Part of a Percent

5 min

Teacher Prep
Setup
Display each problem one at a time. 30 seconds of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on dividing by 100. It encourages students to think about place value and to rely on properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students use decimals to represent percentages that are not whole numbers.

As students use results from the previous expression to help evaluate the next expression, they are making use of repeated reasoning (MP8).

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.

  • 14÷10014 \div 100
  • 7÷1007 \div 100
  • 3.5÷1003.5 \div 100
  • 103.5÷100103.5 \div 100

Sample Response

  • 0.14. Sample reasoning: 0.14 is 1100\frac{1}{100} of 14 because the decimal point is two places to the left.
  • 0.07. Sample reasoning: 7 is 12\frac12 of 14, and 0.07 is 12\frac12 of 0.14.
  • 0.035. Sample reasoning: 3.5 is 12\frac12 of 7, and 0.035 is 12\frac12 of 0.07.
  • 1.035. Sample reasoning: 1.035 is 1100\frac{1}{100} of 103.5 because the decimal point is two places to the left.
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 takeaway is to remind students that numbers can have more than two decimal places. Up to this point, students have mostly worked mostly with percentages that are whole numbers, such as 14% and 7%. Representing these percentages as decimals always gives values with two decimals places, like 0.14 and 0.07. This could lead students to overgeneralize and think that 3.5% would be represented as 0.35. Focusing on the relative size of 3.5% compared to 7% can help students see why the decimal representation is actually 0.035.

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 what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  • 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Building Toward
  • 7.RP.3·Use proportional relationships to solve multistep ratio and percent problems.
  • 7.RP.A.3·Use proportional relationships to solve multistep ratio and percent problems. <span>Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.</span>

15 min

15 min