I’ve been teaching directed numbers to Y7 recently. It’s a topic that always gets me thinking. I am not happy with “two minuses make a plus”, although I was taught it that way (you know, where you circle – – and write + above it). When used as one of the main methods of instruction it can lead to things like -7 + -3 = 10 because the students don’t really know what’s going on. They have an aide-memoire without knowing when to apply it. (And that, for me, is the thing with aides-memoire – I have no issue with them if students know when they are and aren’t applicable or appropriate. When they become the source of the teaching we get problems).
Some people like a vertical number line. I normally prefer a horizontal one, partly because I teach adding and subtracting negatives in terms of “starting number”, “direction” and “how far to move”. My students come from primary school well-versed in moving left when subtracting and moving right when adding, so we begin with a sequence something like:
|2 + 3
1 + 3
0 + 3
-1 + 3
-2 + 3
-3 + 3
-4 + 3
|4 – 3
3 – 3
2 – 3
1 – 3
0 – 3
-1 – 3
-2 – 3
This reinforces the idea of starting on a number and moving right to add/left to subtract, as well as practising going through 0. By only varying the starting number we draw attention to the effect of that aspect only, before moving on to adding/subtracting different amounts (there is an excellent chapter on variation theory in Craig Barton’s new book). We will spend time on this, including using larger numbers, so that in -15 + 26 students get used to moving right 15 to reach 0 then on a further 11.
Once the overarching concept of “starting number”, “direction”, “amount” is secure I introduce the idea that adding/subtracting a negative number makes you reverse your direction. There’s a lot of “what’s the starting number?” “adding sends us right but – ah! – we’re adding a negative number so we’ve got to move left”. I show lots of questions where students call out the starting number and then point in which direction we would have to move. This is one of those times when rapid, whole-class Q&A is particularly handy, for them and me.
The Directed Number Line
The mental image of the number line is one of our strongest allies in teaching directed numbers, and particularly in considering why they are referred to as directed numbers, which is another reason I prefer a horizontal number line to a vertical one: I find the symmetry of the positive and negative numbers easier to see horizontally.
This symmetry is a concept I think I have neglected somewhat over the years and spending more time thinking about it has improved my own mental image of numbers, as well as my ability to convey negatives to students. I now, by default, picture a number line as going in two directions from 0. (Don’t get me wrong, it’s not a new discovery, I just didn’t really give it enough importance in the past).
This image lends an extra dimension to adding and subtracting negative numbers.
More or Less
There is another way of conceiving adding/subtracting negatives, and this symmetry reinforces it, specifically -4 + -3 being adding three negatives to -4, so becoming more negative, or -4 – -3 being subtracting three negatives from -4, so becoming less negative.
The phrases more negative and less negative are ones I use increasingly, because of the water-muddying that comes with “smaller” and “bigger”. Is -1 “smaller than” 1? Not really. As numbers they have the same size (or absolute value), they just point in opposite directions from 0. Is -5 “smaller than” -1? Not really. It is bigger in size, but it is definitely “more negative”. This discussion highlights the subtle but important difference between “less than” and “smaller than”, or “greater than” and “bigger than”. “Less than” indicates to the left on a number line, “greater than” to the right, but these are different to smaller/bigger than.
I find the symmetry of the number line helps with conceptualising multiplication and division of negatives as well. If I take -4 and multiply it by 3 I am stretching -4 (which goes left from 0) to three times its size, so I reach -12. If, however, I take -4 and multiply it by -3 this stretch gets “reflected” about 0 to reach +12.
Subsequent multiplications or divisions by negative numbers make such reflections, so in -4 x -3 x 2 ÷ -6 we know the magnitude of the answer will be 4 (4 x 3 x 2 ÷ 6), but consecutive numbers make our answer go negative, positive, positive, negative, as every time we meet a negative number our stretch is reflected in this imaginary “line” of symmetry about 0.
If you are aware of the negative number tiles used in Singapore, then you will know that this is what those tiles achieve. For those who haven’t seen them, children in Singapore work initially with concrete manipulatives. They have small tiles with +1 written on one side and -1 on the other. To multiply 4 x 3, twelve tiles would be arranged in a 4 x 3 array with their +1 sides up. If the 3 is changed to -3, all three rows of four tiles would be flipped over, revealing twelve -1 faces and an answer of -12. If the 4 is then changed to a -4, all four columns of three tiles would be flipped over, revealing twelve +1 faces and an answer of 12 again.
I am becoming increasingly happy with the idea of reflecting in 0 for multiplying and dividing, and I can see its links to the way this topic is taught in places like Singapore. My Y7s took to it very quickly, and were dealing adeptly with long strings of multiplication and division, considering the magnitude and the sign of the answer separately. Time (and practice) will tell whether or not it is effective as a pedagogical tool rather than just an interesting visualisation.
As always, I’d be very interested to know how other people have tackled directed numbers. Please leave a comment here or on Twitter.