CCSS.Math.Content.K.CC.A.1Know number names and the count sequence.

Count to 100 by ones and by tens.

CCSS.Math.Content.K.CC.A.2Know number names and the count sequence.

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

CCSS.Math.Content.K.CC.A.3Know number names and the count sequence.

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).

CCSS.Math.Content.K.CC.B.4Count to tell the number of objects.

Understand the relationship between numbers and quantities; connect counting to cardinality.

CCSS.Math.Content.K.CC.B.4aCount to tell the number of objects.

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.

CCSS.Math.Content.K.CC.B.4bCount to tell the number of objects.

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.

CCSS.Math.Content.K.CC.B.4cCount to tell the number of objects.

Understand that each successive number name refers to a quantity that is one larger.

CCSS.Math.Content.K.CC.B.5Count to tell the number of objects.

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.

CCSS.Math.Content.K.CC.C.6Compare numbers.

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.

CCSS.Math.Content.K.CC.C.7Compare numbers.

Compare two numbers between 1 and 10 presented as written numerals.

CCSS.Math.Content.K.OA.A.1Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

CCSS.Math.Content.K.OA.A.2Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

CCSS.Math.Content.K.OA.A.3Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

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).

CCSS.Math.Content.K.OA.A.4Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

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.

CCSS.Math.Content.K.OA.A.5Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Fluently add and subtract within 5.

CCSS.Math.Content.K.NBT.A.1Work with numbers 11—19 to gain foundations for place value.

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.

CCSS.Math.Content.K.MD.A.1Describe and compare measurable attributes.

Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

CCSS.Math.Content.K.MD.A.2Describe and compare measurable attributes.

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.

CCSS.Math.Content.K.MD.B.3Classify objects and count the number of objects in each category.

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

CCSS.Math.Content.K.G.A.1Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

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.

CCSS.Math.Content.K.G.A.2Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Correctly name shapes regardless of their orientations or overall size.

CCSS.Math.Content.K.G.A.3Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").

CCSS.Math.Content.K.G.B.4Analyze, compare, create, and compose shapes.

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).

CCSS.Math.Content.K.G.B.5Analyze, compare, create, and compose shapes.

Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

CCSS.Math.Content.K.G.B.6Analyze, compare, create, and compose shapes.

Compose simple shapes to form larger shapes.

CCSS.Math.Content.1.OA.A.1Represent and solve problems involving addition and subtraction.

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.

CCSS.Math.Content.1.OA.A.2Represent and solve problems involving addition and subtraction.

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.

CCSS.Math.Content.1.OA.B.3Understand and apply properties of operations and the relationship between addition and subtraction.

Apply properties of operations as strategies to add and subtract.

CCSS.Math.Content.1.OA.B.4Understand and apply properties of operations and the relationship between addition and subtraction.

Understand subtraction as an unknown-addend problem.

CCSS.Math.Content.1.OA.C.5Add and subtract within 20.

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

CCSS.Math.Content.1.OA.C.6Add and subtract within 20.

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).

CCSS.Math.Content.1.OA.D.7Work with addition and subtraction equations.

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

CCSS.Math.Content.1.OA.D.8Work with addition and subtraction equations.

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

CCSS.Math.Content.1.NBT.A.1Extend the counting sequence.

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.

CCSS.Math.Content.1.NBT.B.2Understand place value.

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

CCSS.Math.Content.1.NBT.B.2aUnderstand place value.

10 can be thought of as a bundle of ten ones — called a "ten."

CCSS.Math.Content.1.NBT.B.2bUnderstand place value.

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

CCSS.Math.Content.1.NBT.B.2cUnderstand place value.

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).

CCSS.Math.Content.1.NBT.B.3Understand place value.

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

CCSS.Math.Content.1.NBT.C.4Use place value understanding and properties of operations to add and subtract.

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.

CCSS.Math.Content.1.NBT.C.5Use place value understanding and properties of operations to add and subtract.

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

CCSS.Math.Content.1.NBT.C.6Use place value understanding and properties of operations to add and subtract.

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.

CCSS.Math.Content.1.MD.A.1Measure lengths indirectly and by iterating length units.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

CCSS.Math.Content.1.MD.A.2Measure lengths indirectly and by iterating length units.

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.

CCSS.Math.Content.1.MD.B.3Tell and write time.

Tell and write time in hours and half-hours using analog and digital clocks.

CCSS.Math.Content.1.MD.C.4Represent and interpret data.

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.

CCSS.Math.Content.1.G.A.1Reason with shapes and their attributes.

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.

CCSS.Math.Content.1.G.A.2Reason with shapes and their attributes.

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.

CCSS.Math.Content.1.G.A.3Reason with shapes and their attributes.

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.

CCSS.Math.Content.2.OA.A.1Represent and solve problems involving addition and subtraction.

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.

CCSS.Math.Content.2.OA.B.2Add and subtract within 20.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

CCSS.Math.Content.2.OA.C.3Work with equal groups of objects to gain foundations for multiplication.

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.

CCSS.Math.Content.2.OA.C.4Work with equal groups of objects to gain foundations for multiplication.

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.

CCSS.Math.Content.2.NBT.A.1Understand place value.

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:

CCSS.Math.Content.2.NBT.A.1aUnderstand place value.

100 can be thought of as a bundle of ten tens — called a "hundred."

CCSS.Math.Content.2.NBT.A.1bUnderstand place value.

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).

CCSS.Math.Content.2.NBT.A.2Understand place value.

Count within 1000; skip-count by 5s, 10s, and 100s.

CCSS.Math.Content.2.NBT.A.3Understand place value.

Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

CCSS.Math.Content.2.NBT.A.4Understand place value.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

CCSS.Math.Content.2.NBT.B.5Use place value understanding and properties of operations to add and subtract.

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

CCSS.Math.Content.2.NBT.B.6Use place value understanding and properties of operations to add and subtract.

Add up to four two-digit numbers using strategies based on place value and properties of operations.

CCSS.Math.Content.2.NBT.B.7Use place value understanding and properties of operations to add and subtract.

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.

CCSS.Math.Content.2.NBT.B.8Use place value understanding and properties of operations to add and subtract.

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

CCSS.Math.Content.2.NBT.B.9Use place value understanding and properties of operations to add and subtract.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

CCSS.Math.Content.2.MD.A.1Measure and estimate lengths in standard units.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

CCSS.Math.Content.2.MD.A.2Measure and estimate lengths in standard units.

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.

CCSS.Math.Content.2.MD.A.3Measure and estimate lengths in standard units.

Estimate lengths using units of inches, feet, centimeters, and meters.

CCSS.Math.Content.2.MD.A.4Measure and estimate lengths in standard units.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

CCSS.Math.Content.2.MD.B.5Relate addition and subtraction to length.

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.

CCSS.Math.Content.2.MD.B.6Relate addition and subtraction to length.

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.

CCSS.Math.Content.2.MD.C.7Work with time and money.

Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

CCSS.Math.Content.2.MD.C.8Work with time and money.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

CCSS.Math.Content.2.MD.D.9Represent and interpret data.

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.

CCSS.Math.Content.2.MD.D.10Represent and interpret data.

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.

CCSS.Math.Content.2.G.A.1Reason with shapes and their attributes.

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.

CCSS.Math.Content.2.G.A.2Reason with shapes and their attributes.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

CCSS.Math.Content.2.G.A.3Reason with shapes and their attributes.

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.

CCSS.Math.Content.3.OA.A.1Represent and solve problems involving multiplication and division.

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

CCSS.Math.Content.3.OA.A.2Represent and solve problems involving multiplication and division.

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.

CCSS.Math.Content.3.OA.A.3Represent and solve problems involving multiplication and division.

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.

CCSS.Math.Content.3.OA.A.4Represent and solve problems involving multiplication and division.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

CCSS.Math.Content.3.OA.B.5Understand properties of multiplication and the relationship between multiplication and division.

Apply properties of operations as strategies to multiply and divide.

CCSS.Math.Content.3.OA.B.6Understand properties of multiplication and the relationship between multiplication and division.

Understand division as an unknown-factor problem.

CCSS.Math.Content.3.OA.C.7Multiply and divide within 100.

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.

CCSS.Math.Content.3.OA.D.8Solve problems involving the four operations, and identify and explain patterns in arithmetic.

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.

CCSS.Math.Content.3.OA.D.9Solve problems involving the four operations, and identify and explain patterns in arithmetic.

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

CCSS.Math.Content.3.NBT.A.1Use place value understanding and properties of operations to perform multi-digit arithmetic.

Use place value understanding to round whole numbers to the nearest 10 or 100.

CCSS.Math.Content.3.NBT.A.2Use place value understanding and properties of operations to perform multi-digit arithmetic.

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.

CCSS.Math.Content.3.NBT.A.3Use place value understanding and properties of operations to perform multi-digit arithmetic.

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.

CCSS.Math.Content.3.MD.A.1Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

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.

CCSS.Math.Content.3.MD.A.2Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

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.

CCSS.Math.Content.3.MD.B.3Represent and interpret data.

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.

CCSS.Math.Content.3.MD.B.4Represent and interpret data.

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.

CCSS.Math.Content.3.MD.C.5Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Recognize area as an attribute of plane figures and understand concepts of area measurement.

CCSS.Math.Content.3.MD.C.5aGeometric measurement: understand concepts of area and relate area to multiplication and to addition.

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.

CCSS.Math.Content.3.MD.C.5bGeometric measurement: understand concepts of area and relate area to multiplication and to addition.

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.

CCSS.Math.Content.3.MD.C.6Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

CCSS.Math.Content.3.MD.C.7Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Relate area to the operations of multiplication and addition.

CCSS.Math.Content.3.MD.C.7aGeometric measurement: understand concepts of area and relate area to multiplication and to addition.

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.

CCSS.Math.Content.3.MD.C.7bGeometric measurement: understand concepts of area and relate area to multiplication and to addition.

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.

CCSS.Math.Content.3.MD.C.7cGeometric measurement: understand concepts of area and relate area to multiplication and to addition.

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.

CCSS.Math.Content.3.MD.C.7dGeometric measurement: understand concepts of area and relate area to multiplication and to addition.

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.

CCSS.Math.Content.3.MD.D.8Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

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.

CCSS.Math.Content.3.G.A.1Reason with shapes and their attributes.

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.

CCSS.Math.Content.3.G.A.2Reason with shapes and their attributes.

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

CCSS.Math.Content.3.NF.A.1Develop understanding of fractions as numbers.

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.

CCSS.Math.Content.3.NF.A.2Develop understanding of fractions as numbers.

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

CCSS.Math.Content.3.NF.A.2aDevelop understanding of fractions as numbers.

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.

CCSS.Math.Content.3.NF.A.2bDevelop understanding of fractions as numbers.

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.

CCSS.Math.Content.3.NF.A.3Develop understanding of fractions as numbers.

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

CCSS.Math.Content.3.NF.A.3aDevelop understanding of fractions as numbers.

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

CCSS.Math.Content.3.NF.A.3bDevelop understanding of fractions as numbers.

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.

CCSS.Math.Content.3.NF.A.3cDevelop understanding of fractions as numbers.

Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

CCSS.Math.Content.3.NF.A.3dDevelop understanding of fractions as numbers.

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.

CCSS.Math.Content.4.OA.A.1Use the four operations with whole numbers to solve problems.

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.

CCSS.Math.Content.4.OA.A.2Use the four operations with whole numbers to solve problems.

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.

CCSS.Math.Content.4.OA.A.3Use the four operations with whole numbers to solve problems.

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.

CCSS.Math.Content.4.OA.B.4Gain familiarity with factors and multiples.

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.

CCSS.Math.Content.4.OA.C.5Generate and analyze patterns.

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.

CCSS.Math.Content.4.NBT.A.1Generalize place value understanding for multi-digit whole numbers.

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.

CCSS.Math.Content.4.NBT.A.2Generalize place value understanding for multi-digit whole numbers.

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.

CCSS.Math.Content.4.NBT.A.3Generalize place value understanding for multi-digit whole numbers.

Use place value understanding to round multi-digit whole numbers to any place.

CCSS.Math.Content.4.NBT.B.4Use place value understanding and properties of operations to perform multi-digit arithmetic.

Fluently add and subtract multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.4.NBT.B.5Use place value understanding and properties of operations to perform multi-digit arithmetic.

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.

CCSS.Math.Content.4.NBT.B.6Use place value understanding and properties of operations to perform multi-digit arithmetic.

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.

CCSS.Math.Content.4.MD.A.1Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

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.

CCSS.Math.Content.4.MD.A.2Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

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.

CCSS.Math.Content.4.MD.A.3Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

CCSS.Math.Content.4.MD.B.4Represent and interpret data.

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.

CCSS.Math.Content.4.MD.C.5Geometric measurement: understand concepts of angle and measure angles.

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

CCSS.Math.Content.4.MD.C.5aGeometric measurement: understand concepts of angle and measure angles.

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.

CCSS.Math.Content.4.MD.C.5bGeometric measurement: understand concepts of angle and measure angles.

An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

CCSS.Math.Content.4.MD.C.6Geometric measurement: understand concepts of angle and measure angles.

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

CCSS.Math.Content.4.MD.C.7Geometric measurement: understand concepts of angle and measure angles.

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.

CCSS.Math.Content.4.G.A.1Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

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

CCSS.Math.Content.4.G.A.2Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

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.

CCSS.Math.Content.4.G.A.3Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

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.

CCSS.Math.Content.4.NF.A.1Extend understanding of fraction equivalence and ordering.

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.

CCSS.Math.Content.4.NF.A.2Extend understanding of fraction equivalence and ordering.

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.

CCSS.Math.Content.4.NF.B.3Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

CCSS.Math.Content.4.NF.B.3aBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

CCSS.Math.Content.4.NF.B.3bBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

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.

CCSS.Math.Content.4.NF.B.3cBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

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.

CCSS.Math.Content.4.NF.B.3dBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

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.

CCSS.Math.Content.4.NF.B.4Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

CCSS.Math.Content.4.NF.B.4aBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand a fraction a/b as a multiple of 1/b.

CCSS.Math.Content.4.NF.B.4bBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

CCSS.Math.Content.4.NF.B.4cBuild fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

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.

CCSS.Math.Content.4.NF.C.5Understand decimal notation for fractions, and compare decimal fractions.

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.

CCSS.Math.Content.4.NF.C.6Understand decimal notation for fractions, and compare decimal fractions.

Use decimal notation for fractions with denominators 10 or 100.

CCSS.Math.Content.4.NF.C.7Understand decimal notation for fractions, and compare decimal fractions.

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 a visual model.

CCSS.Math.Content.5.OA.A.1Write and interpret numerical expressions.

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

CCSS.Math.Content.5.OA.A.2Write and interpret numerical expressions.

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

CCSS.Math.Content.5.OA.B.3Analyze patterns and relationships.

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.

CCSS.Math.Content.5.NBT.A.1Understand the place value system.

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.

CCSS.Math.Content.5.NBT.A.2Understand the place value system.

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.

CCSS.Math.Content.5.NBT.A.3Understand the place value system.

Read, write, and compare decimals to thousandths.

CCSS.Math.Content.5.NBT.A.3aUnderstand the place value system.

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).

CCSS.Math.Content.5.NBT.A.3bUnderstand the place value system.

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

CCSS.Math.Content.5.NBT.A.4Understand the place value system.

Use place value understanding to round decimals to any place.

CCSS.Math.Content.5.NBT.B.5Perform operations with multi-digit whole numbers and with decimals to hundredths.

Fluently multiply multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.5.NBT.B.6Perform operations with multi-digit whole numbers and with decimals to hundredths.

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.

CCSS.Math.Content.5.NBT.B.7Perform operations with multi-digit whole numbers and with decimals to hundredths.

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.

CCSS.Math.Content.5.MD.A.1Convert like measurement units within a given measurement system.

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.

CCSS.Math.Content.5.MD.B.2Represent and interpret data.

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.

CCSS.Math.Content.5.MD.C.3Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

CCSS.Math.Content.5.MD.C.3aGeometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

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.

CCSS.Math.Content.5.MD.C.3bGeometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

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.

CCSS.Math.Content.5.MD.C.4Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

CCSS.Math.Content.5.MD.C.5Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

CCSS.Math.Content.5.MD.C.5aGeometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

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.

CCSS.Math.Content.5.MD.C.5bGeometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

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.

CCSS.Math.Content.5.MD.C.5cGeometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

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.

CCSS.Math.Content.5.G.A.1Graph points on the coordinate plane to solve real-world and mathematical problems.

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).

CCSS.Math.Content.5.G.A.2Graph points on the coordinate plane to solve real-world and mathematical problems.

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.

CCSS.Math.Content.5.G.B.3Classify two-dimensional figures into categories based on their properties.

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

CCSS.Math.Content.5.G.B.4Classify two-dimensional figures into categories based on their properties.

Classify two-dimensional figures in a hierarchy based on properties.

CCSS.Math.Content.5.NF.A.1Use equivalent fractions as a strategy to add and subtract fractions.

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.

CCSS.Math.Content.5.NF.A.2Use equivalent fractions as a strategy to add and subtract fractions.

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.

CCSS.Math.Content.5.NF.B.3Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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.

CCSS.Math.Content.5.NF.B.4Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

CCSS.Math.Content.5.NF.B.4aApply and extend previous understandings of multiplication and division to multiply and divide fractions.

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.

CCSS.Math.Content.5.NF.B.4bApply and extend previous understandings of multiplication and division to multiply and divide fractions.

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.

CCSS.Math.Content.5.NF.B.5Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Interpret multiplication as scaling (resizing), by:

CCSS.Math.Content.5.NF.B.5aApply and extend previous understandings of multiplication and division to multiply and divide fractions.

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.

CCSS.Math.Content.5.NF.B.5bApply and extend previous understandings of multiplication and division to multiply and divide fractions.

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.

CCSS.Math.Content.5.NF.B.6Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

CCSS.Math.Content.5.NF.B.7Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

CCSS.Math.Content.5.NF.B.7aApply and extend previous understandings of multiplication and division to multiply and divide fractions.

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

CCSS.Math.Content.5.NF.B.7bApply and extend previous understandings of multiplication and division to multiply and divide fractions.

Interpret division of a whole number by a unit fraction, and compute such quotients.

CCSS.Math.Content.5.NF.B.7cApply and extend previous understandings of multiplication and division to multiply and divide fractions.

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.

CCSS.Math.Content.6.G.A.1Solve real-world and mathematical problems involving area, surface area, and volume.

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.

CCSS.Math.Content.6.G.A.2Solve real-world and mathematical problems involving area, surface area, and volume.

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.

CCSS.Math.Content.6.G.A.3Solve real-world and mathematical problems involving area, surface area, and volume.

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.

CCSS.Math.Content.6.G.A.4Solve real-world and mathematical problems involving area, surface area, and volume.

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.

CCSS.Math.Content.6.RP.A.1Understand ratio concepts and use ratio reasoning to solve problems.

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

CCSS.Math.Content.6.RP.A.2Understand ratio concepts and use ratio reasoning to solve problems.

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.

CCSS.Math.Content.6.RP.A.3Understand ratio concepts and use ratio reasoning to solve problems.

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.

CCSS.Math.Content.6.RP.A.3aUnderstand ratio concepts and use ratio reasoning to solve problems.

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.

CCSS.Math.Content.6.RP.A.3bUnderstand ratio concepts and use ratio reasoning to solve problems.

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

CCSS.Math.Content.6.RP.A.3cUnderstand ratio concepts and use ratio reasoning to solve problems.

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.

CCSS.Math.Content.6.RP.A.3dUnderstand ratio concepts and use ratio reasoning to solve problems.

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

CCSS.Math.Content.6.NS.A.1Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

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.

CCSS.Math.Content.6.NS.B.2Compute fluently with multi-digit numbers and find common factors and multiples.

Fluently divide multi-digit numbers using the standard algorithm.

CCSS.Math.Content.6.NS.B.3Compute fluently with multi-digit numbers and find common factors and multiples.

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

CCSS.Math.Content.6.NS.B.4Compute fluently with multi-digit numbers and find common factors and multiples.

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.

CCSS.Math.Content.6.NS.C.5Apply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.NS.C.6Apply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.NS.C.6aApply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.NS.C.6bApply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.NS.C.6cApply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.NS.C.7Apply and extend previous understandings of numbers to the system of rational numbers.

Understand ordering and absolute value of rational numbers.

CCSS.Math.Content.6.NS.C.7aApply and extend previous understandings of numbers to the system of rational numbers.

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

CCSS.Math.Content.6.NS.C.7bApply and extend previous understandings of numbers to the system of rational numbers.

Write, interpret, and explain statements of order for rational numbers in real-world contexts.

CCSS.Math.Content.6.NS.C.7cApply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.NS.C.7dApply and extend previous understandings of numbers to the system of rational numbers.

Distinguish comparisons of absolute value from statements about order.

CCSS.Math.Content.6.NS.C.8Apply and extend previous understandings of numbers to the system of rational numbers.

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.

CCSS.Math.Content.6.EE.A.1Apply and extend previous understandings of arithmetic to algebraic expressions.

Write and evaluate numerical expressions involving whole-number exponents.

CCSS.Math.Content.6.EE.A.2Apply and extend previous understandings of arithmetic to algebraic expressions.

Write, read, and evaluate expressions in which letters stand for numbers.

CCSS.Math.Content.6.EE.A.2aApply and extend previous understandings of arithmetic to algebraic expressions.

Write expressions that record operations with numbers and with letters standing for numbers.

CCSS.Math.Content.6.EE.A.2bApply and extend previous understandings of arithmetic to algebraic expressions.

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.

CCSS.Math.Content.6.EE.A.2cApply and extend previous understandings of arithmetic to algebraic expressions.

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).

CCSS.Math.Content.6.EE.A.3Apply and extend previous understandings of arithmetic to algebraic expressions.

Apply the properties of operations to generate equivalent expressions.

CCSS.Math.Content.6.EE.A.4Apply and extend previous understandings of arithmetic to algebraic expressions.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

CCSS.Math.Content.6.EE.B.5Reason about and solve one-variable equations and inequalities.

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.

CCSS.Math.Content.6.EE.B.6Reason about and solve one-variable equations and inequalities.

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.

CCSS.Math.Content.6.EE.B.7Reason about and solve one-variable equations and inequalities.

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.

CCSS.Math.Content.6.EE.B.8Reason about and solve one-variable equations and inequalities.

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.

CCSS.Math.Content.6.EE.C.9Represent and analyze quantitative relationships between dependent and independent variables.

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.

CCSS.Math.Content.6.SP.A.1Develop understanding of statistical variability.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

CCSS.Math.Content.6.SP.A.2Develop understanding of statistical variability.

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.

CCSS.Math.Content.6.SP.A.3Develop understanding of statistical variability.

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.

CCSS.Math.Content.6.SP.B.4Summarize and describe distributions.

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

CCSS.Math.Content.6.SP.B.5Summarize and describe distributions.

Summarize numerical data sets in relation to their context, such as by:

CCSS.Math.Content.6.SP.B.5aSummarize and describe distributions.

Reporting the number of observations.

CCSS.Math.Content.6.SP.B.5bSummarize and describe distributions.

Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

CCSS.Math.Content.6.SP.B.5cSummarize and describe distributions.

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.

CCSS.Math.Content.6.SP.B.5dSummarize and describe distributions.

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.

CCSS.Math.Content.7.G.A.1Draw, construct, and describe geometrical figures and describe the relationships between them.

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.

CCSS.Math.Content.7.G.A.2Draw, construct, and describe geometrical figures and describe the relationships between them.

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.

CCSS.Math.Content.7.G.A.3Draw, construct, and describe geometrical figures and describe the relationships between them.

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

CCSS.Math.Content.7.G.B.4Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

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.

CCSS.Math.Content.7.G.B.5Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

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.

CCSS.Math.Content.7.G.B.6Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

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.

CCSS.Math.Content.7.RP.A.1Analyze proportional relationships and use them to solve real-world and mathematical problems.

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

CCSS.Math.Content.7.RP.A.2Analyze proportional relationships and use them to solve real-world and mathematical problems.

Recognize and represent proportional relationships between quantities.

CCSS.Math.Content.7.RP.A.2aAnalyze proportional relationships and use them to solve real-world and mathematical problems.

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.

CCSS.Math.Content.7.RP.A.2bAnalyze proportional relationships and use them to solve real-world and mathematical problems.

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

CCSS.Math.Content.7.RP.A.2cAnalyze proportional relationships and use them to solve real-world and mathematical problems.

Represent proportional relationships by equations.

CCSS.Math.Content.7.RP.A.2dAnalyze proportional relationships and use them to solve real-world and mathematical problems.

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.

CCSS.Math.Content.7.RP.A.3Analyze proportional relationships and use them to solve real-world and mathematical problems.

Use proportional relationships to solve multistep ratio and percent problems.

CCSS.Math.Content.7.NS.A.1Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

CCSS.Math.Content.7.NS.A.1aApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Describe situations in which opposite quantities combine to make 0.

CCSS.Math.Content.7.NS.A.1bApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

CCSS.Math.Content.7.NS.A.1cApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

CCSS.Math.Content.7.NS.A.1dApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Apply properties of operations as strategies to add and subtract rational numbers.

CCSS.Math.Content.7.NS.A.2Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

CCSS.Math.Content.7.NS.A.2aApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

CCSS.Math.Content.7.NS.A.2bApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

CCSS.Math.Content.7.NS.A.2cApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Apply properties of operations as strategies to multiply and divide rational numbers.

CCSS.Math.Content.7.NS.A.2dApply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

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.

CCSS.Math.Content.7.NS.A.3Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Solve real-world and mathematical problems involving the four operations with rational numbers.

CCSS.Math.Content.7.EE.A.1Use properties of operations to generate equivalent expressions.

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

CCSS.Math.Content.7.EE.A.2Use properties of operations to generate equivalent expressions.

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.

CCSS.Math.Content.7.EE.B.3Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

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.

CCSS.Math.Content.7.EE.B.4Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

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.

CCSS.Math.Content.7.EE.B.4aSolve real-life and mathematical problems using numerical and algebraic expressions and equations.

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.

CCSS.Math.Content.7.EE.B.4bSolve real-life and mathematical problems using numerical and algebraic expressions and equations.

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.

CCSS.Math.Content.7.SP.A.1Use random sampling to draw inferences about a population.

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.

CCSS.Math.Content.7.SP.A.2Use random sampling to draw inferences about a population.

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.

CCSS.Math.Content.7.SP.B.3Draw informal comparative inferences about two populations.

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.

CCSS.Math.Content.7.SP.B.4Draw informal comparative inferences about two populations.

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

CCSS.Math.Content.7.SP.C.5Investigate chance processes and develop, use, and evaluate probability models.

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.

CCSS.Math.Content.7.SP.C.6Investigate chance processes and develop, use, and evaluate probability models.

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.

CCSS.Math.Content.7.SP.C.7Investigate chance processes and develop, use, and evaluate probability models.

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.

CCSS.Math.Content.7.SP.C.7aInvestigate chance processes and develop, use, and evaluate probability models.

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

CCSS.Math.Content.7.SP.C.7bInvestigate chance processes and develop, use, and evaluate probability models.

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

CCSS.Math.Content.7.SP.C.8Investigate chance processes and develop, use, and evaluate probability models.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

CCSS.Math.Content.7.SP.C.8aInvestigate chance processes and develop, use, and evaluate probability models.

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.

CCSS.Math.Content.7.SP.C.8bInvestigate chance processes and develop, use, and evaluate probability models.

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.

CCSS.Math.Content.7.SP.C.8cInvestigate chance processes and develop, use, and evaluate probability models.

Design and use a simulation to generate frequencies for compound events.

CCSS.Math.Content.8.G.A.1Understand congruence and similarity using physical models, transparencies, or geometry software.

Verify experimentally the properties of rotations, reflections, and translations:

CCSS.Math.Content.8.G.A.1aUnderstand congruence and similarity using physical models, transparencies, or geometry software.

Lines are taken to lines, and line segments to line segments of the same length.

CCSS.Math.Content.8.G.A.1bUnderstand congruence and similarity using physical models, transparencies, or geometry software.

Angles are taken to angles of the same measure.

CCSS.Math.Content.8.G.A.1cUnderstand congruence and similarity using physical models, transparencies, or geometry software.

Parallel lines are taken to parallel lines.

CCSS.Math.Content.8.G.A.2Understand congruence and similarity using physical models, transparencies, or geometry software.

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.

CCSS.Math.Content.8.G.A.3Understand congruence and similarity using physical models, transparencies, or geometry software.

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

CCSS.Math.Content.8.G.A.4Understand congruence and similarity using physical models, transparencies, or geometry software.

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.

CCSS.Math.Content.8.G.A.5Understand congruence and similarity using physical models, transparencies, or geometry software.

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.

CCSS.Math.Content.8.G.B.6Understand and apply the Pythagorean Theorem.

Explain a proof of the Pythagorean Theorem and its converse.

CCSS.Math.Content.8.G.B.7Understand and apply the Pythagorean Theorem.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

CCSS.Math.Content.8.G.B.8Understand and apply the Pythagorean Theorem.

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

CCSS.Math.Content.8.G.C.9Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

CCSS.Math.Content.8.NS.A.1Know that there are numbers that are not rational, and approximate them by rational numbers.

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.

CCSS.Math.Content.8.NS.A.2Know that there are numbers that are not rational, and approximate them by rational numbers.

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., π²).

CCSS.Math.Content.8.EE.A.1Work with radicals and integer exponents.

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

CCSS.Math.Content.8.EE.A.2Work with radicals and integer exponents.

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 √2 is irrational.

CCSS.Math.Content.8.EE.A.3Work with radicals and integer exponents.

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.

CCSS.Math.Content.8.EE.A.4Work with radicals and integer exponents.

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.

CCSS.Math.Content.8.EE.B.5Understand the connections between proportional relationships, lines, and linear equations.

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