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

Recent Articles

  1. 45-45-90 Triangle

    May 01, 23 07:00 AM

    45-45-90 triangle
    What is a 45-45-90 triangle? Definition, proof, area, and easy to follow real-world examples.

    Read More

  2. Theoretical Probability - Definition, Explanation, and Examples

    Apr 24, 23 07:02 AM

    Theoretical probability versus experimental probability
    Learn how to compute the likelihood or probability of an event using the theoretical probability formula.

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes