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B2.8 · Multiply and divide fractions by fractions with models

Learn to multiply and divide fractions by fractions with models through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Number

Ontario Grade 7 Mathematics — B2.8

A fraction names equal parts of one whole. In

\frac{3}{5}

, the bottom number tells how many equal parts make the whole. The top number tells how many of those parts are counted. A fraction bar is a picture of a whole split into equal parts. In this lesson, you will use models to show what fraction multiplication and division mean before you write a number sentence.

What you will learn

Show fraction multiplication as finding a part of another part.
Use a model to explain fraction division as finding how many groups fit.
Write a multiplication or division statement that matches a fraction model.
Check whether a fraction answer makes sense.

1. Multiplication finds a part of a part

Imagine a strip that stands for one whole. Split it into four equal parts and mark three. The marked amount is

\frac{3}{4}

. Now find one half of that marked amount. Split every fourth into two equal pieces. The strip now has eight equal pieces. Three of them are both in the marked three-fourths part and in one of the two halves.

The shared part, called the overlap, is

\frac{3}{8}

. The model shows that one half of three fourths is three eighths. Multiplication is finding a part of another amount. In this example, you take one half of the amount

\frac{3}{4}

You can also use a rectangle. Divide it into four equal columns and two equal rows. Mark three columns to show

\frac{3}{4}

, and mark one row to show

\frac{1}{2}

. The shared region contains three of the rectangle’s eight equal small regions. All small regions must be equal in size for the count to represent a fraction correctly.

When both fractions are between zero and one, the product is no larger than either starting amount. That is a useful check: taking only a part of an amount should not make it larger.

\frac{1}{2}\times\frac{3}{4}=\frac{3}{8}

For multiplication, find the region shared by the two marked fractions.
Count the shared equal pieces and all equal pieces in the whole.
The overlap explains what the answer represents.

2. Division counts groups that fit

Division can ask how many groups of a certain size fit inside an amount. The amount you start with is the quantity being divided. The group size is the amount you are dividing by.

Suppose you have

\frac{3}{4}

of a whole and want to know how many groups of

\frac{1}{8}

fit. Split each fourth into two equal pieces. Each new piece is one eighth of the whole. Three fourths now covers six eighths, so six groups fit.

The answer counts groups. It is not a shaded part of the original whole. Sometimes more than one group fits, so a division answer can be greater than one. This makes sense when the starting amount is larger than the size of one group.

To make a division model, draw or imagine the starting amount as a bar. Mark equal pieces that match the group size, then count how many groups fit. If part of a group remains, show that part clearly. Keep the whole and the group size labelled so the count has a clear meaning.

\frac{3}{4}\div\frac{1}{8}=6

Name the starting amount and the size of each group.
Count the groups that fit in the starting amount.
A division answer tells how many groups, not how much of the original whole is shaded.

3. Make models that show equal parts

A model must use equal-sized pieces when you count them. In a multiplication area model, one set of lines can show the first fraction and another set can show the second. The lines together divide the whole into smaller regions. Mark the overlap and count the regions.

For example, dividing a rectangle into three equal columns and four equal rows makes twelve equal small regions. Each small region is one of twelve equal parts of the whole. The divisions work together because every small region has the same size.

In a division bar model, the marked length represents the starting amount. The repeated group marks show the size named in the question. Count complete groups, and show any part of a group that remains. A bar that represents more than one whole may need to extend past the first whole.

Before writing a number sentence, say what the model shows. For multiplication, ask, “What part of the first amount is being found?” For division, ask, “How many groups of this size fit?” Then check that the answer matches the question.

Label the whole and each fraction in a model.
Make the pieces you count equal in size.
Explain what the count means before writing the matching number sentence.

4. Practise reading the models

Try a multiplication model: draw one whole, divide it into fifths, and mark two fifths. Divide the whole into two equal halves as well. Find the shared region. The model shows one half of two fifths. Check that the shared regions are equal before you count them.

Now try a division model: take

\frac{4}{3}

and ask how many groups of

\frac{2}{3}

fit. Think of four pieces, each one third of a whole. Pair the pieces to make groups of two thirds. The pairing shows two groups.

In each case, the model gives meaning to the calculation. Do not choose an operation only because of the numbers. Decide whether you are finding a part of an amount or counting groups of a given size.

Multiplication models show a part of a part.
Division models show how many groups fit.
Use the situation in the model to choose the matching operation.

Worked example

Multiply using an overlap model

Find one half of three fourths. Use an area model and state the multiplication sentence.

One half of three fourths

The whole is divided into 8 equal regions. The overlap contains 3 regions, so it is 3 eighths of the whole.

Set up the whole
Draw one rectangle for the whole. Divide it into four equal columns to show fourths, then into two equal rows to show halves.
Find the overlap
Mark three of the four columns and one of the two rows. The shared region has three small pieces. The whole rectangle has eight equal small pieces.
$\frac{3}{8}$
Write the sentence
The shared region is one half of three fourths, so the model and number sentence show the same amount.
$\frac{1}{2}\times\frac{3}{4}=\frac{3}{8}$

Answer: One half of three fourths is three eighths.

Check: Three eighths is less than three fourths. That makes sense because the question asks for only one half of three fourths.

Worked example

Divide a fraction by a unit fraction

How many groups of one eighth fit in three fourths? Show the groups with a fraction bar.

Six eighths in three fourths

The marked length is 6 eighths, equal to 3 fourths. Each eighth is one group.

Name the amount and group
The starting amount is three fourths of a whole. Each group is one eighth of the whole.
Use equal-sized pieces
Split each fourth into two equal eighths. Three fourths then contains six eighths. Each eighth is one group.
$\frac{3}{4}=\frac{6}{8}$
Answer the question
Count the six groups of one eighth in the starting amount.
$\frac{3}{4}\div\frac{1}{8}=6$

Answer: Six groups of one eighth fit in three fourths.

Check: Six eighths equal three fourths, so the group count matches the starting amount.

Worked example

Divide when the group size is not a unit fraction

How many groups of two thirds fit in four thirds? Draw a bar that extends beyond one whole.

Represent the amount
Four thirds means four pieces, each one third of a whole. The bar extends beyond one whole because it contains four thirds.
$\frac{4}{3}$
Mark groups of two thirds
Pair the four one-third pieces. Each pair makes one group of two thirds. There are two pairs.
$\frac{4}{3}=\frac{2}{3}+\frac{2}{3}$
State the division
The starting amount contains two groups of the stated size.
$\frac{4}{3}\div\frac{2}{3}=2$

Answer: Two groups of two thirds fit in four thirds.

Check: Two groups of two thirds use four one-third pieces, which matches the starting amount.

Common mistakes and how to avoid them

Counting the shaded parts but not the total equal parts in a multiplication model.

Correction: Count the overlap for the numerator and all equal small regions in the whole for the denominator.

Treating a division answer as another shaded fraction.

Correction: Division asks how many groups fit. Count the groups of the stated size.

Using pieces of different sizes in one model.

Correction: Redraw or subdivide the parts so every small piece being counted is equal.

Lesson summary

Fraction multiplication can be shown by finding the overlap of two fractions in one whole.
Fraction division can be shown by counting groups of a given size in a starting amount.
Label the whole and make equal parts before counting.
Check that the answer matches the meaning of the question.

Check your understanding

Question 1

A rectangle is split into 3 equal columns and 4 equal rows. What fraction of the whole is one small region?

One seventh
One twelfth
Three fourths
Seven twelfths

Show answer and explanation

One twelfth

There are 12 equal small regions altogether. One region is one twelfth of the whole.

Question 2

How many groups of one sixth fit in two thirds?

Show answer and explanation

Two thirds is four sixths, so it contains four groups of one sixth.

Question 3

A model shows one half of two fifths. Which statement matches the model?

One half times two fifths equals one fifth
One half times two fifths equals two fifths
One half divided by two fifths equals one fifth
One half times two fifths equals two tenths

Show answer and explanation

One half times two fifths equals one fifth

The overlap is two of ten equal regions. Two tenths is the same amount as one fifth.

Key terms

Whole: The complete amount being considered.
Equal parts: Parts that are the same size.
Area model: A rectangle divided into equal regions to show amounts and their overlap.
Group: One set of the size named in a division question.
Overlap: The part shared by two marked regions.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B2.8. It is a study resource, not an official curriculum publication.

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