Using Fractions to Multiply Decimals

5 min

Narrative

In this Warm-up, students encounter four multiplication equations, each with a variable xx. Students are likely to notice that:

  • The equations all have xx and 10 as their factors.
  • The four products are related by powers of 10, or that each product is one-tenth of the one before it.
  • The products all have the digits 8 and 1, and those digits move to the right, to smaller place values.

They make use of the structure of the equations to determine where the value of xx is the greatest and how the values of the variables are related (MP7).

Launch

Display the four equations for all to see. Give students 1–2 minutes of quiet time to analyze the equations and to answer the questions. Ask them to do so without writing anything and to be prepared to explain their reasoning. Follow that with a whole-class discussion.

Student Task

  1. In which equation is the value of xx the largest? Explain your reasoning.

    \begin{align}  x \boldcdot 10 &= 810\\[2ex]   x \boldcdot 10 &= \phantom{8}81\\[2ex]   x \boldcdot 10 &= \phantom{81}8.1\\[2ex]   x \boldcdot 10 &= \phantom{81}0.81\\  \end{align}

  2. How many times the size of 0.81 is 810?

Sample Response

  1. The xx has the largest value in the first equation. Sample reasoning:
    • When multiplied by 10, the xx in the first equation has the largest product. 
    • Each xx is one tenth of the product, and the largest product is 810.
  2. 810 is 1,000 times the size of 0.81. Sample reasoning:
    • Multiplying 0.81 by 10 moves the digits one place to the left, so multiplying 0.81 by 1,000 moves the digits 8 and 1 three places to the left.
    • 810 is 10 times 81, and 81 is 100 times 0.81, so 810 must be 1,000 times 0.81.

Synthesis

Before discussing the responses to the questions, invite students to share what they noticed about the four equations. Record observations about the structure of the equations, as noted in the Activity Narrative.

Then ask students to share their responses and reasoning. Highlight responses that clarify that multiplying a number by 10 moves the digits one place to the left. So if a number times 10 is 8.1, that number must be 0.81. Discuss how this understanding can help find how many times the size of 0.81 is 810.

Standards
Building On
  • 5.NBT.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  • 5.NBT.A.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Addressing
  • 6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.
  • 6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.

15 min

15 min