The Disturbing Saga of Math Cannibals!!!

I have just started to teach Basic Algebra Concepts to my college mathphobics.  This is where the "rubber meets the road" as they say.  The biggest hurdle for my students is understanding positive and negative numbers.  Multiplying and dividing seem to be no problem, but addition and subtraction are another story.  To state that subtracting a positive number is the same as adding a negative number is considered hieroglyphics to many.  Since many of my students are visual/kinesthetic learners, I needed a strategy that would connect the abstract to the concrete. 

I took film canisters (a Trash to Treasure idea!) and filled them with two sided beans. One side of the bean is red (negative), and the other side is white (positive). Suppose the students have the problem -5 + 2.  They would get out five red beans and two white ones as illustrated on the left. Then the fun begins because suddenly the beans become "cannibalistic".  The red ones begin to "eat" the white ones and vice versa. (In reality, the students are matching each red bean with a white one and moving them aside; see illustration on the right.)  After each bean has been “eaten” by the opposing color, three red beans remain.  As a result, the answer to the problem of -5 + 2 is -3.

Don't stew; study!

If the problem were -2 - 6, the students would lay out two red beans and six red beans.  Since all the beans are the same color and no bean desires to "eat" anyone on their team, the student simply counts all of the red beans.  So  -2 - 6 = - 8.

What happens with a problem such as 5 + ^-3?  At the beginning, I have the students get out five white beans and three red ones; then match them resulting in the answer of 2.  Unfortunately, in our Algebra book, the double signs vanish by about the third page of the chapter; so, the students must recognize what to do. 

 

The first option is to insert a + sign such as in the problem – 4 – 2 = -4 - ⁺2.  This allows them to see that, in reality, they are subtracting a positive number. 

However, what do they do with -4 - ^-2?   I instruct them to circle the two signs, and use the multiplication rule for a negative times a negative to change the double minus signs into a plus sign as seen in the illustration on the left. They can then proceed to use their beans to solve the problem.  This may seem unusual, but it makes sense to my mathphobics.

You might ask, "How long do the students use the beans?"  It’s interesting, but all of my students put them away, just at different times.  A few only need them for the first assignment whereas others need them for many.  I once had a special education student who was mainstreamed into my regular PreAlgebra class.  He was the last one to rely on the beans, but he did eventually put them away.  The important thing was he had a picture in his head that he could use over and over again.  Incidentally, he passed the class with a “C”, completing all of the same work the other students did.

Need a game instead of a worksheet to practice adding and subtracting positive and negative numbers?  Try Bug Mania or Roll and Calculate.  Just click on the name of the game.