Dividing by Unit and Non-Unit Fractions

10 min

Teacher Prep
Setup
Students in groups of 2. Each partner answers one set of problems (A or B). A few minutes of quiet work time, followed by a partner discussion on the last two questions.

Narrative

In this warm-up, students use tape diagrams to revisit the idea that dividing by a whole number is equivalent to multiplying by a unit fraction. Though this is a review of a grade 5 expectation, connecting the division problems to diagrams allows students to see the equivalence in the related division and multiplication problems. It also prepares students to apply the same reasoning and representations to division by non-unit fractions later.

Launch

Arrange students in groups of 2. Ask one person in each group to draw diagrams and answer the questions for Partner A, and the other to take on the questions for Partner B. Give students a few minutes of quiet time to complete the first two questions, and then ask them to compare their responses and collaborate on the last two questions.

Student Task

Work with a partner. One person solves the problems labeled “Partner A” and the other person solves those labeled “Partner B.”

Write an equation for each question. If you get stuck, consider drawing a diagram.

  1. Partner A:

    How many 3s are in 12?

    Division equation:

    A blank grid with height 5 units and length 16 units.

    How many 4s are in 12?

    Division equation:

    A blank grid with height 5 units and length 16 units.

    How many 6s are in 12?

    Division equation:

    A blank grid with height 5 units and length 16 units.

    Partner B:

    What is 12 groups of 13\frac 13?

    Multiplication equation:

    A blank grid with height 5 units and length 16 units.

    What is 12 groups of 14\frac 14?

    Multiplication equation:

    A blank grid with height 5 units and length 16 units.

    What is 12 groups of 16\frac 16?

    Multiplication equation:

    A blank grid with height 5 units and length 16 units.

  2. What do you notice about the diagrams and equations? Discuss with your partner.

  3. Complete this sentence based on what you noticed:

    Dividing by a whole number aa produces the same result as multiplying by ‾\underline{\hspace{.5in}} .

Sample Response

  1.  

    <p>6 fraction bar diagrams.</p>

  2. Sample responses:
    • The quotients in the division equations have the same value as the products in the multiplication equations.
    • Both sets of problems use the number 12. The division problems have 3, 4, and 6 as divisors, and the multiplication problems have a factor that is a fraction with those numbers in the denominator (13, 14, 16\frac13,\,\frac14,\,\frac16). 
    • For each pair, the diagrams are divided into the same number of major parts but they show different information.
    • Dividing by a whole number gives the same result as multiplying by a fraction with that number as the denominator.
  3. Dividing by a whole number aa produces the same result as multiplying by 1a\frac {1}{a}.
Activity Synthesis (Teacher Notes)

Invite a couple of students to share their observations about their group's diagrams and answers. Students should notice that the answers for the three division problems match those for the multiplication ones, even though the questions were not the same, and their diagrams show groups of different sizes. Ask a few students to share their response to the last question.

Consider displaying the following image to reinforce the idea that dividing by a whole number aa has the same effect as multiplying by 1a\frac{1}{a}.

<p>6 fraction bar diagrams.</p>

Standards
Building On
  • 5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
  • 5.NF.B·Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

15 min

10 min