Notoriously difficult for pupils to understand, I think addition and subtraction of negatives is one of the things that one comes to understand after doing lots of practice. HOWEVER, that practice needs to be yielding correct answers from the off. It’s no good sending pupils off to do lots of practice if they’re getting it wrong as often as they’re getting it right.
Heavily influenced by our reading on working memory, here’s how we teach addition and subtraction of negative numbers:
When to start – We start teaching negative numbers at the beginning of year 8.
Introducing – We spend the first lesson introducing negative numbers – touching on their history, real-life applications (briefly), finding them on a number line, ordering them, etc.
No analogies – When teaching addition and subtraction, we NEVER talk about “two negatives make a positive” or use analogies about ice cubes, good/bad people, or use negative/positive tiles.
Air number line – We spend a lot of time up front using an “air number line”, where the desk level represents 0, above the desk is positive and below the desk is negative. It is important they are comfortable moving up and down this number line so there’s a lot of whole class practice that goes like this:
- Adding and subtracting positive numbers with a positive answer.
“Put your hand on 5 and add 3. Which way did you go? How far? Where are you now?”
- Adding and subtracting below the desk where the answer remains below the desk.
“Put your hand on -5 and add 3. Which way did you go? How far? Where are you now?”
- Adding to negative numbers and subtracting from positive numbers such that you have to bridge past 0.
This starts off as “Put your hand on 5. Take away 5. Now take away another 2. Which way did you go? How far? Where did you get to?” Here we’re explicitly practising arriving at zero and then going beyond.
And progresses to “Put your hand on 5. Take away 7. Which way did you go? How far? Where did you get to?”
- Finally, having reinforced that adding goes “up” and subtracting goes “down”, we look at the effect of adding a negative and subtracting a negative. We explicitly teach them, without an analogy, that when adding a negative you go down and when subtracting a negative you go up. Again, lots of whole class practice with the air number line. It’s slow and deliberate to start with but becomes high energy and high stakes as they get more proficient. By high energy, I mean doing lots of questions quickly as a class, call-and-response style and by high stakes I mean playing Simon Says.
START DIRECTION DISTANCE – Then we move to understanding the symbolic form. Not getting the solution but just translating the sum into START-DIRECTION-DISTANCE. They’re usually good at identifying the first number as the START and the last number as the DISTANCE but DIRECTION often needs practice, feedback and support. So to clarify, all we ask them to do is look at each sum and identify the START, the DIRECTION and the DISTANCE.
Solving using SDD – We then use the START-DIRECTION-DISTANCE (SDD) approach to derive solutions. So they have to identify the S, the D and the D and then solve it.
They’ll either use the air number line or move their finger along the number line printed on the page. The number line on the page is printed with fingertip-size circles on it so that pupils can ‘touch’ the number line.
Solving without using SDD – Finally, in the second or third lesson, they get questions to solve without being asked to identify SDD. We see many pupils still using the air number line of their own accord though in order to answer the question. Eventually, most have a mental picture of the number line that they move up and down.
Order of questions – The order of question-type is important:
- Adding/subtracting positive numbers that don’t bridge 0. e.g. 5 – 4
- Adding negative numbers to positive numbers that don’t bridge 0. e.g. 5 + -4
- Adding negative numbers to positive numbers that do bridge 0. e.g. 5 + -6
- Subtracting positive numbers from negative numbers. e.g. -5 – 4
- Adding negative numbers to negative numbers. e.g. -5 + -4
- Subtracting negative numbers from negative numbers that don’t bridge 0. e.g. -5 – -4
- Subtracting negative numbers from negative numbers. e.g. -5 – -6
Getting to mastery – All told this is about 3 or 4 lessons but they’re not experts by this stage so we keep the skills ticking along in Do Nows, quick warm-ups and homeworks for a couple of weeks. They feature regularly on the same throughout the year.
Multiplication & Division – won’t surprise you to know that we save multiplication and division of negative numbers for a little while (two or three weeks separation).
Following year – In year 9, at the beginning of lessons, they will do 50 negative number questions broken into 5 batches of 10. They are given a few minutes to start with but as the days and weeks go by, they are given less and less. They do this every day for about 6 weeks in the Autumn term and then sporadically in Spring and Summer.
That about does it.