Corresponding Parts and Scale Factors

5 min

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

Narrative

This is the first Math Talk activity in the course. See the launch for extended instructions for facilitating this activity successfully.

This Math Talk focuses on multiplying a whole number 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 are identifying scale factors.

Launch

This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.

Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and are to give a signal when they have an answer and a strategy. The teacher then selects students to share different strategies (likely 2–3, given limited time), and might ask questions such as “Who thought about it in a different way?” The teacher then records the responses for all to see, and might ask clarifying questions about the strategies before revealing the next problem.

Consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
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. 
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.

  • 1420\frac14 \boldcdot 20
  • 441444 \boldcdot \frac14
  • 1363\frac13 \boldcdot 63
  •  901690 \boldcdot \frac16

Sample Response

  • 5. Sample reasoning: 20÷4=520\div4=5
  • 11. Sample reasoning: 44÷4=1144 \div 4 = 11
  • 21. Sample reasoning: 60÷3=2060 \div 3 = 20, 3÷3=13 \div 3 = 1, and 20+1=2120 + 1 = 21.
  • 15. Sample reasoning: 90÷3=3090 \div 3 = 30 and 30÷2=1530 \div 2 = 15.
Activity Synthesis (Teacher Notes)

Make sure the connection to division is brought up in the discussion, before moving on to the second expression.

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 takeaway is that these problems all involve multiplying by a unit fraction. One strategy that works in such cases is dividing the other factor by the denominator of the fraction.

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
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.
  • 5.NF.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  • 5.NF.B.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

15 min

15 min