# Teaching Negative Numbers

Discussion in 'Mathematics' started by ChrisMwell, Aug 30, 2009.

1. ### ChrisMwell

Hi folks, had a nice wee lesson I thought I would share.

I have a little S2 bottom set who are generally quite low ability. I opened the course folder last weekend to see that I had to teach them negative numbers. Hmmm - how can I teach these kids negative numbers in an interesting, visual and tactile way?

The answer was a two part lesson. The first part was a powerpoint I found from somewhere which I could put on my smartboard. It was the user interface of an ATM machine. The bank account had £50 in it. I had kids come up and "withdraw" certain amounts of money. We then investigated what happened when people withdraw sums of greater than £50. This led is onto the idea of over overdrafts and negative numbers in context. The calculations of the overdrawn amounts seemed to come to the pupils very easily. Next we took a look at another powerpoint, this time we had a virtual thermometer and discussed temperature rise and fall.

So far, all very well. But what about doing calculations proper, so to speak? I always favour having a number line from -20 to 20 on the kids page and then they can do the "steps" method of counting along. In a bid to break free of jotter work I printed out all of these numbers and lay them along our corridor. I then got kids to answer quesitons such as -5 + 7 and - 2 - 4 by standing on a number and then moving the correct number of steps to get the answer. We then had a girls v boys challenge quiz. With the scores tied at 4-4 and first to five being the winner I opted to throw a spanner in the works, just to see how they would handle it. I asked the girls 5 - (-4). I got all sort of answers some of which were more contrived than others. The girls didn't manage to get the answer, although at this point I didn't explain how to do the question. The boys too didn't get the answer. However, upon me revealing the answer to them some interesting discussion took place.

Kid A: "Hey, does take away not mean difference?"
Me: "Yip". (Although I'm thinking to myself - where is he going with this?)
Kid B pipes in: "Difference, what's that?"
Kid C: "The distance"
At this point I gasp in astonishment as the kids point out that to get the answer of 5 - (-4)
all you have to do is count the number of "hops" between 5 and -4, thus giving the required answer of 9.

Startling as it may be, this was a completely new concept to this particular mathematics graduate. I suppose I'd always just accepted that two negatives next to each other become a plus. The kids on the other hand had no preconception of this and as such were a blank canvas! They made be really think that yeah, by old school definition take away is difference, so getting the answer 9 is perfectly logical! I told my colleagues, some of whom are very experienced, and they had to admit this was something they too had never properly considered.

With some more questions answered, by using our corridor number line the kids managed to spot that the two negatives do indeed become a plus.

I have to admit this lesson exceeded my own ambitions as to how it might go. It gave me a whole new insight into this particular area of maths and how to deliver it in future. It's always been a dry area of the subject for me, with little scope for doing nice problem solving lessons leading to conclusions, like this lesson did. However, in future, I will always be doing this lesson rather than just telling them the rule!

Some of my own preconceptions of the kids have been challenged and that is no bad thing.

If you found this interesting at all please take the time to visit my new little blog on http://scottishmathsteacher.blogspot.com I'm posting all sorts of little reflections over the course of the school year. I'd like to think that some of the things I post will be interesting to younger teachers who are still finding their feet after a couple of years, like myself. I'd also hope that more experienced teachers might be able to criticise and advise too anything they read!

2. ### brookes

Hi, your pupils' consideration of 5-(-4) as being the difference between 5 and -4 is 9 has really made me think! As with your colleagues, I hadn't thought about it like that. Then I thought about -5-4 and wondered if the fact that the answer is not 9 might cause some confusion.
I like to use the "disc" model for introducing subtraction and addition of negatives and positives. Lots of "hands-on" activity for the kids and it can be applied to any combination of adding & subtracting with negatives & positives.
I would also avoid "two negatives do indeed become a plus." because otherwise you'll get pupils giving a positive answer to -10-(-3) (perhaps 7 or 13).

3. ### ChrisMwell

Hi! Glad it isn't just me who finds this as somewhat of an insight!

I did consider the cases you mentioned such -10 - (-3). The idea we are exploring as a class is that "two positives right next to each other become a plus"

We've looked at and discussed -5 - 4 etc. The kids have come up with the descriptions, wording and terminology they want to use. I'm experimenting a wee bit just now in my teaching. I keep reading that the most effective learning is when the teacher does not give "ready made meaning" and the pupil has to generate their own meaning for ideas. I hope that through the approaches I am trying in class, we are getting round these obstacles. Basically anything that has went on the board has been what they have agreed on. Not what I have said. (Although of course I do gently guide them if they are going miles off course with what they are saying!)

So for the above the kids would have to write out:

-10 - (-3) = - 10 + 3 = -7

Remembering this is a bottom set, they seem to be getting the hang of it. I suppose the big test will be tomorrow when i see how much they can recall from before the weekend!!

As a Scottish teacher I never hear much about nrich up here, but I'll be sure to check out the disc model of which you speak!

Cheers for replying by the way!

4. ### ChrisMwell

I have no idea why my blank lines keep vanishing by the way! Sorry it is so unreadable. My blog is more nicely presented!

5. ### stevencarrwork

One analogy I use is a bank balance.
If I have 10 pound in the bank, and I get a 2 pound bank charge, then my bank balance goes down to 8.
If the bank takes away from my statement that 2 pound bank charge, then my bank balance goes back up to 10.

It took the best brains on the planet hundreds of years to work out how to deal with negative numbers. For centuries mathematicians just avoided them wherever possible. I guess our least able pupils will soon figure out negative numbers if we give them a little group work. There might even be time over for a plenary.

6. ### dydx

Moving on to the next stage, how will you convince your pupils that (-5) x (-4) = + 20 ?

5 - (-4) should always be said as the '<u>difference' or 'gap'</u> between positive 5 and and negative 4, giving +9.

Similarly, -5 -(-4) is the difference or 'gap' between negative 5 and negative 4, thus implying -1.

7. ### oscars

Hi Chris, great lesson! I wish you had done it a week ago!!! I am in the middle of an integers topic too!
We were on multiplying today and I introduced that by recapping last weeks addition/subtraction sums on the mini whiteboards, e.g. -2 + (-3), 4 - (-5), etc. The last few questions were like -2 + (-2) + (-2), i.e. examples with several repeated additions. After showing their answers for these I asked if there was a quicker way to do these. This led to 3 lots of -2 equals -6, etc.
I deliberately left it without much further discussion then gave them a Tarsia jigsaw with a mixture of operations. The pupils (in groups of 3/4) were set two tasks, one was to complete the Tarsia and the other was to agree in their group how to multiply with negative numbers.
They decided that 4 x (-3) was -12 due to repeated addition. They then concluded that -4 x (-3) would give -(-12) and thus equalled 12 as "two negatives make a positive".
Most groups figured this out and when shared with the class.
I had always covered the multiplying/dividing in the past by extending a multiplication grid into negatives and deducing results from the patterns of signs. This beat that by a long way!

8. ### HB1

Thank you for sharing this. It has certainly given me some ideas for lessons next week.

9. ### Andrew JeffreyNew commenter

How would others go on to explain about the example mentioned, or why 5 - -4 is 9, but -4 -5 is negative 9?
Also, this is a fabulous example of the benefit of children being given the time and space to work things out from first principles rather than "Here's the method, just reproduce it on your GCSE so I can keep my job".
Finally, I have had some success in the past with teaching students about 'emotions and the barometer of life.' However you are feeling, if someone adds something negative to your life your emotional barometer goes down. If someone takes away something negative from your life, it goes up. Sounds a bit mad, but works very well.

10. ### ChrisMwell

What excellent replies to my original post. I didn't think many people would find it to be anything particularly revealing!

Oscars - I like your idea for the multiplication of negatives etc. I'll be looking into this for delivering the next part of the topic!

Andrew: You've hit the nail on the head for me. There is a great value in kids being allowed to work things out from first principles -it's the biggest change I'm trying to make to my own teacing style, as I enter my fourth year. I'm trying to get rid of the lecture style format followed by text book questions. I'm trying to branch out - although I am only happy to do this if it is effective in helping the kids to learn it. I won't do use gimmicky teaching strategies without consideration of their potential benefits for the kids.

I like your idea of the emotions of barometer of life. I was talking today about my take away pizza at the weekend. The kids asked what type of pizza? I said "it was an + and - pizza" so I took it away. This was a daft wee gimmick to try and help them remember the concepts we'd already discovered, but sometimes a daft thing like this clicks with some kids like you say.

Another shameless plug for my blog: scottishmathsteacher.blogspot.com. One guy has replied to this post with an article about the state of maths teaching today. And not to exaggerate it looks excellent from the first couple of pages I have read. Pop over and have a look!

11. ### bombaysapphireStar commenter

Subtraction has a direction. We accept that 9-6=3 and 6-9=-3. The value is positive if we subtract a lesser value and negative if we subtract a greater value.
5 - -4 is subtracting a lesser value so the answer is +9
-4 - 5 is subtracting the greater value so the answer is -9
Thanks for all the posts on this thread. I always find it difficult to decide which route in to teaching these bits of negative numbers. It will be good to try some tried and tested suggestions.

12. ### oscars

I've never really been happy with using this analogy but I have found that it does help some pupils. Honest!!!

Here's how I sometimes help pupils with 5 - (-4) if they are really stuck:
"Think of positive numbers as warm temperatures, negative numbers as cold temperatures...
Imagine the classroom is 5 degrees ('warm'). What would happen if we could remove 4 degrees of 'cold' from the classroom?"
Pupil usually concludes that the temperature would go up 4 degrees to 9 degrees.
So 5 - (-4) = 5 + 4 = 9.
I know it's dodgy but it usually works!! (holds head in shame )

15. ### GoldMathsNew commenter

When I went to teach this topic I found this;

http://www.teachers.tv/video/23033

Which was very useful - I progressed the idea with positive numbers being a flame and negative numbers being an ice cube.

E.G. The sum 4 + -2. The plus is an "action" so we have four flames we add two ice cubes where are we now? I was very impressed with how well this worked.

16. ### ChrisMwell

Just update the blog today folks:

http://scottishmathsteacher.blogspot.com/

Hope some of you have time to go over and have a wee read at it for me

17. ### Colleen_YoungOccasional commenter

I'd like more headings Chris, I want to be able to glance at a blog and know what it's all about (I look at quite a few), long fairly narrow columns of text are hard to take in.

18. ### stewartyNew commenter

When I was on my teaching practice 4 years ago I was lucky lucky to observe a teacher who I must admit to having copied every year since!
When thinking about working with negative numbers he always made the pupils imagine a number line.
When asked to answer an addition or subtraction a volunteer came to the front and stood on the first number, and faced up the number line (you always face up because we are all happy positive people!!). A '+' means to continue, and a '-' means turn around. So if the question is 5+3 you start on three facing forwards, the + means continue and you take 3 steps to end up on 8. If the question is -2 - 4, you start on -2 facing up the line. The '-' means turn around and then the 4 means to take 4 steps ending at -6. Lastly if the question is 5 - (-3) the pupil starts on 5 facing up the line. The '-' means turn around. The next '-' means turn around again, and then move 3, ending up at 8.
The key thing of course is that the first number is simply the starting point.

Take a breath!
Stewarty

19. ### ChrisMwell

Thanks for the tip. I'm new to this whole blogging thing, I'll try to get it sorted for future posts.

20. ### ChrisMwell

This is fantastic. Far better than what I was doing. I think this is just the very thing I will try out next time! Total genius! Thanks for posting!