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

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


    Multiply and divide a multi-digit number with decimals to the tenths by one tenth and one-hundredth with procedural reliability. Example: The number 12.3 divided by 0.01 can be thought of as ?× 0.01 = 12.3 to determine the quotient is 1,230.

    Benchmark Clarifications:

    Clarification 1: Instruction focuses on the place value of the digit when multiplying or dividing

    Purpose and Instructional Strategies
    The purpose of this benchmark is for students to multiply multi-digit numbers with decimals to the tenths by .1 and by .01 with procedural reliability. Procedural reliability refers to the ability for students to develop an accurate, reliable method that aligns with a student’s understanding and learning style. Fluency of multiplying and dividing multi-digit whole numbers with decimals is not expected until Grade 6 (MA.6.NSO.2.1).

     When multiplying and dividing, students should continue to use the number sense strategies built in MA.5.NSO.2.4 (estimation, rounding, exploring place value relationships). Using these strategies will helps students predict reasonable solutions and determine whether their solutions make sense after solving.

     During instruction, students should see the relationship between multiplying and dividing multi-digit numbers with decimals to multiplying and dividing by whole numbers. Students extend their understanding to generalize patterns that exist when multiplying or dividing by 10 or 100 (MTR.5.1).

     Instruction may include the language that the “digits shift” relative to the position of the decimal point as long as there is an accompanying explanation. An instructional strategy that helps students see this is by putting digits on sticky notes or cards and showing how the values shift (or the decimal point moves) when multiplying by a power of ten. For example, a teacher could show one card with a 3 and another with a 5, and place them on the left and right of a decimal point on a blank place value chart. The teacher could then ask students to multiply by ten and shift both digits one place left to show the equation 3.5 × 10 = 35. They could ask students to multiply by 1/10 and show that 3.5 ×1/10= 0.35. Instruction may also include using the language “moving the decimal point” as long as there is an explanation about what happens to a number when multiplying and dividing by 0.1 and 0.01. Moving the decimal point does not change its meaning; it always indicates the transition from the ones to the tenths place. From either point of view, when the change is made it is important to emphasize the digits have new place values. (MTR.2.1, MTR.4.1,MTR.5.1)


    Common Misconceptions or Errors

     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
    Part A. What is 1/10 times 15?
    Part B. How many dimes are in $1.50?
    Part C. Write an expression to represent how many dimes are in $1.50.

    Instructional Items
    Instructional Item 1
    Which compares the products of 7.8 × 0.1 and 7.8 × 10 correctly?
    a. The product of 7.8 × 0.1 is 100 times less than the product of 7.8 × 10.
    b. The product of 7.8 × 0.1 is 10 times less than the product of 7.8 × 10.
    c. The product of 7.8 × 0.1 is 100 times more than the product of 7.8 × 10.
    d. The product of 7.8 × 0.1 is 10 times more than the product of 7.8 × 10.

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