Number Sense in California

The folling material is taken from The California Mathematics Academic Content Standards for Grades K - 12. Only standards for grades K-7 appear here.

Individual standards relevant to number sense have been isolated and grouped into more specific topic areas. Within each topic area the standards are arranged by grade level. This arrangement makes it easier to see the development of topics across grade levels.

By showing topic development across grade levels, it becomes easier to evaluate any particular piece of mathematics content relative to the California standards. The California grade level of content from textbooks or tests or other sets of mathematics standards can thus be more easily identified. The groupings can also be useful for the study of the California standards themselves.

Users of this document should be aware of the fact that the California standards set high-level achievement objectives, roughly on par with the academic development in some of the most successful countries in international comparisons.

This document focuses on the number sense strand in the California standards. A few of the standards listed come from other strands. The strands are identified by the following codes:

NS Number Sense
AF Algebra and Functions
MG Measurement and Geometry
ST Statistics, Data Analysis, and Probability
MR Mathematical Reasoning

For more detail about the mathematics curriculum in California see:

Mathematics Framework for California Public Schools: Kindergarten through Grade Twelve
[requires Adobe Acrobat reader]

For test items that illustrate the mathematics standards in California see:

Mathematics Assessments from NCITE/LACOE

Basic Number Sense - Count, Read, Write, Compare Whole Numbers Grade K NS1.0 Students understand the relationship between numbers and quantities (i.e., that a set of objects has the same number of objects in different situations regardless of its position or arrangement) NS1.1 Compare two or more sets (up to ten objects in each group) and identify which set is equal to, more than, or less than the other NS1.2 Count, recognize, represent, name, and order a number of objects (up to 30) NS1.3 Know that the larger numbers describe sets with more objects in them than the smaller numbers have Grade 1 NS1.0 Students understand and use numbers up to 100 NS1.1 Count, read, and write whole numbers to 100 NS1.2 Compare and order whole numbers to 100 by using the symbols for less than, equal to, or greater than (<, =, >) NS1.3 Represent equivalent forms of the same number through the use of physical models, diagrams, and number expressions (to 20) (e.g., 8 can be represented as 4+4, 5+3, 2+2+2+2, 10-2, 11-3) NS1.4 Count and group objects in ones and tens (e.g., three groups of 10 and 4 equals 34 or 30+4) Grade 2 NS1.0 Students understand the relationship among numbers, quantities, and place value in whole numbers up to 1,000 NS1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit NS1.2 Use words, models, and expanded forms (e.g., 45 = 4 tens + 5) to represent numbers (to 1,000) NS1.3 Order and compare whole numbers to 1,000 by using the symbols <, =, > Grade 3 NS1.0 Students understand place value of whole numbers NS1.1 Count, read, and write whole numbers to 10,000 NS1.2 Compare and order whole numbers to 10,000 NS1.3 Identify the place value for each digit in numbers to 10,000 NS1.5 Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6) Grade 4 NS1.1 Read and write whole numbers in the millions NS1.2 Order and compare whole numbers and decimals to two decimal places Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers

Counting Grade K NS1.2 Count, recognize, represent, name, and order a number of objects (up to 30) Grade 1 NS1.1 Count, read, and write whole numbers to 100 NS1.4 Count and group objects in ones and tens (e.g., three groups of 10 and 4 equals 34 or 30+4) NS2.4 Count by 2s, 5s, and 10s to 100 Grade 2 NS1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit NS3.1 Use repeated addition, arrays, counting by multiples to do multiplication ST1.1 Record numerical data in systematic ways, keeping track of what has been counted Grade 3 NS1.1 Count, read, and write whole numbers to 10,000 AF2.2 Extend and recognize a linear pattern by its rules (e.g., the number of legs on a given number of horses can be calculated by counting by 4's or by multiplying the number or horses by 4) Grade 4 NS1.8 Use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature, "owing")

Place Value of Whole Numbers Grade K NS3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones and tens places Grade 1 NS1.4 Count and group objects in ones and tens (e.g., three groups of 10 and 4 equals 34 or 30+4) NS3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the one, tens, and hundreds places Grade 2 NS1.0 Students understand the relationship among numbers, quantities, and place value in whole numbers up to 1,000 NS1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit NS6.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones, tens, hundreds, and thousands places Grade 3 NS1.0 Students understand place value of whole numbers NS1.3 Identify the place value for each digit in numbers to 10,000 Grade 4 NS1.0 Students understand place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts negative numbers

Operations with Money Grade 1 NS1.5 Identify and know the value of coins and show different combinations of coins that equal the same value Grade 2 NS5.0 Students model and solve problems by representing, adding, and subtracting amounts of money NS5.1 Solve problems using combinations of coins and bills NS5.2 Know and use the decimal notation and the dollar and cents symbols for money Grade 3 NS2.7 Determine the unit cost when given the total cost and number of units NS3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole number multipliers and divisors NS3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents 1/2 of a dollar, 75 cents is 3/4 of a dollar)

Addition and Subtraction of Whole Numbers Grade K NS2.0 Students understand and describe simple additions and subtractions NS2.1 Use concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10) Grade 1 NS2.0 Students demonstrate the meaning of addition and subtraction and use these operations to solve problems NS2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory NS2.2 Use the inverse relationship between addition and subtraction to solve problems NS2.3 Identify one more than, one less than, 10 more than, and 10 less than a given number NS2.5 Show the meaning of addition (putting together, increasing) and subtraction (taking away, comparing, finding the difference) NS2.6 Solve addition and subtraction problems with one- and two-digit numbers (e.g., 5+58=__) NS2.7 Find the sum of three one-digit numbers AF1.1 Write and solve number sentences from problem situations that express relationships involving addition and subtraction AF1.2 Understand the meaning of the symbols +, -, = AF1.3 Create problem situations that could lead to given number sentences involving addition and subtraction Grade 2 NS2.0 Students estimate, calculate, and solve problems involving addition and subtraction of two- and three-digit numbers NS2.1 Understand and use the inverse relationship between addition and subtraction (e.g., an opposite number sentence for 8+6=14 is 14-6=8) to solve problems and check solutions NS2.2 Find the sum or difference of two whole numbers up to three digits long NS2.3 Use mental arithmetic to find the sum or difference of two two-digit numbers AF1.0 Students model, represent, and interpret number relationships to create and solve problems involving addition and subtraction AF1.1 Use the commutative and associative rules to simplify mental calculations and to check results AF1.2 Relate problem situations and number sentences involving addition and subtraction AF1.3 Solve addition and subtraction problems using data from simple charts, picture graphs, and number sentences Grade 3 NS2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division NS2.1 Find the sum or difference of two whole numbers between 0 and 10,000 Grade 4 NS2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places NS3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations NS3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for addition and subtraction of multidigit numbers

Computational Targets for Addition and Subtraction with Whole Numbers Grade K NS2.1 Use concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10) Grade 1 NS2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory NS2.6 Solve addition and subtraction problems with one- and two-digit numbers (e.g., 5+58=__) NS2.7 Find the sum of three one-digit numbers Grade 2 NS2.2 Find the sum or difference of two whole numbers up to three digits long NS2.3 Use mental arithmetic to find the sum or difference of two two-digit numbers Grade 3 NS2.1 Find the sum or difference of two whole numbers between 0 and 10,000 Grade 4 NS3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for addition and subtraction of multidigit numbers

Multiplication and Division of Whole Numbers Grade 2 NS3.0 Students model and solve simple problems involving multiplication and division NS3.1 Use repeated addition, arrays, counting by multiples to do multiplication NS3.2 Use repeated subtraction, equal sharing, and forming equal groups with remainders to do division NS3.3 Know the multiplication tables of 2s, 5s and 10s (to "times 10") and commit them to memory Grade 3 NS2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division NS2.2 Memorize to automatcity the multiplication table for numbers between 1 and 10 NS2.3 Use the inverse relationship of multiplication and division to compute and check results NS2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 x 3 = _) NS2.5 Solve division problems in which a multidigit number is evenly divided by a one-digit number (135� NS2.6 Understand the special properties of 0 and 1 in multiplication and division NS2.7 Determine the unit cost when given the total cost and number of units NS3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole number multipliers and divisors Grade 4 NS3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations NS3.2 Demonstrate understanding of, and ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results NS3.3 Solve problems involving multiplication of multidigit numbers by two-digit numbers NS3.4 Solve problems involving division of multidigit numbers by one-digit numbers AF1.4 Use and interpret formulas (e.g., area = length X width or A=lw) to answer questions about quantities and their relationships Grade 5 NS1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23x3) NS2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors Grade 6 NS2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative numbers and combinations of these operations

Computational Targets for Multiplication and Division of Whole Numbers Grade 2 NS3.3 Know the multiplication tables of 2s, 5s and 10s (to "times 10") and commit them to memory Grade 3 NS2.5 Solve division problems in which a multidigit number is evenly divided by a one-digit number (135� Grade 4 NS3.2 Demonstrate understanding of, and ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results NS3.3 Solve problems involving multiplication of multidigit numbers by two-digit numbers NS3.4 Solve problems involving division of multidigit numbers by one-digit numbers Grade 5 NS2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors

Decimals Grade 2 NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole NS5.2 Know and use the decimal notation and the dollar and cents symbols for money Grade 3 NS3.0 Students understand the relationship between whole numbers, simple fractions, and decimals NS3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole number multipliers and divisors NS3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents 1/2 of a dollar, 75 cents is 3/4 of a dollar) Grade 4 NS1.0 Students understand place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts negative numbers NS1.2 Order and compare whole numbers and decimals to two decimal places NS1.6 Write tenths and hundredths in decimal and fraction notation and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or 0.50; 7/4 = 1 3/4 = 1.75) NS1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places NS2.0 Students extend their use and understanding of whole numbers to addition and subtraction of simple decimals NS2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places NS2.2 Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS1.2 Interpret percents as part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers NS2.0 Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results NS2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors Grade 6 NS1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages. NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line ST3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that probabilities computed are reasonable; know that if P is related the probability of an event, 1-P is the probability of an event not occurring Grade 7 NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers NS1.3 Convert fractions to decimals and percents and use these representations in estimation, computations, and applications NS1.5 Know that every fraction is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions

Decimal Arithmetic Grade 3 NS3.3 Solve problems involving addition, subtraction, multiplication, and division of money amounts in decimal notation and multiply and divide money amounts in decimal notation by using whole number multipliers and divisors Grade 4 NS2.0 Students extend their use and understanding of whole numbers to addition and subtraction of simple decimals NS2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS2.0 Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results NS2.2 Demonstrate proficiency with division, including division with positive decimals and long division with multidigit divisors Grade 6 NS1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages. Grade 7 NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers

Relationship between Fractions and Decimals Grade 2 NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole Grade 3 NS3.0 Students understand the relationship between whole numbers, simple fractions, and decimals NS3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents 1/2 of a dollar, 75 cents is 3/4 of a dollar) Grade 4 NS1.0 Students understand place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts negative numbers NS1.6 Write tenths and hundredths in decimal and fraction notation and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or 0.50; 7/4 = 1 3/4 = 1.75) NS1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS1.2 Interpret percents as part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers Grade 6 NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line ST3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that probabilities computed are reasonable; know that if P is related the probability of an event, 1-P is the probability of an event not occurring Grade 7 NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers NS1.3 Convert fractions to decimals and percents and use these representations in estimation, computations, and applications NS1.5 Know that every rational number is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions

Fractions Grade 2 NS4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole NS4.1 Recognize, name, and compare unit fractions from 1/12 to 1/2 NS4.2 Recognize fractions of a whole and parts of a group (e.g., one-fourth of a pie, two-thirds of 15 balls) NS4.3 Know that when all fractional parts are included, such as four-fourths, the result is equal to the whole and to one Grade 3 NS3.0 Students understand the relationship between whole numbers, simple fractions, and decimals NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4) NS3.2 Add and subtract simple fractions (e.g. determine that 1/8 + 3/8 is the same as 1/2) NS3.4 Know and understand that fractions and decimals are two different representations of the same concept (e.g., 50 cents 1/2 of a dollar, 75 cents is 3/4 of a dollar) Grade 4 NS1.0 Students understand place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts negative numbers NS1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions NS1.6 Write tenths and hundredths in decimal and fraction notation and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or 0.50; 7/4 = 1 3/4 = 1.75) NS1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line NS1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS1.2 Interpret percents as part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers NS2.0 Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals NS2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less) and express answers in the simplest form NS2.4 Understand the concept of multiplication and division of fractions NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems ST1.3 Use fractions and percentages to compare data sets of different sizes Grade 6 NS1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages. NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line NS1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as multiplication of both sides of an equation by a multiplicative inverse. NS2.1 Solve problems involving addition, subtraction, multiplication, and division of fractions and explain why a particular operation was used for a given situation NS2.2 Explain the meaning of multiplication and division of positive fractions and perform the calculations (e.g., 5/8 �/16 = 5/8 x 16/15 = 2/3) NS2.4 determine the least common multiple and greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator in order to add two fractions or to find the reduced form for a fraction) Grade 7 NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers NS1.3 Convert fractions to decimals and percents and use these representations in estimation, computations, and applications NS1.5 Know that every rational number is either a terminating or repeating decimal and be able to convert terminating decimals into reduced fractions NS2.0 Students use exponents, powers, and roots and use exponents in working with fractions. NS2.2 Add and subtract fractions using factoring to find common denominators NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules

Arithmetic with Fractions Grade 3 NS3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., 1/2 of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than 1/4) NS3.2 Add and subtract simple fractions (e.g. determine that 1/8 + 3/8 is the same as 1/2) Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers NS2.0 Students perform calculations and solve problems involving addition, subtraction and simple multiplication and division of fractions and decimals NS2.3 Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and unlike denominators of 20 or less) and express answers in the simplest form NS2.4 Understand the concept of multiplication and division of fractions NS2.5 Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems Grade 6 NS2.1 Solve problems involving addition, subtraction, multiplication, and division of fractions and explain why a particular operation was used for a given situation NS2.2 Explain the meaning of multiplication and division of positive fractions and perform the calculations (e.g., 5/8�/16 = 5/8 x 16/15 = 2/3) NS2.4 determine the least common multiple and greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator in order to add two fractions or to find the reduced form for a fraction) Grade 7 NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers NS2.0 Students use exponents, powers, and roots and use exponents in working with fractions. NS2.2 Add and subtract fractions using factoring to find common denominators NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules

Estimation and Rounding Grade K NS3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones and tens places NS3.1 Recognize when an estimate is reasonable Grade 1 NS3.0 Students use estimation strategies in computation and problem solving that involve numbers that involve numbers that use the one, tens, and hundreds places NS3.1 Make reasonable estimates when comparing larger or smaller numbers Grade 2 NS2.0 Students estimate, calculate, and solve problems involving addition and subtraction of two- and three-digit numbers NS6.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones, tens, hundreds, and thousands places NS6.1 recognize when an estimate is reasonable in measurements (e.g., closest inch) Grade 3 NS1.4 Round off numbers to 10,000 to the nearest ten, hundred, and thousand MG1.1 Choose appropriate tools and units (metric and U.S.) and estimate and measure length, liquid volume, and weight/mass of given objects MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them MR2.1 Use estimation to verify the reasonableness of calculated results Grade 4 NS1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand NS1.4 Decide when a rounded solution is called for and explain why such a solution may be appropriate NS2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places NS2.2 Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer MR2.1 Use estimation to verify the reasonableness of calculated results Grade 5 NS1.1 Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers MR2.1 Use estimation to verify the reasonableness of calculated results Grade 6 MG1.2 Know common estimates of (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements ST3.2 Use data to estimate the probability for future events (e.g., batting averages or number of accidents per mile driven) MR2.1 Use estimation to verify the reasonableness of calculated results MR2.3 Estimate unknown quantities graphically and solve for them using logical reasoning and arithmetic and algebraic techniques Grade 7 NS1.3 Convert fractions to decimals and percents and use these representations in estimation, computations, and applications MG2.2 Estimate and compute the area of more complex or irregular two- and three-dimensional figures by breaking the figures down into more basic geometric objects MR2.1 Use estimation to verify the reasonableness of calculated results MR2.3 Estimate unknown quantities graphically and solve for them using logical reasoning and arithmetic and algebraic techniques

Negative Numbers Grade 4 NS1.0 Students understand place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts negative numbers NS1.8 Use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature, in "owing") Grade 5 NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers NS2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results Grade 6 NS1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line NS2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations that use positive and negative numbers and combinations of these operations

Integers Grade 3 MG1.3 Find the perimeter of a polygon with integer sides Grade 5 NS1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication NS1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers

Rational Numbers Grade 7 NS1.0 Students know the properties of, and compute with, rational numbers expressed in a variety of forms NS1.1 Read, write and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers NS1.4 Differentiate between rational and irrational numbers

Absolute Value Grade 7 NS2.5 Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers

Multiples, Factors, Primes, Composites Grade 4 NS4.0 Students know how to factor small whole numbers NS4.1 Understand that many whole numbers break down in different ways (e.g., 12 = 4x3 = 2x6 = 2x2x3) NS4.2 Know that numbers such as 2, 3, 5, 7, 11 do not have any factors except 1 and themselves and that such numbers are called prime numbers Grade 5 NS1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23x3) Grade 6 NS2.4 determine the least common multiple and greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator in order to add two fractions or to find the reduced form for a fraction) Grade 7 NS2.2 Add and subtract fractions using factoring to find common denominators

Powers, Exponents, and Roots Grade 5 NS1.3 Understand and compute positive integer powers of nonnegative integers; compute examples as repeated multiplication NS1.4 Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3 = 23x3) Grade 7 NS1.1 Read, write and compare rational numbers in scientific notation (positive and negative powers of 10) with approximate numbers using scientific notation NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers NS2.0 Students use exponents, powers, and roots and use exponents in working with fractions. NS2.1 Understand negative whole-number exponents. Multiply and divide expressions involving exponents with a common base NS2.3 Multiply, divide, and simplify rational numbers by using exponent rules NS2.4 Use the inverse relationship between raising to a power and extracting a root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why AF2.0 Students interpret and evaluate expressions involving integer powers and simple roots. AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent

Percentages Grade 5 NS1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions and percents. They understand the relative magnitudes of numbers NS1.2 Interpret percents as part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number ST1.3 Use fractions and percentages to compare data sets of different sizes Grade 6 NS1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages. NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips ST3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that probabilities computed are reasonable; know that if P is related the probability of an event, 1-P is the probability of an event not occurring Grade 7 NS1.3 Convert fractions to decimals and percents and use these representations in estimation, computations, and applications NS1.6 Calculate percentage of increases and decreases of a quantity NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest

Ratios, Rates, and Proportions Grade 6 NS1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages. NS1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b) NS1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as multiplication of both sides of an equation by a multiplicative inverse. AF2.0 Students analyze and use tables, graphs, and rules to solve problems involving rates and proportions. AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches) AF2.2 Demonstrate understanding that rate is a measure of one quantity per unit value of another quantity AF2.3 Solve problems involving rates, average speed, distance, and time ST3.3 Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that probabilities computed are reasonable; know that if P is related the probability of an event, 1-P is the probability of an event not occurring Grade 7 AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation MG1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer