Lesson 6

Using Diagrams to Find the Number of Groups

Let’s draw tape diagrams to think about division with fractions.

6.1: How Many of These in That?

  1. We can think of the division expression \(10 \div 2\frac12\) as the question: “How many groups of \(2\frac 12\) 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.”
  2. 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.”

6.2: Representing Groups of Fractions with Tape Diagrams

To make sense of the question “How many \(\frac 23\)s are in 1?,” Andre wrote equations and drew a tape diagram.

\(\displaystyle {?} \boldcdot \frac 23 = 1\)

\(\displaystyle 1 \div \frac 23 = {?}\)

A tape diagram. 

  1. In an earlier task, we used pattern blocks to help us solve the equation \(1 \div \frac 23 = {?}\). Explain how Andre’s tape diagram can also help us solve the equation.

  2. Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.

    1. How many \(\frac 34\)s are in 1?
      A blank grid with a height of 7 units and length of 16 units.
    2. How many \(\frac23\)s are in 3?
      A blank grid with a height of 7 units and length of 16 units.
    3. How many \(\frac32\)s are in 5?
      A blank grid with a height of 7 units and length of 16 units.

6.3: Finding Number of Groups

  1. Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.

    1. How many \(\frac38\)-inch thick books make a stack that is 6 inches tall?

    2. How many groups of \(\frac12\) pound are in \(2\frac 34\) pounds?

  2. Write a question that can be represented by the division equation \(5 \div 1\frac12 = {?}\). Then, find the answer and explain or show your reasoning.

Summary

A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for \(\frac25\) kilogram of flour per batch. How many batches did she make?

We can think of the question as: “How many groups of \(\frac25\) kilogram make 2 kilograms?” and represent that question with the equations:

\(\displaystyle {?} \boldcdot \frac25=2\)

\(\displaystyle 2 \div \frac25 = {?}\)

To help us make sense of the question, we can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.

Fraction bar diagram. 10 equal parts. Each part labeled "the fraction 1 over 5." 

We can see there are 5 groups of \(\frac 25\) in 2. Multiplying 5 and \(\frac25\) allows us to check this answer: \(5 \boldcdot \frac 25 = \frac{10}{5}\) and \(\frac {10}{5} = 2\), so the answer is correct.

Notice the number of groups that result from \(2 \div \frac25\) is a whole number. Sometimes the number of groups we find from dividing may not be a whole number. Here is an example:

Suppose one serving of rice is \(\frac34\) cup. How many servings are there in \(3\frac12\) cups?

 
\(\displaystyle {?}\boldcdot \frac34 = 3\frac12\)

\(\displaystyle 3\frac12 \div \frac34 = {?}\)

Fraction bar diagram. 16 equal parts. 14 parts shaded.  

Looking at the diagram, we can see there are 4 full groups of \(\frac 34\), plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make \(\frac 23\) of a group. So the number of servings (the “?” in each equation) is \(4\frac23\). We can check this by multiplying \(4\frac23\) and \(\frac34\).

\(4\frac23 \boldcdot \frac34 = \frac{14}{3} \boldcdot \frac34\), and \(\frac{14}{3} \boldcdot \frac34 = \frac{14}{4}\), which is indeed equivalent to \(3\frac12\).