Standards in this Framework
Standard  Lessons 

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

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

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). 

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

K.CC.4a
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. (onetoone correspondence) 

K.CC.4b
Understand that the last number name said tells the number of objects counted (cardinality). The number of objects is the same regardless of their arrangement or the order in which they were counted. 

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

K.CC.5
Count to answer “how many?” questions. 

K.CC.5a
Count to answer “how many?” questions about as many as 20 things arranged in a variety of ways (a line, a rectangular array, or a circle), or as many as 10 things in a scattered configuration. 

K.CC.5b
Given a number from 120, count out that many objects 

K.CC.5c
Identify and be able to count pennies within 20. (Use pennies as manipulatives in multiple mathematical contexts.) 

K.CC.6
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies 

K.CC.7
Compare two numbers between 1 and 10 presented as written numerals. 

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

K.OA.2
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem 

K.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. (drawings need not include an equation). 

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

K.OA.5
Fluently add and subtract within 5 

K.NBT.1
Compose and decompose numbers from 11 to 19 into ten ones and some further ones to understand that these numbers are composed of ten ones and one, two, three, four, five, six , seven, eight, or nine ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8). 

K.MD.1
Describe several measurable attributes of an object, such as length or weight. For example, a student may describe a shoe as, “This shoe is heavy! It is also really long!” 

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. 

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

K.G.1
Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. 

K.G.2
Correctly name shapes regardless of their orientations or overall size. 

K.G.3
Identify shapes as twodimensional (lying in a plane, “flat”) or threedimensional (“solid”). 

K.G.4
Analyze and compare two and threedimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length). 

K.G.5
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing 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?” 

1.OA.1
Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem 

1.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. 

1.OA.3
Apply properties of operations as strategies to add and subtract.5 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.) 

1.OA.4
Understand subtraction as an unknownaddend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. 

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

1.OA.6
Add and subtract within 20. 

1.OA.6a
Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). 

1.OA.6b
Fluently add and subtract within 10. 

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

1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating to 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 = ∆. 

1.NBT.1
Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. 

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

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

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

1.NBT.2c
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). 

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 <. 

1.NBT.4
Add within 100, including adding a twodigit number and a onedigit number and adding a twodigit number and a multiple of ten (e.g., 24 + 9, 13 + 10, 27 + 40), using concrete models or drawings and strategies based on place value, properties of operations, and/or relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 

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. 

1.NBT.6
Subtract multiples of 10 in the range 1090 from multiples of 10 in the range of 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. (e.g.,70 – 30, 30 – 10, 60 – 60) 

1.NBT.7
Identify dimes, and understand ten pennies can be thought of as a dime. (Use dimes as manipulatives in multiple mathematical contexts.) 

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

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. (Iteration) 

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

1.MD.4
Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. 

1.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. 

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. 

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. 

2.OA.1
Use addition and subtraction within 100 to solve one and twostep word problems by using drawings and equations with a symbol for the unknown number to represent the problem. Problems include contexts that involve adding to, taking from, putting together/taking apart (part/part/whole) and comparing with unknowns in all positions. 

2.OA.2
Fluently add and subtract within 20 using mental strategies.8 By end of Grade 2, know from memory all sums of two onedigit numbers. 

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. 

2.OA.4
Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 

2.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. 

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

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

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

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

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. 

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

2.NBT.6
Add up to four twodigit numbers using strategies based on place value and properties of operations. 

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. 

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

2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties of operations. 

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

2.MD.2
Measure the length of an object twice, using length units of different measurements; describe how the two measurements relate to the size of the unit chosen. Understand the relative size of units in different systems of measurement. 

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

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. 

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. 

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. 

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

2.MD.8
Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. 

2.MD.9
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in wholenumber units. 

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 puttogether, takeapart, and compare problems10 using information presented in a bar graph. 

2.G.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.11 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 

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

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. 

3.OA.1
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. 

3.OA.2
Interpret whole number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares (How many in each group?), or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each (How many groups can you make?). 

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

3.OA.4
Determine the unknown whole number in a multiplication or division equation relating three whole numbers using the inverse relationship of multiplication and division. 

3.OA.5
Apply properties of operations as strategies to multiply and divide.13 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 

3.OA.6
Understand division as an unknownfactor problem. 

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

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. 

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

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

3.NBT.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 

3.NBT.3
Multiply onedigit 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. 

3.NF.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a frction a/b as the quantity formed by a parts of size 1/b. For example, 3/4 means there are three 1/4 parts, so 3/4 = 1/4+1/4+1/4. 

3.NF.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram. 

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

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

3.NF.3
Explain equivalence of fractions through reasoning with visual fraction models. Compare fractions by reasoning about their size. 

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

3.NF.3b
Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and 8, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent. 

3.NF.3c
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. 

3.NF.3d
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 

3.MD.1
Tell and write time to the nearest minute and measure elapsed 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, drawing a pictorial representation on a clock face, etc. 

3.MD.2
2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).17 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. 

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. 

3.MD.4
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters. 

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

3.MD.5a
A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. 

3.MD.5b
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. 

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

3.MD.7
Relate area to the operations of multiplication and addition. 

3.MD.7a
Find the area of a rectangle with wholenumber side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. 

3.MD.7b
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 wholenumber products as rectangular areas in mathematical reasoning. 

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

3.MD.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. 

3.G.1
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. 

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

4.OA.1
Understand that a multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity 

4.OA.1a
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. 

4.OA.1b
Represent verbal statements of multiplicative comparisons as multiplication equations. 

4.OA.2
Multiply or divide to solve word problems involving multiplicative comparison. Use drawings and equations with a symbol or letter for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 

4.OA.3
Solve multistep word problems 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 symbol or letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 

4.OA.4
Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1–100 is prime or composite. 

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. Explain informally why the pattern will continue to develop in this way. 

4.NBT.1
Recognize that in a multidigit whole number, a digit in any one place represents ten times what it represents in the place to its right 

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 

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

4.NBT.4
Fluently add and subtract multidigit whole numbers using the standard algorithm 

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. 

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. 

4.NF.1
Explain why two or more fractions are equivalent a/b = n x a/ n x b by using visual fraction models. Focus attention on how the number and size of the parts differ even though the fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 

4.NF.2
Compare two fractions with different numerators and different denominators, e.g., by using visual fraction models, 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. 

4.NF.3
Understand a fraction a/b with a numerator >1 as a sum of unit fractions 1/b. 

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

4.NF.3b
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. 

4.NF.3c
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. 

4.NF.3d
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 

4.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number e.g., by using a visual such as a number line or area model. 

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

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

4.NF.4c
Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. 

4.NF.5
Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100 

4.NF.6
Use decimal notation for fractions with denominators 10 or 100. 

4.NF.7
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. 

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. 

4.MD.1a
Understand the relationship between gallons, cups, quarts, and pints 

4.MD.1b
Express larger units in terms of smaller units within the same measurement system. 

4.MD.1c
Record measurement equivalents in a two column table. 

4.MD.2
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 

4.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 

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 with common denominators by using information presented in line plots. 

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

4.MD.5a
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. 

4.MD.5b
An angle that turns through n onedegree angles is said to have an angle measure of n degrees. 

4.MD.6
Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. 

4.MD.7
Recognize angle measure as additive. When an angle is decomposed into nonoverlapping 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 or letter for the unknown angle measure. 

4.MD.8
Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems. 

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

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. 

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. 

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

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 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. 

5.OA.3
Generate two numerical patterns using a given rule. Identify apparent relationships between corresponding terms by completing a function table or input/output table. Using the terms created, form and graph ordered pairs on a coordinate plane. 

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. 

5.NBT.2
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use wholenumber exponents to denote powers of 10. 

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

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

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

5.NBT.4
Use place value understanding to round decimals up to the hundredths place. 

5.NBT.5
Fluently multiply multidigit whole numbers using the standard algorithm (or other strategies demonstrating understanding of multiplication) up to a 3 digit by 2 digit factor. 

5.NBT.6
Fluently divide up to 4digit dividends and 2digit divisors by using at least one of the following methods: 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 or concrete models. (e.g., rectangular arrays, area models) 

5.NBT.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 

5.NF.1
Add and subtract fractions and mixed numbers with unlike denominators by finding a common denominator and equivalent fractions to produce like denominators. 

5.NF.2
Solve word problems involving addition and subtraction of fractions, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. 

5.NF.3
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 

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

5.NF.4a
Apply and use understanding of multiplication to multiply a fraction or whole number by a fraction. 

5.NF.4b
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. 

5.NF.5
Interpret multiplication as scaling (resizing), by: 

5.NF.5a
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. Example: 4 x 10 is twice as large as 2 x 10. 

5.NF.5b
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. 

5.NF.6
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 

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

5.NF.7a
Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. 

5.NF.7b
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4 

5.NF.7c
Solve real world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3cup servings are in 2 cups of raisins? 

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

5.MD.2
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. 

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

5.MD.3a
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. 

5.MD.3b
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. 

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

5.MD.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 

5.MD.5a
Find the volume of a right rectangular prism with wholenumber 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 wholenumber products as volumes, e.g., to represent the associative property of multiplication. 

5.MD.5b
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. 

5.MD.5c
Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the nonoverlapping parts, applying this technique to solve real world 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). 

5.G.2
Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. 

5.G.3
Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category. 

5.G.4
Classify twodimensional figures in a hierarchy based on properties (polygons, triangles, and quadrilaterals). 
