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.

A few questions …

Why is your number line vertical rather than horizontal?

In the section about order of questions 1) adding/subtracting positive numbers that bridge 0 your example does not bridge zero is that correct?

I agree with not using analogies as I think these can cause misconctions that we as teachers aren’t even aware that we cause and if a student has 5 teachers they may face 5 different analogies over the course of KS3 and KS4- not surprising then that students get confused. I really dislike when students tell me two negatives make a positive and it’s almost impossible to undo that once they think it’s the case….

Hi Dancingscotty,

The number line is vertical to mirror the air number line – desk level is 0, above the desk is positive, below is negative. We also put horizontal lines on the worksheets.

I’ve edited the first one – the example was right, the description was wrong. Just start with ‘plain’ adding and subtracting…as a warm up.

Glad you agree with the analogies. π

All the best,

Bruno

Just discovered your post so I know my reply is out of date but I can see another reason for having a vertical as opposed to horizontal number line: left-right confusion, which I have a feeling might contribute to the confusion…

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Thank you for this progression of lessons to teach addition and subtraction of negative numbers. It is well thought-out and sensible.

I do have a question – What about larger numbers? For example, 24 + (-59). Conceptually, this is the same but the magnitude of the numbers seems to give my students problems. Do you have any thoughts about this?

Thank you.

Seth Leavitt

Hi Seth,

Thanks for the comments – very kind. You’re right that there’s an extra step if you want to start working with bigger numbers. Depends on the question…

For questions like -24 + (-59) they need to be really comfortable with taking away from negatives to the point where they see it would be useful to add 24 and 59 together.

For questions like the one you’ve suggested, 24 + (-59), they need to be comfortable bridging past zero. So I would have activities set up where we break the subtrahend into two parts. Then the subtraction becomes 24 – 24 – 35.

If I was really using these large numbers for practice, which I have to say I don’t, as long as they understand START, DIRECTION, DISTANCE then I think/hope they’ll develop their own robust strategies.

Thanks,

Bruno

brilliant post Bruno, especially the No analogies bit, really echos the way I teach this. The amount of times I’ve heard “-5-7 can’t be -12 because so and so taught me that two negatives make a positive” or similar is extremely frustrating.

So true.

Thanks Stephen.

A great post – making me analyse my own approach. I have long searched for an analogy that works for all students and had finally settled on the ‘ice cube and temperature’ one. I can see where the confusion comes when students apply the same rules/analogy to multiplying and dividing. Can I ask what is the main motivation for not using analogies? Do they not help some students who need an image to hold on to? I like to use visual or oral ‘hooks’ to help my students recall ideas and procedures and I often use one of the head teacher in the bath. May I ask what you tell students when they ask why do we go up when we subtract a negative and down when we add a negative?

I also wonder how students would respond to this approach if they are not following a mastery curriculum and have already met negative numbers, successfully or not? Would they appreciate going back to basics I’ll let you know when I try this method with my year 8’s next year! Fantastic food for thought! Many thanks,

Hi Anna,

Thanks for the feedback and questions. It’s given me the chance to think more about my position.

Analogies– Oral hooks are great and they have their place in teaching and learning.However, in the case of negative numbers when pupils are learning to calculate with them for the first time, the analogies add to the workload that the brain has to do. We’re asking them to grapple with the language of the analogy and then translate the analogy to the symbolic form. That takes a fair amount of mental processing.

To use the ice cube one as an example, they need to…

Each of those steps require a degree of mental processing and combined they can overwhelm a substantial portion of the class. As much as the hooks are fun, I have never had success with the

wholeclass in the way I’ve seen with SDD.For me, the time to use the analogies is once they can work fluently with the numbers using the SDD method to the point where it’s automatic. Then they’re ready to hear why it works. Of all the analogies the ice cube is probably best at getting across why subtracting a negative is equivalent to adding.

What do I tell them?To start with, simply that “When we’re working out which direction to go, ‘normally’ with subtraction which way do you go? [Down.] Correct. However, if you’re subtracting a negative, you move in the opposite direction.”If they really press for why this is the case, then I’d need to judge the timing, nature of the question, the person who’s asking the question, etc. and make a decision as to whether it’s the right time to explain using the ice cube analogy. In every case, I’d prefer for them to be utterly proficient in the computation before I explain why.

I want to clarify that I place a great deal of importance on ‘understanding the maths’ – I’m far from being a ‘just do maths’ teacher. When it comes to negative numbers, I happen to have had more success driving understanding by doing

first. It raises the question of which topics are ‘better understood by doing first’ and which are ‘better done by understanding first’. Any thoughts?How would they respond if they are not following a mastery curriculum?I’m not entirely sure what you mean by the question. Please can you elaborate on the difference you think the mastery curriculum makes?I would definitely appreciate your feedback on how it works with year 8s next year! Thank you π

Bruno

Not sure I can see much of a difference between saying ‘two negatives make a positive’ and ‘ if you’re subtracting a negative, move inn the opposite direction’. Personally I really love integer discs and laughed out loud when I first saw it explained- especially when I finally ‘got’ why -x- really is a +, rather than just believing it as an article of faith received from my maths teacher. I don’t see why they are confusing. Most schools have some sort of merit/ demerit system- we give green positive points and red negative ones so there’s a context the kids are already familiar with and already know that if you’ve got 3 negative points then get 5 positive then you end up with 2 overall. So it’s a small step to add in the ‘grumpy teacher’ who says ‘ I was too grumpy yesterday so I am going to take away the 3 negative points I gave you yesterday…if I take away negatives ( modelled using integer discs) it’s like I’m adding positives. I do this alongside using a number line.

But I concede some of them say ‘ yeah, whatever, just tell me the rule’ and don’t want to know why.

Personally I get really excited when you have to make a zero +- pair. It’s just such a neat trick!

Hi Clare,

Thanks for your comments. You’re right, on the surface there’s not much between them but there is a subtle difference.

“Two negatives makes a positive” isn’t clear, for addition and subtraction at least, what you mean by two negatives. There are two negatives in -3 + -4, for example. Also two “negatives” in 3 – -4. What happens with *three* negatives, like -3 – -4?! The second problem is that “makes a positive” doesn’t clarify what you’re actually supposed to do.

“If youβre subtracting a negative, move in the opposite direction [to subtracting a positive]” does what it what it says on the tin.

Integer discs –You loved integer discs once you already knew the rule, that’s the difference. You were already fluent in the computation and you were ready to understand why it worked.Merits/demerits –Another good analogy but as with all the others, when you’re first starting out with negative numbers as a pupil, it’s cognitive overload to translate backwards and forwards between the analogy and the symbolic form. Too much overload to be able to solve correctly on a consistent basis. So pupils start to look for patterns and end up dangerously telling themselves “You just add them together and make the answer negative.” and other such shortcuts.I stand by doing the number line work first, developing the computational fluency and *then* layering on the understanding.

You’re right..making zero from a +- pair is a neat trick π

Thanks,

Bruno

Hi Bruno,

A fascinating blog and along with the comments and responses below it really starts to unpick the reasons students struggle and how we can introduce addition and subtraction with negatives.

I have always introduced addition and subtraction with negatives using patterns:

1 + 3 = 4

1 + 2 = 3

1 + 1 = 2

1 + 0 = 1

1 + -1 = …

The whole class would chant and find they could correctly predict what happened when adding the negative. I feel that it provides a hook and a hint of why whilst avoiding the use of analogies. I was wondering if you would still recommend using something like this after the skill has been mastered?

Thanks

Sam

Hello Sam!!

Ah, the old pattern trick. Generally a fan of this kind of thing (minimally different examples, etc.)

The pattern certainly highlights to the pupils that something strange is going on when they get to 1 + -1. They know the pattern shouldn’t break so there’s a “woh, hold on a minute” moment when they see that. You’re right there is no analogy in sight. I’m not averse to the analogies *after* they’re fluent though.

However, as a way to teach how to add and subtract negatives the pattern doesn’t sit long in the memory. It’s not easy for them to infer what to do everytime from a single pattern and therefore it’s hard to remember how to apply it. Will they need to write out the pattern each time to remind themselves what to do? Does it help conceptualise what happens with a negative starting point (like -1 + -1) if they are not secure with the number line?

Then there’s the need for a separate pattern for subtracting negatives…

1 – 3 = -2

1 – 2 = -1

1 – 1 = 0

1 – 0 = 1

1 – -1 = …

Should they remember that too?

There are too many variations.

The thing with START-DIRECTION-DISTANCE is that no matter what the sum, it’s universally true that START is the first number and DISTANCE the second number. That leaves DIRECTION as the only thing pupils need to make a decision on. I try to keep that decision making as simple as possible:

I know your question actually relates to whether this is a way to explain the why after the skill has been mastered so I went off on a tangent there. I would say it helps them reconcile what they used to know about adding and subtracting before you blew their world away with the revelation that adding doesn’t always lead to an increase. With this pattern, you settle their minds that the old world and the new world can live together. That said, the more convincing answers to the ‘why’ question come from the ice cubes or demerit analogies.

In summary, there’s a place for this pattern particularly in combination with positive/negative tiles. The sweetspot is probably between fluent computations and using an analogy to explain why negative calculations are the way they are.

Thanks Sam π

When referring to a calculation, the language used would confuse me. For, say -5 a 2, I would hear “minus five minus two”. Confusing. Would you refer to it as “Negative five minus two”?

Definitely not “minus five minus two”. Louise suggests the best approach – see comment below.

We were told at uni (last year in PGCE) to avoid the use of “minus” at all. Use “subtract” or “takeaway” for the operation and “Negative” for the sign.

I have found recently that using the word “minus” instead of “take away” or “subtract” definitely causes confusion, as soon as I change to one of the latter uses they understand what I am asking. This goes for both bottom and top sets.

Hi Louise, sounds like eminently sensible advice. I’ll try that. Thanks, Bruno

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Here is how you should teach addition and subtraction and negative numbers. You tell the following story:

A bus driver starts its route by driving his bus with no passengers on it from the depot to the first stop. 3 people get on the bus. At the second stop 2 more people get on the bus. At the third stop 1 person gets on the bus and 2 get off. At the fourth stop 2 people get on the bus, 3 get off. At the fifth stop 2 people get off the bus and 1 person gets on the bus. At the sixth stop 4 people get off the bus.

QUESTION: How many people need to GET ON the bus at the seventh and final stop so that the bus is EMPTY of passengers?

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Thanks for this post and the comments. I have been looking for something like this for over a year. I found the ideas and discussion very useful: vertical number line, 0 being desk level, no analogy, careful with terms using take-away and negative, introducing more complex sums in a controlled fashion.

I particularly like the Start-Direction-Distance analysis of the activity and it has prompted an idea that I’d like to check with you.

Imagine a counter that would fit over (and stick lightly to) the circles you use on your vertical number line. Imagine the counter had an arrow on it.

For the sum 5 – 3, the counter is put onto the 5 circle, the Start, then when the Direction is worked out, the counter is turned to point down. And is moved a Distance of 3.

For the sum -5 – -3, we start the same way, counter on -5, counter turned to face down – the Direction, and we move -3 units. Backwards. The third – meaning that the distance is negative reversing away from the direction that the counter is pointing.

I’m not sure that this idea would help, or hinder. Hinder because each of the “minus” signs has different meanings – the first determining location on the number line, the second meaning direction of travel on the number line and the third giving a distance, but including a directional element (forward or reverse). It works for me intuitively, but I haven’t thought through the consequences of the explanation.

I’d very much like to see how you explain multiplication and division involving negative numbers. I have been using the magic cauldron (ice cubes and fire cubes) and multiplication works fairly naturally by showing that multiplication is just multiple additions. Division is another matter.

Hi Nick,

Thanks for the feedback and for sharing your idea with the little arrows. My instinct is that you’ve come up with a good one. I like the physical and visual cues it’s providing. Two things I’d be conscious of:

1. Using them in a small group setting, e.g. a tutorial/intervention. Novel, fiddly resources in a whole class setting are prone to being used unfaithfully to their purpose (because it’s hard to check that all pupils are using them as intended every time).

2. After some success with your “discy arrows”, I’d probably step quickly to the hand above/below the desk because it’s a more efficient model and I want them to be more wedded to this in the long-term.

I do like it though. Must try it at the next opportunity.

Thanks again,

Bruno

ps on multiplication and division, I have nothing half as well thought out because there are fewer decisions pupils have to make. Ben probably says it most succinctly in the comments on Dan Meyer’s post. The whole discussion is worth a read.

I also use the pattern method especially when it comes to x and division.

With adding and subtracting I tried an X factor idea when students had to score an act from -5 to +5, we then used selected to students to show how adding a negative score resulted in a decrease and taking away a negative resulted in an increase.

I too am having to deal with misconceptions arising from 2 minuses making a plus, set in place in other classes- it’s like trying to de-programme year 7 that a prime can only be divided by 1 and itself!

π

A further comment on analogies – the first time I taught addition of negative numbers I found the analogy of adding ice cubes effective for students, but then realized I had left an unfortunate legacy for the physics department – students who think that cold is an actual entity, not just an absence of heat!

Very good point Lesley!

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Lesley,

As a physics teacher currently trying to teach my Year 12 class about thermal physics and a maths teacher currently trying to teach my Year 7 class negative number operations, your comment is highly relevant for me right now!

In trying to resolve this contradiction, I think I’m going to tell my students to imagine that the ice cubes are actually objects that don’t melt with their own internal power source (imagine they can somehow extract heat from their surroundings and then keep it insulated inside of them), that have the effect of decreasing the temperature whatever body of fluid they are added to by one degree. And we add them or take them away one at a time, waiting until one has done its job of lowering the temperature by one degree before adding the next one. And fire cubes are just heat sources, again with their own internal power source.

OK, that explanation is getting a bit long-winded for Year 7 – but basically I’ll tell them that they’re magic ice cubes, and just like the fire cubes, don’t really exist, but we’ll pretend they do for this thought experiment. And they know from experience that once you put the ice cube in, you can’t take the ice cube out again – part of it will have melted. Now that would be a good entropy lesson!

Benson

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Hi Bruno,

I’ve just found this, and your process sounds like it makes a lot of sense! Do you happen to have any worksheets for the SDD method available?

Sophie

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