Modeling multiplication

Modeling multiplication of integers with number counters is the goal of this lesson.

The number counters are shown below.

A few examples showing how modeling multiplication with counters work

A single red counter represents -1 and a single blue counter represents +1.

For some of the models, you will see how the concept of repeated addition can be used to do the modeling of multiplication.

Number counters

Example #1: +3  × +2

For example, how would you model +3  × +2? Well, first, you need to understand the meaning of +3  × +2. The meaning of +3  × +2 is to put three models of 2 blue counters down. Since there are 6 blue counters now and they are all positive, the answer is + 6.

Modeling of 3 times 2 with number counters

Make the following important observations:

  • The meaning of the "+ sign" next to 3 is to "put down" 
  • The meaning of the "× sign" is "of"
  • The meaning of the "+ sign" next to 2 is "blue counters" 

Example #2: +4  × -2

How about  +4  × -2 ? Well, it means to put down four models of negative two.

How to multiply 4 by -2
Modeling of 4 times -2 with number counters

Since there are 8 red counters now and they are all negative, the answer is - 8.

In some cases, as opposed to adding counters, you will need to take out some counters. The last example is particularly interesting and will show how to do this. 

Example #3: -4 × + 2

For example, how do we model -4 × + 2 ?  This one is a tricky math problem! First, you need to understand the meaning of -4 × + 2.

As you can see in the illustration below, it means to take out four models of 2 (or take out four models of two blue counters).

How to multiply -4 by +2

So far though you do not have any blue counters to take. Therefore, you need to put it down. The way you do that is by putting a zero down. The zero will have 8 blue counters and 8 red counters.

Model of negative four times positive two

Notice that by putting a zero down, you are not changing the problem since a zero is nothing.

Again, did you notice the following?

  • The meaning of the "- sign" next to 4 is to "take out" 
  • The meaning of the "× sign" is "of"
  • The meaning of the "+ sign" next to 2 is "blue counters" 

After removing the 8 blue counters, you are left with 8 red counters. Therefore, the answer is -8.

Model of -8

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