Fractional Lengths

5 min

Teacher Prep
Setup
2 minutes of quiet think time, followed by a whole-class discussion.

Narrative

This Warm-up prompts students to carefully analyze and compare four situations that involve multiplication or division and a fraction. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology related to multiplication and division in the context of length and talk about characteristics of the items in comparison to one another.

Launch

Arrange students in groups of 2–4. Display the four items for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three items that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.

Student Task

Which three go together? Why do they go together?

  1. A string that is 34\frac{3}{4} meter long is cut into 15 equal pieces. How long is each piece?

  2. ?⋅34=15? \boldcdot \frac{3}{4}= 15

  3. A driver drove 34\frac{3}{4} km from home to a gas station and then drove 15 times as far to go to work. What is the distance between the gas station and his work?
  4. Mai built a tower that is 21 inches tall by stacking 34\frac{3}{4}-inch tall cubes. How many cubes did she use?

Sample Response

Sample responses:

  • A, B, and C go together because they all involve 34\frac{3}{4}, 15, and an unknown number.
  • A, B, and D go together because the answer can be found by dividing the given numbers.
  • A, C, and D go together because they are all word problems.
  • B, C, and D go together because to find the answer, the 34\frac{3}{4} is being multiplied by another number and not being divided.
Activity Synthesis (Teacher Notes)

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology that they use to describe the relationship between the known and unknown quantities, such as “34\frac{3}{4} split into 15 parts,“ “15 times as long as 34\frac{3}{4},“ and “21 is some number times 34\frac{3}{4}.” Ask students to clarify their reasoning as needed. Consider asking:

  • “How do you know . . . ?”
  • “What do you mean by . . . ?”
  • “Can you say that in another way?”
Speaking: 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.
Design Principle(s): Optimize output (for explanation)
Standards
Addressing
  • 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>

20 min