Introducing Double Number Line Diagrams

10 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on multiplication of a whole number and a decimal. It encourages students to observe the impact of adjusting a factor and to rely on structure, patterns, and the properties of operations to find products (MP7).

Each expression is designed to elicit slightly different reasoning. In explaining their strategies, students need to be precise in their word choice and use of language (MP6). Although many ways of reasoning may emerge, it may not be feasible to discuss every strategy. Consider gathering only 2–3 different strategies per expression. As students explain their strategies, ask them how the factors affected their approach.

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 product mentally.

  • 2⋅452 \boldcdot 45
  • 2⋅(4.5)2 \boldcdot (4.5)
  • 6⋅(4.5)6 \boldcdot (4.5)
  • (0.2)⋅45(0.2) \boldcdot 45

Sample Response

  • 90. Sample reasoning: 
    • (2⋅40)+(2⋅5)=80+10=90(2 \boldcdot 40) + (2 \boldcdot 5) = 80 + 10 = 90
    • Two times 50 is 100, so 2 times 45 is 2 times 5, or 10, less than 100, which is 90.
  • 9. Sample reasoning:
    • The first factor, 4.5, is one-tenth of 45, so 2⋅(4.5)2 \boldcdot (4.5) is one-tenth of 2⋅452 \boldcdot 45 or one-tenth of 90, which is 9.
    • (2⋅4)+(2⋅(0.5))=8+1=9(2 \boldcdot 4) + (2 \boldcdot (0.5)) = 8 + 1 = 9
  • 27. Sample reasoning: 
    • Six is 3 times 2, so 6⋅(4.5)6 \boldcdot (4.5) is 3 times the result of 2⋅(4.5)2 \boldcdot (4.5), or 3 times 9, which is 27.
    • 6⋅4=246 \boldcdot 4 = 24 and 6⋅(0.5)=36 \boldcdot (0.5) = 3, so 6⋅(4.5)6 \boldcdot (4.5) is the sum of 24 and 3, which is 27.
  • 9. Sample reasoning: 
    • The first factor, 0.2, is one-tenth of 2, so (0.2)⋅45(0.2) \boldcdot 45 is one-tenth of 2⋅452 \boldcdot 45, or one-tenth of 90, which is 9.
    • Two-tenths is one-fifth, and one-fifth of 45 is 9.
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 have 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 _____ 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

15 min

15 min