Making leftovers count

You might have learned how to multiply whole numbers before, but how can we use those same strategies to multiply fractions? Think about when you’ve had unfinished parts of something. Imagine your favourite TV series has 12 episodes, and you got through $\frac{1}{2}$ of them in one sitting – how many episodes was that? Or maybe your friends ordered a 24-slice pizza and you ate $\frac{1}{4}$ of it – how many slices were left?

What are some of the other fractions you might encounter in your day-to-day life? Can you think of any ways you could figure out these fractions out?

Multiplication and division strategies

Let’s review some important tools and strategies to support us as we multiply and divide fractions.

Multiplying fractions

When you multiply fractions, you can go right ahead and multiply the numerators and the denominators. You will then need to simplify the fraction by finding the greatest common factor.

The following image shows 3 steps for multiplying two fractions. Step 1 is the question represented by two fractions. Step 2 represents multiplying the numerators together and the denominators together. The third step is the answer in the form of a fraction and, if possible, to simplify it.

Greatest common factor

When a fraction is large, we want to simplify the fraction. Finding the greatest common factor is necessary because we want to find the smallest equivalent fraction. To do this, we have to find numbers that are divisible with both the numerator and denominator. In the example provided, the fraction to simplify is $\frac{16}{24}.$

First, we need to list all the factors (numbers that can be divided into the numerator and denominator). For example, the numerator 16 can be divided into 1, 2, 4, 8, 16. Then, we do the same for the denominator.

Once you have listed all of the factors, we compare the lists to recognize which number is the greatest common number. In this case, both 16 and 24 can be divided by 8.

Now, we need to figure out what the smallest equivalent fraction is. We will do this by dividing both the numerator and denominator by the greatest common factor. 16 divided by 8 is 2, and 24 divided by 8 is 3. Therefore, if we were to simplify$\frac{16}{24},$ the simplest form would be $\frac{2}{3}.$

Multiplying using a number line

You can also multiply a fraction using a number line. In the image above we use $\frac{3}{4}\times \frac{1}{2}$ as an example. Think of the question as asking what is $\frac{3}{4}$ of $\frac{1}{2}$.

Create a number line from 0 to 1 and divide it into halves. This will represent the $\frac{1}{2}$ fraction.

Remember, you want to find $\frac{3}{4}$ of $\frac{1}{2}$. Divide each half of your number line into fourths. You’ve just divided your halves into quarters.

Using your number line, show $\frac{3}{4}$ of the first half.

There are 8 parts on the whole number line. Three parts were used, or $\frac{3}{8}$ of the number line.

Therefore, $\frac{3}{4}$ $\times$$\frac{1}{2}$ = $\frac{3}{8}$.

Dividing fractions

The first thing you need to do is record your division question. Then, you need to flip the second fraction you are dividing by; this is called the reciprocal fraction. Now, instead of dividing, you will multiply as you normally would when you multiply fractions. The resulting answer is often an improper fraction. This means that the numerator is larger than the denominator. If so, you will have to convert your fraction into a mixed number. Examine the computation in the following image:

Dividing using a number line

You can also divide fractions using a number line. Let’s use$\frac{3}{4}$$÷$$\frac{1}{2}$ as our example. The following images represent this example.

Create a number line showing $\frac{3}{4}$

Take the denominator of the second fraction $\frac{1}{2}$, which is 2. Divide each section of your number line into two parts.

Find the common denominator of the two fractions by multiplying the two denominators. In this case, 2 × 4 = 8. Eight (8) is the common denominator. Express both fractions $\frac{3}{4}$ and $\frac{1}{2}$ as equivalent fractions using the common denominator 8.

$\frac{3}{4}$ = $\frac{6}{8}$     $\frac{1}{2}$ = $\frac{4}{8}$

Using the numerator of the second fraction, which is 4, show the groups of 4 that exist on the number line.

There is only 1 complete group of four parts, and a group of 2 parts. Expressed as a fraction, it is 1$\frac{2}{4}$. Reduced to lowest terms, it is 1$\frac{1}{2}$.

Therefore, $\frac{3}{4}$$÷$$\frac{1}{2}$ = 1$\frac{1}{2}$.

Converting improper fractions to mixed numbers

As we solve division problems with fractions our answers may result in an improper fraction, which means that the numerator is larger than the denominator. This means that we have created a whole or multiple wholes when we divided. In our example of $\frac{7}{3}$, 7 is larger than 3 so we need to convert the fraction. The first thing we need to do is 7 divided by 3. When we do that, our answer is 2 with a remainder of 1. The 2 represents the number of whole groups so we record it to the left of the fraction. Then, we keep the denominator the same and the remainder becomes our numerator. The following image represents this computation.

Task: Real life application

If you would like, you can complete the next series of word problems using TVO Mathify. You can also use your notebook or the following fillable and printable document.

This hearty soup recipe was shared from a family cookbook and serves 2 people. How would you figure out how much would be needed if 10 people needed to be fed? What if the number of people who needed to be fed doubled each month, what would the measurement of the ingredients be in a month? Show your thinking using two different strategies.

Examine the following chart for information about the list of ingredients and the measurements needed for this recipe:

Part 2: Individual serving

If this recipe was used to feed 4 people, could you figure out the measurement of the ingredients per serving? Show your thinking using two different strategies.

Task: Fraction challenge

Think about the ways that you encounter multiplying and/or dividing fractions in your life. Create two situations based on day-to-day life and create a word problem for each of them. If possible, have a partner or a family member solve and compare the solutions to the problems and justify their thinking. Here are some conditions for creating your problems:

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

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