Skip to 0 minutes and 7 secondsPAULA KELLY: So let's have a look now at how we can defend subtraction with negative numbers.

Skip to 0 minutes and 13 secondsMICHAEL ANDERSON: OK so we'll use the counters again. So let's start with, for example, a positive 3. So the yellows are worth positive 1 each, and the reds are worth negative 1. So that still has a value of positive 3, because this negative and this positive pair up and cancel each other out. And it's useful to think that with any number, we can have any amount of matching negatives and positives that we can use if we need. So this still has a value of 3. We've got three more yellows counters than we have red counters. So what we're going to do here is we're going to take our 3 and we're going to take away negative 2.

Skip to 0 minutes and 55 secondsNow often students are taught a rule, when you take away a negative, two negatives make a plus. But they never really consider why that works. With this diagram, it can help us to understand what's going on a little bit. So if we have three, we're going to take away negative 2. I'm going to just remove these two counters from our diagram.

Skip to 1 minute and 17 secondsPAULA KELLY: OK so we had our 3, and then we subtracted, when we took away our negative 2. So again, I put that into brackets to be clear that's an operation. That's our negative 2. OK.

Skip to 1 minute and 32 secondsMICHAEL ANDERSON: So let's see what happens when we take these two away. Now we can pair off the reds and the yellows, so we can get rid of those. We can get rid of those. But these two here that were paired with the reds that we've taken away, we can't match them up so they're just left. So in the end, we have five yellow counters, so that's plus five or positive 5.

Skip to 1 minute and 52 secondsPAULA KELLY: OK, fantastic. So can we have another example just be really clear?

Skip to 1 minute and 58 secondsMICHAEL ANDERSON: So often it can get really tricky when you're dealing with negatives, and then we're going to take away a negative. So let's have a look at say negative 4. And then again, I've got my whole army of paired up positives and negatives that I can use. So that's still negative 4 and I can use as many of these as I like, positive and a negative pairing up. So this is negative 4. And where we're going to do is take away say negative 1.

Skip to 2 minutes and 27 secondsPAULA KELLY: So we're going to have our negative 4, so a negative 4. Then we're going to take away some of our reds. We're going to subtract. How many should we take away?

Skip to 2 minutes and 36 secondsMICHAEL ANDERSON: We'll just take that one away. So negative 4, take away negative 1. So it doesn't matter which negative I take away, let's take that one, so that's now gone from our picture. But we can still do a lot of pairing up. So this yellow and this red, this negative and this positive, we can move them out. They match up. They cancel each other out as well, and they cancel each other out. But we still got one yellow left, so I'm going to match it up with one of our original reds. So this yellow say with this red, and they cancel each other out as well. So what we're now left with is--

Skip to 3 minutes and 11 secondsPAULA KELLY: Our negative 3, OK.

Skip to 3 minutes and 15 secondsMICHAEL ANDERSON: So hopefully these counters can see where these rules apply and help us when we're taking away negative numbers.

Subtraction with negative numbers: using counters

We have seen how double sided counters are useful to develop an understanding of adding two numbers where at least one of the numbers is negative. In the last step we saw how a number line can be useful when subtraction involving negative numbers is thought of as the difference between two numbers.

In this video, Michael and Paula demonstrate how double sided counters, where each side of the counter is a different colour, can be used to aid understanding when performing subtraction calculations which involve negative numbers.

Task

Try these calculations using double sided counters. If you do not have any double sided counters you can make your own out of card by printing off this counter template and gluing the counters back to back.

a) (+5) – (-3) =
b) (-3) – 4 =
c) (-2) – (-5) =

Problem worksheet

Now complete questions 1 and 2 from this week’s worksheet.

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This video is from the free online course:

Maths Subject Knowledge: Understanding Numbers

National STEM Learning Centre