Finding the Percentage

10 min

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

Narrative

This Math Talk focuses on fraction and decimal equivalence. It encourages students to think about equivalent fractions and to rely on what they know about the relationship between fractions and decimals to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students generalize the process of finding what percentage a number is of another number and express it with one or more expressions.

To determine whether two fractions (or a fraction and a decimal) are equivalent, students need to look for and make use of structure (MP7). In explaining their reasoning, students need to be precise in their word choice and use of language (MP6).

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

Decide mentally if each equation is true or false.

  • 15=210\frac{1}{5} = \frac{2}{10}
  • 35=0.35\frac{3}{5} = 0.35
  • 65= 120100\frac{6}{5} =  \frac{120}{100}
  • 1120=0.55\frac{11}{20} = 0.55

Sample Response

  1. True. Sample reasoning:
    • They are equivalent fractions. Multiplying the numerator and denominator of 15\frac{1}{5} by 2 gives 210\frac{2}{10}.
    • Both are equivalent to 0.2.
    • Both are in the same location on the number line.
  2. False. Sample reasoning:
    • 35\frac{3}{5} is equivalent to 610\frac{6}{10} or 0.6.
    • 35=60100\frac{3}{5} = \frac{60}{100}, which is 60 hundredths and 0.35 is 35 hundredths.
  3. True. Sample reasoning:
    • 65=1210\frac{6}{5} = \frac{12}{10}, which is equivalent to 120100\frac{120}{100}.
    • 6÷5=1.26 \div 5 = 1.2 and 120÷10120 \div 10 is also 1.2.
  4. True. Sample reasoning:
    • 1120\frac{11}{20} is equivalent to 55100\frac{55}{100}, which is 0.55.
    • 110÷20=5.5110 \div 20 = 5.5, so 11÷20=0.5511 \div 20 = 0.55.
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?”

If not brought up in students’ explanations, emphasize the following points:

  • Two fractions are equivalent if multiplying the numerator and denominator of one fraction by the same number gives the numerator and denominator of the other fraction. (For example, 1120=55100\frac{11}{20} = \frac{55}{100} because multiplying 11 and 20 each by 5 gives 55 and 100.)
  • Dividing the numerator of a fraction by the denominator gives an equivalent number in decimal form. (For example, 11÷20= 0.5511 \div 20=  0.55) 
  • Writing an equivalent fraction with a power of 10 in the denominator is another way to express a fraction in decimal form. (For example, 1120=55100\frac{11}{20} = \frac{55}{100}, which is 0.55.)
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
Addressing
  • 6.RP.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
  • 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

20 min