• MA.5.AR.2 Algebraic Reasoning

    Demonstrate an understanding of equality, the order of operations and equivalent numerical expressions



    Translate written real-world and mathematical descriptions into numerical expressions and numerical expressions into written mathematical descriptions. Example: The expression 4.5 + (3 × 2) in word form is 𝑓𝑜𝑢𝑟 𝑎𝑛𝑑 𝑓𝑖𝑣𝑒 𝑡𝑒𝑛𝑡ℎ𝑠 plus the quantity 3 times 2.

    Benchmark Clarifications:
    Clarification 1: Expressions are limited to any combination of the arithmetic operations, including parentheses, with whole numbers, decimals and fractions.
    Clarification 2: Within this benchmark, the expectation is not to include exponents or nested grouping symbols. 


    Purpose and Instructional Strategies
    The purpose of this benchmark is for students to translate between numerical and written mathematical expressions. This builds from previous work where students wrote equations with unknowns in any position of the equation in Grade 4 (MA.4.AR.2.2). Algebraic expressions are a major theme in Grade 6 starting with MA.6.AR.1.1.

     During instruction, teachers should model how to translate numerical expressions into words using correct vocabulary. This includes naming fractions and decimals correctly. Students should use diverse vocabulary to describe expressions. For example, in the expression 4.5 + (3 × 2) could be read in multiple ways to show its operations. Students should explore them and find connections between their meanings (MTR.3.1, MTR.4.1, MTR.5.1).

    o 4 𝑎𝑛𝑑 𝑓𝑖𝑣𝑒 𝑡𝑒𝑛𝑡ℎ𝑠 𝑝𝑙𝑢𝑠 𝑡ℎ𝑒 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 3 𝑡𝑖𝑚𝑒𝑠 2
    o 4 𝑎𝑛𝑑 5 𝑡𝑒𝑛𝑡ℎ𝑠 𝑝𝑙𝑢𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 3 𝑎𝑛𝑑 2
    o 𝑇ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 4 𝑎𝑛𝑑 5 𝑡𝑒𝑛𝑡ℎ𝑠 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 3 𝑡𝑖𝑚𝑒𝑠 2
    o 𝑇ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 4 𝑎𝑛𝑑 5 𝑡𝑒𝑛𝑡ℎ𝑠 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 3 𝑎𝑛𝑑 2

     The expectation of this benchmark is to not use exponents or nested grouping symbols. Nested grouping symbols refer to grouping symbols within one another in an expression, like in 3 + [5.2 + (4 × 2)].
     Instruction of this benchmark helps students understand the order of operations, the expectation of MA.5.AR.2.2.

    Common Misconceptions or Errors
     Students can misrepresent decimal and fraction numbers in words. This benchmark helps students practice naming numbers according to place value.

     Some students can confuse the difference between what is expected in the expressions 5(9 + 3) and 5 + (9 + 3). Students need practice naming the former as multiplication (e.g., 5 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 9 𝑎𝑛𝑑 3) and understanding that in that expression, both 5 and 9 + 3 are factors.

    Instructional Tasks
    Instructional Task 1
    Nadia sees the numerical expression 6.5 + 1/2 (4 − 2). She translates the expression as,
    “6 𝑎𝑛𝑑 𝑓𝑖𝑣𝑒 𝑡𝑒𝑛𝑡ℎ𝑠 𝑝𝑙𝑢𝑠 1 ℎ𝑎𝑙𝑓 𝑡𝑖𝑚𝑒𝑠 4, 𝑚𝑖𝑛𝑢𝑠 2.” Part A: Is her translation correct? Explain. Part B: Evaluate the expression.

    Instructional Task 2
    Translate the written mathematical description below into a numerical expression: 𝐷𝑖𝑣𝑖𝑑𝑒 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 20 𝑎𝑛𝑑 5 𝑏𝑦 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 4 𝑎𝑛𝑑 1.

    Instructional Items
    Instructional Item 1
    Translate the numerical expression below into a written mathematical description. 2(53.8 + 4 − 22.9)

    Instructional Item 2
    Translate the written mathematical description into a numerical expression. “one half the difference of 6 and 8 hundredths and 2” 

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