Iowa K–6 mathematics standards

By the end of kindergarten, flexibly, efficiently and accurately find all sums within 5. Note: Fluency of this standard is critical by the end of grade level.

Count to 100 by ones and by tens.

Count forward beginning from any given number within the range of 0–100.

Write numbers from 0 to 20. Given a set of 0–20 objects, write a numeral to represent the quantity.

Number names must be said in the standard order (sequencing).

Each object must be paired with one and only one number name and each number name with one and only one object (one-to-one correspondence).

The last number name said tells the number of objects counted, objects may be counted in any order (cardinality).

The number of objects is the same regardless of their arrangement or the order in which they were counted (conservation of number).

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

Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration. Given a number from 1–20, count out that many objects.

Determine whether the number of objects in one group of 1– 10 objects is greater than, less than, or equal to the number of objects in another group of 1–10 objects. For example, using matching and counting strategies.

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

Count backwards by ones from 20 to 0.

Count backwards beginning from any given number within the range of 0–20.

Quickly recognize and name the quantity of up to 5 objects briefly shown in structured or unstructured arrangements without counting (perceptual subitizing).

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

Correctly name shapes regardless of their orientations or overall size.

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

Analyze and compare two- and three-dimensional shapes, in varied sizes and orientations, using informal language to describe their similarities, differences, parts and other attributes. For example, number of sides and vertices/corners and having sides of equal length.

Model shapes in the world by building shapes from components and drawing shapes. For example, sticks and clay balls.

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

Describe several measurable attributes (for example, length, width, weight) of objects by using words such as short, long, small, big, heavy, light.

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.

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. Limit category counts to be less than or equal to 10.

Identify the penny and know the value is one cent. Count pennies up to 20.

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

Represent addition and subtraction situations in a variety of ways. For example, with objects, fingers, mental images, drawings, sounds (claps), acting out situations, verbal explanations, expressions, or equations.

Add-to with result unknown.

Take-from with result unknown.

Put-together/take-apart with total unknown.

Put-together with both addends unknown.

Decompose numbers less than or equal to 10 in more than one way. For example, by using objects or drawings, and record each decomposition by a drawing or equation, as in 5 = 2 + 3, 5 = 4 + 1, and 5 = 2 + 2 + 1.

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

Counting on.

Counting back.

Using the relationship between addition and subtraction.

Creating equivalent, but easier or known sums.

By the end of Grade 1, flexibly, efficiently, and accurately find all sums within 10. Note: Fluency of this standard is critical by the end of grade level.

Distinguish between defining attributes (for example, triangles are closed and three-sided) versus non-defining attributes (for example, color, orientation, overall size); build and draw shapes to possess defining attributes.

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or threedimensional shapes (cubes, rectangular prisms, cones and cylinders) to create a composite shape, and compose new shapes from the composite shape. Students do not need to learn formal names for these shapes.

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.

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

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

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

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.

Identify pennies and dimes and their values. Count a mixed collection of dimes and pennies to determine the cent value (total not to exceed 100 cents).

Count forward and backward starting with any given number within the range of 0–120, In this range, read and write numerals and represent a number of objects with a written numeral.

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

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

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).

Compare two two-digit numbers based on meanings of the tens and ones digits, using phrases as greater than, less than or equal to, connecting to the use of >, =, and < symbols.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

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

Subtract multiples of 10 in the range 10 to 90 from multiples of 10 in the range 10 to 90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; explain the reasoning used.

Adding to.

Taking from.

Putting together.

Taking apart.

Comparing.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20. For example, by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Apply properties of operations, (commutative and associative), as strategies to add and subtract. For example, Commutative property of addition, if 8 + 3 = 11 is known then, 3 + 8 = 11 is also known. Associative 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.

Understand subtraction as an unknown-addend problem. For example, subtract 10 − 8 by finding the number that makes 10 when added to 8.

Relate counting forward and backward to addition and subtraction, add or subtract 1 or 2.

Counting on.

Making ten (for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14).

Decomposing a number leading to a ten (for example, 13 − 4 = 13 − 3 − 1 = 10 − 1 = 9).

Using the relationship between addition and subtraction (for example, knowing that 8 + 4 = 12, one knows 12 − 8 = 4).

Creating equivalent but easier or known sums (for example, adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Counting up to subtract.

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 is true and which is false? 6 = 6, 7 = 8 − 1, 5 + 2 = 2 + 5, 4 + 3 = 5 + 2.

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 = ⎕.

Relate counting to addition and subtraction (for example, by counting on 2 to add 2).

Use conceptual subitizing in unstructured arrangements with totals up to 10 and structured arrangements anchored to 5 or 10 (for example, 10 frames, double ten frames, math rack) with totals up to 20 to relate the compositions and decompositions to addition and subtraction.

Counting on.

Making ten.

Decomposing a number leading to a ten.

Using the relationship between addition and subtraction.

Creating equivalent, but easier or known sums.

Counting up to subtract.

By the end of Grade 2, flexibly, efficiently and accurately find all sums of two one-digit numbers. Note: Fluency of this standard is critical by the end of grade.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify twodimensional shapes: triangles, quadrilaterals, rectangles, squares, trapezoids, pentagons, hexagons, circles, half-circles and quarter-circles, and three-dimensional figures: cubes, right rectangular prisms, right circular cones, and right circular cylinders. (Sizes are compared directly or visually, not compared by measuring.)

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

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.

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

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

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

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

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units. For example, by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

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

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. For example, if you have 3 quarters, 2 dimes and 4 pennies, how many cents do you have? For this standard, it may be appropriate to record amounts using decimals but does not include adding and subtracting with decimals.

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

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple problems: put-together, take-apart, and compare, using information presented in a bar graph.

Describe the relationship among standard units of time: minutes, hours, days, weeks, months and years (such as 7 days in a week, 60 minutes in an hour, etc.).

Identify nickels, quarters and dollars and know their values.

Use interviews, surveys, and observations to collect data that answer questions about students' interests and/or their environment.

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

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 zero tens and zero ones). For example: 706 equals 7 hundreds, 0 tens, and 6 ones.

Count forward and backward within 1,000; skip-count forward and backward by 5s, 10s, and 100s.

Read and write numbers to 1,000 using base-ten numerals, number names, and expanded form.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using terms “greater than”, “less than”, and “equal to”, connecting to the use of >, =, and < symbols.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Note: Fluency of this standard is critical by the end of grade level.

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

Add and subtract within 1,000, 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.

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

Explain why addition and subtraction strategies work, using place value and the properties of operations. Explanations may be supported by drawings or objects.

Adding to.

Taking from.

Putting together.

Taking apart.

Comparing.

Counting on.

Counting back.

Making ten.

Decomposing a number leading to a ten.

Using the relationship between addition and subtraction.

Creating equivalent, but easier or known sums.

Adding up to subtract.

Determine whether a group of objects (up to 20) has an odd or even number of members; write an equation to express an even number as a sum of two equal addends. For example, by pairing objects or counting them by 2s.

Use repeated addition to find the total number of objects arranged in equal groups and rectangular arrays; write an equation to express the total as a sum of equal addends.

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

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.

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. For example, by representing the problem on a number line.

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l) (Excludes compound units such as cubic centimeters and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve onestep word problems involving measured quantities (masses and liquid volumes). Excludes multiplicative comparison problems involving notions of "times as much"; problems do not require unit conversion.

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs.

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.

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.

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.

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

Find the area of a rectangle with whole-number side lengths by tiling it and show that the area is the same as would be found by multiplying the side lengths.

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems.

Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths 𝑎𝑎 and 𝑏𝑏 + 𝑐𝑐 is the sum of 𝑎𝑎 × 𝑏𝑏 and 𝑎𝑎 × 𝑐𝑐 . Use area models to represent the distributive property in mathematical reasoning.

Recognize area as additive. Find areas of figures that can be decomposed into non-overlapping rectangles and add the areas of the non-overlapping parts, applying this technique to solve real world problems.

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.

Round whole numbers to the nearest 10 or 100 within the range of 0–1,000. For example, rounding 643 to the nearest 10 would be 640; to the nearest 100 would be 600.

Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. For example, 412 - 13 =412 -12 -1 = 400 -1 =399; 505+70 = 575. Note: Fluency of this standard is critical by the end of grade level.

Use place value and properties of operations to multiply onedigit whole numbers by multiples of 10 in the range 10–90. For example, 9 × 80, 5 × 60.

Understand a fraction 1 𝑏𝑏 as the quantity formed by 1 part when a whole is partitioned into 𝑏𝑏 equal parts; understand a fraction 𝑎𝑎 𝑏𝑏 as the quantity formed by a part of size 1 𝑏𝑏 .

Represent a fraction 1 𝑏𝑏 on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into 𝑏𝑏 equal parts. Recognize that each part has size 1 𝑏𝑏 and that the endpoint of the part based at 0 locates the number 1 𝑏𝑏 on the number line.

Represent a fraction 𝑎𝑎 𝑏𝑏 on a number line diagram by marking off 𝑎𝑎 lengths 1 𝑏𝑏 from 0. Recognize that the resulting interval has size 𝑎𝑎 𝑏𝑏 and that its endpoint locates the number 𝑎𝑎 𝑏𝑏 on the number line.

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

Recognize and generate simple equivalent fractions. For example, 1 2 = 2 4 , 4 6 = 2 3 . Explain why the fractions are equivalent. For example, by using a visual fraction model.

Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers. For example, express 3 in the form 3 = 3 1 ; recognize that 6 1 = 6; locate 4 4 and 1 at the same point of a number line diagram.

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 conclusion. For example, by using a visual fraction model.

Interpret products of whole numbers. For example, interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each; describe a context in which a total number of objects can be expressed as 5 × 7.

Interpret whole-number quotients of whole numbers as the number of groups or the number in each group in situations of equal groups. For example, describe a context involving equal groups of objects in which the number of groups or the number in each group can be expressed as 56 ÷ 8.

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, with unknowns in all positions. For example, by using drawings and equations with a symbol for the unknown number to represent the problem.

Be able to represent a word problem by writing an equation with a symbol for the unknown whole number and determine the unknown whole number in a multiplication or division equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ⎕ = 48, 5 = ⎕ ÷ 3, 6 × 6 = ⎕.

Use properties of operations as strategies to multiply and divide. For example, 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.)

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding (or remembering) the number that makes 32 when multiplied by 8 (▯ × 8 = 32).

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. For example, knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8. By the end of Grade 3, flexibly, efficiently, and accurately find all products of two one-digit numbers. Note: Fluency of this standard is critical by the end of grade level.

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Note: this standard is limited to problems posed with whole numbers and have whole number answers; students should know how to perform operations in conventional order when there are no parentheses to specify a particular order (Order of Operations).

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.

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 pencils in a collection.

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

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category and identify right triangles.

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Know relative sizes of measurement units within one system of measurement, 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 by using multiplication. For example, record measurement equivalents in a two-column table, know that 1 ft is 12 times as long as 1 in or express the length of a 4 ft snake as 48 in.

Use the four operations to solve word problems involving distances, intervals of time (including elapsed 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.

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.

8 , 2 8 , 3 8 , 4 8 , 5 8 , 6 8 , 7 8 , 8 8

4 , 2 4 , 3 4 , 4 4

1 2 , 2 2

An angle is measured with reference to a circle with its center at the common endpoint of the angle’s rays. An angle that turns through 1 360 of a circle is called a "one-degree angle," and can be used to measure angles.

An angle that turns through n one-degree angles is said to have an angle measure of n° . For example, an angle that turns through 45 one-degree angles has an angle measure of 45 degrees.

Draw and measure angles in whole-number degrees (1–180°) using a protractor. Sketch angles of specified measure.

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems. For example, by using an equation with a symbol for the unknown angle measure.

Recognize that in a multi-digit whole number, a digit in one place represents ten times what that same digit represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Read and write whole multi-digit numbers using base-ten numerals (standard form), number names (word form), and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Use place value understanding to round multi-digit whole numbers to any place. For example, 435,450 rounded to the nearest tenthousands place is 440,000 because it is more than halfway between 430,000 and 440,000.

Fluently add and subtract multi-digit whole numbers up to 1,000,000 using an algorithm. Algorithms may include the standard algorithm, partial sums, partial differences, counting or adding up in increments. Note: Fluency of this standard is critical by the end of grade level.

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Be able to illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Find whole-number quotients and remainders with up to fourdigit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Illustrate and explain numerical statements of fraction equivalence 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 write equivalent fractions.

Compare two fractions with different numerators and different denominators, by creating common denominators or numerators, comparing to a benchmark fraction such as 1 2 and/or by using a visual fraction model. 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.

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

Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Be able to justify decompositions. For example, by using a visual fraction model.

Add and subtract mixed numbers with like denominators and show sums and differences of mixed numbers on a number line diagram.

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, by using visual fraction models and or equations to represent the problem.

Using a visual fraction model, understand a fraction with a numerator greater than 1 is a multiple of a unit fraction. For example, using a number line to show 5 4 as the product of 5 × 1 4 .

Multiply a fraction by a whole number using the principle that the product is the whole number times the numerator of the fraction with the same denominator.

Solve word problems involving multiplication of a fraction by a whole number. Use visual fraction models and/or equations to represent the problem.

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. For example, express 3 10 𝑎𝑎𝑎𝑎 30 100, and add 3 10 + 4 100 = 34 100. Note: Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators is not a requirement at this grade.

Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62 100 and locate 0.62 on a number line.

Compare two decimals to hundredths by reasoning about their size, recording the results of comparisons with the symbols >, =, or <. Recognize that comparisons are valid only when the two decimals refer to the same whole. Show decimals on a number line diagram and be able to justify numerical statements of decimal comparison by using a visual fraction model.

Interpret a multiplication equation as a comparison and represent verbal statements of multiplicative comparisons as multiplication equations. For example, write 35 = 7 x 5 to represent the statement that a 35-foot-long whale shark is 7 times as long as a 5-footlong reef shark.

Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison. Be able to use drawings and equations with a variable for the unknown number to represent the problem. For example, Tom’s pencil is 4 times as long as Julie’s pencil. Tom’s pencil is 8 inches long. How long is Julie’s pencil? (multiplicative comparison) For example, Julie’s pencil is 2 inches long. Tom’s pencil is 8 inches long. How much longer is Tom’s pencil than Julie’s pencil? (additive comparison)

Solve multistep word problems posed with whole numbers and whole-number answers using the four operations, including problems in which remainders must be interpreted. Be able to represent word problems with mathematical diagrams and with equations in which a letter stands for an unknown quantity and be able to assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Be able to find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

Given the rule for a sequence of numbers, identify apparent features of the sequence that were not explicit in the rule itself; explain informally why the numbers will continue to alternate in this way. For example, given the rule "Add 3" and the number sequence 1, 4, 7, 10, 13 observe that the terms appear to alternate between odd and even numbers;

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. Plot points in the first quadrant of a coordinate plane. Understand that the first number indicates how far to travel from the origin in the direction of the x-axis, and the second number indicates how far to travel in the direction of the yaxis, with the convention that the names of the two axes and the coordinates correspond (𝑥𝑥, 𝑦𝑦).

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.

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.

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

Convert among different-sized standard measurement units within a given measurement system and use these conversions in solving multi-step, real world problems. For example, (convert 5 cm to 0.05 m).

1 8 , 2 8 , 3 8 , 4 8 , 5 8 , 6 8 , 7 8 , 8 8

1 4 , 2 4 , 3 4 , 4 4

1 2 , 2 2

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.

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.

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

Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes and show that the volume is the same as would be if found by multiplying the edge lengths or equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes to represent the associative property of multiplication.

Apply the formulas 𝑉𝑉 = 𝑙𝑙 × 𝑤𝑤 × ℎ and 𝑉𝑉 = 𝐵𝐵 × ℎ (where 𝐵𝐵 stands for the area of the base) 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.

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts (composite figures), applying this technique to solve real world problems. For example, find the volume of composite figures.

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1 10 of what it represents in the place to its left.

Explain and use patterns in the number of zeros of the product when multiplying a number by powers of 10 and use patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. For example, 347.392 = 300 + 40 + 7 + 0.3 + 0.09 + 0.002.

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

Use place value understanding to round decimals to any place. For example, 5.43 rounded to the tenths is 5.4 because the last digit must be in the place the decimal is rounded to. Note: 5.40 would not be correct as it is rounded to the hundredths, not tenths.

Fluently multiply whole multi-digit numbers including using an algorithm. Algorithms may include the standard algorithm, partial products, area model. Note: Fluency of this standard is critical by the end of grade level.

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division, including the standard algorithm. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Add, subtract, multiply, and divide decimals to hundredths. Be able to illustrate and explain using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2 3 + 5 4 = 8 12 + 15 12 = 23 12. (In general, 𝑎𝑎 𝑏𝑏 + 𝑐𝑐 𝑑𝑑 = (𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏) 𝑏𝑏𝑏𝑏 ).

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. For example, 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. For example, my friend and I each have some lemons. We need 1 cup of lemon juice to make lemonade. If I squeeze 1 2 cup of lemon juice and my friend squeezes 2 5 a cup of lemon juice how much lemon juice do we have? Is it enough?

Interpret that a fraction is the division of the numerator by the denominator (𝑎𝑎 𝑏𝑏 = 𝑎𝑎 ÷ 𝑏𝑏). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, by using visual fraction models or equations to represent the problem. For example, if 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Interpret the product ( 𝑎𝑎 𝑏𝑏 ) × 𝑞𝑞 as a part of a partition of q into b equal parts; equivalently, as the result of a sequence of operations 𝑎𝑎 × 𝑞𝑞 ÷ 𝑏𝑏. Recognize that 1 𝑏𝑏 × 𝑞𝑞 = 𝑞𝑞 ÷ 𝑏𝑏 (dividing by a whole is the same as multiplying by the reciprocal. For example, use a visual fraction model to show ( 2 3 ) × 4 = 8 3 , and create a story context for this equation. Do the same with ( 2 3 ) × (4 5 ) = 8 15. (In general, ( 𝑎𝑎 𝑏𝑏 ) × (𝑐𝑐 𝑑𝑑 ) = 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 .)

Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles and represent fraction products as rectangular areas.

Comparing the size of a product to the size of one factor based on the size of the other factor, without performing the indicated multiplication.

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 𝑎𝑎 𝑏𝑏 = 𝑛𝑛×𝑎𝑎 𝑛𝑛×𝑏𝑏 to the effect of multiplying 𝑎𝑎 𝑏𝑏 by 1.

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

Interpret division of a unit fraction by a non-zero 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 .

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.

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

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols, including expressions in which whole numbers and fractions appear.

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 1 2 as 1 2 × (8 + 7). Recognize that 3 × ( 18 19 + 2 3 ) is three times as large as 18 19 +2 3 , without having to calculate the indicated sum or product.

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; explain informally why this is so. 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.

Write and evaluate numerical expressions involving wholenumber exponents.

Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 − 𝑦𝑦.

Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas 𝑉𝑉 = 𝑠𝑠3 and 𝐴𝐴 = 6𝑠𝑠2 to find the volume and surface area of a cube with sides of length 𝑠𝑠 = 1 2 .

Apply the properties of operations to generate equivalent expressions. Know that expressions are called equivalent when they name the same number regardless of which value is substituted into them. For example, apply the distributive property to the expression 3(2 + 𝑥𝑥) to produce the equivalent expression 6 + 3𝑥𝑥; apply the distributive property to the expression 24𝑥𝑥 + 18𝑦𝑦 to produce the equivalent expression 6(4𝑥𝑥 + 3𝑦𝑦); apply properties of operations to 𝑦𝑦 + 𝑦𝑦 + 𝑦𝑦 to produce the equivalent expression 3𝑦𝑦.

Describe the properties of operations used to show two expressions are equivalent. For example, show that 3𝑐𝑐 + 3𝑐𝑐𝑐𝑐 and 3𝑐𝑐(1 + 𝑑𝑑) are equivalent.

Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Solving an equation or inequality is a process of answering a question: Which values from a specified set, if any, make the equation or inequality true?

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or depending on the purpose at hand, any number in a specified set.

Solve real-world and mathematical problems by writing and solving equations of the form 𝑥𝑥 + 𝑝𝑝 = 𝑞𝑞 and 𝑝𝑝𝑝𝑝 = 𝑞𝑞 for cases in which 𝑝𝑝, 𝑞𝑞 and 𝑥𝑥 are all nonnegative rational numbers.

Write an inequality of the form 𝑥𝑥 > 𝑐𝑐 or 𝑥𝑥 < 𝑐𝑐 to represent a constraint or condition in a real-world or mathematical problem. Recognize that an inequality of the form 𝑥𝑥 > 𝑐𝑐 or 𝑥𝑥 < 𝑐𝑐 has infinitely many solutions; use a number line diagram to represent infinitely many solutions of such an inequality.

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity in terms of the other quantity. Analyze the relationship between the dependent and independent variables using graphs and tables and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation 𝑑𝑑 = 65𝑡𝑡 to represent the relationship between distance and time.

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas 𝑉𝑉 = 𝑙𝑙𝑙𝑙ℎ and 𝑉𝑉 = 𝐵𝐵ℎ (where 𝐵𝐵 stands for the area of the base) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.

Use and interpret models to compute quotients of fractions. Solve word problems involving division of fractions by fractions. Be able to use 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, ( 𝑎𝑎 𝑏𝑏 ) ÷ (𝑐𝑐 𝑑𝑑 ) = 𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 ). If 2 3 of a shoelace is 1 2 meter long, how many meters long is the shoelace? How many 3 4 cup 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-mile?

Divide multi-digit numbers using the standard algorithm. For at least 4 digits by 1-digit division by hand; more complicated cases using technology. For example, 6,389 7 .

Add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. For more complex cases, use technology.

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

Describe quantities having opposite directions or values using positive and negative numbers: temperature above/below zero, elevation above/below sea level, credits/debits, and positive/negative electric charge. Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself. For example, – (– 3) = 3, and that 0 is its own opposite.

Describe locations in the coordinate plane using signed numbers in ordered pairs; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

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.

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret – 3 > – 7 as a statement that – 3 is located to the right of – 7 on a number line oriented from left to right.

Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write – 3°𝐶𝐶 > – 7°𝐶𝐶 to express the fact that −3° 𝐶𝐶 is warmer than – 7°𝐶𝐶.

Describe the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of −30 dollars, write |−30| = 30 to describe the size of the debt in dollars.

Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than −30 dollars represents a debt greater than 30 dollars.

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

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. ”

Understand the concept of a unit rate a/b associated with a ratio a:b with 𝑏𝑏 ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3 4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Expectations for unit rates in this grade are limited to non-complex fractions.

Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Use unit rates and scaling to solve problems about proportional relationships, including problems involving unit pricing and constant speed.

Find a percentage of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percentage. For example, 30% of a quantity means 30 100 times the quantity.

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

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

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.

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

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

Reporting the number of observations.

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

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.

Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.