by Kathy Kuhl

IMG_0984When we teach math, we must show why things are so, or better yet, help our children to discover them for themselves. But how can we show negative numbers with manipulatives—counters or some other tokens that help our kids discover what’s true?

It’s easy to show addition and subtraction with positive numbers. Here at left is 5 – 2 = 3, illustrated with beans.

With positive and negative numbers, it’s easier to start with integers, the whole numbers and their opposites, {…-3, -2, -1, 0, 1, 2, 3, 4….}. Save -3 1/2 and -7.2936 for another day. Always teach new concepts with examples that only need simple arithmetic computations.

How to model negative numbers? You can’t have negative beans, can you? Harold Jacobs’ book, Elementary Algebra, gave me an idea. He illustrates positive integers as small white circles, and negative integers as small black circles, and asks students to pretend that the white circles are matter, and the black circles are anti-matter.

At our homeschool group classes, I brought white navy beans to represent positive integers, and black beans to stand for negative integers. I tell them, “These black beans are anti-matter, small but powerful.” (I say that because they are slightly smaller than white beans.) “What happens when matter and anti-matter meet?” I ask.

“They destroy each other,” someone will say.

“Right!” I answer, and make an explosive sound as I bring a white and black bean together. (The volume and theatrics depend on the age of my students.) “It’s like adding one and subtracting one. After I go up a step on the stairs and down one step, how much has my position changed?”


“Right, zero! Zip!” I go on to explain how if I add two positive beans and two negative beans, they cancel each other out. It’s like adding zero.

So what happens when I add 5 + -2?

5 + -2 = ?

5 + -2 = ?

The two negative beans “annihilate” two of the positive beans, leaving 3 positive beans.

IMG_0986Another possible manipulative is poker chips. When I use them, I like the red chips to represent negative numbers, since being “in the red,” means having a negative balance.”

What tools do you use to teach integer arithmetic? Share your ideas in the comment section below, please.

This 4-part series continues here:
Part 2,
Part 3, and
Part 4.