8.2: Multiplication

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You will need: Positive and Negative Counters (Material Cards 18A and 18B)

Now, we'll explore how to multiply positive and negative numbers using the counters. Let's look again at a definition of multiplication for whole numbers.

Definition: Multiplication of Whole Numbers using the Repeated-Addition Approach

If \(m\) and \(n\) are whole numbers, then \[m \times n = n + n + n + n + ... + n, \nonumber \] where there are \(m\) add ends of \(n\) in this sum.

This definition makes sense as long as \(m\) and \(n\) are positive numbers. In fact, we can even make sense of and modify this definition using counters as long as m is positive by rewording the definition slightly, as stated below:

Definition: Multiplication of a Whole Number times an Integer using the Repeated-Addition Approach, using positive and negative counters

If \(m\) is a whole number and \(n\) is any integer, \( m \times n\) is obtained by combining \(m\) subsets of a collection of counters representing \(n\). The number that the resulting collection represents is the answer to the problem \( m \times n\).

Exercise 1

This exercise shows you one way to use the above definition to multiply \(3 \times -4\). In this problem, the definition can be used because 3 is a whole number and -4 is an integer. We need to combine 3 subsets of a collection of counters that represent -4.

Use your positive and negative counters to represent a collection of counters that represent -4. For this exercise, choose a collection of 5 negatives and 1 positive. Show what your collection looks like below:
Now, form two more collections (for a total of 3 subsets) of counters that you had for part a. Combine the counters together and show what the large collection looks like below:
After removing any red-green pairs (zero) from your collection in part b, show the collection that remains below. What number does this represent? _____

Exercise 2

Do \(3 \times -4\) again, using a different collection of counters to represent -4.

Use your positive and negative counters to represent a collection of counters that represent -4. This time, choose a collection of 6 negatives and 2 positives. Show what your collection looks like below:
Now, form two more collections (for a total of 3 subsets) of counters that you had for part a. Combine the counters together and show what the large collection looks like below:
After removing any red-green pairs (zero) from your collection in part b, show the collection that remains below. What number does this represent? _____

Exercise 3

Okay, let's do \(3 \times -4\) one more time, choosing the easiest way to represent -4.

Form a collection of counters to represent -4. Do it the easy, natural way, using the least number of counters possible. Show what your collection looks like below:
Now, form two more collections (for a total of 3 subsets) of counters that you had for part a. Combine the counters together and show what the large collection looks like below:
What number does the collection in part b represent? _____

Well, I hope you got the answer of -12 for exercises 1, 2 and 3, since \(3 \times -4 = -12\)! This illustrates that it doesn't matter exactly which collection of counters you use to represent -4, as long as the collection really is -4. To compute \(3 \times -4\), you could combine 3 subsets of a collection of 8 reds and 4 greens, or you could combine 3 subsets of a collection of 7 reds and 3 greens, etc. You'll always end up with a collection that represents -12. For exercise 3, did you choose 4 negatives as your representation? If so, did you notice you didn't have to remove any red-green pairs to answer part c? From now on, let's do it the easy way, using the simplest collection possible.

Exercise 4

Use your counters to do each of the following multiplication problems using the definition of multiplying a whole number by an integer. Then, explain what the multiplication problem given means in terms of the counters, and explain and show the individual steps. Use the following example as a model.

Example

Multiply \(\bf 2 \times -6\).

Solution

Multiplying \(2 \times -6\) means to combine 2 subsets of 6 red counters. The number that the resulting collection represents is the product (answer). \(2 \times -6\) = RRRRRR + RRRRRR = RRRRRRRRRRRR = -12

a. Multiply \(4 \times -2\)

This means to

b. Multiply \(3 \times 5\)

This means to

c. Multiply \(5 \times -3\)

This means to

d. Multiply \(7 \times 2\)

This means to

e. Multiply \(0 \times -3\)

This means to

Okay, now that you've mastered how to multiply a whole number by an integer, let's work on how we can use the counters to multiply a negative integer by an integer. Let's look once more at the definition for multiplying \(m \times n\), when m is a whole number.

Definition: Multiplication of a Whole Number times an Integer using the Repeated-Addition Approach, using positive and negative counters

If m is a whole number and n is any integer, \(m \times n\) is obtained by combining m subsets of a collection of counters representing n. The number that the resulting collection represents is the answer to the problem \(m \times n\).

If m is negative, this definition doesn't make sense since you certainly can't combine a negative number of subsets! The way we'll revise this definition to include the possibility that m may be negative is to agree that if m is negative, we REMOVE \(m\) subsets of a collection of counters representing \(n\). The trick to doing this is to remove the subsets from a collection of counters representing zero. So, here is the comprehensive definition for multiplying any two integers, using positive and negative counters.

Multiplication of two Integers, using positive and negative counters

Case 1: If \(m\) is a whole number and \(n\) is any integer, \(m \times n\) is obtained by combining \(m\) subsets of a collection of counters representing \(n\). The product of m and \(n\), \(m \times n\), is the number that the resulting collection represents.
Case 2: If \(m\) is negative and \(n\) is any integer, \(m \times n\) is obtained by removing |m| subsets of a collection of counters representing \(n\) from a collection of counters representing zero. The product of \(m\) and \(n\), \(m \times n\), is the number that the resulting collection represents.

Exercise 5

This exercise shows you how to use the above definition to multiply \(-4 \times 3\). For this problem, we need to remove 4 subsets of a collection of counters that represents 3 from a collection of counters representing zero. The simplest collection to represent 3 is 3 positives, or 3 green counters. (We could use a more complicated collection and still arrive at the same answer. We did this kind of an exercise in exercises 1 - 3. Convince yourself it wouldn't matter here, either.)

We first need to form a collection of counters that represents zero so that it will be possible to remove 4 subsets of 3 green counters. For this example, make a collection of 14 red and 14 green counters. Write down what your collection looks like here:
From your collection, remove a subset of 3 green counters. Then, remove 3 more subsets of 3 green counters. You have just removed 4 subsets of 3 green counters from zero. To show it on paper, circle a subset of 3 green counters in the collection above in part a. Then, circle 3 more subsets of 3 green counters so that four separate subsets are circled. After you remove the counters (which are shown by what you circled in part a), show what is left in your collection below.
Remove any red-green pairs (zero) from your remaining collection. Show this on paper by crossing off or circling any red-green pairs (zero) from your collection shown in part b. Show the collection that remains below. What number does this collection of counters represent?

Exercise 6

Use the definition again to multiply \(-4 \times 3\). Remember, you need to remove 4 subsets of a collection of counters that represents 3.

We first need to form a collection of counters that represents zero so that it will be possible to remove 4 subsets of 3 green counters. This time, put in the smallest collection of counters possible so that you'll be able to remove 4 subsets of 3 green counters. Write down what your collection looks like here:
From your collection, remove four subsets of 3 green counters (take out one subset of 3 green counters at a time). You have just removed 4 subsets of 3 green counters from zero. To show it on paper, circle 4 different subsets of 3 green counters in the collection shown in part a. After you remove the counters (which are shown by what you circled in part a), show what is left in your collection below.
What number does the above collection represent?

Did you notice that if you start out with a minimal collection to represent zero, you don't have to remove any red-green pairs when you get to part c? That is the easiest way to do it because you know how to calculate what to take out before you start the problem. However, if you were teaching this to someone who couldn't figure that out ahead of time, you might always start out with zero being 20 (or some other agreed upon number) of each counter.

On the next page, the steps are shown for the example below– how to use your counters to do a multiplication problem when the first number is a negative integer. It's necessary to explain what the multiplication problem given means in terms of the counters, and then explain and show the individual steps. Use the example as a model for the exercises that follow.

Example

Multiply \(-4 \times -3\) by using the definition of multiplying integers.

Solution

Multiplying \(-4 \times -3\) means to remove 4 subsets of 3 red counters from a representation of zero. The number that the resulting collection represents is the answer.

Since there are 12 greens remaining, the answer is +12. Didn't you always wonder why a negative number times a negative number equaled a positive number? Using the definition of multiplying integers with counters, you can really see why it's true. After you do a few more problems, feel free to go out and show all of your friends who still don't understand it or just memorized the rules because someone told them that's the way it is and to just accept it. Okay, enough of that unbridled enthusiasm from me for now.

It's time for you to work a few problems. Some have negative numbers before the multiplication sign (start with a representation of zero and remove subsets as in the last example) and some have whole numbers before the multiplication sign (so just combine subsets together like you did in the earlier exercises of this exercise set). It's the number before the multiplication sign that will indicate which case of the definition you will use.

Exercise 7

Use your counters to do each of the following multiplication problems using the definition of multiplying two integers with positive and negative counters. Then, explain what the multiplication problem given means in terms of the counters, and explain and show each of the individual steps. Use the example above as a model when the first number is negative.

a. \(-5 \times 3\) ____ This means to ____________________________________________________