Number Relationships and Operations -- Operations and Algebraic Thinking

Goal 1: Children develop and use concepts, properties, and representations of number that extend to other number systems, to measures, and to algebra.

By the end of each age group or grade level, most children will have met prior age group or grade level standards in this domain.  Infants Through Older Toddlers

Infants

(0-12 Months)

1. No standards in this age group for this Element.

Young Toddlers

(9-18 Months)

1. No standards in this age group for this Element.

Older Toddlers

(18-36 Months)

1. No standards in this age group for this Element.  Younger Preschoolers Through Kindergartners

Younger Preschoolers

(36-48 months)

1. No standards in this age group for this Element.

Older Preschoolers

(48-60 months)

1. No standards in this age group for this Element.

Kindergartners

1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps) acting out situations, verbal explanations, expressions, or equations.
2. Solve addition and subtraction word problems, and add and subtract within 10 (e.g., by using objects or drawings to represent the problem).
3. 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)
4. 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).
5. Fluently add and subtract within 5  First Graders Through Third Graders

Represent and solve problems involving addition and subtraction.

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
2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20

Understand and apply properties of operations and the relationship between addition and subtraction.

1. Apply properties of operations as strategies to add and subtract.2 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
2. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.

1. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
2. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

Work with addition and subtraction equations.

1. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
2. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

Represent and solve problems involving addition and subtraction.

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, with unknowns in all positions (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem)

1. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Work with equal groups of objects to gain foundations for multiplication.

1. 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.
2. 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.

Represent and solve problems involving multiplication and division.

1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each
2. 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)
3. 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)
4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

Understand properties of multiplication and the relationship between multiplication and division.

1. Apply properties of operations as strategies to multiply and divide.
2. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Multiply and divide within 100.

1. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division. By the end of Grade 3, know from memory all products of two one-digit numbers.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

1. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3
2. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. Vermont Agency of Education
Secretary Daniel M. French
219 North Main Street, Suite 402
Barre, VT 05641

(802) 479-1030 | aoe.edinfo@vermont.gov
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