Idaho K–6 mathematics standards

Count to 100 by ones and by tens.

Starting at a given number, count forward within 100 and backward within 20.

Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).

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.

Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

Understand that each successive number name refers to a quantity that is one larger. Recognize the "one more" pattern of counting using objects.

Given a group of up to 20 objects, count the number of objects in that group and state the number of objects in a rearrangement of that group without recounting. Given a verbal or written number from zero to 20, count out that many objects.

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group for groups with up to ten objects.

Compare two numbers between one and ten presented as written numerals.

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."

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 different sizes and orientations, using informal language to describe their similarities, differences, parts, and other attributes.

Model shapes in the world by building shapes from components/materials and drawing shapes.

Compose simple shapes to form larger two-dimensional shapes.

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

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.

Classify objects into given categories; count the numbers of objects in each category (up to and including ten) and sort the categories by count.

Compose (put together) and decompose (break apart) numbers from 11 to 19 into ten ones and some further ones, and record each composition or decomposition by using physical, visual, or symbolic representations; 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 of two whole numbers within ten. Use objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Solve addition and subtraction word problems within ten by using physical, visual, and symbolic representations.

Decompose whole numbers from one to ten into pairs in more than one way by using physical, visual, or symbolic representations.

For a given whole number from one to nine, find the number that makes ten when added to the number by using physical, visual, or symbolic representations.

Fluently add and subtract within five, including zero.

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Compare defining attributes and non-defining attributes of two- and three-dimensional shapes; build and draw shapes that possess defining attributes.

Compose two-dimensional (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

Describe the shares using the words "halves," "fourths," and "quarters," and use the phrases "half of," "a fourth of," and "a quarter of."

Describe the whole as two of, or four of, the 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.

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 quarters, dimes, and nickels and relate their values to pennies. Find equivalent values (e.g., a nickel is equivalent to five pennies).

Starting at a given number, count forward and backwards within 120 by ones. Skip count by twos to 20, by fives to 100, and by tens to 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 zero ones).

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

Add a two-digit number and a one-digit number.

Add a two-digit number and a multiple of ten.

Understand that when adding two-digit numbers, combine like base-ten units such as tens and tens, ones and ones, and sometimes it is necessary to compose a ten.

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

Subtract multiples of ten in the range 10 – 90 from multiples of ten in the range 10 – 90 by using physical, visual, and symbolic representations, with an emphasis on place value, properties of operations, and/or the relationships between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Solve addition and subtraction word problems within 20 involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, by using physical, visual, and symbolic representations.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 by using physical, visual, and symbolic representations.

Apply properties of operations to add.

Restate a subtraction problem as a missing addend problem using the relationship between addition and subtraction.

Relate counting to addition and subtraction.

Demonstrate fluency for addition and subtraction within ten, use strategies to add and subtract within 20.

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

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers, with the unknown in any position.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, squares, rectangles, rhombi, trapezoids, pentagons, hexagons, octagons, and cubes.

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

Describe the shares using the words "halves," "thirds," "fourths," and "quarter," and use the phrases "half of," "a third of," "a fourth of," and "quarter of."

Describe the whole as two of, three of, or four of the shares.

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.

Represent whole numbers as lengths from zero 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 (up to $10), using $ and ¢ symbols appropriately and whole-dollar amounts.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Organize and record data on a line plot (dot 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 put-together, take-apart, and compare problems using information presented in the graph.

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

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

Count within 1,000; skip-count by fives, tens, and 100s. Identify patterns in skip counting starting at any number.

Read and write numbers from 0 to 1,000 using standard form, expanded form, and word form.

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

Fluently add and subtract whole numbers within 100 using understanding of place value and properties of operations.

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

Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones.

Understand that sometimes it is necessary to compose or decompose tens or hundreds.

Use mental strategies to add or subtract a number that is ten more, ten less, one hundred more, and one hundred less than a given three-digit number.

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

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, by using physical, visual, and symbolic representations.

Demonstrate fluency for addition and subtraction within 20 using mental strategies. By the end of grade two, recall basic facts to add and subtract within 20 with automaticity.

Determine whether a group of objects (up to 20) has an odd or even number of members and write an equation to express an even number as a sum of two equal addends.

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.

Understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category. Compare and classify shapes by their sides and angles. Recognize rhombi, rectangles, squares, and trapezoids as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Partition two-dimensional figures into equal areas, and express the area of each part as a unit fraction of the whole.

Tell and write time to the nearest minute within the same hour and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes.

Identify and use the appropriate tools and units of measurement, both customary and metric, to solve one-step word problems using the four operations involving weight, mass, liquid volume, and capacity (within the same system and unit).

Draw a scaled picture graph and 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 of objects using rulers marked with halves and fourths of an inch. Record and show the data by making a line plot (dot plot), where the horizontal scale is marked off in appropriate units— whole numbers, halves, or fourths.

A square with side length one 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 nonstandard 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, and represent whole-number products as rectangular areas in mathematical reasoning.

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

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

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 a whole number to the tens or hundreds place, using place value understanding or a visual representation.

Fluently add and subtract whole numbers within 1,000 using understanding of place value and properties of operations.

Multiply one-digit whole numbers by multiples of ten in the range 10–90 using understanding of place value and properties of operations.

Understand a fraction 1/b as the quantity formed by one part when a whole (a single unit) is partitioned into b equal parts; understand a/b as the quantity formed by a parts of size 1/b.

Represent a unit fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the fraction 1/b is located 1/b of a whole unit from 0 on the number line.

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

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, and explain why the fractions are equivalent, such as by using a visual fraction model.

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

Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize the comparisons are valid only when the two fractions refer to the same whole. Record the results of the comparisons with the symbols >, = , and <, and justify the conclusion using visual representations and/or verbal reasoning.

Interpret a product of whole numbers as a grouping of sets, e.g., 5 × 7 as five groups of seven objects each.

Interpret a quotient of whole numbers as equal sharing, e.g., 56 ÷ 8 as the number in each share when 56 objects are split into 8 equal shares, or as the number of shares when 56 objects are split into equal shares of 8 objects each.

Use multiplication and division within 100 to solve word problems involving equal groups, arrays, and measurements by using visual and symbolic representations, with a symbol for an unknown number.

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

Apply the properties of operations to multiply and divide.

Understand division as determining an unknown factor in a multiplication problem.

Demonstrate understanding of strategies that make use of the relationship between multiplication and division or properties of operations.

Demonstrate fluency for multiplication within 100. Know from memory all products of two single-digit numbers and related division facts.

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.

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

Draw points, lines, line segments, rays, angles (right, acute, 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.

Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit.

Record measurement equivalents in a two-column table.

Include problems involving simple fractions or decimals.

Include 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.

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

Make a line plot (dot plot) to show a set of measurements in fractions of a unit ½, ¼, 1/8. Solve problems involving addition and subtraction of fractions by using information presented in line plots (dot plots).

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 n one-degree angles is said to have an angle measure of n degrees.

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

Use an equation with a symbol for the unknown angle 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.

Recognize that in a multi-digit whole number, a digit in any place represents ten times as much as it represents in the place to its right.

Read and write multi-digit whole numbers using standard form, expanded form, and word form. Compare two multi-digit numbers based on meanings of the digits and each place, recording the results of comparisons with the symbols >, =, and <.

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

Fluently use the standard algorithm for multi-digit whole-number addition and subtraction.

Use strategies based on place value and the properties of operations.

Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

Illustrate and explain the calculation by using rectangular arrays, area models, and/or equations.

Explain why a fraction a/b is equivalent to a fraction n × a/n × b by using visual fraction models, with attention to how the numbers and sizes of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions, including fractions greater than 1.

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, by using a visual fraction model and/or verbal reasoning.

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. Justify the conclusions by using a visual fraction model and/or verbal reasoning.

Add and subtract mixed numbers with like denominators by replacing the mixed number with an equivalent fraction and/or by using properties of operations and the relationship between addition and subtraction.

Solve word problems involving addition and subtraction of fractions, including mixed numbers, with the same denominator. Justify the conclusions using a visual fraction model and/or verbal reasoning.

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

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

Solve word problems involving multiplication of a fraction by a whole number e.g., by using 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.

Use decimal notation to represent fractions with denominators 10 or 100.

Recognize that comparisons are valid only when the two decimals refer to the same whole.

Record the results of the comparisons with the symbols >, =, and <, and justify the conclusions using visual representations and/or verbal reasoning.

Interpret a multiplication equation as a comparison, e.g., 35 = 5 × 7, as 35 is 5 times as many as 7. Represent verbal multiplicative comparisons as equations.

Multiply or divide to solve word problems involving multiplicative comparison.

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.

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.

Generate a number or shape pattern that follows a given rule. Identify and explain features of the pattern that were not explicit in the rule itself.

Use a pair of perpendicular number lines (axes) with the intersection of the lines (the origin (0,0)) 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 x-coordinate, the first number in an ordered pair, indicates movement parallel to the x-axis starting at the origin; and the y-coordinate, the second number, indicates movement parallel to the y-axis starting at the origin.

Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane (x and y both have positive values), 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 of the subcategories of that category.

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

Convert among different-sized standard measurement units within a given measurement system. Use conversions in solving multi-step, real-world problems.

Interpret numerical data, with whole-number values, represented with tables or line plots.

Use graphic displays of data (line plots (dot plots), tables, etc.) to solve real-world problems using fractional data.

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.

Use concrete and/or visual models to measure the volume of rectangular prisms in cubic units by counting cubic cm, cubic in, cubic ft, and nonstandard units.

Find the volume of a right rectangular prism with whole-number edge 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.

Apply the formulas V = l × w × h and V = B × h (where B stands for the area of the base) for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths, and in the context of solving real-world and mathematical problems.

Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts.

Apply this technique to solve real-world problems.

Recognize that in a multi-digit number, including decimals, a digit in any place represents ten 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 patterns in the number of zeros of the product when multiplying a number by powers of ten, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of ten. Use whole-number exponents to denote powers of ten.

Read and write decimals to thousandths using standard form, expanded form, and word from.

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

Use place value understanding to round decimals to any place.

Demonstrate fluency for multiplication of multi-digit whole numbers using the standard algorithm. Include two-digit × four-digit numbers and three-digit × three-digit numbers.

Use 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.

Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction and between multiplication and division.

Relate the strategy to a written method and explain the reasoning used.

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions to produce an equivalent sum or difference of fractions with like denominators.

Justify the conclusions by using visual fraction models and/or equations to represent the problem.

Use benchmark fractions and number sense of fraction to estimate mentally and assess the reasonableness of answers.

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 by using visual fraction models and/or equations to represent the problem.

Interpret the product (a/b) × q as a parts of a partitions of q into b equal parts, and equivalently, as the result of the sequence of operations a × q ÷ b.

Tile it with unit squares of the appropriate unit fraction side lengths.

Show that the area is the same by tiling 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 fractional product to the size of one factor on the basis of 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, 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.

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

Represent division of a unit fraction by a nonzero whole number and compute such quotients using a visual fraction model. Use the relationship between multiplication and division to explain that 1/b ÷ c = 1/bc because 1/bc × c = 1/b.

Represent division of a whole number by a unit fraction, and compute such quotients using a visual fraction model. Use the relationship between multiplication and division to explain that a ÷ 1/b because ab × 1/b = a.

Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions by using visual fraction models and/or equations to represent the problem.

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

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

Identify apparent relationships between corresponding terms.

Form ordered pairs consisting of corresponding terms from the two patterns.

Graph the ordered pairs on a coordinate plane.

Write and evaluate numerical expressions involving whole-number exponents.

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

Identify parts of an expression using mathematical terms (e.g. sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.

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).

Apply the properties of operations to generate equivalent expressions.

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

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

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 x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

Recognize that inequalities of the form x > c or x < c have infinitely many solutions.

Represent solutions of such inequalities on number line diagrams.

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write equations to represent the relationship between the two quantities. Analyze the relationship using graphs and tables and relate these to the equations. Include an understanding of independent and dependent variables.

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 V = lwh and V = Bh 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 and area by 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 real-world and mathematical problems.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

Fluently divide multi-digit numbers using the standard algorithm.

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

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.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative numbers (including fractions and decimals) to represent quantities in real-world contexts, explaining the meaning of zero 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, e.g., −(−3)=3, and that 0 is its own opposite.

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

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.

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

Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

Distinguish comparisons of absolute value from statements about order.

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.

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

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

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

Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent.

Use ratio reasoning to convert measurement units within and between measurement systems; 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.

Understand that a set of data collected to answer a statistical question has a distribution, which can be described by its center (median and/or mean), spread (range, interquartile range, and/or mean absolute deviation), and overall shape. The focus of mean absolute deviation (MAD) is visualizing deviations from the mean as a measure of variability as opposed to a focus on calculating MAD.

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 (range, 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.