Lesson 10
Dividing by Unit and NonUnit Fractions
Let’s look for patterns when we divide by a fraction.
10.1: Dividing by a Whole Number
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.

Partner A:
How many 3s are in 12?
Division equation:
How many 4s are in 12?
Division equation:
How many 6s are in 12?
Division equation:
Partner B:
What is 12 groups of \(\frac 13\)?
Multiplication equation:
What is 12 groups of \(\frac 14\)?
Multiplication equation:
What is 12 groups of \(\frac 16\)?
Multiplication equation:

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

Complete this sentence based on what you noticed:
Dividing by a whole number \(a\) produces the same result as multiplying by ________.
10.2: Dividing by Unit Fractions
To find the value of \(6 \div \frac 12\), Elena thought, “How many \(\frac 12\)s are in 6?” and then she drew this tape diagram. It shows 6 ones, with each one partitioned into 2 equal pieces.
\(6 \div \frac 12\)
 For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression.

\(6 \div \frac 13\)Value of the expression: ____________

\(6 \div \frac 14\)Value of the expression: ____________

\(6 \div \frac 16\)Value of the expression: ____________


Examine the expressions and answers more closely. Look for a pattern. How could you find how many halves, thirds, fourths, or sixths were in 6 without counting all of them? Explain your reasoning.

Use the pattern you noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.
 \(6 \div \frac 18\)
 \(6 \div \frac {1}{10}\)
 \(6 \div \frac {1}{25}\)
 \(6 \div \frac {1}{b}\)

Find the value of each expression.
 \(8 \div \frac 14\)
 \(12 \div \frac 15\)
 \(a \div \frac 12\)
 \(a \div \frac {1}{b}\)
10.3: Dividing by Nonunit Fractions
 To find the value of \(6 \div \frac 23\), Elena started by drawing a diagram the same way she did for \(6 \div \frac 13\).
 Complete the diagram to show how many \(\frac 23\)s are in 6.
 Elena says, “To find \(6 \div \frac23\), I can just take the value of \(6 \div \frac13\) and then either multiply it by \(\frac 12\) or divide it by 2.” Do you agree with her? Explain your reasoning.

For each division expression, complete the diagram using the same method as Elena. Then, find the value of the expression. Think about how you could find that value without counting all the pieces in your diagram.

\(6 \div \frac 34\)Value of the expression:___________

\(6 \div \frac 43\)Value of the expression:___________

\(6 \div \frac 46\)Value of the expression:___________


Elena examined her diagrams and noticed that she always took the same two steps to show division by a fraction on a tape diagram. She said:
“My first step was to divide each 1 whole into as many parts as the number in the denominator. So if the expression is \(6 \div \frac 34\), I would break each 1 whole into 4 parts. Now I have 4 times as many parts.
My second step was to put a certain number of those parts into one group, and that number is the numerator of the divisor. So if the fraction is \(\frac34\), I would put 3 of the \(\frac 14\)s into one group. Then I could tell how many \(\frac 34\)s are in 6.”
Which expression represents how many \(\frac 34\)s Elena would have after these two steps? Be prepared to explain your reasoning.
 \(6 \div 4 \boldcdot 3\)
 \(6 \div 4 \div 3\)
 \(6 \boldcdot 4 \div 3\)
 \(6 \boldcdot 4 \boldcdot 3\)

Use the pattern Elena noticed to find the values of these expressions. If you get stuck, consider drawing a diagram.
 \(6 \div \frac27\)
 \(6\div\frac{3}{10}\)
 \(6 \div \frac {6}{25}\)
Find the missing value.
Summary
To answer the question “How many \(\frac 13\)s are in 4?” or “What is \(4 \div \frac 13\)?”, we can reason that there are 3 thirds in 1, so there are \((4\boldcdot 3)\) thirds in 4.
In other words, dividing 4 by \(\frac13\) has the same result as multiplying 4 by 3.
\(\displaystyle 4\div \frac13 = 4 \boldcdot 3\)
In general, dividing a number by a unit fraction \(\frac{1}{b}\) is the same as multiplying the number by \(b\), which is the reciprocal of \(\frac{1}{b}\).
How can we reason about \(4 \div \frac23\)?
We already know that there are \((4\boldcdot 3)\) or 12 groups of \(\frac 13\)s in 4. To find how many \(\frac23\)s are in 4, we need to put together every 2 of the \(\frac13\)s into a group. Doing this results in half as many groups, which is 6 groups. In other words:
\(\displaystyle 4 \div \frac23 = (4 \boldcdot 3) \div 2\)
or
\(\displaystyle 4 \div \frac23 = (4 \boldcdot 3) \boldcdot \frac 12\)
In general, dividing a number by \(\frac{a}{b}\), is the same as multiplying the number by \(b\) and then dividing by \(a\), or multiplying the number by \(b\) and then by \(\frac{1}{a}\).
Glossary Entries

reciprocal
Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac25\) is \(\frac52\).