How Many Groups? (Part 1)

5 min

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

Narrative

This Warm-up prompts students to interpret depictions of equal-size groups and write multiplication and division equations that represent them. In two cases, the size of one group is a fraction. Prior to this point, students have written multiplication equations with one or more fractional factors. Here, students see that division equations can have a fraction as a divisor or a quotient.

The reasoning here further builds students’ understanding, from earlier lessons, of the relationship between the number of groups and the size of each group. It also activates what students know, from earlier grades, about multiplying whole numbers and unit fractions.

Launch

Give students 2 minutes of quiet think time, followed by a whole-class discussion.

Student Task

Write a multiplication equation and a division equation for each sentence or diagram.

  1. Eight $5 bills are worth $40.
  2.  

    A tape diagram of 5 equal parts. Each part is labeled one fifth. Above the bar is a bracket, labeled 1, that spans the entire length of the bar.

  3. There are 9 thirds in 3 ones.

Sample Response

  1. 8⋅5=408 \boldcdot 5 = 40 (or 5 ⋅8 =405 \boldcdot 8 = 40) and 40÷5=840 \div 5 = 8 (or 40÷8=540 \div 8 = 5)
  2. 5⋅15=15 \boldcdot \frac15 = 1 (or 15⋅5=1\frac15 \boldcdot 5 = 1) and 1÷5=151 \div 5 = \frac15 (or 1÷15=51 \div \frac15 = 5)
  3. 9⋅13=39 \boldcdot \frac 13 = 3 (or 13 ⋅9=3\frac 13 \boldcdot 9 = 3) and 3÷9=133 \div 9 = \frac13 (or 3÷13=93 \div \frac13 =9)
Activity Synthesis (Teacher Notes)

Select 1–2 students to share their responses. Record the responses for all to see.

As students present the equations for each problem, connect the pieces in each equation to the idea of equal-size groups. Ask questions such as:

  • “Which number in the multiplication equation refers to the number of groups?”
  • “Which number in the multiplication equation refers to how much is in each group?”
  • “In this case, what does the division equation 1÷15=51 \div \frac{1}{5} = 5 tell us?”
  • “In this case, what does the division equation 3÷9=133 \div 9 = \frac{1}{3} tell us?”
Anticipated Misconceptions

If students recognize the equal-size groups but represent them with repeated addition (such as 15+15+15+15+15=1\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = 1) instead of multiplication, remind them about the connections between the two. Refer to one of their addition statements and ask questions such as:

  • “How many same-size groups are being added? What is in each group?”
  • “How can you express the same information using multiplication?”
Standards
Building On
  • 5.NF.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  • 5.NF.7·Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
  • 5.NF.B.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
  • 5.NF.B.7·Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.<span>Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.</span>

25 min