Challenge Level

This problem offers a context for the first explorations into the addition of positive and negative numbers, using the familiar idea of deposits and withdrawals from a bank account. It is intended to be used in a sequence of lessons together with the game Up, Down, Flying Around and Strange Bank Account (part 2), which introduce subtraction of positive and
negative numbers. The article Adding and Subtracting Positive and Negative Numbers offers a variety of models to refer to when teaching this topic.

Explain the rules for Charlie's Bank: "you are only allowed to deposit £2 at a time and withdraw £3 at a time" If necessary, show an example (perhaps the one in the video), and then invite students to choose a number of deposits and withdrawals and work out the combined effect.

Choose a few students to share their answers. Two possible lines of enquiry emerge:

Is it possible to change the account balance by **any** amount?

Can each amount be made in more than one way?

The second of these two questions is explored in the follow-up problem Strange Bank Account (part 2), so focus on the first question.

One possible way to explore is to invite students to try to make all possible amounts from £1 to £30 using only combinations of + £2 and - £3. Efficient notation naturally emerges; rather than writing down $+ £2 + £2+ £2+ £2+ £2$ it is much quicker to write $5 \times (+ £2)$.

Once students have had some time to work on this, take some time to discuss what they have found out. While students are working, you may wish to look out for insights into why it is possible to make every number. There are at least two nice methods for students to convince themselves that all totals are possible:

**Method 1**

All even totals are possible just by repeatedly adding + £2. I can make £1: (+ £2) + (+ £2) + (- £3) so then I can make all odd totals by repeatedly adding + £2 to my £1.

**Method 2**

I can make £1: (+ £2) + (+ £2) + (- £3)

I can make £2 by adding £1 and £1: {(+ £2) + (+ £2) + (- £3)} + {(+ £2) + (+ £2) + (- £3)}

So I can keep adding on (+ £2) + (+ £2) + (- £3) to increase the total by £1, and hence make any number.

Next: "The bank wants to change its rules, so that instead of depositing £2 and withdrawing £3, you need to decide the deposit and withdrawal amounts. You need to make sure that it's possible to make every possible total, so experiment with some different amounts. In a while, I'm going to choose a deposit and a withdrawal amount, and you need to be able to tell me straight away whether all
amounts will be possible."

You may wish to bring the class together for a mini-plenary to share what they have found out along the way, or collect together on the board observations about which pairs of deposit/withdrawal amounts give rise to all possible amounts and which ones don't.

Here are some pairs you might wish to offer students at the end of the lesson to check their understanding:

+5 and - £8

+8 and - £5

+5 and - £15

+ £6 and - £9

+ £4 and - £10

+11 and - £17

For a similar investigation that only requires consideration of positive numbers, see How Much Can We Spend?

*This problem is based on Don Steward's ideas. His resources can be found here.*

Up, Down, Flying Around and Strange Bank Account (part 2) follow on from this problem.