Calculating Products of Decimals

5 min

Narrative

This Math Talk focuses on multiplication of a whole number and a decimal. It encourages students to think about properties of operations and to rely on what they know about place value to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students decompose and multiply decimals beyond tenths later in the lesson.

To multiply decimals to the hundredths, students need to look for and make use of the structure of base-ten numbers (MP7).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies, and record and display their responses for all to see.
  • Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each expression mentally.

  • 20⋅520 \boldcdot 5
  • 20⋅(0.8)20 \boldcdot (0.8)
  • 20⋅(0.04)20 \boldcdot (0.04)
  • 20⋅(5.84)20 \boldcdot (5.84)

Sample Response

  • 100. Sample reasoning:
    • 2⋅5=102\boldcdot 5 =10, so 20⋅520 \boldcdot 5 is 10 times 10, which is 100.
    • 20⋅10=20020 \boldcdot 10 = 200, so 20⋅520 \boldcdot 5 is half of 200, which is 100.
  • 16. Sample reasoning:
    • 20⋅8=16020\boldcdot 8 =160 and 0.8 is one-tenth of 8, so 20⋅(0.8)20 \boldcdot (0.8) is one-tenth of 160, which is 16.
  • 0.8. Sample reasoning:
    • 20⋅4=8020\boldcdot 4 =80 and 0.04 is one-hundredth of 4, so 20⋅(0.04)20 \boldcdot (0.04) is one-hundredth of 80, which is 0.8.
    • The decimal 0.4 is half of 0.8, so 20⋅(0.4)20 \boldcdot (0.4) is half of 16, which is 8. The decimal 0.04 is a tenth of 0.4, so 20⋅(0.04)20 \boldcdot (0.04) is a tenth of 8, or 0.8.
  • 116.8. Sample reasoning:
    • 5.84 is 5+0.8+0.045 + 0.8 + 0.04, so 20⋅(5.84)20 \boldcdot (5.84) is the sum of, 20⋅520 \boldcdot 5, 20⋅(0.8)20 \boldcdot (0.8), and 20⋅(0.04)20 \boldcdot (0.04), or 100+16+0.8100+16+0.8, which is 116.8.

Synthesis

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} so I . . . .” Some students may benefit from the opportunity to rehearse with a partner what they will say before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 5.NBT.7·Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
  • 5.NBT.B.7·Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Building Toward
  • 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.

25 min

15 min