Visualizing Integer Multiplication

Let's extend our integer operation visuals to multiplying positive and negative values using coloured square tiles and number lines!

Here’s another post to add to theĀ integer operation series. Here, we look at multiplying positive by positive, negative by positive, positive by negative and negative by negative to help students generate their own rules for integer multiplication beginning with the symbolic expressions.


See the instructional guide below aimed to help teachers and/or parents use this activity with their own students/children.

Visual Prompt #1

In the video above, we start with whole number multiplication as a low floor start:

3 x 2 = 6

3 groups of 2 = 6

The viewer should be comfortable with multiplying and representing multiplication with arrays in order to do this entire activity:

Integer Multiplication Symbolically.016 3 x 2 is 6


Visual Prompt #2

In the second visual prompt, pause the video at this frame where the viewers are given an opportunity to again show their understanding of multiplication using arrays and number lines:

Integer Multiplication Symbolically.019 3 x 4 prompt

The following frames animate the solution:

Integer Multiplication Symbolically.029 3 x 4 solution

Visual Prompt #3

Now we ask the viewer to show what 4 groups of (-2) looks like.


 Integer Multiplication Symbolically.034 4 x -2 prompt

4 groups of (-2) should yield a result of -8:

Integer Multiplication Symbolically.045 4 x -2 solution


Visual Prompt #4

Viewers are then given another opportunity to work with multiplying a positive by a negative:

3 groups of (-3) = ?


Integer Multiplication Symbolically.050 3 x -3 prompt


Then, you can show the animated solution:

Integer Multiplication Symbolically.054 3 x -3 solution

Visual Prompt #5

Now, present the viewer with an opportunity to tinker with what negative groups means:

-2 x 5 = ?

The goal here is for students to gain an understanding that -2 x 5 is the same as subtracting 2 groups of 5.

Integer Multiplication Symbolically.059 -2 x 5 prompt

In order to take away 2 groups of 5, we must employ the zero principle (i.e.: 1 + (-1) = 0, so introduce 2 groups of 5 zero pairs). Then, we’ll take away the 2 groups of 5 and be left with 2 groups of -5 or -10:

Integer Multiplication Symbolically.071 -2 x 5 solution

Visual Prompt #6

Here’s another opportunity for the viewer to tinker with this idea:

-3 x 4 = ?

Remove 3 groups of 4 = ?

Integer Multiplication Symbolically.075 -3 x 4 prompt

The video animates the solution to show that -3 x 4 = -12:

Integer Multiplication Symbolically.088 -3 x 4 solution

Visual Prompt #7

Then, we move on to the least intuitive of them all. Multiplying a negative by a negative:

-2 x (-3) = ?

Remove 2 groups of (-3) = ?

Integer Multiplication Symbolically.091 -2 x -3 prompt

Again, we will use the zero principle, except this time we are removing 2 groups of -3 which will leave 2 groups of positive 3 or 6.

Integer Multiplication Symbolically.103 -2 x -3 solution

Visual Prompt #8

Finally, the viewer is given an opportunity to try multiplying a negative by a negative to drive home the point:

-1 x (-5) = ?

Remove 1 groups of (-5) = ?

Using the zero principle, we will introduce 1 group of 5 zero pairs in order to remove 1 group of (-5) which will leave 1 group of (+5).

Integer Multiplication Symbolically.117 -1 x -5 solution

Math IS Visual. Let’s teach it that way.



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