Visualizing math

If possible, share and compare your description or visualization with a partner. What did you do the same? What did you do differently? Alternatively, record your thinking using a method of your choice.

Visualizing dividing fractions

Let’s visualize an image for the following division of whole numbers $12÷4$

Could we do something similar for fractions?

It turns out this question is more challenging for division, but if we understand what division means, we can still visualize the division of fractions. Consider the following:

$12÷4$   is asking you: how many "4"s can you fit into 12?

Let us take this same logic to a division question with fractions in it:

$2÷\frac{1}{4}$   is asking you: how many " $\frac{1}{4}$"s can you fit into 2?

To do this, use fraction strips. We draw 2 and count how many quarters we can “fit” into it:

There are eight quarters in the picture, therefore: $2÷\frac{1}{4}=8$

Practice

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

Solve the following division problems by drawing or describing a model. When drawing your model:

• Draw the first number as a fraction strip.
• Draw the second number as a fraction strip underneath the first.
• Your fraction strips must be the same length.

What patterns do you notice from the problems you just solved?

• Use the patterns you find to construct a general rule for dividing any fractions.
• Hint: Division is related to multiplication very closely. How is it related here?

Putting it all together

Visuals of math problems help us understand why math operations work. Let’s explore another example to help us:

Practice

Solve the following fraction division problems. Show every step.

Fill in the blanks!

For the following problems, fill in the blank. If there is more than one possible answer, then suggest at least 2 possible solutions. When completing these problems, consider:

• Work through your predictions. Did your answer work?
• Show your steps.
• How can you tell if a question could have more than one possible answer?

Dividing fractions in the real world

If you ever need to find parts of a fraction, you are dividing by fractions:

Example: With only $\frac{1}{2}$ of jug of juice left and 4 people to share it with, what fraction of the juice should go to each person?

Solution:

$\frac{1}{2}÷4$

$=\frac{1}{2}\times \frac{1}{4}$

$=\frac{1}{8}$

So, each person is going to get $\frac{1}{8}$ of the juice.

Would anyone realistically measure out $\frac{1}{8}$ of the juice for each person? Probably not but knowing this would allow them to better estimate how much juice each person should get. So even if you are not doing the exact calculations, you are still using the concept of division with fractions to help you!

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.

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