Integers and problem solving

The following is a table of the minimum and maximum surface temperatures of some planets in our solar system. For each planet, how could you determine the following?

• The difference between the minimum and maximum surface temperatures.
• The average surface temperatures.

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

Multiplying integers

Recall that an “integer” is a positive or negative whole number. We explored examples of integers in our chart of surface temperatures of different planets. We will revisit this later. For now, we are going to talk about multiplying and dividing integers.

Task 1: Multiplying a positive and a negative integer

When we multiply two numbers together, we are just doing repeated addition:

The same is true when one of the integers in negative. Since multiplication works in either order (for example 5 × 4 = 4 × 5), you can change the order so that you are always adding the negative integer. Complete Repeated Addition Using Negative Integers in your notebook or using the following fillable worksheet.

Reflection

• What do you notice about the answers in the table?
• Is it always true that a positive integer times a negative integer is negative? Why or why not?

Task 2: Multiplying a negative integer by a negative integer

Using the idea of repeated addition no longer works when multiplying two negative integers. Consider this problem:

(−2) × (−3) = ?

We cannot add (−3) to itself “−2 times.” So how will this work? Make a prediction for how you think it might work and try to explain why your prediction is the appropriate method.

To answer this, consider this multiplication problem:

18 × 17 = 306

One way we understand how this multiplication works is by using a base-ten rectangle model (recall that the area of a rectangle is length × width):

The rectangle can be split into 4 sections:

10 × 10 = 100 (red)

10 × 8 = 80 (pink)

10 × 7 = 70 (pink)

7 × 8 = 56 (blue)

So 18 × 17 = 100 + 80 + 70 + 56 = 306.

But, what if instead of 18 we wrote 20 − 2 and instead of 17 we wrote 20 − 3:

20 × 20 = 400

20 × (−2) = −40

20 × (−3) = −60

(−3) × (−2) = ???

Add up what we know:

400 + (−40) + (−60) = (Blank)

This is supposed to be 17 × 18 = 306

So, what are we missing?

Therefore, (−3) × (−2) = (Blank)

Ever come across the expression: two wrongs don’t make a right? Well, that is not true when it comes to multiplying negative integers! Two “wrongs” (negatives) do make a right (positive). Two “wrongs” (negatives) do make a right (positive).

Reflection

Consider the following multiplication problems:

(−2) × (−2) × (−2) =

(−2) × (−2) × (−2) × (−2) =

(−2) × (−2) × (−2) × (−2) × (−2) =

What do you notice about the number of (−2)’s in each question and the sign of the answer?

Task 3: Dividing integers

Let's take a look at what happens when we multiply a positive integer by a negative integer in the following chart:

Describe the pattern in the chart

• how come when you multiply two negative numbers the product is a positive number?

Dividing with integers is similar to multiplying with integers. It follows the same type of pattern.

Describe the pattern.

Try making your own chart by changing Positive (+) to an action or feeling you associate to something positive and similar for Negative (−).

Example:

Reflection

Consider the following six questions. What do you notice about the chart? What patterns do you find?

Negative measurements

Practice

Solve the following word problems. Be sure to explain your thinking for each problem and use the strategies you developed to multiply and/or divide integers.

1. A submarine is resting at sea level. It starts to descend at a rate of 26 meters per minute. Where is the submarine after descending for 9.5 minutes?
2. Mr. X is designing a biology test for their students. The test has 30 multiple choice questions. Mr. X has decided that students will be awarded 5 points for a correct answer, will lose 3 points for an incorrect answer, and will lose 1 point for not answering the question. Here are one student’s results:
   • 20 questions correct
   • 3 questions unanswered
   • 7 questions incorrect
   What was the student’s final score?

3. Arnie is an elevator repairperson working on a stuck elevator in a giant apartment complex. The apartment marks its floors as follows:
   … ∣ 3 ∣ 2 ∣ 1 ∣ G ∣ −1 ∣ −2 ∣ −3 ∣ …
   The floors with negative numbers are the parking garage.
   Arnie is currently working on the elevator on the 85th floor. Suddenly, the elevator starts to fall! Arnie watches the floors and quickly notices that the elevator is falling at a rate of 7 floors every 5 seconds. The parking garage goes to the -35th floor. How many seconds does Arnie have to fix the elevator before it crashes?

Build your own

• Construct your own word problem involving the multiplication and/or division of integers.
• Write a complete solution to your problem.
• If possible, share your problem with a partner or with someone at home and try to solve their problem. Alternatively, record the solution to your problem using your method of choice.

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.

Press ‘Discover More’ to extend your skills.

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