# Directed Numbers

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.

1. Thank you for this. I grapple with this myself. I like the reflection idea for multiplication and actually saw that used in my algebra book from the 60’s (can send you a scan of the page that discusses this if you’re interested).

For subtracting negatives, I often use examples about finding the difference in temperature between 40 degrees and -10 degrees: how much did the temperature climb from -10 to get to 40? Or a football (US football that is) example: a team loses 5 yards, and then makes a “first down” on the next play (that is, they are 10 yards ahead of their initial starting point). How many yards did they travel? 15 yards, since they gained the 5 yards that the lost and then ran 10 more. So it’s an example of 10 – (-5) = 15.

The book I’m using for my 7th graders (yr 8’s in UK) has good examples for multiplying negatives. If 3 minutes ago is denoted by -3, and the rate of losing 2 degrees C per minute is denoted as -2, then students are asked: If the temperature now is 0 deg C, what is the temperature 3 minutes from now? 3 x -2 = -6, or 6 degrees C below zero. Then: What WAS the temperature 3 minutes ago? -3 x -2 = 6 deg C. Intuitively they see it has to be positive.

Of course, the mathematical proof can be done via the distributive property which I also do.

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• My only problem with these contextual analogies is that they can add a layer of complexity, especially with weaker students.

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• So can the number lines. I have a class full of weak students. The analogies about the temperature drop/climb (and a similar mountain descent) worked rather well.

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• To add to what I said below, one has to learn to trust the math. One girl did not grasp the mental image and so she just trusted the math to work the problem. Over the weekend, she did grasp it. There are, of course, aspects of math that do not translate into mental images and then one has to trust the math. I recall in 2nd year algebra, my teacher showing us the proof using the distributive property. I also had the symmetrical number lines from the first year algebra book. I understood the proof and trusted the math but I needed a mental image so came up with the idea of filming someone riding a bike backwards, and then running the film backwards. That helped me.

So different things work for different people. Including teachers.

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• Absolutely. I like the phrase “trust the math”.

I’d love to see the excerpt from the 60s book you mentioned.

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2. I like the idea of multiplication being “stretching” a number in certain direction, this obviously ties in with positive and negative enlargements which they will encounter later on and so may have benefits there. It is a tricky topic. I have sometimes used analogies of temperature (by removing cold air, it gets warmer??) but I have never been convinced that they really help. I agree that the mental image of the number line is the best tool for adding and subtracting, although with multiplying and dividing I’m much more comfortable providing “the rules” because, so long as they are remembered correctly, they are very straightforward.

Just as another idea for showing why those rules work, and providing some self-check practice, I have used these multiplication grids involving negative numbers quite effectively.
https://mhorley.wordpress.com/2016/06/23/negative-number-grids/

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• Thanks for the grids, and I agree on mental images – tend to steer clear. As for stretching, I think it’s a much better representation of multiplication than repeated addition is, as it deals with negatives and fractions quite well.

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3. Kevin Higginson says:

I teach resit GCSE at a sixth form college, and I am forever getting frustrated with students who use the – – is +, as they get so confused with it. I have used a number line for a while with addition and subtraction, but I love your idea of reflecting for multiplying and dividing as there is a strong connection with the tiles concept of reflecting them as well.

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4. Richard Deakin says:

Before you do directed numbers I always think more work needs doing with the number line. I like this activity from Don http://donsteward.blogspot.co.uk/2016/02/number-line.html
Here students encounter sums like 5-19, I want them away from counting asking themselves 5-? = 0, okay 5, how much have I still got to take away, okay 14, so -14. This then allows students to have the strategy to do 5-26 or 5-1000 further on.
I would want this covered and questions involving decimals and fractions upto, but maybe not quite as hard as 1/4 – 1.3 before moving onto directed numbers.

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• Interesting. I agree with what you say first, but I’m inclined to leave mixed calcs with fractions and decimals until after working with directed integers, unless they stay on the simpler end like the example you give of 1/4-0.3.

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5. Hi Jemma. Thank you for your interesting blog posts. The use of the horizontal number line by teachers to introduce the idea of negative numbers being ‘less than zero’ is the WORST pedagogical mistake in the history of mathematics. Without it, we might have colonized Mars by now. In 1685 John Wallis (of number line fame) KNEW the idea negative numbers being less than zero would break mathematical logic. And so, here we are today with WORSE elementary mathematical understandings than those that existed 1400 years ago.

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