• MA.5.NSO.2 Number Sense and Operations

    Add, subtract, multiply and divide multi-digit numbers.


    Explore the multiplication and division of multi-digit numbers with decimals to the hundredths using estimation, rounding and place value. Example: The quotient of 23 and 0.42 can be estimated as a little bigger than 46 because 0.42 is less than one-half and 23 times 2 is 46.

    Benchmark Clarifications:
    Clarification 1: Estimating quotients builds the foundation for division using a standard algorithm.
    Clarification 2: Instruction includes the use of models based on place value and the properties of operations.

    Purpose and Instructional Strategies
    The purpose of this benchmark is for students to explore multiplication division of multi-digit numbers with decimals using estimation, rounding, place value, and exploring the relationship between multiplication and division. This benchmark connects to the work students did in Grade 4 with addition and subtraction of decimals (MA.4.NSO.2.7). Students achieve procedural fluency with multiplying and dividing multi-digit numbers with decimals in Grade 6 (MA.6.NSO.2.1)

     Instruction of this benchmark will focus on number sense to help students develop procedural reliability while multiplying and dividing multi-digit numbers with decimals.

     During instruction, students should explore how the products and quotients of whole numbers relate to decimals. For example, if students know the product of 8 x 7 and the quotient of 56 ÷ 4, then they can reason through 0.08 x 7 or 5.6 ÷ 0.4 through place value relationships. Classroom discussions should allow for students to explore these patterns and use them to estimate products and quotients.

     Teachers should connect what students know about place value and fractions. For example, because students know that multiplying a number by one-fourth will result in a product that is smaller, multiplying a number by 0.25 (its decimal equivalence) will also result in a smaller product. In division, dividing a number by one-fourth and 0.25 will result in a larger quotient. Continued work in this benchmark will help students to generalize patterns in multiplication and division of whole numbers and fractions (K12.MTR.5.1).

     Models that help students explore the multiplication and division of multi-digit numbers with decimals include base ten representations (e.g., blocks) and place value mats.

    Common Misconceptions or Errors
     Students may not understand the reasoning behind the placement of the decimal point in the product. Modeling and exploring the relationships between place value will help students gain understanding.

     Students can confuse that multiplication always results in a larger product, and that division always results in a smaller quotient. Through classroom discussion, estimation and modeling, classroom work should address this misconception.

    Instructional Tasks
    Instructional Task 1
    What is the same about the products of these expressions? What is different? Explain.
    14 × 5           0.14 × 0.05

    Instructional Task 2
    What is the same about the quotients of these expressions? What is different? Explain.
    50 ÷ 25           50 ÷ 0.25

    Instructional Task 3
    How can you use 2 × 12 = 24 to help you find the product of 2 × 1.2? Explain.

    Instructional Items
    Instructional Item 1
    Raul reasons that the product of 82 × 0.56 will be greater than 41 and less than 82. Explain whether or not his conclusion is reasonable.


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