Math Curriculum - Numbers & Operations

math graphic

Number and Operations - K-3, 4-6, 7-12
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Number and Operations K-3 (printable PDF)
Essential Skills:
1. Understand numbers, ways of representing numbers, relationships among numbers and number systems
2. Understand meanings of operations and how they relate to one another
3. Compute fluently.
4. Make reasonable estimates and relate them to solutions.
.
Kindergarten
Important to Know
• Write with accuracy (0-10)
• Using concrete materials, count forward by rote through 20 and backwards from 10 through 0
• Using concrete materials, relate quantities to numerals 0 through 10
• Using concrete materials, separate, join, and order sets of objects 0 through 10
• Using concrete materials, compare number of objects in two or more sets through 10
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Grade 1
Important to Know
• Use ordinal numbers through 10th orally
• Skip count forwards and backwards by twos through 20 and fives and tens through 100
• Order and compare quantities through 100 with manipulatives
• Develop place value concepts through a variety of manipulative grouping experiences (through 100)
• Associate standard word names with 1-digit and 2-digit numerals through 100 using oral and written language experiences
• Round to 10's place using concrete materials
• Demonstrate the concepts of addition and subtraction from real life situations including joining, separating, and comparing sets of objects 0 through 20
• Use the following strategies for addition and subtraction of whole numbers: counting on, counting back, one more, one less, two more, two less, doubles, doubles plus or minus one; doubles plus or minus two
• Describe the operations of addition and subtraction in words (oral or written)
• Use conventional symbols (+, -, =) to represent the operations of addition and subtraction, both vertically and horizontally, up to 2-digit numbers without regrouping
• Memorize addition sums through ten
• Use oral vocabulary and manipulatives to communicate relative quantity for numbers up through 100; vocabulary includes "more than," "less than," and "equal to"
• Orally identify fractional parts (1/2's and 1/4's) of whole objects or sets of objects
• Estimate a number prior to counting and count objects by twos, fives, and tens
.
Grade 2
Important to Know
• Develop the concept of odd and even using concrete materials
• Use ordinal numbers through 31st
• Read and write numerals up through 1,000
• Develop the concept of place value with concrete models of 100's, 10's, and 1's through 1,000’s. Include written and oral experiences
• Round whole numbers to nearest 10. Begin rounding to 100's using concrete materials
• Translate real world situations involving addition and/or subtraction into conventional numerals and symbols
• Investigate regrouping for addition and subtraction of 2-digit numbers using concrete materials
• Regroup in addition on paper up to 3 digits
• Memorize: addition sums through 18 and subtraction differences through 10 (10 is top number)
• Begin to explore multiplication and division situations using stories and concrete materials
• Place whole numbers on a number line 0 through 1,000
• Compare whole numbers through 1,000 using <, >, or =
• Use concrete models of fractions (1/2's, 1/3's, and 1/4's)
• Demonstrate fractional parts of whole objects or sets of objects
• Develop skill in estimating sums and differences
.
Grade 3
Important to Know
• Identify even and odd numbers
• Translate accurately and efficiently between words and symbols
• Demonstrate the concept of place value through 1,000's using manipulatives and written and oral language
• Investigate and read numbers through 100,000
• Round whole numbers to 10's and 100's; round to 1,000's using manipulatives
• Add and subtract accurately and efficiently to the 1,000's place with regrouping
• Memorize subtraction differences through 18 (18 is top number)
• Memorize multiplication facts through 6 x 10
• Develop and use models of multiplication and division (using pictures, arrays, and manipulatives)
• Describe multiplication and division in oral or written language
• Use conventional symbols to represent multiplication and division
• Translate real life situations involving multiplication and division into conventional mathematical symbols
• Relate multiplication to skip counting
• Multiply using the multiplication charts through 10 x 10
• Find multiplication and division facts through 10 x 10 using the following strategies: commutative property, anchor facts and squares (students do not need to know these terms in this grade)
• Demonstrate remainders in division with concrete objects
• Use the symbols >, <, and = to compare whole numbers up through 10,000
• Represent tenths as fractions, decimals, and in words
• Demonstrate an understanding of numerator and denominator using concrete objects
• Use concrete materials to order fractions (1/2's, 1/3's, 1/4's, and 1/10's)
• Order whole numbers or fractions with like denominators on the number line (1/2's, 1/3's, 1/4's, and 1/10's)

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Number and Operations 4-6 (printable PDF)
Essentials Skills:
1. Understand numbers, ways of representing numbers, relationships among numbers and number systems.
2. Understand meanings of operations and how they relate to one another.
3. Compute fluently.
4. Make reasonable estimates and relate them to solutions
.
Grade 4
Important to Know
• Demonstrate an understanding of place value by reading, writing, and decomposing whole numbers through the ten thousands place.
• Order numbers by placing them correctly on a number line through the 100,000 place.
• Compare any two whole numbers using < , >, or = through 100,000.
• Round whole numbers to 10’s, 100’s, 1,000’s, and 10,000’s places
• Demonstrate an understanding and identify odd and even numbers
• Demonstrate an understanding of numbers less than 0 using concrete materials (thermometers, coins, debt, depth)
• Demonstrate an understanding of numbers less than zero by extending the number line and accurately labeling through negative 20
• Add and subtract accurately and efficiently any 2 whole numbers through 100,000 when the problems are written horizontally or vertically
• Add and subtract decimals involving money
• Demonstrate that multiplication is repeated addition or is making equal groups of a number
• Memorize multiplication facts through 10 x 10
• Demonstrate that division is making equal groups from a set of objects or a number
• Demonstrate an understanding that multiplication of whole numbers greater than 1 makes the product bigger than either factor
• Demonstrate an understanding that when a whole number is divided by a number greater than 1, the quotient is smaller than the dividend
• Multiply a monetary amount by a whole number
• Write a monetary amount less than $1.00 using both decimal point and "$" sign or cent sign
• Using pictures/models, show 2 ways to solve a multiplication problem (2 digits X 1 digit) using the distributive property (do not have to identify it as the distributive property.)
• Accurately and efficiently, multiply any two 2-digit numbers when written vertically or horizontally.
• Divide with 1-digit divisor to obtain a 1-digit or 2 -digit quotient with or without remainders.
• Express quotients as mixed numbers or as whole numbers with remainder “R”
• Divide by a multiple of 10 to obtain a 1-digit or 2-digit quotient with or without a remainder
• Use estimation strategies to judge the reasonableness of the answer to whole number computation
• Recognize and generate equivalent representations for a given common fraction (1/2's, 1/3's, 1/4's, 1/8's, and 1/10's) using pictures and manipulatives.
• Name and identify a fraction given a physical representation (any fraction)
• Order fractions from least to greatest using pictures, manipulatives, or a number line (common fractions 1/2’s, 1/3’s, 1/4’s, 1/8’s, 1/10’s )
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Grade 5
Important to Know
• Demonstrate an understanding of place value by reading, writing, and decomposing whole numbers to the millions place and decimals to the hundredths place
• Demonstrate an understanding of composite and prime numbers and identify prime numbers through 50.
• Order integers (positive and negative whole numbers) given concrete materials from lowest to highest
• Compare any 2 whole numbers using <, >,or = through 1,000,000
• Round whole numbers to the nearest millions and decimals to the nearest tenth.
• Demonstrate division as repeated subtraction or equal grouping from a set of numbers
• Memorize division facts through 100 divided by 10
• Use strategies to estimate the results of division problems with 2-digit divisors
• Solve long division problems accurately and efficiently using 2-digit divisors with and without remainders, showing the quotient as a mixed number
• Add, subtract, multiply and divide efficiently sets of whole numbers written vertically and horizontally
• Demonstrate an understanding of the associative and commutative properties of addition and multiplication to solve problems involving 3 or more whole numbers
• Demonstrate an understanding of the distributive property by being able to write at least 2 ways of grouping and solving a multiplication problem such as 12 X 24 as [(6 X 24) + (6 X 24)] or [(10 X 24) +
(2 X 24)]
• Explain why the commutative property works in addition and multiplication, but not in subtraction and division
• Given a model, compare fractions with denominators 2 through 10.
• Convert fractions in tenths and hundredths to decimals
• Recognize and convert commonly used percents to fractions and decimals (10%, 20%, 25%, 50%, 75%. 100%)
• Use strategies to estimate results. Analyze the results to judge the reasonableness of answers
.
Grade 6
Important to Know
• Compare common fractions using the symbols < , >, or = or by placing them on a number line.
• Compare and order decimals and find their approximate location on a number line.
• Round whole numbers to the nearest millions place; decimals to the thousandths place.
• Demonstrate that when multiplying 2 decimal numbers less than 1, the product will be smaller than either original decimal.
• Solve computational and word problems involving decimals using all four operations
• Read, write, and decompose decimals to the thousandths place
• Convert decimals to fractions and vice versa through the millionths
• Use factors, multiples and prime factorization to solve problems
• Write two ways to solve a problem involving decimals using the distributive property
Ex. (24 X .12; see 5th grade example)
• Demonstrate an understanding of the associative and commutative properties of addition and multiplication by solving problems involving a set of three decimals
• Use strategies to estimate results. Analyze the results to judge the reasonableness of the answer
• Recognize and convert commonly used percents to fractions and decimals (10%, 20%, 25%, 50%, 75%. 100%)
• Given a pair of numbers find the greatest common factor (GCF) and the least common multiple (LCM) of the numbers
• Recognize and generate equivalent representations for any fraction

.
Number and Operations 7-12 (printable PDF)
Essentials Skills:
1. Understand numbers, ways of representing numbers, relationships among numbers and number systems
2. Understand meanings of operations and how they relate to one another
3. Compute fluently.
4. Make reasonable estimates and relate the estimates to solutions
.
Grade 7
Important to Know
• Solve word problems using decimals, fractions and mixed numbers
• Compare and order decimals, fractions, and mixed numbers efficiently and find their
approximate locations on a number line
• Use factors, multiples, and prime factorizations to solve problems
• Explain the effects of arithmetic operations with decimals, fractions, and mixed numbers
• Use the associative and commutative properties of addition and multiplication and distributive property of multiplication over addition to simplify computations with decimals, fractions, and mixed numbers
• Use integers to represent and compare quantities
• Explain the effects of arithmetic operations with integers
• Use inverse relationships of addition /subtraction, multiplication/division to simplify computations and solve problems
• Use a variety of methods and tools for computing with decimals, fractions, and mixed numbers that include: mental computation, estimation, calculators or computers, and paper and pencil
• Evaluate any whole number to any whole number exponent
• Compute efficiently with decimals, fractions, mixed numbers, and integers
• Recognize and generalize equivalent representations for any fraction
.
Grade 8
Important to Know
• Solve word problems using percents
• Compare and order percents efficiently and find their approximate location on a number line
• Demonstrate an understanding of percents greater than 100% and less than 1%
• Use ratios and proportions to represent quantitative relationships
• Recognize and represent large numbers in exponential, scientific, and calculator notation for powers of ten
• Use factors, multiples, prime factorizations and relatively prime numbers to solve word problems
• Use integers to represent and compare quantities
• Explain the effects of arithmetic operations with integers
• Use associative and commutative properties of addition and multiplication and distributive property of multiplication over addition to simplify computations with integers
• Understand square roots and compute them using a calculator
• Use inverse relationships of addition/subtraction, multiplication/division, and squaring /square roots to simplify computations and to solve word problems
• Use a variety of methods and tools for computing with decimals, fractions, mixed numbers, percents, and integers
• Develop and use strategies to estimate the results of rational number computations and judge the reasonableness of results
• Develop, analyze and explain methods for solving problems involving proportions
• Compute efficiently with decimals, fractions, mixed numbers, percents and integers
.
Grade 9
Important to Know
• Perform operations with scientific notation
• Explain the effects of multiplication , division, and computing powers and roots on the magnitudes of quantities
• Perform operations with real numbers accurately and efficiently, using mental computation or paper and pencil calculations for simple cases, and technology for more complicated cases
• Develop and use strategies to estimate the results of rational number computations and judge the reasonableness of those results
.
Grade 10
Important to Know
• Compare and contrast the properties of numbers and number systems
• Identify the properties of matrices and compare to other number systems
• Use permutations and combinations as counting techniques
• Perform operations with matrices using mental computation or paper and pencil calculations for simple cases and technology for more complicated cases
.
Grade 11-12
Important to Know
• Use complex numbers as solutions to quadratic equations that do not have real solutions
• Identify the properties of vectors
• Perform vector addition and multiplication (dot and cross)
• Use permutations and combinations as counting techniques

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		Number and Operations - K-3, 4-6, 7-12 . Number and Operations K-3 (printable PDF) Essential Skills: 1. Understand numbers, ways of representing numbers, relationships among numbers and number systems 2. Understand meanings of operations and how they relate to one another 3. Compute fluently. 4. Make reasonable estimates and relate them to solutions. . Kindergarten Important to Know • Write with accuracy (0-10) • Using concrete materials, count forward by rote through 20 and backwards from 10 through 0 • Using concrete materials, relate quantities to numerals 0 through 10 • Using concrete materials, separate, join, and order sets of objects 0 through 10 • Using concrete materials, compare number of objects in two or more sets through 10 . Grade 1 Important to Know • Use ordinal numbers through 10th orally • Skip count forwards and backwards by twos through 20 and fives and tens through 100 • Order and compare quantities through 100 with manipulatives • Develop place value concepts through a variety of manipulative grouping experiences (through 100) • Associate standard word names with 1-digit and 2-digit numerals through 100 using oral and written language experiences • Round to 10's place using concrete materials • Demonstrate the concepts of addition and subtraction from real life situations including joining, separating, and comparing sets of objects 0 through 20 • Use the following strategies for addition and subtraction of whole numbers: counting on, counting back, one more, one less, two more, two less, doubles, doubles plus or minus one; doubles plus or minus two • Describe the operations of addition and subtraction in words (oral or written) • Use conventional symbols (+, -, =) to represent the operations of addition and subtraction, both vertically and horizontally, up to 2-digit numbers without regrouping • Memorize addition sums through ten • Use oral vocabulary and manipulatives to communicate relative quantity for numbers up through 100; vocabulary includes "more than," "less than," and "equal to" • Orally identify fractional parts (1/2's and 1/4's) of whole objects or sets of objects • Estimate a number prior to counting and count objects by twos, fives, and tens . Grade 2 Important to Know • Develop the concept of odd and even using concrete materials • Use ordinal numbers through 31st • Read and write numerals up through 1,000 • Develop the concept of place value with concrete models of 100's, 10's, and 1's through 1,000’s. Include written and oral experiences • Round whole numbers to nearest 10. Begin rounding to 100's using concrete materials • Translate real world situations involving addition and/or subtraction into conventional numerals and symbols • Investigate regrouping for addition and subtraction of 2-digit numbers using concrete materials • Regroup in addition on paper up to 3 digits • Memorize: addition sums through 18 and subtraction differences through 10 (10 is top number) • Begin to explore multiplication and division situations using stories and concrete materials • Place whole numbers on a number line 0 through 1,000 • Compare whole numbers through 1,000 using <, >, or = • Use concrete models of fractions (1/2's, 1/3's, and 1/4's) • Demonstrate fractional parts of whole objects or sets of objects • Develop skill in estimating sums and differences . Grade 3 Important to Know • Identify even and odd numbers • Translate accurately and efficiently between words and symbols • Demonstrate the concept of place value through 1,000's using manipulatives and written and oral language • Investigate and read numbers through 100,000 • Round whole numbers to 10's and 100's; round to 1,000's using manipulatives • Add and subtract accurately and efficiently to the 1,000's place with regrouping • Memorize subtraction differences through 18 (18 is top number) • Memorize multiplication facts through 6 x 10 • Develop and use models of multiplication and division (using pictures, arrays, and manipulatives) • Describe multiplication and division in oral or written language • Use conventional symbols to represent multiplication and division • Translate real life situations involving multiplication and division into conventional mathematical symbols • Relate multiplication to skip counting • Multiply using the multiplication charts through 10 x 10 • Find multiplication and division facts through 10 x 10 using the following strategies: commutative property, anchor facts and squares (students do not need to know these terms in this grade) • Demonstrate remainders in division with concrete objects • Use the symbols >, <, and = to compare whole numbers up through 10,000 • Represent tenths as fractions, decimals, and in words • Demonstrate an understanding of numerator and denominator using concrete objects • Use concrete materials to order fractions (1/2's, 1/3's, 1/4's, and 1/10's) • Order whole numbers or fractions with like denominators on the number line (1/2's, 1/3's, 1/4's, and 1/10's) . Number and Operations 4-6 (printable PDF) Essentials Skills: 1. Understand numbers, ways of representing numbers, relationships among numbers and number systems. 2. Understand meanings of operations and how they relate to one another. 3. Compute fluently. 4. Make reasonable estimates and relate them to solutions . Grade 4 Important to Know • Demonstrate an understanding of place value by reading, writing, and decomposing whole numbers through the ten thousands place. • Order numbers by placing them correctly on a number line through the 100,000 place. • Compare any two whole numbers using < , >, or = through 100,000. • Round whole numbers to 10’s, 100’s, 1,000’s, and 10,000’s places • Demonstrate an understanding and identify odd and even numbers • Demonstrate an understanding of numbers less than 0 using concrete materials (thermometers, coins, debt, depth) • Demonstrate an understanding of numbers less than zero by extending the number line and accurately labeling through negative 20 • Add and subtract accurately and efficiently any 2 whole numbers through 100,000 when the problems are written horizontally or vertically • Add and subtract decimals involving money • Demonstrate that multiplication is repeated addition or is making equal groups of a number • Memorize multiplication facts through 10 x 10 • Demonstrate that division is making equal groups from a set of objects or a number • Demonstrate an understanding that multiplication of whole numbers greater than 1 makes the product bigger than either factor • Demonstrate an understanding that when a whole number is divided by a number greater than 1, the quotient is smaller than the dividend • Multiply a monetary amount by a whole number • Write a monetary amount less than $1.00 using both decimal point and "$" sign or cent sign • Using pictures/models, show 2 ways to solve a multiplication problem (2 digits X 1 digit) using the distributive property (do not have to identify it as the distributive property.) • Accurately and efficiently, multiply any two 2-digit numbers when written vertically or horizontally. • Divide with 1-digit divisor to obtain a 1-digit or 2 -digit quotient with or without remainders. • Express quotients as mixed numbers or as whole numbers with remainder “R” • Divide by a multiple of 10 to obtain a 1-digit or 2-digit quotient with or without a remainder • Use estimation strategies to judge the reasonableness of the answer to whole number computation • Recognize and generate equivalent representations for a given common fraction (1/2's, 1/3's, 1/4's, 1/8's, and 1/10's) using pictures and manipulatives. • Name and identify a fraction given a physical representation (any fraction) • Order fractions from least to greatest using pictures, manipulatives, or a number line (common fractions 1/2’s, 1/3’s, 1/4’s, 1/8’s, 1/10’s ) . Grade 5 Important to Know • Demonstrate an understanding of place value by reading, writing, and decomposing whole numbers to the millions place and decimals to the hundredths place • Demonstrate an understanding of composite and prime numbers and identify prime numbers through 50. • Order integers (positive and negative whole numbers) given concrete materials from lowest to highest • Compare any 2 whole numbers using <, >,or = through 1,000,000 • Round whole numbers to the nearest millions and decimals to the nearest tenth. • Demonstrate division as repeated subtraction or equal grouping from a set of numbers • Memorize division facts through 100 divided by 10 • Use strategies to estimate the results of division problems with 2-digit divisors • Solve long division problems accurately and efficiently using 2-digit divisors with and without remainders, showing the quotient as a mixed number • Add, subtract, multiply and divide efficiently sets of whole numbers written vertically and horizontally • Demonstrate an understanding of the associative and commutative properties of addition and multiplication to solve problems involving 3 or more whole numbers • Demonstrate an understanding of the distributive property by being able to write at least 2 ways of grouping and solving a multiplication problem such as 12 X 24 as [(6 X 24) + (6 X 24)] or [(10 X 24) + (2 X 24)] • Explain why the commutative property works in addition and multiplication, but not in subtraction and division • Given a model, compare fractions with denominators 2 through 10. • Convert fractions in tenths and hundredths to decimals • Recognize and convert commonly used percents to fractions and decimals (10%, 20%, 25%, 50%, 75%. 100%) • Use strategies to estimate results. Analyze the results to judge the reasonableness of answers . Grade 6 Important to Know • Compare common fractions using the symbols < , >, or = or by placing them on a number line. • Compare and order decimals and find their approximate location on a number line. • Round whole numbers to the nearest millions place; decimals to the thousandths place. • Demonstrate that when multiplying 2 decimal numbers less than 1, the product will be smaller than either original decimal. • Solve computational and word problems involving decimals using all four operations • Read, write, and decompose decimals to the thousandths place • Convert decimals to fractions and vice versa through the millionths • Use factors, multiples and prime factorization to solve problems • Write two ways to solve a problem involving decimals using the distributive property Ex. (24 X .12; see 5th grade example) • Demonstrate an understanding of the associative and commutative properties of addition and multiplication by solving problems involving a set of three decimals • Use strategies to estimate results. Analyze the results to judge the reasonableness of the answer • Recognize and convert commonly used percents to fractions and decimals (10%, 20%, 25%, 50%, 75%. 100%) • Given a pair of numbers find the greatest common factor (GCF) and the least common multiple (LCM) of the numbers • Recognize and generate equivalent representations for any fraction . Number and Operations 7-12 (printable PDF) Essentials Skills: 1. Understand numbers, ways of representing numbers, relationships among numbers and number systems 2. Understand meanings of operations and how they relate to one another 3. Compute fluently. 4. Make reasonable estimates and relate the estimates to solutions . Grade 7 Important to Know • Solve word problems using decimals, fractions and mixed numbers • Compare and order decimals, fractions, and mixed numbers efficiently and find their approximate locations on a number line • Use factors, multiples, and prime factorizations to solve problems • Explain the effects of arithmetic operations with decimals, fractions, and mixed numbers • Use the associative and commutative properties of addition and multiplication and distributive property of multiplication over addition to simplify computations with decimals, fractions, and mixed numbers • Use integers to represent and compare quantities • Explain the effects of arithmetic operations with integers • Use inverse relationships of addition /subtraction, multiplication/division to simplify computations and solve problems • Use a variety of methods and tools for computing with decimals, fractions, and mixed numbers that include: mental computation, estimation, calculators or computers, and paper and pencil • Evaluate any whole number to any whole number exponent • Compute efficiently with decimals, fractions, mixed numbers, and integers • Recognize and generalize equivalent representations for any fraction . Grade 8 Important to Know • Solve word problems using percents • Compare and order percents efficiently and find their approximate location on a number line • Demonstrate an understanding of percents greater than 100% and less than 1% • Use ratios and proportions to represent quantitative relationships • Recognize and represent large numbers in exponential, scientific, and calculator notation for powers of ten • Use factors, multiples, prime factorizations and relatively prime numbers to solve word problems • Use integers to represent and compare quantities • Explain the effects of arithmetic operations with integers • Use associative and commutative properties of addition and multiplication and distributive property of multiplication over addition to simplify computations with integers • Understand square roots and compute them using a calculator • Use inverse relationships of addition/subtraction, multiplication/division, and squaring /square roots to simplify computations and to solve word problems • Use a variety of methods and tools for computing with decimals, fractions, mixed numbers, percents, and integers • Develop and use strategies to estimate the results of rational number computations and judge the reasonableness of results • Develop, analyze and explain methods for solving problems involving proportions • Compute efficiently with decimals, fractions, mixed numbers, percents and integers . Grade 9 Important to Know • Perform operations with scientific notation • Explain the effects of multiplication , division, and computing powers and roots on the magnitudes of quantities • Perform operations with real numbers accurately and efficiently, using mental computation or paper and pencil calculations for simple cases, and technology for more complicated cases • Develop and use strategies to estimate the results of rational number computations and judge the reasonableness of those results . Grade 10 Important to Know • Compare and contrast the properties of numbers and number systems • Identify the properties of matrices and compare to other number systems • Use permutations and combinations as counting techniques • Perform operations with matrices using mental computation or paper and pencil calculations for simple cases and technology for more complicated cases . Grade 11-12 Important to Know • Use complex numbers as solutions to quadratic equations that do not have real solutions • Identify the properties of vectors • Perform vector addition and multiplication (dot and cross) • Use permutations and combinations as counting techniques







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