Operations on Integers – Subtraction

We have discussed about addition of integers and its representation as chips.  Recall in our previous discussion that a single positive chip added up to a single negative chip is equal to 0. The addition below as we have discussed will give us a notion of how to add signed numbers.

Now let us discuss on how use signed chips can be used as a strategy to subtract integers. Let us represent the following using chips.

  1. +7 – +4
  2. +3+5
  3. +5 – (3)
  4. 3 – (4)
  5. -5+2

Before discussing subtraction of integers using signed chips, let us recall some concepts.  First, we know that any number added to 0 is equal to that number. That is, if n + 0 = n for any number n. Second, if we add the same number of positive and negative chips to n, then we are just adding 0; therefore, it does not change the value of n.

We will also agree on the following. First, we can think of subtraction as “taking away,” and second, when we say positive chip, we mean+1 chip, unless stated otherwise. This means that we when we say 3 positive chips, we mean three pieces of +1 chips.


Case 1: +7 – +3

The solution for case 1 is very obvious. +7 – +4 is equal to +3. But if we represent it using the chips, we have 7 positive chips and we want to take away 3 positive chips. So, the answer is 3 chips.  Again, +7 +3 = +4.


Case 2: +3 – +5

In the second problem, we have three positive chips, and we want to take away 5 positive chips. We have a slight problem; we don’t have enough chips to take away!

Hmm…. but we can use the second concept that we have recalled above. .  In A (in the second figure), we need 5 chips so we can add 2 more positive chips, but that will change our original number of chips to +5. To remedy this problem, we add another two negative chips to cancel out the +2 that we have added. Note that we added +2 and added 2, which means that we added 0. In B, we represented +3 with +5 and 2 chips. Now, we can take away +5 and we are left with -2.  Therefore, +3+5 = 2.


Case 3: +5 – (3)

In the third case, we have 5 positive chips, and we want to take away 3 negative chips. Again we don’t have negative chips. Just like in case 2, we add -3 and +3 chips which mean that just added 0. Therefore +5 – (3) =  +8


 

Case 4: 3 – (4)

In A, we have 3 negative chips and we take away 4 negative chips, so we add 1 negative chip and 1 positive chip which is equivalent to adding 0. In B, we already have four negative chips and 1 positive chip; we take away four negative chips and we are left with 1 positive chip. Therefore,3 – (4) = +1


 

Case 5: 5+2

In A, we have 5 negative chips and we want to take away 2 negative chips. So we add 0 by adding 2 positive chips and 2 negative chips. In B, we have 7 negative chips and 2 positive chips. We take away +2. We are left with 7 negative chips. Therefore, 5+2 = 7


The five cases above also review us of what we have learned in addition of integers.  In case 2, we have +3 = +5 + (2); in case 3, we have +8 + (3) = +5; in case 4, we have 3 = 4 + +1; and in case 5, we have 5 = 7 + +2.

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9 thoughts on “Operations on Integers – Subtraction

  1. This is a great idea for teaching integers. I use a similar idea based off of the “Hands on Equation” model. Sometimes I use the number line to show students how to add and subtract integers. It simply means that the students move left and right on the number line according to the sign of the integer: Left for negative and Right for positive. Both methods seem to work well, especially for visual learners. I imagine the “chip method” you suggest would also have a positive result in the classroom.

  2. When I was teaching 6th grade and just teaching the kids about Algebra, we also used the “Hands on Equation” set. I actually loved it, because I find that the students do better with actually “seeing” what is going on in solving these equations. I moved to a different school, which don’t use the HOE model and I had to come up with different ways of explaining the process. This way with the chips seem like a great idea to use in the classroom.

    • @Tiffany: Integers is one of the hardest topic to teach, if you want to teach it conceptually. Sometimes, teachers just give the rules on adding/subtracting (especially in our country) without giving the conceptual idea behind. Glad you liked it.

  3. I find this “red chip/black chip” method to be good except for when kids start using the chips as projctiles! I did my entire graduate thesis on integers and the reasons why [my] students in eleventh grade cannot work with them. Many students were telling me that “-22 + 5 = -27” because they never learned the relationship between positives and negatives [in seventh grade]. As part of my thesis I tested a manipulative that I invented called the ZeroSum ruler. It reduced my students’ error rate by 62% on integer addition and subtraction, which is essential when it comes to higher math like equation solving.

  4. Pingback: Discovering multiplication of integers through patterns « Mathematics and Multimedia

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