Integer Operations with a Flexible Number Line

Integer operations manipulative for adding and subtracting negative numbers

It really doesn't seem to matter whether a student is in 7th grade or in 12th grade, integer operations are always tricky. When I taught Algebra 2 in Boston, working with negative integers was a major sticking point when we'd be solving equations. Students would get all the way to the end of a difficult equation... and this would happen:

They'd freeze. They'd guess. They'd get the strict rules of multiplication confused with the much looser rules of addition and subtraction. I'd get a lot of -19's and 5's. The problem was that the signs are opposite. With same signs, we just add and keep the sign. With opposite signs, it can feel like we're in a whole other ballgame.

Integer operations manipulative for adding and subtracting negative numbers

To say this issue consumed me is an understatement. I became obsessed with finding a simple, intuitive way to help my students successfully add and subtract integers so that their confidence would no longer suffer. Because this is what happens - kids think they are "dumb" when they can't evaluate "simple" problems - even after successfully moving past the tough steps of a quadratic, logarithm or radical. It was these very last steps of equation solving that were hanging them up.

Thankfully, since they were all I could think about anyway, my graduate thesis proposal to research negative numbers was accepted. Part of my research involved creating and using a tool, which I called the "Zerosum Ruler", to see if it had any effect on my students' ability to work with integers. 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:

Integer operations manipulative for adding and subtracting negative numbers

If we go back to our problem 7 - 12 from the equation at the top, 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 permanent. 

Integer operations manipulative for adding and subtracting negative numbers

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

Integer operations manipulative for adding and subtracting negative numbers

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

Integer operations manipulative for adding and subtracting negative numbers

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 works 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 manipulative is available for download here.  You can also check out a video on how the tool works.

I like to laminate the tool it so that students can use dry erase markers. This helps some students keep an accurate count of the spaces between numbers. 

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. 

Adding and subtracting integers pennant

You can find this integer operations manipulative in this Integers mini-bundle for learning and practicing integers. 

integers mini bundle for adding, subtracting, multiplying, dividing, ordering integers


  1. I do something similar.
    I play a game with a videoscore
    Pos. Vs. Neg.
    7 - 12
    I ask ... Who is the winner?
    They say,. Negative
    I ask... How many points of difference?
    They say ... 5
    So ... solution is ... -5

    1. I love that! Yes, so similar. I feel this is the way we manipulate debts and payments as adults so it makes a ton of sense to teach it this way. Thanks for your comment!