Integers

Integers represent the collection of positive (+) and negative (−) whole numbers. Here is a number line with integers:

How would you get to −4 from 2 on this number line? Add a coloured dot on the number line at −4.

If you would like, you can complete the following activity using TVO Mathify. You can also use your notebook or the following fillable worksheet.

Brainstorm

Brainstorm examples of integers in real life. Try to come up with as many examples as possible. Record your ideas.

Integers and investments

Here is an example of an investment made in the stock market at a moment in time:

Adding and subtracting integers using tiles

Let's learn about integers by completing an activity. All you need is two sets of identical items. One set will represent positive numbers and one set will represent negative numbers. We are going to use red circles for positive numbers and blue circles for negative numbers, but feel free to use whatever you prefer (for example: paper clips for +1 and staples for −1, black socks for +1 and white socks for −1, etc.). We will always call them “tiles.”

You will be using zero models to help visualize the addition and subtraction of integers. A zero model is a pair of numbers whose sum is zero, e.g. +1, −1, with positive and negative integers.

Question: Using your tiles, make a total of zero in at least three different ways. What do you notice about the number of tiles?

To make a zero pair, (+1) + (−1) = 0.

Making zero by combining one of each tile is essential to understand how operations on integers work. Keep this in mind when solving integer problems.

Adding integers

Add a positive integer:

Here is an example of using the tiles to solve: −5 + 3 = ?

Question: Try it for yourself and solve: −3 + 7 = ?

Add a negative integer:

Question: What happens when we add a negative number? Use your tiles to consider the problem:

4 + (−5) = ?

Hint: Make zero!

Subtracting integers

Subtract a positive integer:

Let’s explore the problem −4 − 3 = ?

In this case, we only need to count the number of tiles present. There are 7 blue tiles so the final answer is −7.

In other words, −4 − 3 = −4 + (−3)

Question: Use your tiles to solve: −2 − 5 = ?

Subtract a negative integer:

In this question you have need to add 3 zero pairs so that you have three negative blue tiles to take away. This will result in -7. Let’s explore the problem: 4 − (−3) = ?

The issue with our model in the picture is that there are no negative (blue) tiles to subtract or “take away” from the positive (red) tiles.

When we add the zero pairs do we change the question? Why or why not? Explain your thinking.

Now there are three blue tiles on the right.

• What do you notice about the left-hand side? Even though we added tiles, did we change the question? Explain.

We can now subtract the three blue tiles and here is what we have left:

Question: Use your tiles to solve: 7 − (−4) = ?

Experimenting with integer tiles

You are going to use your tiles to show the following list of outcomes. But there is a twist! Some of these items might be impossible to create. If it is impossible, explain why.

Practicing with integers and negative numbers

Using any strategy, solve the following integer problems. Be sure to explain any steps or strategies you used.

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the correlating 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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