Using Manipulatives to Teach Addition and Subtraction of Negative Integers and Variables

Zero

I believe that I have previously mentioned that Gohan has a language-based learning disability. When Gohan was originally diagnosed, he was quite advanced for math, but we had been warned that as he got into more advanced math, he might start exhibiting signs of dyscalculia also. Sure enough, in 6th grade, math started to become more of a challenge for Gohan. This year I switched to a Montessori curriculum for Gohan’s math, mid-year, and it has been working well. One thing he has struggled with, however, is subtracting negative integers. Recently, the program also started covering subtraction of negative variables and Gohan just kind of fell apart. So I did some research online and came up with a method of teaching this concept using manipulatives. We tried it today and it went fairly well. To do this, you need two colors of beads, beans, buttons, gems, etc. for integers and two colors of the item of your choice for the variables (if you want to teach variables also). We used glass gems on the light box. The yellow gems stand for negative units and the red gems stand for positive units. We started out with five negative units and five positive units, which offset each other and thereby equal zero (you can start with any number, as long as you have an equal number of positive and negative and your child must understand that –5 + 5 = 0).

Zero Plus Two

The next step shows the very first integer in our equation, 2. We have the original 5 yellow and 5 red, plus 2 more red, resulting in a total of +2.

Zero Plus Two Minus Negative Two

The next step shows 2 – (-2) = 4. We literally took away two of the negative units. So now we have three negative units and 7 positive units, for a total of 4.

Zero Plus Two Minus Negative Two Minus Four

We then take away 4 positive units: 2 – (-2) – 4 = 0.

Zero Plus Two Minus Negative Two Minus Four Plus Negative Three

Finally, we add (-3). The end equation being: 2 – (-2) – 4 + (-3) = (-3).

Zero Plus ZeroX

I then moved on to using turquoise gems to represent negative “x’s” and purple gems to represent positive “x’s” (sorry the photos don’t show the purple better Sad smile). Once again, I started with 5 positive and 5 negative gems. So, the above photo shows 0 + 0x.

Zero Plus ZeroX Plus X

The next photo shows 0 + x = x, since I added one positive purple “x” gem, but no unit gems.

Zero Plus ZeroX Plus X Minus Negative Two

We then subtracted (-2), taking away two negative unit gems, resulting in: 0 + x – (-2) = x + 2.

Zero Plus ZeroX Plus X Minus Negative Two Minus Negative Four X

The next step was to subtract four negative “x’s”. So the equation now reads 0 + x – (-2) – (-4x) = 5x + 2.

Zero Plus ZeroX Plus X Minus Negative Two Minus Negative Four X Plus Three

The next step was to add 3 positive unit gems, resulting in: 0 + x – (-2) – (-4x) + 3 = 5x + 5

Zero Plus ZeroX Plus X Minus Negative Two Minus Negative Four X Plus Three Plus Negative Two X

We’re almost done! We now add back some of those negative x’s, getting: 0 + x – (-2) – (-4x) + 3 + (-2x) = 3x + 5

Zero Plus ZeroX Plus X Minus Negative Two Minus Negative Four X Plus Three Plus Negative Two X Minus Three X

Our final step was to subtract some of the positive “x’s” (yes, even my head was spinning at this point!). So we end up with: 0 + x – (-2) – (-4x) + 3 + (-2x) – 3x = 0x + 5, or simply, 5.

Hopefully, this makes some sense written out. Working with the gems in person, Gohan was able to understand the concept much better. By the end of the day, he was getting about 90% of these types of equations correct. Now, I just need to cross my fingers that it sticks (one of the joys of learning disabilities is that what a child knows today and what a child knows tomorrow are not always one and the same).

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Labels: Math
Posted by Maureen Sklaroff