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Visual Help in Multiplying Positive and Negative Integers

by Beth Waltman

As children begin to practice operations using both positive and negative numbers, they may need extra help in illustrating the concepts. Some children easily memorize that, in multiplication,

two negatives make a positive,
two positives make a positive.
one positive and a negative make a negative

Other students need to see a visual aid to help them remember that multiplying is creating “sets.” I have found an approach that is easily recreated by using either crayons or construction paper.

A lesson would consist of the following steps:
Tell students, "We're going to multiply positive and negative integers. How can we decide whether the answer will be positive or negative?"
(Find out what rules students already apply, and whether there are misconceptions.)

Continue:
"Let's use blue for negative numbers and yellow for positive numbers. Whenever you end up with the same color of background and circles, whether it is blue or yellow, your answer will be positive. It will be positively blue or positively yellow. However, if your final product is mixed, like having yellow circles on a blue background, or blue circles on a yellow background, the answer will be negative."

First, demonstrate that the first number will determine the number of sets, and the second number will determine how many circles will be in each set. The number of sets is the number of background squares. In 4 X 2 there are four background squares with two circles on each square.

Say, "We will look at the sign of the first number to tell us what color of background to use."

Demonstrate 4 X 2 (see below)

Do the problem step by step, and allow your children to supply the answers. In fact, the more that they handle the manipulatives, the more they will “own” the activity and be able to reconstruct it when they are working independently.

As you work through the examples with your children, ask questions and have them form logical responses. Explaining their thinking helps to solidify the concept in their minds.

"Are these positive or negative? We're using the first number for the number of sets and the color of the background. How many sets will there be?" (4) "Will they be yellow for + or blue for -?" (yellow) "Okay, now look at the second number. How many circles will there be in each set?" (2)

Some students will jump ahead mentally and already be picturing examples where the first number is negative. Ask for prediction before beginning. Say, "Will this next problem be on a blue background or yellow? What color do you think the dots will be?" Carry the problem through.

Ask students to multiply -3x-5.

“If yellow is positive and blue is negative, what color should the background be? How many shapes?” (3, both blue)

“If the second number tells us how many circles and what color, how many circles should we put in each one?” (5) “What color will we make them?" (blue, because they are negative.) “There, now we have 5 blue circles on 3 blue shapes. Is this product positive or negative? What did we say makes a negative product? Do the circles match the background shape? Right, so the answer is positive, because the product is positively blue."

Finally, give students a number of related problems. Ask them to simplify the process by just placing two circles, one for each integer, beside the problems. They can either use crayons to make circles, or make circles from construction paper using a hole punch. Then ask them to write the + or - sign to indicate the sign of the answer. They should then complete the problem by writing the answer.

-4X-5=? (blue, blue, +) -4X-5=20

4X5=? (yellow, yellow, +) 4X5=20

10X-6=? (yellow, blue, -) 10X-6=-60

-10X6=? (blue, yellow, -) -10X6=-60

After students show that they understand how to get a correct product, allow them to do an individual practice. The results should look like the following sheet:

Ask students what they notice about the problems. When one number is positive and one is negative, does it matter whether the first or second is negative? See the attached "shortcut" sheet for an example. For individual practice, the students can use yellow and blue crayons to place dots beside similar problems. Soon, they can mentally picture the dots without drawing them. And hopefully, the concepts will stick. I often have great success with color connections, especially if children are ADHD.

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