Modeling multiplication

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

The number counters are shown above. 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.

A few examples showing how modeling multiplication with counters work

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.

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.

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).

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.

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.