Using Diagrams to Find the Number of Groups

5 min

Teacher Prep
Setup
Students in groups of 2. 2 minutes of quiet work time, a minute of partner discussion.

Narrative

This Warm-up gives students a chance to create tape diagrams to represent equal-size groups and division expressions in a scaffolded way. Each tape is started on a grid and pre-labeled with the known quantity. Each grid square represents 1.

Launch

Arrange students in groups of 2. Give students 2 minutes of quiet work time and another minute to share their diagrams with their partner.

Student Task

We can think of the division expression 10÷21210 \div 2\frac12 as the question: “How many groups of 2122\frac 12 are
in 10?” Complete the tape diagram to represent this question. Then find the answer.

Tape diagram on a grid. 10 equal parts. Each part is 1 unit. Total labeled “10.”

Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.

Tape diagram on a grid. 7 equal parts. Each part is 1 unit. Total labeled “7.”

Sample Response

  1. There are 4 groups of 2122\frac12 in 10. 

    <p>Fraction bar diagram on a grid. </p>

  2. There are 3123\frac12 groups of 2 in 7.

    <p>Fraction bar diagram on a grid. </p>

Activity Synthesis (Teacher Notes)

The goal of this discussion is for students to articulate how they represent the quantities in division questions on the blank tape diagrams. Select a few students to share their diagrams and answers. Discuss questions such as:

  • “In the first question, how did you know how large each part of the diagram should be?” (The length of the tape represents 10, and there are 10 grid squares, so each grid represents 1. Because the size of each group is 2122\frac{1}{2}, each part needs to have 2122\frac{1}{2} squares.)
  • “In the second question, we see 3 groups of 2 and an extra square of 1. How did you know that the 1 is 12\frac{1}{2} of a group and not 17\frac{1}{7} of a group?” (The question asks “How many groups of 2 . . . ,” so the size of each group is 2, not 7.)
  • “How can you tell the answers to the questions from the completed diagrams?” (We can count the number of full groups and partial groups.)
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

10 min