Ratios and Rates with Fractions

5 min

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

Narrative

This Math Talk focuses on dividing by a fraction. It encourages students to think about the meaning of 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 calculate unit rates for quantities with fractional values.

To apply reasoning from previous expressions to help evaluate the next expression, students need to look for and make use of structure (MP7).

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 each answer mentally.

  • How many 13\frac13s are there in 5?
  • What is 2÷132\div \frac13?
  • What is 12÷13\frac12 \div \frac13?
  • What is 212÷132\frac12 \div \frac13?

Sample Response

  • 15. Sample reasoning:
    • There are three 13\frac13s in 1. There are 5 times as many 13\frac13s in 5, and 5⋅3=155 \boldcdot 3 = 15,

  • 6. Sample reasoning: There are three 13\frac13s in 1. There are twice as many 13\frac13s in 2, and 2⋅3=62 \boldcdot 3 = 6.
  • 1121\frac12. Sample reasoning: Multiplying a number by 3 gives the same answer as dividing it by 13\frac13, so 12⋅3=112\frac12 \boldcdot 3 = 1\frac12.
  • 7127\frac12. Sample reasoning:
    • 212⋅3=7122\frac12 \boldcdot 3 = 7\frac12

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 for students to have several methods for explaining why it makes sense that dividing by a fraction gives the same answer as multiplying by its reciprocal.

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
Anticipated Misconceptions

Students may get stuck trying to remember a procedure to divide fractions. Help students reason about the meaning of division by asking “How many 13\frac13s are there in ___?”

Standards
Building On
  • 6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <em>For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?</em>
  • 6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. <span>For example, create a story context for <span class="math">\((2/3) \div (3/4)\)</span> and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that <span class="math">\((2/3) \div (3/4) = 8/9\)</span> because <span class="math">\(3/4\)</span> of <span class="math">\(8/9\)</span> is <span class="math">\(2/3\)</span>. (In general, <span class="math">\((a/b) \div (c/d) = ad/bc\)</span>.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? </span>
Building Toward
  • 7.RP.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. <em>For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction <sup>1/2</sup>/<sub>1/4</sub> miles per hour, equivalently 2 miles per hour.</em>
  • 7.RP.A.1·Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. <span>For example, if a person walks <span class="math">\(1/2\)</span> mile in each <span class="math">\(1/4\)</span> hour, compute the unit rate as the complex fraction <span class="math">\(\frac{1/2}{1/4}\)</span> miles per hour, equivalently <span class="math">\(2\)</span> miles per hour.</span>

10 min

10 min