Can you see them?

No?

That's because it's a pretend story.

And there's a heap of bananas over here (gesture). Can you see it?

No? That's because it's a potential pile at the moment....

You are only allowed to do two things:

A: add 3 bananas

B: take 7 bananas

A:

B is the same as adding 7 anti-bananas

an anti-banana:

[to begin more simply, it is easier to use A : + 2 and B : - 3

NRICH do this in their 'Strange Bank Account' resources with a fine introductory video clip]

You can do either of these things (A or B) as many times as you like.

How can you end up with just 1 banana in the pile?

What other numbers of bananas can be made?"

1 = 5A + 2B

2 = 3A + B

3 = A

etc.

Students continue, finding other (possibly simpler) versions of expressions.

They can detect patterns in a sequence of these expressions.

"Hang on a minute, if 1 = 5A + 2B how could you get 2? [10A + 4B]

But we've found that 2 = 3A + B.

So that means there are different ways of making some of these..."

"How did anyone make 5?"

4A + B = 5

11A + 4B = 5

18A + 7B = 5

how does this pattern continue?

why does this work?

what happens if we put the pattern into reverse? what is the expression before 4A + B?

why does this work?

what is the next one up from this? etc.

Let's try to prove that you can make all positive numbers [ they can either be derived from 1 or since 1, 2 and 3 can be made so can 4 (1 + 3) etc]

How can we make -1? [ an anti-banana pile]? Etc.

What is the connection between a positive number and its opposite?

1 = 5A + 2B

-1 = 2A + B

-2 = 4A +2B and 2 = 3A + B

-3 = 6A + 3B and 3 = A

Why does the connection work?

What could A - B mean?

But this is the same as 8A + 2B? Why?

Extend to other pairs of operations for A and B...

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