Up and down the integer line

Krittika Hazra

When I started teaching integers in class 6, the very first question that crossed my mind was, how do I teach that minus and minus should actually be added? I was taught by my senior colleague how Shri PK Srinivasan beautifully explained integer operations using a game with black and white cards (see box on how to play this game). I decided to take the help of this activity to introduce the concept of integers.

All was fine to begin with, but as I went on to explain sums like a – (-b) or a + (-b), I realized that there was a problem with the language that I had to use. However, not arriving at an alternative I continued using the phrase ‘minus minus is plus’ and plus minus is minus’.

I wanted to find a less confusing way of explaining integers to my students though. So my efforts to find a better method continued until I found it in a Cambridge School Mathematical Project (SMP) booklet. This booklet introduced integers with the help of a temperature scale (FIG 1). As I read through the book, it occurred to me that the language I had been using was truly insufficient to explain integer sums effectively. I stopped using it altogether. In fact, Shri Srinivasan’s game of black and white cards also made sense only when the language ‘minus minus is plus’ was not used!

I observed that even as the children did memorize the phrase, it did not help them understand the concept. On the contrary, they recalled the memorized phrase frequently used by the teacher and made random mistakes more often than not!

What and where was the problem, I wondered? What could be a better way of teaching the concept? What if I could do away with the problematic phrase? What could be a suitable/better alternative?

Below is a description of my experiments in the classroom. These are the answers I discovered to my questions. And I hope the following methods are a better way of teaching the concept of addition and subtraction of integers.

Here are a few examples.
Table 1: Frequently made mistakes

With the black and white card game, their understanding of the operations was perfect. But when problems like 3 – (-4) or 3 + (-4) arose, my students became greatly confused with the previous concepts as well. And they started making the errors discussed in Table 1.

I tried introducing the idea of a – (-b) and a + (-b) in the following manner, using the idea I discovered in the SMP booklet.

Introducing integer addition and subtraction
Below is a temperature scale (a vertical number line) to help the students observe patterns of numbers and deduce the rule by themselves.

The following questions can be asked:
Take the help of the temperature scale and solve the questions (by this stage it is expected that the students know, for addition we go above zero and for subtraction we go below zero)
a) 5 – 5
b) 5 – 4
c) 5 – 3
d) 5 – 2
e) 5 – 1
f) 5 – 0
g) 5 – (-1)
h) 5 – (-2)
i) 5 – (-3) and so on.

While solving the above exercise, the students saw that the answers were in ascending order. So even if they did not know what 5 – (-1) was, they knew by the number pattern that the answer had to be 6 and answer of 5 – (-2) would be 7. At this point I asked the children, “Instead of 5 – (-1), can we write 5 + 1?” (as both these number statements will yield the same result.)

I then stressed the fact that from the above observation we could say that –

Subtracting a negative number is the same as adding that positive number.
The students readily agreed to this.

The second number pattern was as follows
a) 5 + 5
b) 5 + 4
c) 5 + 3
d) 5 + 2
e) 5 + 1
f) 5 + 0
g) 5 + (-1)
h) 5 + (-2) and so on.

In this number pattern the students observed that the answers were in descending order and they quickly deduced 5 + (-1) was 4. So in a similar manner I emphasized that 5 + (-1) was the same as 5 – 1, and generalized that –

Adding a negative number is same as subtracting that positive number.
Students wrote these observations in their notebooks.

Teachers can give similar problems to their students for practise. But remember, once you use this method of teaching you should not go back to saying, “minus minus is plus” so as to avoid any confusion.

Black and white card game*

This game is played to introduce the operations of integers. The rule of the game is that one black card cancels one white card, so together they form nothing. For e.g., ask the students to place 4 black cards on the table and then add 3 white cards to this. As per the rule, after cancellation they have 1 black card left. This way addition of integers is introduced.

For subtraction ask the students to bring 4 black cards to the table and take away 3 white cards. But there are no white cards on the table. Therefore, they have to bring in 3 white cards along with 3 black cards (in order not to change the values in the problem or in other words bring in 0 to the table). Now, they can take way 3 white cards from the system and they are left with 7 black cards.

Later they learn how to write this down. First using B(black) and W(white), then replacing B and W with N(negative) and P(positive), and in the end replacing N and P with – and +.

*Readers who wish to know more about this game can look up At Right Angles, Vol 6, No. 3, November 2017 edition, the pull out, titled “Teaching Integers” written by padmapriya Shirali or watch this video on YouTube – https://www.youtube.com/watch?v=7x7H09-Cca4.

The author is working with the Future Foundation School, Kolkata, as a middle school mathematics teacher. She has worked with KFI Pune earlier and has done her MPhil in Mathematics from the Ramanujan Institute for Advanced Study in Mathematics, Chennai. She can be reached at krittika.hazra7@gmail.com.