Creating Scale Drawings

5 min

Teacher Prep
Setup
Display one problem at a time. Up to 2 minutes of quiet think time per problem, followed by a whole-class discussion.

Narrative

This Math Talk focuses on multiplying a decimal by a unit fraction. It encourages students to think about the relationship between multiplication and division and to rely on properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students find equivalent scales involving decimals.

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 access to sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each expression mentally.

  • 13 ⋅15\frac 13  \boldcdot 15
  • 15⋅ 1415\boldcdot  \frac 14
  • (2.6)⋅12(2.6) \boldcdot \frac 12
  • 15 ⋅(8.5)\frac 15  \boldcdot (8.5)

Sample Response

  • 5. Sample reasoning: 15÷3=515 \div 3 = 5
  • 3.75. Sample reasoning: 15÷2=7.515 \div 2 = 7.5 and 7.5÷2=3.757.5 \div 2 = 3.75
  • 1.3. Sample reasoning:
    • distributive property: 2÷2=12 \div 2 = 1 and 0.6÷2=0.30.6 \div 2 = 0.3
    • place value: 26÷2=1326 \div 2 = 13 and 13÷10=1.313 \div 10 = 1.3
  • 1.7. Sample reasoning:
    • distributive property: 5÷5=15 \div 5 = 1 and 3.5÷5=0.73.5 \div 5 = 0.7
    • place value: 8.5÷10=0.858.5 \div 10 = 0.85 and 0.85⋅2=1.70.85 \boldcdot 2 = 1.7
Activity Synthesis (Teacher Notes)

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?”

The key takeaways are:

  • Multiplying a number by a unit fraction is the same as dividing that number by the denominator of the fraction.
  • We can use various strategies to reason about division, including place value and the distributive property.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Anticipated Misconceptions

Students may misinterpret the last question as 15⋅ 1315 \boldcdot \frac{1}{3} or 15⋅1415 \boldcdot \frac{1}{4}. Point out that one way to interpret the first expression is “How many one-thirds are there in 15?”

Standards
Building On
  • 3.NF.3·Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
  • 3.NF.A.3·Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
  • 5.NBT.3·Read, write, and compare decimals to thousandths.
  • 5.NBT.A.3·Read, write, and compare decimals to thousandths.

10 min

20 min