A manipulative for integer operations

Integer Operations Manipulative

Working with negative integers always seems to be a giant sticking point for 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 looser. 

I developed this tool to help kids 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. Please excuse the background noise. Bonus points if you know what cartoon my daughter is watching:) 

This post on integer operations has more information or, if you have an hour or so to spare, here is my thesis with data showing the tool works to help fix misconceptions, even in high school students. 

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. 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 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!