Common Core State Standards

Math

Content

Kindergarten

Counting And Cardinality

Know Number Names And The Count Sequence.

K.CC.1
Count to 100 by ones and by tens.Examples

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

K.CC.3
Write numbers from 0 to 20. Represent a number of objects with a written numeral 020 (with 0 representing a count of no objects).Examples

K.CC.1

Count To Tell The Number Of Objects.

K.CC.4
Understand the relationship between numbers and quantities; connect counting to cardinality.Examples

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 120, count out that many objects.Examples

K.CC.4

Compare Numbers.

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.1Examples

K.CC.6

Know Number Names And The Count Sequence.

Operations And Algebraic Thinking

Understand Addition As Putting Together And Adding To, And Under Stand Subtraction As Taking Apart And Taking From.

K.OA.1
Represent addition and subtraction with objects, fingers, mental images, drawings2, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Examples

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.Examples

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).Examples

K.OA.5
Fluently add and subtract within 5.Examples

K.OA.1

Understand Addition As Putting Together And Adding To, And Under Stand Subtraction As Taking Apart And Taking From.

Measurement And Data

Describe And Compare Measurable Attributes.

K.MD.1
Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.Examples

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. For example, directly compare the heights of two children and describe one child as taller/shorter.Examples

K.MD.1

Classify Objects And Count The Number Of Objects In Each Category.

K.MD.3
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.3Examples

K.MD.3

Describe And Compare Measurable Attributes.

Geometry
 Identify And Describe Shapes (Squares, Circles, Triangles, Rectangles, Hexagons, Cubes, Cones, Cylinders, And Spheres).

Analyze, Compare, Create, And Compose Shapes.

K.G.6
Compose simple shapes to form larger shapes. For example, 'Can you join these two triangles with full sides touching to make a rectangle?'Examples

K.G.6

Counting And Cardinality

Grade 1

Operations And Algebraic Thinking

Represent And Solve Problems Involving Addition And Subtraction.

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.2Play Activities 228Examples

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.Examples

1.OA.1

Understand And Apply Properties Of Operations And The Relationship Between Addition And Subtraction.

1.OA.3
Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)Examples

1.OA.3

Add And Subtract Within 20.

1.OA.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).Examples

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).Examples

1.OA.5

Work With Addition And Subtraction Equations.

1.OA.7
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8  1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.Examples

1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _  3, 6 + 6 = _.Examples

1.OA.7

Represent And Solve Problems Involving Addition And Subtraction.

Number And Operations In Base Ten

Extend The Counting Sequence.

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.Examples

1.NBT.1

Understand Place Value.

1.NBT.2
Understand that the two digits of a twodigit number represent amounts of tens and ones. Understand the following as special cases:Examples

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.Examples

1.NBT.3
Compare two twodigit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.Examples

1.NBT.2

Use Place Value Understanding And Properties Of Operations To Add And Subtract.

1.NBT.4
Add within 100, including adding a twodigit number and a onedigit number, and adding a twodigit 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 twodigit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Examples

1.NBT.5
Given a twodigit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Examples

1.NBT.6
Subtract multiples of 10 in the range 1090 from multiples of 10 in the range 1090 (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.Examples

1.NBT.4

Extend The Counting Sequence.

Measurement And Data

Measure Lengths Indirectly And By Iterating Length Units.

1.MD.1
Order three objects by length; compare the lengths of two objects indirectly by using a third object.Examples

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 samesize length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.Examples

1.MD.1

Tell And Write Time.

1.MD.3
Tell and write time in hours and halfhours using analog and digital clocks.Examples

1.MD.3

Represent And Interpret Data.

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.Examples

1.MD.4

Measure Lengths Indirectly And By Iterating Length Units.

Geometry

Reason With Shapes And Their Attributes.

1.G.1
Distinguish between defining attributes (e.g., triangles are closed and threesided) versus nondefining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.Examples

1.G.2
Compose twodimensional shapes (rectangles, squares, trapezoids, triangles, halfcircles, and quartercircles) or threedimensional 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.4Examples

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.Examples

1.G.1

Reason With Shapes And Their Attributes.

Operations And Algebraic Thinking

Grade 2

Operations And Algebraic Thinking

Represent And Solve Problems Involving Addition And Subtraction.

2.OA.1
Use addition and subtraction within 100 to solve one and twostep 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.1Play Activities 170Examples

2.OA.1

Add And Subtract Within 20.

2.OA.2
Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two onedigit numbers.Examples

2.OA.2

Work With Equal Groups Of Objects To Gain Foundations For Multiplication.

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.Examples

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.Examples

2.OA.3

Represent And Solve Problems Involving Addition And Subtraction.

Number And Operations In Base Ten

Understand Place Value.

2.NBT.1
Understand that the three digits of a threedigit 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:Examples

2.NBT.2
Count within 1000; skipcount by 5s, 10s, and 100s.Examples

2.NBT.3
Read and write numbers to 1000 using baseten numerals, number names, and expanded form.Play Activities 102Examples

2.NBT.4
Compare two threedigit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.Examples

2.NBT.1

Use Place Value Understanding And Properties Of Operations To Add And Subtract.

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.Examples

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.Examples

2.NBT.8
Mentally add 10 or 100 to a given number 100900, and mentally subtract 10 or 100 from a given number 100900.Examples

2.NBT.5

Understand Place Value.

Measurement And Data

Measure And Estimate Lengths In Standard Units.

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

2.MD.3
Estimate lengths using units of inches, feet, centimeters, and meters.Examples

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.Examples

2.MD.1

Relate Addition And Subtraction To Length.

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.Examples

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 wholenumber sums and differences within 100 on a number line diagram.Examples

2.MD.5

Work With Time And Money.

2.MD.7
Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Examples

2.MD.8
Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and c symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?Examples

2.MD.7

Represent And Interpret Data.

2.MD.10
Draw a picture graph and a bar graph (with singleunit scale) to represent a data set with up to four categories. Solve simple put together, takeapart, and compare problems4 using information presented in a bar graph.Examples

2.MD.10

Measure And Estimate Lengths In Standard Units.

Geometry

Reason With Shapes And Their Attributes.

2.G.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Examples

2.G.2
Partition a rectangle into rows and columns of samesize squares and count to find the total number of them.Examples

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.Examples

2.G.1

Reason With Shapes And Their Attributes.

Operations And Algebraic Thinking

Grade 3

Operations And Algebraic Thinking

Represent And Solve Problems Involving Multiplication And Division.

3.OA.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.Examples

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.1Play Activities 127Examples

3.OA.4
Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 x ? = 48, 5 = _ / 3, 6 x 6 = ?.Examples

3.OA.1

Understand Properties Of Multiplication And The Relationship Between Multiplication And Division.

3.OA.5
Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)Examples

3.OA.5

Multiply And Divide Within 100.

3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 / 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers.Examples

3.OA.7

Solve Problems Involving The Four Operations, And Identify And Explain Patterns In Arithmetic.

3.OA.8
Solve twostep 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.3Play Activities 130Examples

3.OA.9
Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.Examples

3.OA.8

Represent And Solve Problems Involving Multiplication And Division.

Number And Operations In Base Ten

Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic.4

3.NBT.1
Use place value understanding to round whole numbers to the nearest 10 or 100.Examples

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.Play Activities 429Examples

3.NBT.1

Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic.4

Number And OperationsFractions5

Develop Understanding Of Fractions As Numbers.

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.Examples

3.NF.1

Develop Understanding Of Fractions As Numbers.

Measurement And Data

Solve Problems Involving Measurement And Estimation Of Intervals Of Time, Liquid Volumes, And Masses Of Objects.

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.Examples

3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve onestep 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.7Examples

3.MD.1

Represent And Interpret Data.

3.MD.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one and twostep 'how many more' and 'how many less' problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.Examples

3.MD.3
 Geometric Measurement: Understand Concepts Of Area And Relate Area To Multiplication And To Addition.

Solve Problems Involving Measurement And Estimation Of Intervals Of Time, Liquid Volumes, And Masses Of Objects.

Geometry

Reason With Shapes And Their Attributes.

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.Examples

3.G.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.Examples

3.G.1

Reason With Shapes And Their Attributes.

Operations And Algebraic Thinking

Grade 4

Operations And Algebraic Thinking

Use The Four Operations With Whole Numbers To Solve Problems.

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.1Examples

4.OA.3
Solve multistep word problems posed with whole numbers and having wholenumber 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.Play Activities 213Examples

4.OA.2

Gain Familiarity With Factors And Multiples.

4.OA.4
Find all factor pairs for a whole number in the range 1100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1100 is prime or composite.Play Activities 102Examples

4.OA.4

Generate And Analyze Patterns.

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. For example, given the rule 'Add 3' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.Play Activities 132Examples

4.OA.5

Use The Four Operations With Whole Numbers To Solve Problems.

Number And Operations In Base Ten2

Generalize Place Value Understanding For MultiDigit Whole Numbers.

4.NBT.1
Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 / 70 = 10 by applying concepts of place value and division.Examples

4.NBT.2
Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Play Activities 188Examples

4.NBT.3
Use place value understanding to round multidigit whole numbers to any place.Examples

4.NBT.1

Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic.

4.NBT.4
Fluently add and subtract multidigit whole numbers using the standard algorithm.Play Activities 516Examples

4.NBT.5
Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit 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.Examples

4.NBT.6
Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit 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.Examples

4.NBT.4

Generalize Place Value Understanding For MultiDigit Whole Numbers.

Number And OperationsFractions3

Extend Understanding Of Fraction Equivalence And Ordering.

4.NF.1
Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Examples

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.Examples

4.NF.1

Build Fractions From Unit Fractions By Applying And Extending Previous Understandings Of Operations On Whole Numbers.

4.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.Examples

4.NF.4

Understand Decimal Notation For Fractions, And Compare Decimal Fractions.

4.NF.6
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.Examples

4.NF.6

Extend Understanding Of Fraction Equivalence And Ordering.

Measurement And Data

Solve Problems Involving Measurement And Conversion Of Measurements From A Larger Unit To A Smaller Unit.

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. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...Examples

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.Play Activities 202Examples

4.MD.3
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.Examples

4.MD.1

Represent And Interpret Data.

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. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.Examples

4.MD.4

Geometric Measurement: Understand Concepts Of Angle And Measure Angles.

4.MD.5
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:Examples

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 'onedegree angle,' and can be used to measure angles.Examples

4.MD.5

Solve Problems Involving Measurement And Conversion Of Measurements From A Larger Unit To A Smaller Unit.

Geometry

Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles.

4.G.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures.Examples

4.G.2
Classify twodimensional 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.Examples

4.G.3
Recognize a line of symmetry for a twodimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify linesymmetric figures and draw lines of symmetry.Examples

4.G.1

Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles.

Operations And Algebraic Thinking

Grade 5

Operations And Algebraic Thinking

Write And Interpret Numerical Expressions.

5.OA.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Examples

5.OA.2
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation 'add 8 and 7, then multiply by 2' as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.Examples

5.OA.1

Analyze Patterns And Relationships.

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. For example, given the rule 'Add 3' and the starting number 0, and given the rule 'Add 6' and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.Examples

5.OA.3

Write And Interpret Numerical Expressions.

Number And Operations In Base Ten

Understand The Place Value System.

5.NBT.1
Recognize that in a multidigit 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.Examples

5.NBT.3
Read, write, and compare decimals to thousandths.Examples

5.NBT.3.a
Read and write decimals to thousandths using baseten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000).Examples

5.NBT.4
Use place value understanding to round decimals to any place.Examples

5.NBT.1

Perform Operations With MultiDigit Whole Numbers And With Decimals To Hundredths.

5.NBT.5
Fluently multiply multidigit whole numbers using the standard algorithm.Examples

5.NBT.6
Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit 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.Examples

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.Play Activities 232Examples

5.NBT.5

Understand The Place Value System.

Number And OperationsFractions

Use Equivalent Fractions As A Strategy To Add And Subtract Fractions.

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 fractionsExamples

5.NF.2

Apply And Extend Previous Understandings Of Multiplication And Division To Multiply And Divide Fractions.

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. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?Examples

5.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Examples

5.NF.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1Examples

5.NF.3

Use Equivalent Fractions As A Strategy To Add And Subtract Fractions.

Measurement And Data

Convert Like Measurement Units Within A Given Measurement System.

5.MD.1
Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real world problems.Examples

5.MD.1

Geometric Measurement: Understand Concepts Of Volume And Relate Volume To Multiplication And To Addition.

5.MD.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.Examples

5.MD.4

Convert Like Measurement Units Within A Given Measurement System.

Geometry

Graph Points On The Coordinate Plane To Solve RealWorld And Mathematical Problems.

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., xaxis and xcoordinate, yaxis and ycoordinate).Examples

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.Examples

5.G.1

Classify TwoDimensional Figures Into Categories Based On Their Properties.

5.G.3
Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.Examples

5.G.3

Graph Points On The Coordinate Plane To Solve RealWorld And Mathematical Problems.

Operations And Algebraic Thinking

Grade 6

Ratios And Proportional Relationships

Understand Ratio Concepts And Use Ratio Reasoning To Solve Problems.

6.RP.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, 'The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.' 'For every vote candidate A received, candidate C received nearly three votes.'Examples

6.RP.3
Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Play Activities 158Examples

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.Examples

6.RP.1

Understand Ratio Concepts And Use Ratio Reasoning To Solve Problems.

The Number System

Apply And Extend Previous Understandings Of Multiplication And Division To Divide Fractions By Fractions.

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. For example, create a story context for (2/3) / (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) / (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) / (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?Examples

6.NS.1

Compute Fluently With MultiDigit Numbers And Find Common Factors And Multiples.

6.NS.2
Fluently divide multidigit numbers using the standard algorithm.Examples

6.NS.3
Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.Play Activities 202Examples

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 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).Examples

6.NS.2

Apply And Extend Previous Understandings Of Numbers To The System Of Rational Numbers.

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.Examples

6.NS.7
Understand ordering and absolute value of rational numbers.Examples

6.NS.8
Solve realworld 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.Examples

6.NS.6.c

Apply And Extend Previous Understandings Of Multiplication And Division To Divide Fractions By Fractions.

Expressions And Equations

Apply And Extend Previous Understandings Of Arithmetic To Algebraic Expressions.

6.EE.1
Write and evaluate numerical expressions involving wholenumber exponents.Examples

6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.Examples

6.EE.2.a
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation 'Subtract y from 5' as 5  y.Examples

6.EE.3
Apply the properties of operations to generate equivalent expressions.Examples

6.EE.1

Reason About And Solve OneVariable Equations And Inequalities.

6.EE.6
Use variables to represent numbers and write expressions when solving a realworld 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.Examples

6.EE.7
Solve realworld 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.Examples

6.EE.6

Apply And Extend Previous Understandings Of Arithmetic To Algebraic Expressions.

Geometry

Solve RealWorld And Mathematical Problems Involving Area, Surface Area, And Volume.

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 realworld and mathematical problems.Examples

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 realworld and mathematical problems.Examples

6.G.4
Represent threedimensional 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 realworld and mathematical problems.Examples

6.G.1

Solve RealWorld And Mathematical Problems Involving Area, Surface Area, And Volume.

Statistics And Probability

Develop Understanding Of Statistical Variability.

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.Examples

6.SP.2

Summarize And Describe Distributions.

6.SP.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.Examples

6.SP.5
Summarize numerical data sets in relation to their context, such as by:Examples

6.SP.5.a
Reporting the number of observations.Examples

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.Examples

6.SP.4

Develop Understanding Of Statistical Variability.

Ratios And Proportional Relationships

Grade 7

Ratios And Proportional Relationships

Analyze Proportional Relationships And Use Them To Solve RealWorld And Mathematical Problems.

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. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.Examples

7.RP.2
Recognize and represent proportional relationships between quantities.Play Activities 194Examples

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.Examples

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

7.RP.2.c
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.Examples

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.Examples

7.RP.3
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.Play Activities 192Examples

7.RP.1

Analyze Proportional Relationships And Use Them To Solve RealWorld And Mathematical Problems.

The Number System

Apply And Extend Previous Understandings Of Operations With Fractions To Add, Subtract, Multiply, And Divide Rational Numbers.

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.Play Activities 423Examples

7.NS.1.d
Apply properties of operations as strategies to add and subtract rational numbers.Play Activities 186Examples

7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Play Activities 198Examples

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 realworld contexts.Examples

7.NS.2.b
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero 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.Examples

7.NS.2.c
Apply properties of operations as strategies to multiply and divide rational numbers.Play Activities 162Examples

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.Examples

7.NS.3
Solve realworld and mathematical problems involving the four operations with rational numbers.Play Activities 174Examples

7.NS.1

Apply And Extend Previous Understandings Of Operations With Fractions To Add, Subtract, Multiply, And Divide Rational Numbers.

Expressions And Equations

Use Properties Of Operations To Generate Equivalent Expressions.

7.EE.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Examples

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. For example, a + 0.05a = 1.05a means that 'increase by 5%' is the same as 'multiply by 1.05.'Examples

7.EE.1

Solve RealLife And Mathematical Problems Using Numerical And Algebraic Expressions And Equations.

7.EE.3
Solve multistep reallife 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. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.Play Activities 204Examples

7.EE.4
Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Examples

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. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?Examples

7.EE.3

Use Properties Of Operations To Generate Equivalent Expressions.

Geometry

Draw, Construct, And Describe Geometrical Figures And Describe The Relationships Between Them.

7.G.3
Describe the twodimensional figures that result from slicing three dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Examples

7.G.3

Solve RealLife And Mathematical Problems Involving Angle Measure, Area, Surface Area, And Volume.

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.Examples

7.G.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.Examples

7.G.6
Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Play Activities 103Examples

7.G.4

Draw, Construct, And Describe Geometrical Figures And Describe The Relationships Between Them.

Statistics And Probability

Use Random Sampling To Draw Inferences About A Population.

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. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.Examples

7.SP.2

Draw Informal Comparative Inferences About Two Populations.

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. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.Examples

7.SP.3

Investigate Chance Processes And Develop, Use, And Evaluate Probability Models.

7.SP.6
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.Examples

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.Examples

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. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.Examples

7.SP.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Examples

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.Examples

7.SP.6

Use Random Sampling To Draw Inferences About A Population.

Ratios And Proportional Relationships

Grade 8

The Number System

Know That There Are Numbers That Are Not Rational, And Approximate Them By Rational Numbers.

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.Examples

8.NS.1

Know That There Are Numbers That Are Not Rational, And Approximate Them By Rational Numbers.

Expressions And Equations

Work With Radicals And Integer Exponents.

8.EE.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 x 35 = 33 = 1/33 = 1/27.Examples

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 _2 is irrational.Examples

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. For example, estimate the population of the United States as 3 x 108 and the population of the world as 7 x 109, and determine that the world population is more than 20 times larger.Examples

8.EE.1

Understand The Connections Between Proportional Relationships, Lines, And Linear Equations.

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. For example, compare a distancetime graph to a distancetime equation to determine which of two moving objects has greater speed.Examples

8.EE.5

Analyze And Solve Linear Equations And Pairs Of Simultaneous Linear Equations.

8.EE.7
Solve linear equations in one variable.Examples

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.Examples

8.EE.8
Analyze and solve pairs of simultaneous linear equations.Examples

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.Examples

8.EE.7

Work With Radicals And Integer Exponents.

Functions

Define, Evaluate, And Compare Functions.

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.1Examples

8.F.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.Examples

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. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.Examples

8.F.1

Use Functions To Model Relationships Between Quantities.

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.Examples

8.F.4

Define, Evaluate, And Compare Functions.

Geometry

Understand Congruence And Similarity Using Physical Models, Trans Parencies, Or Geometry Software.

8.G.1
Verify experimentally the properties of rotations, reflections, and translations:Examples

8.G.3
Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.Examples

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 angleangle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.Examples

8.G.1

Understand And Apply The Pythagorean Theorem.

8.G.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.Examples

8.G.7

Solve RealWorld And Mathematical Problems Involving Volume Of Cylinders, Cones, And Spheres.

8.G.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.Examples

8.G.9

Understand Congruence And Similarity Using Physical Models, Trans Parencies, Or Geometry Software.

Statistics And Probability

Investigate Patterns Of Association In Bivariate Data.

8.SP.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway 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. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?Examples

8.SP.4

Investigate Patterns Of Association In Bivariate Data.

The Number System

High School

The Real Number System

Extend The Properties Of Exponents To Rational Exponents.

N.RN.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.Examples

N.RN.2

Extend The Properties Of Exponents To Rational Exponents.

The Complex Number System

Represent Complex Numbers And Their Operations On The Complex Plane.

N.CN.6
(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.Examples

N.CN.6

Represent Complex Numbers And Their Operations On The Complex Plane.

Seeing Structure In Expressions

Interpret The Structure Of Expressions

A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4  y4 as (x2)2  (y2)2, thus recognizing it as a difference of squares that can be factored as (x2  y2)(x2 + y2).Examples

A.SSE.2

Write Expressions In Equivalent Forms To Solve Problems

A.SSE.3.c
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t _ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.Examples

A.SSE.3.c

Interpret The Structure Of Expressions

Interpreting Functions

Understand The Concept Of A Function And Use Function Notation

F.IF.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n _ 1.Examples

F.IF.3

Interpret Functions That Arise In Applications In Terms Of The Context

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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity._Examples

F.IF.4
 Analyze Functions Using Different Representations

Understand The Concept Of A Function And Use Function Notation

Congruence

Experiment With Transformations In The Plane

G.CO.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Examples

G.CO.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Examples

G.CO.1

Understand Congruence In Terms Of Rigid Motions

G.CO.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Examples

G.CO.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Examples

G.CO.6

Prove Geometric Theorems

G.CO.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.Examples

G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.Examples

G.CO.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.Examples

G.CO.9

Experiment With Transformations In The Plane

Similarity, Right Triangles, And Trigonometry

Understand Similarity In Terms Of Similarity Transformations

G.SRT.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.Examples

G.SRT.3

Prove Theorems Involving Similarity

G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.Examples

G.SRT.4

Understand Similarity In Terms Of Similarity Transformations

Circles

Understand And Apply Theorems About Circles

G.C.2
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.Examples

G.C.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.Examples

G.C.4
(+) Construct a tangent line from a point outside a given circle to the circle.Examples

G.C.2

Find Arc Lengths And Areas Of Sectors Of Circles

G.C.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.Examples

G.C.5

Understand And Apply Theorems About Circles

Expressing Geometric Properties With Equations

Use Coordinates To Prove Simple Geometric Theorems Algebraically

G.GPE.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula._Examples

G.GPE.7

Use Coordinates To Prove Simple Geometric Theorems Algebraically

Geometric Measurement And Dimension

Explain Volume Formulas And Use Them To Solve Problems

G.GMD.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.Examples

G.GMD.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems._Examples

G.GMD.1

Explain Volume Formulas And Use Them To Solve Problems

Interpreting Categorical And Quantitative Data

Summarize, Represent, And Interpret Data On A Single Count Or Measurement Variable

S.ID.4
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.Examples

S.ID.4

Interpret Linear Models

S.ID.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Examples

S.ID.7

Summarize, Represent, And Interpret Data On A Single Count Or Measurement Variable

Conditional Probability And The Rules Of Probability

Use The Rules Of Probability To Compute Probabilities Of Compound Events In A Uniform Probability Model

S.CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B)  P(A and B), and interpret the answer in terms of the model.Examples

S.CP.8
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model.Examples

S.CP.9
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.Examples

S.CP.7

Use The Rules Of Probability To Compute Probabilities Of Compound Events In A Uniform Probability Model

The Real Number System

Kindergarten

Content

ELA/Literacy

Kindergarten

Reading Foundational Skills
 Print Concepts

Phonological awareness

K.RF.2
Demonstrate understanding of spoken words, syllables, and sounds (phonemes).Examples

K.RF.2.a
Recognize and produce rhyming words.Examples

K.RF.2.c
Blend and segment onsets and rimes of singlesyllable spoken words.Examples

K.RF.2.d
Isolate and pronounce the initial, medial vowel, and final sounds (phonemes) in threephoneme (consonentvowelconsonent, or CVC) words.* (This does not include CVCs ending with /l/, /r/, or /x/.)Examples

K.RF.2.e
Add or substitute individual sounds (phonemes) in simple, onesyllable words to make new words.Examples

K.RF.2

Phonics and Word recognition

K.RF.3
Know and apply gradelevel phonics and word analysis skills in decoding words.Examples

K.RF.3.c
Read common highfrequency words by sight (e.g., the, of, to, you, she, my, is, are, do, does).Examples

K.RF.3.d
Distinguish between similarly spelled words by identifying the sounds of the letters that differ.Play Activities 128Examples

K.RF.3

Language

Conventions of Standard English

K.L.1
Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.Examples

K.L.1.e
Use the most frequently occurring prepositions (e.g., to, from, in, out, on, off, for, of, by, with).Examples

K.L.2
Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.Examples

K.L.2.c
Write a letter or letters for most consonant and shortvowel sounds (phonemes).Examples

K.L.2.d
Spell simple words phonetically, drawing on knowledge of soundletter relationships.Examples

K.L.1

Vocabulary Acquisition and Use

K.L.5.a
Sort common objects into categories (e.g., shapes, foods) to gain a sense of the concepts the categories represent.Examples

K.L.5.a

Conventions of Standard English

Reading Literature

Key Ideas and Details

K.RL.3
With prompting and support, identify characters, settings, and major events in a story.Examples

K.RL.3

Key Ideas and Details

Reading Foundational Skills

Grade 1

Reading Informational

Craft and Structure

1.RI.6
Distinguish between information provided by pictures or other illustrations and information provided by the words in a text.Examples

1.RI.6

Integration of Knowledge and Ideas

1.RI.9
Identify basic similarities in and differences between two texts on the same topic (e.g., in illustrations, descriptions, or procedures).Examples

1.RI.9

Craft and Structure

Reading Foundational Skills
 Print Concepts
 Phonological awareness

Phonics and Word recognition

1.RF.3
Know and apply gradelevel phonics and word analysis skills in decoding words.Play Activities 283Examples

1.RF.3.a
Know the spellingsound correspondences for common consonant digraphs.Examples

1.RF.3.b
Decode regularly spelled onesyllable words.Play Activities 172Examples

1.RF.3.g
Recognize and read gradeappropriate irregularly spelled words.Examples

1.RF.3

Fluency

1.RF.4.c
Use context to confirm or selfcorrect word recognition and understanding, rereading as necessary.Examples

1.RF.4.c

Language

Conventions of Standard English

1.L.1
Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.Examples

1.L.1.b
Use common, proper, and possessive nouns.Examples

1.L.1.e
Use verbs to convey a sense of past, present, and future (e.g., Yesterday I walked home; Today I walk home; Tomorrow I will walk home).Examples

1.L.1.f
Use frequently occurring adjectives.Examples

1.L.1.i
Use frequently occurring prepositions (e.g., during, beyond, toward).Examples

1.L.2
Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.Examples

1.L.2.a
Capitalize dates and names of people.Examples

1.L.2.e
Spell untaught words phonetically, drawing on phonemic awareness and spelling conventions.Examples

1.L.1

Vocabulary Acquisition and Use

1.L.5.a
Sort words into categories (e.g., colors, clothing) to gain a sense of the concepts the categories represent.Examples

1.L.5.a

Conventions of Standard English

Reading Informational

Grade 2

Reading Literature

Craft and Structure

2.RL.5
Describe the overall structure of a story, including describing how the beginning introduces the story and the ending concludes the action.Examples

2.RL.5

Integration of Knowledge and Ideas

2.RL.7
Use information gained from the illustrations and words in a print or digital text to demonstrate understanding of its characters, setting, or plot.Examples

2.RL.7

Craft and Structure

Reading Informational

Key Ideas and Details

2.RI.1
Ask and answer such questions as who, what, where, when, why, and how to demonstrate understanding of key details in a text.Examples

2.RI.1

Key Ideas and Details

Reading Foundational Skills

Phonics and Word recognition

2.RF.3
Know and apply gradelevel phonics and word analysis skills in decoding words.Play Activities 276Examples

2.RF.3.a
Distinguish long and short vowels when reading regularly spelled onesyllable words.Examples

2.RF.3.b
Know spellingsound correspondences for additional common vowel teams.Examples

2.RF.3.c
Decode regularly spelled twosyllable words with long vowels.Examples

2.RF.3.e
Identify words with inconsistent but common spellingsound correspondences.Examples

2.RF.3.f
Recognize and read gradeappropriate irregularly spelled words.Examples

2.RF.3

Fluency

2.RF.4.c
Use context to confirm or selfcorrect word recognition and understanding, rereading as necessary.Examples

2.RF.4.c

Phonics and Word recognition

Language

Conventions of Standard English

2.L.1
Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.Examples

2.L.1.d
Form and use the past tense of frequently occurring irregular verbs (e.g., sat, hid, told).Examples

2.L.1.e
Use adjectives and adverbs, and choose between them depending on what is to be modified.Examples

2.L.2
Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.Examples

2.L.1

Knowledge of Language

2.L.3
Use knowledge of language and its conventions when writing, speaking, reading, or listening.Examples

2.L.3

Vocabulary Acquisition and Use

2.L.4.a
Use sentencelevel context as a clue to the meaning of a word or phrase.Examples

2.L.4.c
Use a known root word as a clue to the meaning of an unknown word with the same root (e.g., addition, additional).Examples

2.L.4.d
Use knowledge of the meaning of individual words to predict the meaning of compound words (e.g., birdhouse, lighthouse, housefly; bookshelf, notebook, bookmark).Examples

2.L.5
Demonstrate understanding of figurative language, word relationships and nuances in word meanings.Examples

2.L.4.a

Conventions of Standard English

Reading Literature

Grade 3

Reading Literature

Key Ideas and Details

3.RL.1
Ask and answer questions to demonstrate understanding of a text, referring explicitly to the text as the basis for the answers.Examples

3.RL.1

Key Ideas and Details

Reading Informational

Key Ideas and Details

3.RI.1
Ask and answer questions to demonstrate understanding of a text, referring explicitly to the text as the basis for the answers.Examples

3.RI.1

Integration of Knowledge and Ideas

3.RI.7
Use information gained from illustrations (e.g., maps, photographs) and the words in a text to demonstrate understanding of the text (e.g., where, when, why, and how key events occur).Examples

3.RI.8
Describe the logical connection between particular sentences and paragraphs in a text (e.g., comparison, cause/effect, first/second/third in a sequence).Examples

3.RI.7

Key Ideas and Details

Reading Foundational Skills

Phonics and Word recognition

3.RF.3
Know and apply gradelevel phonics and word analysis skills in decoding words.Play Activities 229Examples

3.RF.3.a
Identify and know the meaning of the most common prefixes and derivational suffixes.Examples

3.RF.3.c
Decode multisyllable words.Examples

3.RF.3.d
Read gradeappropriate irregularly spelled words.Examples

3.RF.3
 Fluency

Phonics and Word recognition

Language

Conventions of Standard English

3.L.1
Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.Examples

3.L.1.a
Explain the function of nouns, pronouns, verbs, adjectives, and adverbs in general and their functions in particular sentences.Examples

3.L.1.g
Form and use comparative and superlative adjectives and adverbs, and choose between them depending on what is to be modified.Examples

3.L.2
Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.Examples

3.L.2.f
Use spelling patterns and generalizations (e.g., word families, positionbased spellings, syllable patterns, ending rules, meaningful word parts) in writing words.Examples

3.L.1

Vocabulary Acquisition and Use

3.L.4.a
Use sentencelevel context as a clue to the meaning of a word or phrase.Examples

3.L.4.b
Determine the meaning of the new word formed when a known affix is added to a known word (e.g., agreeable/disagreeable, comfortable/uncomfortable, care/careless, heat/preheat).Examples

3.L.5.b
Identify reallife connections between words and their use (e.g., describe people who are friendly or helpful).Examples

3.L.4.a

Conventions of Standard English

Reading Literature

Grade 4

Reading Literature
 Key Ideas and Details

Craft and Structure

4.RL.4
Determine the meaning of words and phrases as they are used in a text, including those that allude to significant characters found in mythology (e.g., Herculean).Examples

4.RL.4

Range of Reading and Complexity of Text

4.RL.10
By the end of the year, read and comprehend literature, including stories, dramas, and poetry, in the grades 45 text complexity band proficiently, with scaffolding as needed at the high end of the range.Examples

4.RL.10

Reading Informational

Key Ideas and Details

4.RI.1
Refer to details and examples in a text when explaining what the text says explicitly and when drawing inferences from the text.Examples

4.RI.1

Craft and Structure

4.RI.5
Describe the overall structure (e.g., chronology, comparison, cause/effect, problem/solution) of events, ideas, concepts, or information in a text or part of a text.Examples

4.RI.5

Integration of Knowledge and Ideas

4.RI.7
Interpret information presented visually, orally, or quantitatively (e.g., in charts, graphs, diagrams, time lines, animations, or interactive elements on Web pages) and explain how the information contributes to an understanding of the text in which it appears.Examples

4.RI.7

Key Ideas and Details

Reading Foundational Skills

Phonics and Word recognition

4.RF.3
Know and apply gradelevel phonics and word analysis skills in decoding words.Examples

4.RF.3.a
Use combined knowledge of all lettersound correspondences, syllabication patterns, and morphology (e.g., roots and affixes) to read accurately unfamiliar multisyllabic words in context and out of context.Examples

4.RF.3
 Fluency

Phonics and Word recognition

Writing

Text Types and Purposes

4.W.3
Write narratives to develop real or imagined experiences or events using effective technique, descriptive details, and clear event sequences.Examples

4.W.3

Text Types and Purposes

Language

Conventions of Standard English

4.L.1
Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.Examples

4.L.1.a
Use relative pronouns (who, whose, whom, which, that) and relative adverbs (where, when, why).Examples

4.L.1.b
Form and use the progressive (e.g., I was walking; I am walking; I will be walking) verb tenses.Examples

4.L.1.g
Correctly use frequently confused words (e.g., to, too, two; there, their).*Examples

4.L.2
Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing.Examples

4.L.2.d
Spell gradeappropriate words correctly, consulting references as needed.Examples

4.L.1
 Knowledge of Language

Vocabulary Acquisition and Use

4.L.4.a
Use context (e.g., definitions, examples, or restatements in text) as a clue to the meaning of a word or phrase.Examples

4.L.5
Demonstrate understanding of figurative language, word relationships, and nuances in word meanings.Examples

4.L.5.c
Demonstrate understanding of words by relating them to their opposites (antonyms) and to words with similar but not identical meanings (synonyms).Examples

4.L.4.a

Conventions of Standard English

Reading Literature

Kindergarten