Course: Math Placement Test

Standards for Mathematical Practice
Education Level: 12
Subject: math

Make sense of problems and persevere in solving them.
Education Level: 12
Subject: math

Reason abstractly and quantitatively.
Education Level: 12
Subject: math

Construct viable arguments and critique the reasoning of others.
Education Level: 12
Subject: math

Model with mathematics
Education Level: 12
Subject: math

Use appropriate tools strategically
Education Level: 12
Subject: math

Attend to precision.
Education Level: 12
Subject: math

Look for and make use of structure.
Education Level: 12
Subject: math

Look for and express regularity in repeated reasoning.
Education Level: 12
Subject: math

Counting and Cardinality
Education Level: K
Subject: math

Know number names and the count sequence.
Education Level: K
Subject: math

Count to 100 by ones and by tens and identify as a growth pattern.
Education Level: K
Subject: math

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
Education Level: K
Subject: math

Read and write numerals from 0 to 20.
Education Level: K
Subject: math

Count to tell the number of objects.
Education Level: K
Subject: math

Understand the relationship between numbers and quantities; connect counting to cardinality.
Education Level: K
Subject: math

When counting objects, say each number's name in sequential order, pairing each object with one and only one number name and each number name with one and only one object.
Education Level: K
Subject: math

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.
Education Level: K
Subject: math

Understand that each successive number name refers to a quantity that is one larger.
Education Level: K
Subject: math

Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
Education Level: K
Subject: math

Count to answer "how many?" up to20 concrete or pictorial objects arranged in a line, a rectangular array, or a circle, or as many as 10 objects in a scattered configuration (subitizing); given a number from 1 to 20, count out that many objects.
Education Level: K
Subject: math

Compare numbers.
Education Level: K
Subject: math

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, (e.g. by using matching and counting strategies.) Include groups with up to ten objects.
Education Level: K
Subject: math

Compare two numbers between 1 and 10 presented as written numerals.
Education Level: K
Subject: math

Operations and Algebraic Thinking
Education Level: 5
Subject: math

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
Education Level: K
Subject: math

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

Solve addition and subtraction word problems, and add and subtract within 10, (e.g. by using objects or drawings to represent the problem.)
Education Level: K
Subject: math

Decompose numbers less than or equal to 10 into pairs in more than one way, (e.g. by using objects or drawings, and record each decomposition by a drawing or equation (e.g. 5 = 2 + 3 and 5 = 4 + 1 )).
Education Level: K
Subject: math

For any number from 1 to 9, find the number that makes 10 when added to the given number, (e.g. by using objects or drawings, and record the answer with a drawing or equation.).
Education Level: K
Subject: math

Fluently (efficiently, accurately, and flexibly) add and subtract within 5.
Education Level: K
Subject: math

Represent and solve problems involving addition and subtraction.
Education Level: 2
Subject: math

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, (e.g. by using objects, drawings, and situation equations and/or solution equations with a symbol for the unknown number to represent the problem.)
Education Level: 1
Subject: math

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, (e.g. by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.)
Education Level: 1
Subject: math

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, (e.g. by using drawings and situation equations and/or solution equations with a symbol for the unknown number to represent the problem.)
Education Level: 2
Subject: math

Understand and apply properties of operations and the relationship between addition and subtraction.
Education Level: 1
Subject: math

Apply (not necessary to name) properties of operations as strategies to add and subtract.
Education Level: 1
Subject: math

Understand subtraction as an unknown-addend problem. For example, subtract 10-8 by finding the number that makes 10 when added to 8.
Education Level: 1
Subject: math

Add and subtract within 20.
Education Level: 2
Subject: math

Relate counting to addition and subtraction (e.g. by counting on 2 to add 2, counting back 1 to subtract 1).
Education Level: 1
Subject: math

Add and subtract within 20, demonstrating fluency (efficiently, accurately, and flexibly) for addition and subtraction within 10. Use mental strategies such as counting on; making ten (e.g. 8+6 = 8+2+4 = 10+4 = 14); decomposing a number leading to a ten (e.g. 13-4 = 13-3-1 = 10-1 = 9); using the relationship between addition and subtraction (e.g. knowing that 8+4 = 12, one knows 12-8 = 4); and creating equivalent but easier or known sums (e.g. adding 6+7 by creating the known equivalent 6+6+1 = 12+1 = 13).
Education Level: 1
Subject: math

Fluently (efficiently, accurately, and flexibly) add and subtract within 20 using mental strategies (counting on, making a ten, decomposing a number, creating an equivalent but easier and known sum, and using the relationship between addition and subtraction).
Education Level: 2
Subject: math

Work with addition and subtraction equations.
Education Level: 1
Subject: math

Understand the meaning of the equal sign (the value is the same on both sides of the equal sign), and determine if equations involving addition and subtraction are true or false.
Education Level: 1
Subject: math

Using related equations, Determine the unknown whole number in an addition or subtraction equation.
Education Level: 1
Subject: math

Work with equal groups of objects to gain foundations for multiplication.
Education Level: 2
Subject: math

Determine whether a group of objects (up to 20) has an odd or even number of members, (e.g. by pairing objects or counting them by 2s); write an equation to express an even number as a sum of two equal addends.
Education Level: 2
Subject: math

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.
Education Level: 2
Subject: math

Represent and solve problems involving multiplication and division.
Education Level: 3
Subject: math

Interpret products of whole numbers, (e.g. interpret 5·7 as the total number of objects in 5 groups of 7 objects each.)
Education Level: 3
Subject: math

Interpret whole-number quotients of whole numbers, (e.g. interpret 56÷8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.)
Education Level: 3
Subject: math

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

Determine the unknown whole number in a multiplication or division equation by using related equations.
Education Level: 3
Subject: math

Understand properties of multiplication and the relationship between multiplication and division.
Education Level: 3
Subject: math

Apply properties of operations as strategies to multiply and divide.
Education Level: 3
Subject: math

Understand division as an unknown-factor problem.
Education Level: 3
Subject: math

Multiply and divide within 100 (basic facts up to 10 x 10).
Education Level: 3
Subject: math

Fluently (efficiently, accurately, and flexibly) multiply and divide with single digit multiplications and related divisions using strategies (e.g. relationship between multiplication and division, doubles, double and double again, half and then double, etc.) or properties of operations.
Education Level: 3
Subject: math

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Education Level: 3
Subject: math

Solve two-step word problems using any of the four operations. Represent these problems using both situation equations and/or solution equations with a letter or symbol standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers.
Education Level: 3
Subject: math

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.
Education Level: 3
Subject: math

Use the four operations with whole numbers to solve problems.
Education Level: 4
Subject: math

Interpret a multiplication equation as a comparison, (e.g. interpret 35 = 5·7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.) Represent verbal statements of multiplicative comparisons as multiplication equations.
Education Level: 4
Subject: math

Multiply or divide to solve word problems involving multiplicative comparison, (e.g. by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.)
Education Level: 4
Subject: math

Solve multi-step word problem posed with whole numbers and having whole number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using situation equations and/or solution equations with a letter or symbol standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Education Level: 4
Subject: math

Gain familiarity with factors and multiples.
Education Level: 4
Subject: math

Find all factor pairs for a whole number in the range 1 to 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 to 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 to 100 is prime or composite.
Education Level: 4
Subject: math

Generate and analyze patterns.
Education Level: 4
Subject: math

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
Education Level: 4
Subject: math

Write and interpret numerical expressions.
Education Level: 5
Subject: math

Use parentheses in numerical expressions and evaluate expressions with these symbols.
Education Level: 5
Subject: math

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
Education Level: 5
Subject: math

Number and Operations in Base Ten
Education Level: 5
Subject: math

Work with numbers 11–19 to gain foundations for place value.
Education Level: K
Subject: math

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, (e.g. by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g. 10 + 8 = 18 and 19 = 10 + 9); ); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
Education Level: K
Subject: math

Extend the counting sequence.
Education Level: 1
Subject: math

Count to 120 (recognizing growth and repeating patterns), starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
Education Level: 1
Subject: math

Understand place value.
Education Level: 2
Subject: math

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
Education Level: 1
Subject: math

10 can be thought of as a grouping of ten ones—called a "ten."
Education Level: 1
Subject: math

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
Education Level: 1
Subject: math

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).
Education Level: 1
Subject: math

Show flexibility in composing and decomposing tens and ones (e.g. 20 can be composed from 2 tens or 1 ten and 10 ones, or 20 ones.)
Education Level: 1
Subject: math

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the relational symbols >,
Education Level: 1
Subject: math

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; (e.g. 706 equals 7 hundreds, 0 tens, and 6 ones.) Understand the following as special cases:
Education Level: 2
Subject: math

100 can be thought of as a bundle of ten tens—called a "hundred."
Education Level: 2
Subject: math

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds.
Education Level: 2
Subject: math

Show flexibility in composing and decomposing hundreds, tens and ones (e.g. 207 can be composed from 2 hundreds 7 ones OR 20 tens 7 ones OR 207 ones OR 1 hundred 10 tens 7 ones OR 1 hundred 9 tens 17 ones, etc.)
Education Level: 2
Subject: math

Count within 1000; skip-count by 2s, 5s, 10s, and 100s; explain and generalize the patterns.
Education Level: 2
Subject: math

Read and write numbers within 1000 using base-ten numerals, number names, expanded form, and unit form.
Education Level: 2
Subject: math

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >,
Education Level: 2
Subject: math

Use place value understanding and properties of operations to add and subtract.
Education Level: 2
Subject: math

Add within 100 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used including:
Education Level: 1
Subject: math

Adding a two-digit number and a one-digit number
Education Level: 1
Subject: math

Adding a two-digit number and a multiple of 10
Education Level: 1
Subject: math

Understanding 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.
Education Level: 1
Subject: math

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
Education Level: 1
Subject: math

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; relate the strategy to a written method and explain the reasoning used.
Education Level: 1
Subject: math

Fluently (efficiently, accurately, and flexibly) add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (e.g. composing/decomposing by like base-10 units, using friendly or benchmark numbers, using related equations, compensation, number line, etc.).
Education Level: 2
Subject: math

Add up to four two-digit numbers using strategies based on place value and properties of operations.
Education Level: 2
Subject: math

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, like base-ten units such as hundreds and hundreds, tens and tens, ones and ones are used; and sometimes it is necessary to compose or decompose tens or hundreds.
Education Level: 2
Subject: math

Mentally add 10 or 100 to a given number 100 – 900, and mentally subtract 10 or 100 from a given number 100 – 900.
Education Level: 2
Subject: math

Explain why addition and subtraction strategies work using place value and the properties of operations. The explanations given may be supported by drawings or objects.
Education Level: 2
Subject: math

Use place value understanding and properties of operations to perform multi-digit arithmetic.
Education Level: 4
Subject: math

Use place value understanding to round whole numbers to the nearest 10 or 100.
Education Level: 3
Subject: math

Fluently (efficiently, accurately, & flexibly) add and subtract within 1000 using strategies (e.g. composing/decomposing by like base-10 units, using friendly or benchmark numbers, using related equations, compensation, number line, etc.) and algorithms (including, but not limited to: traditional, partial-sums, etc.) based on place value, properties of operations, and/or the relationship between addition and subtraction.
Education Level: 3
Subject: math

Multiply one-digit whole numbers by multiples of 10 in the range 10 to 90 (e.g. 9⋅80, 5⋅60) using strategies based on place value and properties of operations.
Education Level: 3
Subject: math

Fluently (efficiently, accurately, and flexibly) add and subtract multi-digit whole numbers using an efficient algorithm (including, but not limited to: traditional, partial-sums, etc.), based on place value understanding and the properties of operations.
Education Level: 4
Subject: math

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Education Level: 4
Subject: math

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Education Level: 4
Subject: math

Generalize place value understanding for multi-digit whole numbers.
Education Level: 4
Subject: math

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
Education Level: 4
Subject: math

Read and write multi-digit whole numbers using base-ten numerals, number names, expanded form, and unit form. Compare two multi-digit numbers based on meanings of the digits in each place, using >,
Education Level: 4
Subject: math

Use place value understanding to round multi-digit whole numbers to any place.
Education Level: 4
Subject: math

Understand the place value system.
Education Level: 5
Subject: math

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.
Education Level: 5
Subject: math

Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Education Level: 5
Subject: math

Read, write, and compare decimals to thousandths.
Education Level: 5
Subject: math

Read and write decimals to thousandths using base-ten numerals, number names, expanded form, and unit form.
Education Level: 5
Subject: math

Compare two decimals to thousandths based on meanings of the digits in each place, using >,
Education Level: 5
Subject: math

Use place value understanding to round decimals to any place
Education Level: 5
Subject: math

Perform operations with multi-digit whole numbers and with decimals to hundredths.
Education Level: 5
Subject: math

Fluently (efficiently, accurately, and flexibly) multiply multi-digit whole numbers using an efficient algorithm (ex., traditional, partial products, etc.) based on place value understanding and the properties of operations.
Education Level: 5
Subject: math

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Education Level: 5
Subject: math

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

Measurement and Data
Education Level: 5
Subject: math

Describe and compare measurable attributes.
Education Level: K
Subject: math

Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
Education Level: K
Subject: math

Directly compare two objects, with a measureable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference.
Education Level: K
Subject: math

Classify objects and count the number of objects in each category.
Education Level: K
Subject: math

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).
Education Level: K
Subject: math

Measure lengths indirectly and by iterating length units.
Education Level: 1
Subject: math

Order three objects by length; compare the lengths of two objects indirectly by using a third object.
Education Level: 1
Subject: math

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.
Education Level: 1
Subject: math

Tell and write time.
Education Level: 1
Subject: math

Tell and write time in hours and half-hours using analog and digital clocks.
Education Level: 1
Subject: math

Represent and interpret data.
Education Level: 4
Subject: math

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.
Education Level: 1
Subject: math

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object using different units. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
Education Level: 2
Subject: math

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph
Education Level: 2
Subject: math

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.
Education Level: 3
Subject: math

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.
Education Level: 3
Subject: math

Make a data display (line plot, bar graph, pictograph) to show a set of measurements in fractions of a unit (½, ¼, 1/8, 1/16). Solve problems involving addition and subtraction of fractions by using information presented in the data display.
Education Level: 4
Subject: math

Convert like measurement units within a given measurement system.
Education Level: 5
Subject: math

Convert among different-sized standard measurement units within a given measurement system (e.g. convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
Education Level: 5
Subject: math

Represent and interpret data.
Education Level: 5
Subject: math

Make a data display (line plot, bar graph, pictograph) to show a data set of measurements in fractions of a unit (½, ¼, 1/8). Use operations (add, subtract, multiply) on fractions for this grade to solve problems involving information presented in the data display.
Education Level: 5
Subject: math

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Education Level: 5
Subject: math

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
Education Level: 5
Subject: math

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.
Education Level: 5
Subject: math

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.
Education Level: 5
Subject: math

Measure volumes by counting unit cubes such as cubic cm, cubic in, cubic ft. or non-standard cubic units.
Education Level: 5
Subject: math

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
Education Level: 5
Subject: math

Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent three-dimensional whole-number products as volumes, (e.g. to represent the associative property of multiplication.)
Education Level: 5
Subject: math

Apply the formulas V = l·w·h and V = B·h (B represents 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.
Education Level: 5
Subject: math

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Education Level: 5
Subject: math

Measure and estimate lengths in standard units.
Education Level: 2
Subject: math

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
Education Level: 2
Subject: math

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.
Education Level: 2
Subject: math

Estimate lengths using whole units of inches, feet, centimeters, and meters.
Education Level: 2
Subject: math

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit (inches, feet, centimeters, and meters).
Education Level: 2
Subject: math

Relate addition and subtraction to length.
Education Level: 2
Subject: math

Use addition and subtraction within 100 to solve one- and two-step word problems involving lengths that are given in the same units, e.g. by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
Education Level: 2
Subject: math

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.
Education Level: 2
Subject: math

Work with time and money.
Education Level: 2
Subject: math

Tell and write time from analog and digital clocks to the nearest five minutes.
Education Level: 2
Subject: math

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately (Do not use decimal point, if showing 25 cents, use the word cents or ¢).
Education Level: 2
Subject: math

Identify coins and bills and their values.
Education Level: 2
Subject: math

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
Education Level: 3
Subject: math

Tell and write time to the nearest minute using a.m. and p.m. and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, (e.g. by representing the problem on a number line diagram.)
Education Level: 3
Subject: math

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l) (Excludes cubed units such as cm³ and finding the geometric volume of a container).
Education Level: 3
Subject: math

Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, (e.g. by using drawings (such as a beaker with a measurement scale) to represent the problem.)
Education Level: 3
Subject: math

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Education Level: 4
Subject: math

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
Education Level: 4
Subject: math

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

Apply the area and perimeter formulas for rectangles in real world and mathematical problems explaining and justifying the appropriate unit of measure.
Education Level: 4
Subject: math

Geometry
Education Level: 8
Subject: math

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
Education Level: K
Subject: math

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.
Education Level: K
Subject: math

Correctly gives most precise name of shapes regardless of their orientations (position and direction in space) or overall size.
Education Level: K
Subject: math

Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").
Education Level: K
Subject: math

Analyze, compare, create, and compose shapes.
Education Level: K
Subject: math

Analyze and compare two- and three-dimensional shapes, in different sizes and orientations (position and direction in space), using informal language to describe their similarities, differences, parts (e.g. number of sides and vertices/"corners") and other attributes (e.g. having sides of equal length).
Education Level: K
Subject: math

Model shapes in the world by building shapes from components (e.g. sticks and clay balls) and drawing shapes.
Education Level: K
Subject: math

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

Reason with shapes and their attributes.
Education Level: 3
Subject: math

Distinguish between defining attributes (e.g. triangles are closed and three-sided) versus non-defining attributes (e.g. color, orientation, overall size); build and draw shapes that possess defining attributes.
Education Level: 1
Subject: math

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. Students do not need to learn formal names such as "right rectangular prism."
Education Level: 1
Subject: math

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.
Education Level: 1
Subject: math

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
Education Level: 2
Subject: math

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
Education Level: 2
Subject: math

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. Note: fraction notation 1/2, 1/3, 1/4 is not expected at this grade level. Recognize that equal shares of identical wholes need not have the same shape.
Education Level: 2
Subject: math

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

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
Education Level: 3
Subject: math

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
Education Level: 4
Subject: math

Draw points, lines, line segments, rays, angles (right, acute, obtuse, straight, reflex), and perpendicular and parallel lines. Identify these in two-dimensional figures.
Education Level: 4
Subject: math

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles (right, acute, obtuse, straight, reflex). Recognize and categorize triangles based on angles (acute, obtuse, equiangular, and right) and/or sides (scalene, isosceles, and equilateral).
Education Level: 4
Subject: math

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.
Education Level: 4
Subject: math

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Education Level: 3
Subject: math

Recognize area as an attribute of plane figures and understand concepts of area measurement.
Education Level: 3
Subject: math

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 (does not require standard square units).
Education Level: 3
Subject: math

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 (does not require standard square units).
Education Level: 3
Subject: math

Measure areas by counting unit squares (square cm, square m, square in, square ft, and non-standard square units).
Education Level: 3
Subject: math

Relate area to the operations of multiplication and addition
Education Level: 3
Subject: math

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.
Education Level: 3
Subject: math

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.
Education Level: 3
Subject: math

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.
Education Level: 3
Subject: math

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.
Education Level: 3
Subject: math

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Education Level: 3
Subject: math

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.
Education Level: 3
Subject: math

Graph points on the coordinate plane to solve real-world and mathematical problems.
Education Level: 5
Subject: math

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g. x-axis and x-coordinate, y-axis and y-coordinate).
Education Level: 5
Subject: math

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. (e.g. plotting the relationship between two positive quantities such as maps, coordinate grid games (such as Battleship), time/temperature, time/distance, cost/quantity, etc.).
Education Level: 5
Subject: math

Classify two-dimensional figures into categories based on their properties
Education Level: 5
Subject: math

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
Education Level: 5
Subject: math

Classify two-dimensional figures in a hierarchy based on properties.
Education Level: 5
Subject: math

Solve real-world and mathematical problems involving area, surface area, and volume.
Education Level: 7
Subject: math

Find the area of all triangles, special quadrilaterals (including parallelograms, kites and trapezoids), and polygons whose edges meet at right angles (rectilinear figure 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.
Education Level: 6
Subject: math

Find the volume of a right rectangular prism with fractional edge lengths by applying the formulas V=lwh and V=Bh (B is the area of the base and h is the height) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
Education Level: 6
Subject: math

Draw polygons whose edges meet at right angles (rectilinear figure 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.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

.Use the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Education Level: 7
Subject: math

Investigate the relationship between three-dimensional geometric shapes;
Education Level: 7
Subject: math

Generalize the volume formula for prisms and cylinders (V = Bh where B is the base and h is the height).
Education Level: 7
Subject: math

Generalize the surface area formula for prisms and cylinders (SA = 2B + Ph where B is the area of the base, P is the perimeter of the base, and h is the height (in the case of a cylinder, perimeter is replaced by circumference)).
Education Level: 7
Subject: math

Solve real-world and mathematical problems involving area of two-dimensional objects and volume and surface area of three-dimensional objects including cylinders and right prisms. (Solutions should not require students to take square roots or cube roots.
Education Level: 7
Subject: math

Draw, construct, and describe geometrical figures and describe the relationships between them.
Education Level: 7
Subject: math

Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Education Level: 7
Subject: math

Identify three-dimensional objects generated by rotating a two-dimensional (rectangular or triangular) object around one edge.
Education Level: 7
Subject: math

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right cylinder.
Education Level: 7
Subject: math

Geometric measurement: understand concepts of angle and measure angles.
Education Level: 8
Subject: math

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
Education Level: 8
Subject: math

An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
Education Level: 8
Subject: math

An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
Education Level: 8
Subject: math

Measure angles in whole-number degrees using a protractor. Draw angles of specified measure using a protractor and straight edge.
Education Level: 8
Subject: math

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g. by using an equation with a symbol for the unknown angle measure.
Education Level: 8
Subject: math

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an unknown angle in a figure.
Education Level: 8
Subject: math

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
Education Level: 8
Subject: math

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on drawing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Education Level: 8
Subject: math

Understand and apply the Pythagorean Theorem.
Education Level: 8
Subject: math

Explain a proof of the Pythagorean Theorem and its converse.
Education Level: 8
Subject: math

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Education Level: 8
Subject: math

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Education Level: 8
Subject: math

Solve real-world and mathematical problems involving measurement.
Education Level: 8
Subject: math

Use the formulas or informal reasoning to find the arc length, areas of sectors, surface areas and volumes of pyramids, cones, and spheres.
Education Level: 8
Subject: math

Investigate the relationship between the formulas of three dimensional geometric shapes;
Education Level: 8
Subject: math

Generalize the volume formula for pyramids and cones (V = 1/3Bh).
Education Level: 8
Subject: math

Generalize surface area formula of pyramids and cones (SA = B + ½Pl).
Education Level: 8
Subject: math

Solve real-world and mathematical problems involving arc length, area of two-dimensional shapes including sectors, volume and surface area of three-dimensional objects including pyramids, cones and spheres.
Education Level: 8
Subject: math

Number and Operations—Fractions
Education Level: 5
Subject: math

Develop understanding of fractions as numbers(Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.)
Education Level: 3
Subject: math

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Education Level: 3
Subject: math

Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Education Level: 3
Subject: math

Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
Education Level: 3
Subject: math

Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line (a is the countable units of 1/b that determines the place on the number line).
Education Level: 3
Subject: math

Explain equivalence of fractions, and compare fractions by reasoning about their size (it is a mathematical convention that when comparing fractions, the whole is the same size).
Education Level: 3
Subject: math

Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Education Level: 3
Subject: math

Recognize and generate simple equivalent fractions, (e.g. ½ = 2/4, 4/6 = 2/3.) Explain why the fractions are equivalent, e.g. by using a visual fraction model.
Education Level: 3
Subject: math

Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
Education Level: 3
Subject: math

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 relational symbols >,
Education Level: 3
Subject: math

Extend understanding of fraction equivalence and ordering.
Education Level: 4
Subject: math

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

Compare two fractions with different numerators and different denominators, (e.g. by creating common numerators or denominators, or by comparing to a benchmark fraction such as ½.) Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with relational symbols >,
Education Level: 4
Subject: math

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Education Level: 4
Subject: math

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
Education Level: 4
Subject: math

Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
Education Level: 4
Subject: math

Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g. by using a visual fraction model.
Education Level: 4
Subject: math

Add and subtract mixed numbers with like denominators, e.g. by replacing each mixed number with an equivalent fraction (simplest form is not an expectation), and/or by using properties of operations and the relationship between addition and subtraction.
Education Level: 4
Subject: math

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g. by using visual fraction models and equations to represent the problem.
Education Level: 4
Subject: math

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Education Level: 4
Subject: math

Understand a fraction a/b as a multiple of 1/b.
Education Level: 4
Subject: math

Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
Education Level: 4
Subject: math

Solve word problems involving multiplication of a fraction by a whole number.
Education Level: 4
Subject: math

Understand decimal notation for fractions, and compare decimal fractions.
Education Level: 4
Subject: math

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.
Education Level: 4
Subject: math

Use decimal notation for fractions with denominators 10 or 100.
Education Level: 4
Subject: math

Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the relational symbols >, < =, or ≠, and justify the conclusions, (e.g. by using a visual model.).
Education Level: 4
Subject: math

Use equivalent fractions as a strategy to add and subtract fractions.
Education Level: 5
Subject: math

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.
Education Level: 5
Subject: math

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

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Education Level: 5
Subject: math

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

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Education Level: 5
Subject: math

Interpret the product a/b·q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a·q÷b.
Education Level: 5
Subject: math

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.
Education Level: 5
Subject: math

Interpret multiplication as scaling (resizing), by:
Education Level: 5
Subject: math

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 (e.g. They see (½·3) as half the size of 3.).
Education Level: 5
Subject: math

Explain 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); explain 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 = na/nb to the effect of multiplying a/b by 1.
Education Level: 5
Subject: math

Solve real world problems involving multiplication of fractions and mixed numbers, (e.g. by using visual fraction models or equations to represent the problem).
Education Level: 5
Subject: math

Apply and extend previous understandings of division, to divide unit fractions by whole numbers and whole numbers by unit fractions. Division of a fraction by a fraction is not a requirement at this grade.
Education Level: 5
Subject: math

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
Education Level: 5
Subject: math

Interpret division of a whole number by a unit fraction, and compute such quotients.
Education Level: 5
Subject: math

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g. by using visual fraction models and equations to represent the problem.
Education Level: 5
Subject: math

Ratios and Proportional Relationships
Education Level: 7
Subject: math

Understand ratio concepts and use ratio reasoning to solve problems.
Education Level: 6
Subject: math

Use ratio language to describe a relationship between two quantities. Distinguish between part-to-part and part-to-whole relationships.
Education Level: 6
Subject: math

Use unit rate language ("for each one", "for every one" and "per") and unit rate notation to demonstrate understanding the concept of a unit rate a/b associated with a ratio a:b with b≠0,
Education Level: 6
Subject: math

Use ratio and rate reasoning to solve real-world and mathematical problems, (e.g. by reasoning about tables of equivalent ratios, tape diagrams, double number line diagram, or using calculations.)
Education Level: 6
Subject: math

Make tables of equivalent ratios relating quantities with whole-number measurements, find the 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.
Education Level: 6
Subject: math

Find a percent of a quantity as a rate per 100 (e.g. 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Education Level: 6
Subject: math

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Education Level: 6
Subject: math

Analyze proportional relationships and use them to solve real-world and mathematical problems.
Education Level: 7
Subject: math

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
Education Level: 7
Subject: math

Recognize and represent proportional relationships between quantities:
Education Level: 7
Subject: math

Determine whether two quantities are in a proportional relationship, e.g. by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Education Level: 7
Subject: math

Analyze a table or graph and recognize that, in a proportional relationship, every pair of numbers has the same unit rate (referred to as the "m").
Education Level: 7
Subject: math

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

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Education Level: 7
Subject: math

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Education Level: 7
Subject: math

The Number System
Education Level: 8
Subject: math

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Education Level: 6
Subject: math

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, requiring multiple exposures connecting various concrete and abstract models.
Education Level: 6
Subject: math

Compute fluently (efficiently, accurately, and flexibly) with multi-digit numbers and find common factors and multiples.
Education Level: 6
Subject: math

Fluently (efficiently, accurately, and flexibly) divide multi-digit numbers using an efficient algorithm.
Education Level: 6
Subject: math

Fluently (efficiently, accurately, and flexibly) add, subtract, multiply, and divide multi-digit decimals using an efficient algorithm for each operation.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

Apply and extend previous understandings of numbers to the system of rational numbers.
Education Level: 6
Subject: math

Understand positive and negative numbers to describe quantities having opposite directions or values (e.g. temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge);
Education Level: 6
Subject: math

Use positive and negative numbers to represent quantities in real-world contexts,
Education Level: 6
Subject: math

Explaining the meaning of 0 in each situation.
Education Level: 6
Subject: math

Understand a rational number as a point on the number line and a coordinate pair as a location on a coordinate plane.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

Recognize signs of numbers in ordered pairs indicate 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.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

Understand ordering and absolute value of rational numbers.
Education Level: 6
Subject: math

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
Education Level: 6
Subject: math

Write, interpret, and explain statements of order for rational numbers in real-world contexts.
Education Level: 6
Subject: math

Explain 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.
Education Level: 6
Subject: math

Distinguish comparisons of absolute value from statements about order.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

Apply and extend previous understandings of operations with positive rational numbers to add, subtract, multiply, and divide all rational numbers.
Education Level: 7
Subject: math

Represent addition and subtraction on a horizontal or vertical number line diagram.
Education Level: 7
Subject: math

Describe situations in which opposite quantities combine to make 0. Show that a number and its opposite have a sum of 0 (are additive inverses).
Education Level: 7
Subject: math

Show p+q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative.
Education Level: 7
Subject: math

Model subtraction of rational numbers as adding the additive inverse, p-q=p+(-q).
Education Level: 7
Subject: math

Model subtraction as the distance between two rational numbers on the number line where the distance is the absolute value of their difference.
Education Level: 7
Subject: math

Apply properties of operations as strategies to add and subtract rational numbers.
Education Level: 7
Subject: math

Apply and extend previous understandings of multiplication and division of positive rational numbers to multiply and divide all rational numbers.
Education Level: 7
Subject: math

Describe how multiplication is extended from positive rational numbers to all rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1)=1 and the rules for multiplying signed numbers.
Education Level: 7
Subject: math

Explain that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. Leading to situations such that if p and q are integers, then -(p/q)=(-p)/q=p/(-q).
Education Level: 7
Subject: math

Apply properties of operations as strategies to multiply and divide rational numbers.
Education Level: 7
Subject: math

Convert a rational number in the form of a fraction to its decimal equivalent using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Education Level: 7
Subject: math

Solve and interpret real-world and mathematical problems involving the four operations with rational numbers.
Education Level: 7
Subject: math

Know that there are numbers that are not rational, and approximate them by rational numbers.
Education Level: 8
Subject: math

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Education Level: 8
Subject: math

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g. π²).
Education Level: 8
Subject: math

Expressions and Equations
Education Level: 8
Subject: math

Apply and extend previous understandings of arithmetic to algebraic expressions.
Education Level: 6
Subject: math

Write and evaluate numerical expressions involving whole-number exponents.
Education Level: 6
Subject: math

Write, read, and evaluate expressions in which letters stand for numbers.
Education Level: 6
Subject: math

Write expressions that record operations with numbers and with letters standing for numbers.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

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).
Education Level: 6
Subject: math

Apply the properties of operations and combine like terms, with the conventions of algebraic notation, to identify and generate equivalent expressions.
Education Level: 6
Subject: math

Reason about and solve one-variable equations and inequalities.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

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.
Education Level: 6
Subject: math

Solve one-step equations involving non-negative rational numbers using addition, subtraction, multiplication and division.
Education Level: 6
Subject: math

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Education Level: 6
Subject: math

Represent and analyze quantitative relationships between dependent and independent variables.
Education Level: 6
Subject: math