A manipulative for integer operations

Help adding and subtracting integers

Working with negative integers has always been a sticking point for my students, whether they are just learning about negative integers or they are solving equations or factoring quadratics. Why? 

The relationship between positive and negative numbers is a slippery one with loose rules. When we multiply and divide, we have strict rules that concrete thinkers really seem to like. With adding and subtracting, the "rules" are a lot more abstract.

Integer Operations Manipulative

I developed this tool as part of my thesis to help my students see the relationship between negative and positive numbers and it significantly cut down on the errors they were making. 

Here is a video explanation for how the manipulative works. Bonus points if you know what cartoon my daughter is watching in the background:) 

This manipulative was a flexible numberline with a pivot at zero. Essentially, it allowed students to use the absolute values of the numbers they were adding or subtracting to find differences.

Here is an example: 

If we use 7 - 12 as an example, we can think of the 7 and the -12 each as integers (nouns), instead of two integers with a subtraction (verb) between them. This usually goes over pretty well with students because it makes the numbers feel more concrete. 

We find 7 and -12 on the numberline and we figure out which integer is farther from zero. Since -12 is farther, our answer will be negative. 

Next comes the fun part. Once we identify that our answer will be negative, we fold the numberline in half.

And then we count the spaces between the two numbers to get 5. This, combined with our previous step, gives us 7 - 12 = -5. 

My students' ability to work with negative integers, even after the manipulative was removed, improved by 62% and my thesis was [eventually, after many, many edits] accepted in May 2011.

When I'm at the board and a problem like "7 - 12" comes up, I don't always have the time to stop and show students on the ruler how we get -5. So, I ask this string of questions:

"[In 7 - 12] Which number is farther from zero?"


"OK, so our answer will be negative. How much farther?"


"OK, so our answer will be?..."

These questions work really well in the moment. When students are working on classwork and there is more time, I like to show them the problem on the ruler if they are having trouble. This usually happens on a 1-to-1 basis. 

Because manipulatives are so hands-on, they allow students to see and feel the numbers, which always seems to stick with them better.

integer operations manipulative

If you find that your students are struggling with integer operations, this integer operations manipulative is available for download. I like to laminate it so that students can use dry erase markers. This helps some students keep an accurate count of the spaces between numbers. 

I also made this free integers graphic organizer that works about the same way and is good for kids who don't need quite as much support at the manipulative offers.

Adding and subtracting integers pennant

For a fun integer operations activity, you may like this integers pennant

Students evaluate three problems on each pennant. The three problems are related to keep the focus on those pesky signs. 

My friend Kara from Learning Made Radical and I have been working on a new set of activities called "around the clock" partner scavenger hunts. This adding and subtracting integers partner scavenger hunt is a newer addition. 

Integers can be tricky, but they are one of those topics that will eventually click. 

There are more ideas for teaching integers in the blog post Can We Really Teach Number Sense?

Scaffolded Math and Science top blog posts


  1. I like the idea of the integer manipulative. How do you use the ruler when finding the same number? Ex. -8 - 8 on the ruler?

    1. There are directions on the back for same-sign problems. I explain it to my students this way... when we were kids and were adding 8 + 8, we didn't think about signs because they were the same. Now with -8 + -8 the signs are still the same, so we add and keep the sign like we did when we were younger.

  2. Fantastic idea. Have you done any further research with your hinged number line?
    You might be interested in a Stanford study that was done in 2015 in neuroscience regarding understanding negative numbers and how the brain uses symmetry to understand abstract concepts. https://news.stanford.edu/2015/07/06/symmetry-math-schwartz-070615/
    Thanks for sharing this and your thesis!

    1. Thank you for sharing, Linda! I concluded my research in 2010 and agree that symmetry is how our brains work with integers. This is an interesting article. Thanks again!

  3. Thank you for sharing. I love the idea with the ruler.

    I'm gonna be doing a Math Madness Session soon with nothing but Math for two hours.

    I'll even get them to make their own ruler.

    Thanks again,

    1. Having them make their own rulers is such a great idea. I feel it will help the kids become familiar with it before using it on integer problems. And Math Madness sounds so awesome! I would love to be a visitor in your class that day!

  4. How do you subtract a negative with the ruler? EX. 4 - (-6)?

    1. I teach -(-6) as "the opposite of (-) negative 6 (-6), which then turns to +6. We'd then get the 4 back in as 4 + 6. There are sign examples on the back of the ruler for situations when signs are the same (or the same in disguise like your example).