1. Kindergarten
1.1. Know number names and the count sequence
1.1.1. count to 100 by ones and tens
1.1.2. Count forward beginning from a given number within the known sequence
1.1.3. Write numbers from 0 to 20
1.2. Count to tell the numbers of object
1.2.1. Understand the relationship between numbers and quantities; connect counting to cardinality
1.2.1.1. say the number names in the standard order
1.2.1.2. Understand that the last number name said tells the number of objects counted
1.2.1.3. Understand that each successive number name refers to a quantity that is one larger
1.2.2. Count to answer "how many?" questions about as many as 20 things
1.3. Compare numbers
1.3.1. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group
1.3.2. Compare two numbers between 1 and 10 presented as written numerals
1.4. Understand addition as putting together and adding to and understand subtraction as taking apart and taking from
1.4.1. Represent addition and subtraction with objects, fingers, mental images
1.4.2. Solve addition and subtraction word problems
1.4.3. Decompose numbers less than or equal to 10 into pairs in more than one way
1.4.4. For any number from 1 to 9, find the number that makes 10 when added to the given number
1.4.5. Fluently add and subtract within 5
1.5. Work with numbers 11-19 to gain foundations for place value
1.5.1. Compose and decompose numbers from 11 to 19 into ten ones and some further ones (such as 10+8=18)
2. Fourth grade
2.1. Use the four operations with whole numbers to solve problems
2.1.1. Interpret a multiplication equation as a comparison
2.1.2. Multiply or divide to solve word problems involving multiplicative comparison
2.1.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations
2.2. Gain familiarity with factors and multiples
2.2.1. Find all factor pairs for a whole number in the range 1-100
2.3. Generate and analyze patterns
2.3.1. Generate a number or shape pattern that follows a given rule
2.4. Generalize place value understanding for multi-digit whole numbers
2.4.1. 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
2.4.2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form
2.4.3. Use place value understanding to round multi-digit whole numbers to any place
2.5. Use place value understanding and properties of operations to perform multi-digit arithmetic
2.5.1. Fluently add and subtract multi-digit whole numbers using the standard algorithm
2.5.2. 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
2.5.3. 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
2.6. Extend understanding of fraction equivalence and ordering
2.6.1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models
2.6.2. Compare two fractions with different numerators and different denominators
2.7. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers
2.7.1. Understand a fraction a/b with a > 1 as a sum of fractions 1/b
2.7.1.1. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole
2.7.1.2. Decompose a fraction into a sum of fractions with the same denominator in more than one way
2.7.1.3. Add and subtract mixed numbers with like denominators
2.7.1.4. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators
2.7.2. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number
2.7.2.1. Understand a fraction a/b as a multiple of 1/b
2.7.2.2. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number
2.7.2.3. Solve word problems involving multiplication of a fraction by a whole number
2.8. Understand decimal notation for fractions and compare decimal fractions
2.8.1. 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
2.8.2. Use decimal notation for fractions with denominators 10 or 100
2.8.3. Compare two decimals to hundredths by reasoning about their size
3. Fifth grade
3.1. Write and interpret numerical expressions
3.1.1. Use parenthesis, brackets, or braces in numerical expressions
3.1.2. write simple expressions that record calculations with numbers
3.2. Analyze patterns and relationships
3.2.1. Generate two numerical patterns using two given rules (ex.given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence
3.3. Understand the place value system
3.3.1. 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
3.3.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10
3.3.3. Read, write, and compare decimals to thousandths
3.3.3.1. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form
3.3.3.2. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and <
3.3.4. Use place value understanding to round decimals to any place
3.4. Perform operations with multi-digit whole numbers and with decimals to hundredths
3.4.1. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors
3.4.2. Fluently multiply multi-digit whole numbers using the standard algorithm
3.4.3. Add, subtract, multiply, and divide decimals to hundredths
3.5. Use equivalent fractions as a strategy to add and subtract fractions
3.5.1. Add and subtract fractions with unlike denominators
3.5.2. Solve word problems involving addition and subtraction of fractions referring to the same whole
3.6. Apply and extend previous understandings of multiplication and division to multiply and divide fractions
3.6.1. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)
3.6.2. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction
3.6.2.1. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts
3.6.2.2. 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
3.6.3. Interpret multiplication as scaling (resizing)
3.6.3.1. Comparing the size of a product to the size of one factor on the basis of the size of the other factor
3.6.3.2. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number
3.6.4. Solve real world problems involving multiplication of fractions and mixed numbers
3.6.5. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions
3.6.5.1. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients
3.6.5.2. Interpret division of a whole number by a unit fraction, and compute such quotients
3.6.5.3. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions
4. Second grade
4.1. Represent and solve problems involving addition and subtraction
4.1.1. 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
4.2. Add and subtract within 20
4.2.1. Fluently add and subtract within 20 using mental strategies
4.3. Work with equal groups of objects to gain foundations for multiplication
4.3.1. Determine whether a group of objects (up to 20) has an odd or even number
4.3.2. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns
4.4. Understand place value
4.4.1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones
4.4.2. Count within 1000; skip-count by 5s, 10s, and 100s
4.4.3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form
4.4.4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons
4.5. Use place value understanding and properties of operations to add and subtract
4.5.1. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction
4.5.2. Add up to four two-digit numbers using strategies based on place value and properties of operations
4.5.3. Add and subtract within 1000, using concrete models or drawings and strategies based on place value
4.5.4. Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900
4.5.5. Explain why addition and subtraction strategies work
5. First Grade
5.1. Represent and solve problems involving addition and subtraction
5.1.1. 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 equations with a symbol for the unknown number to represent the problem
5.1.2. 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
5.2. Understand and apply properties of operations and the relationship between addition and subtraction
5.2.1. Apply properties of operations as strategies to add and subtract
5.2.2. Understand subtraction as an unknown-addend problem
5.3. Add and subtract within 20
5.3.1. Relate counting to addition and subtraction
5.3.2. Demonstrate fluency for addition and subtraction within 10
5.4. Work with addition and subtraction equations
5.4.1. Understand the meaning of the equal sign and determine if equations involving addition an subtraction are true or false
5.4.2. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers
5.5. Extend the counting sequence
5.5.1. Count to 120
5.6. Understand place value
5.6.1. Understand that the two digits of a two-digit number represent amounts of tens and ones
5.6.2. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >,<,=
5.7. Use place value understanding and properties of operations to add and subtract
5.7.1. Add within 100
5.7.2. Mentally find 10 more or 10 less than the number
5.7.3. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90
6. Third grade
6.1. Represent and solve problems involving multiplication and division
6.1.1. Interpret products of whole numbers
6.1.2. Interpret whole-number quotients of whole numbers
6.1.3. Use multiplication and division within 100 to solve word problems
6.1.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers
6.2. Understand properties of multiplication and the relationship between multiplication and division
6.2.1. Apply properties of operations as strategies to multiply and divide
6.2.2. Understand division as an unknown-factor problem
6.3. Multiply and divide within 100
6.3.1. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division
6.4. Solve problems involving the four operations and identify and explain patterns in arithmetic
6.4.1. Solve two-step word problems using the four operations
6.4.2. Identify arithmetic patterns and explain them using properties of operations
6.5. Use place value understanding the properties of operations to perform multi-digit arithmetic
6.5.1. Use place value understanding to round whole numbers to the nearest 10 or 100
6.5.2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction
6.5.3. Multiply one-digit whole numbers by multiples of 10 in the range 10-90
6.6. Develop understanding of fractions as numbers
6.6.1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts
6.6.2. Understand a fraction as a number on the number line
6.6.2.1. 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
6.6.2.2. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0
6.6.3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size
6.6.3.1. Understand two fractions as equivalent if they are the same size, or the same point on a number line
6.6.3.2. Recognize and generate simple equivalent fractions
6.6.3.3. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers
6.6.3.4. Compare two fractions with the same numerator or the same denominator by reasoning about their size