Negative number reasoning with patterns

English: Sandcastle Español: Castillo de arena

English: Sandcastle Español: Castillo de arena (Photo credit: Wikipedia)

How do you explain that subtracting a negative has the same effect as adding the positive corresponding number? I’ve seen teachers use many analogies including:

  • Thinking of sandcastles as +1s and holes as -1s… a sandcastle and a hole makes nothing (1 + -1 = 0).
  • Playing golf and you score -2 on a hole, taking your score from -4 to -6. However, you get disqualified on your last hole so you need to subtract the negative score (-6 – -2 = -4) takes you back to -4.
  • Stand on a numberline at your starting number looking towards the positives. Add by walking forwards, subtract by walking backwards. However if the number you’re adding or subtracting is negative, first turn around 180 degrees then walk forwards or backwards…

I have tried all these analogies with various degrees of success with different classes. One thing I have learned is that if you can show why something works using logic, rather than analogy it often seems to stick better with pupils. An analogy doesn’t explain why something works… An approach to showing how adding and subtracting with negatives works using a logical argument is to show them the following patterns:

Adding a negative

2 + 2 = 4

2 + 1 = 3

2 + 0 = 2

2 + -1 = 1 same as 2 – 1

2 + -2 = 0 same as 2 – 2

Subtracting a negative

-2 – 2 = -4

-2 – 1 = -3

-2 – 0 = -2

-2 – -1 = -1 same as -2 + 1

-2 – -2 = 0 same as -2 + 2

Build each pattern up so they see the answers rising or falling by 1 each time and then when you get to the adding or subtracting a negative part the pupils see what the answer must be through logical reasoning. I have found this a nice approach because it allows pupils to see that when calculating -2 – -2 that you are doing an easier equivalent calculation that returns the same result -2 + 2, because it makes it easier. It has meaning based on logic rather than them having to remember and apply some obscure analogy.

You can take a similar approach to show the effect of multiplying and dividing with negatives…

One to have up your sleeve!

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3 Responses

  1. Steve Croft says:

    Always a tricky topic to teach. I like this approach because it is simple and uncluttered.

  2. Shaun says:

    I agree that seeing the pattern form is a useful way of teaching how to work with negatives. It might also be a useful point to demonstrate that 2 + 2 is really (+2) + (+2), and the simplification causes the signs to combine. With this angle, that might help to explain why (+2) + (-2) winds up being 2 – 2.

  3. Sharon says:

    This is an excellent method to teaching adding and subtracting negative numbers. It also cements that positive numbers are in fact directed numbers yet as a convention we don’t write the + sign. Brilliant thanks for posting this.

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