involving multiplication and division.
- 3.3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each, or 7 groups of 5 objects each.
- 3.3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.
- 3.3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
- 3.3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
3 Understand properties of multiplication and the relationship between multiplication and division.
- 3.3.OA.5 Apply properties of operations as strategies to multiply and divide.
- 3.3.OA.6 Understand division as an unknown-factor problem.
  - Relate multiplication and division for groups (3-I.3)
  - Relate multiplication and division for arrays (3-I.5)
3 Multiply and divide within 100.
- 3.3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3 Solve problems involving the four operations, and identify and explain patterns in arithmetic.
- 3.3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
- 3.3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.

3.3.NBT Number and Operations in Base Ten

3 Use place value understanding and properties of operations to perform multi-digit arithmetic.
- 3.3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.
  - Rounding - nearest ten or hundred only (3-P.1)
- 3.3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
- 3.3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
  - Multiply by 10 (3-F.11)
  - Multiply by a multiple of ten (3-H.1)

3.3.NF Number and Operations-Fractions

3 Develop understanding of fractions as numbers.
- 3.3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
- 3.3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
  - 3.3.NF.2.a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
  - 3.3.NF.2.b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
- 3.3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
  - 3.3.NF.3.a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line
  - 3.3.NF.3.b Recognize and generate simple equivalent fractions, (e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
  - 3.3.NF.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
  - 3.3.NF.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

3.3.MD Measurement and Data

3 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
- 3.3.MD.1 Tell 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., by representing the problem on a number line diagram.
- 3.3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
3 Represent and interpret data.
- 3.3.MD.3 Draw 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.
- 3.3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.
  - Create line plots with fractions (3-U.10)
  - Measure using an inch ruler (3-BB.3)
3 Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
- 3.3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.
  - 3.3.MD.5.a A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.
  - 3.3.MD.5.b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
    - Create figures with a given area (3-FF.9)
- 3.3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
- 3.3.MD.7 Relate area to the operations of multiplication and addition.
  - 3.3.MD.7.a Find 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.
  - 3.3.MD.7.b Multiply 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.
    - Find the area of rectangles and squares (3-FF.10)
    - Find the area of rectangles: word problems (3-FF.13)
  - 3.3.MD.7.c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
  - 3.3.MD.7.d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
    - Find the area of complex figures (3-FF.14)
3 Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
- 3.3.MD.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

3.3.G Geometry

3 Reason with shapes and their attributes.
- 3.3.G.1 Understand 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 of quadrilaterals that do not belong to any of these subcategories.
- 3.3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
  - Identify equal parts (3-W.1)
  - Match unit fractions to models (3-W.7)

4.4.OA Operations and Algebraic Thinking

4 Use the four operations with whole numbers to solve problems.
- 4.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
  - Multiplication facts up to 12: find the missing factor (4-D.2)
  - Compare numbers using multiplication (4-D.3)
- 4.4.OA.2 Multiply or divide 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.
  - Addition, subtraction, multiplication, and division word problems (4-F.2)
- 4.4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
4 Gain familiarity with factors and multiples.
- 4.4.OA.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.
4 Generate and analyze patterns.
- 4.4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

4.4.NBT Number and Operations in Base Ten

4 Generalize place value understanding for multi-digit whole numbers.
- 4.4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
  - Value of a digit (4-A.1)
- 4.4.NBT.2 Read 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.
- 4.4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.
4 Use place value understanding and properties of operations to perform multi-digit arithmetic.
- 4.4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
- 4.4.NBT.5 Multiply 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.
- 4.4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.4.NF Number and Operations-Fractions

4 Extend understanding of fraction equivalence and ordering.
- 4.4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × 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 recognize and generate equivalent fractions.
  - Find equivalent fractions using area models (4-P.5)
  - Graph equivalent fractions on number lines (4-P.6)
- 4.4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
4 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
- 4.4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
  - 4.4.NF.3.a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
    - Add fractions with like denominators (4-Q.5)
    - Subtract fractions with like denominators (4-Q.7)
  - 4.4.NF.3.b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.
    - Decompose fractions into unit fractions (4-Q.1)
    - Decompose fractions multiple ways (4-Q.3)
  - 4.4.NF.3.c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  - 4.4.NF.3.d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
    - Add and subtract fractions with like denominators in recipes (4-Q.12)
- 4.4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
  - 4.4.NF.4.a Understand a fraction a/b as a multiple of 1/b.
    - Multiply unit fractions by whole numbers using models (4-S.2)
  - 4.4.NF.4.b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
  - 4.4.NF.4.c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.
4 Understand decimal notation for fractions, and compare decimal fractions.
- 4.4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.
  - Add fractions with denominators of 10 and 100 (4-R.5)
- 4.4.NF.6 Use decimal notation for fractions with denominators 10 or 100.
- 4.4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using the number line or another visual model.

4.4.MD Measurement and Data

4 Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
- 4.4.MD.1 Know relative sizes of measurement units within one system of units including 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. Record measurement equivalents in a two-column table.
- 4.4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
- 4.4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
4 Represent and interpret data.
- 4.4.MD.4 Make 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.
  - Create and interpret line plots with fractions (4-J.8)
4 Geometric measurement: understand concepts of angle and measure angles.
- 4.4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
  - 4.4.MD.5.a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
    - Angles of 90, 180, 270, and 360 degrees (4-Z.2)
  - 4.4.MD.5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
    - Angles of 90, 180, 270, and 360 degrees (4-Z.2)
    - Estimate angle measurements (4-Z.4)
- 4.4.MD.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
  - Measure angles with a protractor (4-Z.3)
- 4.4.MD.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
  - Adjacent angles (4-Z.5)

4.4.G Geometry

4 Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
- 4.4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
- 4.4.G.2 Classify 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. Recognize right triangles as a category, and identify right triangles. (Two dimensional shapes should include special triangles, e.g., equilateral, isosceles, scalene, and special quadrilaterals, e.g., rhombus, square, rectangle, parallelogram, trapezoid.)
- 4.4.G.3 Recognize 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.

5.5.OA Operations and Algebraic Thinking

5 Write and interpret numerical expressions.
- 5.5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
  - Evaluate numerical expressions (5-O.4)
- 5.5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
  - Write numerical expressions (5-O.3)
- 5.5.OA.2.1 Express a whole number in the range 2-50 as a product of its prime factors.
  - Prime factorization (5-F.2)
5 Analyze patterns and relationships.
- 5.5.OA.3 Generate two numerical patterns using two given rules. Identify 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.

5.5.NBT Number and Operations in Base Ten

5 Understand the place value system.
- 5.5.NBT.1 Recognize 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.
- 5.5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
- 5.5.NBT.3 Read, write, and compare decimals to thousandths.
  - 5.5.NBT.3.a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
  - 5.5.NBT.3.b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
- 5.5.NBT.4 Use place value understanding to round decimals to any place.
  - Round decimals (5-G.7)
  - Estimate sums and differences of decimals (5-H.8)
5 Perform operations with multi-digit whole numbers and with decimals to hundredths.
- 5.5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.
- 5.5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
- 5.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

5.5.NF Number and Operations-Fractions

5 Use equivalent fractions as a strategy to add and subtract fractions.
- 5.5.NF.1 Add 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.
- 5.5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
5 Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
- 5.5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
- 5.5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  - 5.5.NF.4.a Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
  - 5.5.NF.4.b Find 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.
- 5.5.NF.5 Interpret multiplication as scaling (resizing), by:
  - 5.5.NF.5.a Comparing 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.
  - 5.5.NF.5.b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
    - Multiply two fractions using models (5-M.14)
- 5.5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
- 5.5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
  - 5.5.NF.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
    - Divide unit fractions by whole numbers (5-N.1)
    - Divide fractions by whole numbers (5-N.5)
  - 5.5.NF.7.b Interpret division of a whole number by a unit fraction, and compute such quotients.
  - 5.5.NF.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.
    - Divide whole numbers and unit fractions (5-N.4)
    - Divide unit fractions by whole numbers: word problems (5-N.7)

5.5.MD Measurement and Data

5 Convert like measurement units within a given measurement system.
- 5.5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
5 Represent and interpret data.
- 5.5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.
5 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
- 5.5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
  - 5.5.MD.3.a A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
    - Volume of rectangular prisms made of unit cubes (5-EE.11)
  - 5.5.MD.3.b A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
    - Volume of rectangular prisms made of unit cubes (5-EE.11)
- 5.5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
  - Volume of rectangular prisms made of unit cubes (5-EE.11)
- 5.5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
  - 5.5.MD.5.a Find 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.
    - Volume of rectangular prisms made of unit cubes (5-EE.11)
    - Volume of cubes and rectangular prisms (5-EE.13)
  - 5.5.MD.5.b Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
    - Volume of cubes and rectangular prisms (5-EE.13)
  - 5.5.MD.5.c Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
    - Volume of irregular figures made of unit cubes (5-EE.12)

5.5.G Geometry

5 Graph points on the coordinate plane to solve real-world and mathematical problems.
- 5.5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
- 5.5.G.2 Represent 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 Classify two-dimensional figures into categories based on their properties.
- 5.5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
- 5.5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

6.6.RP Ratios and Proportional Relationships

6 Understand ratio concepts and use ratio reasoning to solve problems.
- 6.6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
  - Write a ratio: word problems (6-R.2)
- 6.6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship.
  - Unit rates and equivalent rates (6-R.7)
- 6.6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  - 6.6.RP.3.a Make 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.
  - 6.6.RP.3.b Solve unit rate problems including those involving unit pricing and constant speed.
  - 6.6.RP.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
  - 6.6.RP.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

6.6.NS The Number System

6 Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
- 6.6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
6 Compute fluently with multi-digit numbers and find common factors and multiples.
- 6.6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
- 6.6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
- 6.6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
6 Apply and extend previous understandings of numbers to the system of rational numbers.
- 6.6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
  - Understanding integers (6-M.1)
- 6.6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
  - 6.6.NS.6.a Recognize 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, e.g., -(-3) = 3, and that 0 is its own opposite.
    - Rational numbers: find the sign (6-P.6)
  - 6.6.NS.6.b Understand 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.
    - Quadrants (6-X.3)
  - 6.6.NS.6.c Find 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.
- 6.6.NS.7 Understand ordering and absolute value of rational numbers.
  - 6.6.NS.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
    - Compare integers (6-M.5)
  - 6.6.NS.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts.
  - 6.6.NS.7.c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
    - Absolute value and opposite integers (6-M.3)
    - Absolute value of rational numbers (6-P.3)
  - 6.6.NS.7.d Distinguish comparisons of absolute value from statements about order.
    - Put rational numbers in order (6-P.2)
    - Absolute value of rational numbers (6-P.3)
- 6.6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

6.6.EE Expressions and Equations

6 Apply and extend previous understandings of arithmetic to algebraic expressions.
- 6.6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.
- 6.6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.
  - 6.6.EE.2.a Write expressions that record operations with numbers and with letters standing for numbers.
    - Write variable expressions (6-Y.1)
  - 6.6.EE.2.b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
    - Identify terms and coefficients (6-Y.6)
    - Sort factors of expressions (6-Y.7)
  - 6.6.EE.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
- 6.6.EE.3 Apply the properties of operations to generate equivalent expressions.
- 6.6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).
  - Identify equivalent expressions (6-Y.15)
6 Reason about and solve one-variable equations and inequalities.
- 6.6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
- 6.6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
  - Write variable expressions: word problems (6-Y.2)
- 6.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
- 6.6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
  - Graph inequalities on number lines (6-AA.2)
  - Write inequalities from number lines (6-AA.3)
6 Represent and analyze quantitative relationships between dependent and independent variables.
- 6.6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

6.6.G Geometry

6 Solve real-world and mathematical problems involving area, surface area, and volume.
- 6.6.G.1 Find 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 in the context of solving real-world and mathematical problems.
- 6.6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
  - Volume of cubes and rectangular prisms (6-FF.14)
- 6.6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
  - Graph triangles and quadrilaterals (6-CC.7)
- 6.6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
  - Surface area of cubes and rectangular prisms (6-FF.15)

6.6.SP Statistics and Probability

6 Develop understanding of statistical variability.
- 6.6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
  - Identify representative, random, and biased samples (6-HH.5)
- 6.6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
- 6.6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
  - Mean, median, mode, and range: find the missing number (6-HH.3)
6 Summarize and describe distributions.
- 6.6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
  - Create line plots (6-GG.4)
  - Interpret box-and-whisker plots (6-GG.19)
- 6.6.SP.5 Summarize numerical data sets in relation to their context, such as by:
  - 6.6.SP.5.a Reporting the number of observations.
    - Interpret line plots (6-GG.3)
    - Interpret histograms (6-GG.11)
  - 6.6.SP.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
    - Identify representative, random, and biased samples (6-HH.5)
  - 6.6.SP.5.c Giving 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.
  - 6.6.SP.5.d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

7.7.RP Ratios and Proportional Relationships

7 Analyze proportional relationships and use them to solve real-world and mathematical problems.
- 7.7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
  - Unit prices (7-M.3)
- 7.7.RP.2 Recognize and represent proportional relationships between quantities.
  - 7.7.RP.2.a Decide 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.
  - 7.7.RP.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
    - Find the constant of proportionality from a table (7-K.1)
    - Find the constant of proportionality from a graph (7-K.4)
  - 7.7.RP.2.c Represent proportional relationships by equations.
  - 7.7.RP.2.d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
    - Interpret graphs of proportional relationships (7-K.7)
- 7.7.RP.3 Use proportional relationships to solve multistep ratio and percent problems.

7.7.NS The Number System

7 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
- 7.7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
  - 7.7.NS.1.a Describe situations in which opposite quantities combine to make 0.
    - Absolute value and opposite integers (7-B.4)
  - 7.7.NS.1.b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
    - Integer addition and subtraction rules (7-C.1)
  - 7.7.NS.1.c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
    - Integer addition and subtraction rules (7-C.1)
  - 7.7.NS.1.d Apply properties of operations as strategies to add and subtract rational numbers.
- 7.7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
  - 7.7.NS.2.a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
    - Integer multiplication and division rules (7-C.6)
    - Apply multiplication and division rules (7-H.9)
  - 7.7.NS.2.b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
    - Integer multiplication and division rules (7-C.6)
    - Apply multiplication and division rules (7-H.9)
  - 7.7.NS.2.c Apply properties of operations as strategies to multiply and divide rational numbers.
  - 7.7.NS.2.d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
    - Classify numbers (7-A.10)
    - Convert between decimals and fractions or mixed numbers (7-H.1)
- 7.7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

7.7.EE Expressions and Equations

7 Use properties of operations to generate equivalent expressions.
- 7.7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
- 7.7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.
7 Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
- 7.7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
- 7.7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
  - 7.7.EE.4.a Solve 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.
  - 7.7.EE.4.b Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

7.7.G Geometry

7 Draw, construct, and describe geometrical figures and describe the relationships between them.
- 7.7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
  - Scale drawings: word problems (7-J.7)
- 7.7.G.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
  - Graph triangles and quadrilaterals (7-W.6)
- 7.7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
  - Bases of three-dimensional figures (7-Z.1)
  - Cross-sections of three-dimensional figures (7-Z.4)
7 Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
- 7.7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- 7.7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
  - Identify complementary, supplementary, vertical, adjacent, and congruent angles (7-W.12)
  - Find measures of complementary, supplementary, vertical, and adjacent angles (7-W.13)
- 7.7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

7.7.SP Statistics and Probability

7 Use random sampling to draw inferences about a population.
- 7.7.SP.1 Understand 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.
  - Identify representative, random, and biased samples (7-CC.6)
- 7.7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
  - Estimate population size using proportions (7-J.12)
7 Draw informal comparative inferences about two populations.
- 7.7.SP.3 Informally 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.
- 7.7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
7 Investigate chance processes and develop, use, and evaluate probability models.
- 7.7.SP.5 Understand 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.
  - Probability of simple events (7-DD.1)
- 7.7.SP.6 Approximate 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.
  - Make predictions (7-DD.4)
- 7.7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
  - 7.7.SP.7.a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
    - Probability of simple events (7-DD.1)
    - Probability of opposite, mutually exclusive, and overlapping events (7-DD.2)
  - 7.7.SP.7.b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
    - Experimental probability (7-DD.3)
- 7.7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
  - 7.7.SP.8.a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
  - 7.7.SP.8.b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.
    - Compound events: find the number of outcomes (7-DD.5)
    - Counting principle (7-DD.10)
  - 7.7.SP.8.c Design and use a simulation to generate frequencies for compound events.

8.8.NS The Number System

8 Know that there are numbers that are not rational, and approximate them by rational numbers.
- 8.8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
  - Convert between decimals and fractions or mixed numbers (8-D.4)
  - Identify rational and irrational numbers (8-D.5)
- 8.8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi²).
  - Estimate positive and negative square roots (8-F.16)
  - Estimate cube roots (8-F.21)

8.8.EE Expressions and Equations

8 Work with radicals and integer exponents.
- 8.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.
- 8.8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the square root of 2 is irrational.
- 8.8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
  - Convert between standard and scientific notation (8-G.1)
  - Compare numbers written in scientific notation (8-G.2)
- 8.8.EE.4 Perform 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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
8 Understand the connections between proportional relationships, lines, and linear equations.
- 8.8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
  - Graph proportional relationships (8-I.7)
- 8.8.EE.6 Use 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 b.
8 Analyze and solve linear equations and pairs of simultaneous linear equations.
- 8.8.EE.7 Solve linear equations in one variable.
  - 8.8.EE.7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
    - Find the number of solutions (8-W.14)
    - Create equations with no solutions or infinitely many solutions (8-W.15)
  - 8.8.EE.7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
- 8.8.EE.8 Analyze and solve pairs of simultaneous linear equations.
  - 8.8.EE.8.a Understand 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.
  - 8.8.EE.8.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
  - 8.8.EE.8.c Solve real-world and mathematical problems leading to two linear equations in two variables.

8.8.F Functions

8 Define, evaluate, and compare functions.
- 8.8.F.1 Understand 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.
- 8.8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
- 8.8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
  - Graph a line from an equation in slope-intercept form (8-Y.6)
  - Identify linear and nonlinear functions (8-Z.14)
8 Use functions to model relationships between quantities.
- 8.8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- 8.8.F.5 Describe 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.
  - Write linear functions: word problems (8-Z.12)

8.8.G Geometry

8 Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.8.G.1 Verify experimentally the properties of rotations, reflections, and translations:
  - 8.8.G.1.a Lines are taken to lines, and line segments to line segments of the same length.
  - 8.8.G.1.b Angles are taken to angles of the same measure.
  - 8.8.G.1.c Parallel lines are taken to parallel lines.
- 8.8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
- 8.8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
- 8.8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
  - Similar and congruent figures (8-Q.1)
  - Side lengths and angle measures of similar figures (8-Q.5)
- 8.8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
8 Understand and apply the Pythagorean Theorem.
- 8.8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
  - Converse of the Pythagorean theorem: is it a right triangle? (8-R.5)
- 8.8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
- 8.8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
  - Find the distance between two points (8-N.4)
8 Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
- 8.8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

8.8.SP Statistics and Probability

8 Investigate patterns of association in bivariate data.
- 8.8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
  - Scatter plots (8-CC.14)
  - Outliers in scatter plots (8-DD.8)
- 8.8.SP.2 Know 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.
  - Scatter plots: line of best fit (8-DD.9)
- 8.8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
  - Constant rate of change (8-Z.5)
  - Write linear functions: word problems (8-Z.12)
- 8.8.SP.4 Understand 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.

9-12.A. Algebra

9-12.A-SSE Seeing Structure in Expressions
- 9-12. Interpret the structure of expressions
  - 9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
    - 9-12.A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
      - Polynomial vocabulary (A1-Z.1)
      - Polynomial vocabulary (A2-K.1)
    - 9-12.A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
      - Factor using a quadratic pattern (A2-I.4)
      - Factor using a quadratic pattern (PC-D.14)
  - 9-12.A-SSE.2 Use the structure of an expression to identify ways to rewrite it.
- 9-12. Write expressions in equivalent forms to solve problems
  - 9-12.A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
    - 9-12.A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines.
    - 9-12.A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
    - 9-12.A-SSE.3.c Use the properties of exponents to transform expressions for exponential functions.
9-12.A-APR Arithmetic with Polynomials and Rational Expressions
- 9-12. Perform arithmetic operations on polynomials
  - 9-12.A-APR.1 Understand 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.
9-12.A-CED Creating Equations
- 9-12. Create equations that describe numbers or relationships
  - 9-12.A-CED.1 Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.
  - 9-12.A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  - 9-12.A-CED.3 Represent 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.
  - 9-12.A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
9-12.A-REI Reasoning with Equations and Inequalities
- 9-12. Understand solving equations as a process of reasoning and explain the reasoning
  - 9-12.A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- 9-12. Solve equations and inequalities in one variable
  - 9-12.A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  - 9-12.A-REI.3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.
  - 9-12.A-REI.4 Solve quadratic equations in one variable.
    - 9-12.A-REI.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.
      - Complete the square (A1-BB.8)
      - Complete the square (A2-J.7)
    - 9-12.A-REI.4.b Solve quadratic equations by inspection (e.g., for x² = 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.
- 9-12. Solve systems of equations
  - 9-12.A-REI.5 Prove 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.
  - 9-12.A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  - 9-12.A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
    - Systems of linear and quadratic equations (A1-BB.13)
    - Solve a system of linear and quadratic equations (A2-E.15)
- 9-12. Represent and solve equations and inequalities graphically
  - 9-12.A-REI.10 Understand 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).
  - 9-12.A-REI.11 Explain 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 cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  - 9-12.A-REI.12 Graph 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.

9-12.F Functions

9-12.F-IF Interpreting Functions
- 9-12. Understand the concept of a function and use function notation
  - 9-12.F-IF.1 Understand 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).
  - 9-12.F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  - 9-12.F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- 9-12. Interpret functions that arise in applications in terms of the context
  - 9-12.F-IF.4 For 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.
  - 9-12.F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - 9-12.F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- 9-12. Analyze functions using different representations
  - 9-12.F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
    - 9-12.F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
    - 9-12.F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
      - Graph an absolute value function (A1-DD.2)
    - 9-12.F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
  - 9-12.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - 9-12.F-IF.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
    - 9-12.F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
      - Match exponential functions and graphs (A1-X.2)
      - Match exponential functions and graphs (A2-S.3)
  - 9-12.F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
9-12.F-BF Building Functions
- 9-12. Build a function that models a relationship between two quantities
  - 9-12.F-BF.1 Write a function that describes a relationship between two quantities.
    - 9-12.F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - 9-12.F-BF.1.b Combine standard function types using arithmetic operations.
  - 9-12.F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- 9-12. Build new functions from existing functions
  - 9-12.F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
  - 9-12.F-BF.4 Find inverse functions.
    - 9-12.F-BF.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
9-12.F-LE Linear, Quadratic, and Exponential Models
- 9-12. Construct and compare linear, quadratic, and exponential models and solve problems
  - 9-12.F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
    - 9-12.F-LE.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    - 9-12.F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
    - 9-12.F-LE.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  - 9-12.F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  - 9-12.F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
- 9-12. Interpret expressions for functions in terms of the situation they model
  - 9-12.F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.
  - 9-12.F-LE.6 Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.

9-12.N Number and Quantity

9-12.N-RN The Real Number System
- 9-12. Extend the properties of exponents to rational exponents.
  - 9-12.N-RN.1 Explain how the definition of the meaning 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.
  - 9-12.N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- 9-12. Use properties of rational and irrational numbers.
  - 9-12.N-RN.3 Explain 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.
9-12.N-Q Quantities
- 9-12. Reason quantitatively and use units to solve problems.
  - 9-12.N-Q.1 Use 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.
  - 9-12.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
  - 9-12.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

9-12.S Statistics and Probability

9-12.S-ID Interpreting Categorical and Quantitative Data
- 9-12. Summarize, represent, and interpret data on a single count or measurement variable
  - 9-12.S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
    - Create bar graphs, line graphs, and histograms (A1-N.2)
  - 9-12.S-ID.2 Use 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.
  - 9-12.S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
- 9-12. Summarize, represent, and interpret data on two categorical and quantitative variables
  - 9-12.S-ID.5 Summarize 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.
  - 9-12.S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
    - Outliers in scatter plots (A1-KK.9)
    - Outliers in scatter plots (A2-EE.5)
    - Outliers in scatter plots (PC-Z.5)
    - 9-12.S-ID.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
    - 9-12.S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.
      - Interpret a scatter plot (A1-KK.8)
    - 9-12.S-ID.6.c Fit a linear function for a scatter plot that suggests a linear association.
- 9-12. Interpret linear models
  - 9-12.S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
  - 9-12.S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
  - 9-12.S-ID.9 Distinguish between correlation and causation.