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Negative Dice

Weekly Problem 16 - 2012
If the odd numbers on two dice are made negative, which of the totals cannot be achieved?

Strange Bank Account (part 2)

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem builds on the problem Strange Bank Account and the game Up, Down, Flying Around, to explore both addition and subtraction of positive and negative numbers. We suggest teachers read the article Adding and Subtracting Positive and Negative Numbers to see a variety of contexts that can be used to develop understanding of operations with directed numbers.

Possible approach

Remind students of the rules from Strange Bank Account.
"Can you find a couple of different ways of increasing the amount in Alison's account by £5?"
Tabulate suggestions on the board:

Deposits $(+£2)$ Withdrawals $(-£3)$ Calculation Outcome
... ... ... ...
$4$ $1$ $4 \times (+£2) + 1 \times (-£3)$ $+£5$
$7$ $3$ $7 \times (+£2) + 3 \times (-£3)$ $+£5$
$10$ $5$ $10 \times (+£2) + 5 \times (-£3)$ $+£5$
... ... ... ...

"Have a look at the suggestions so far. What do you notice?"
"Are there any patterns that we could continue?"
"Could we continue the table upwards as well as downwards?"

To answer this last question, it may be necessary to introduce the idea of cancelling transactions. For example, $1 \times (+£2) - 1 \times (-£3)$ can be interpreted as a deposit of £2 and then cancelling a previous £3 withdrawal, leading to a £5 increase in balance.

Students could then investigate making other amounts in many different ways.

Key questions

Why does the number of withdrawals increase/decrease by 2 as the number of deposits increases/decreases by 3, if we keep the total outcome the same?
Can all outcomes be made in many ways?

Possible extension

Once students are confident at manipulating positive and negative numbers, Weights offers an interesting investigation.

Possible support

Play the variants of Up, Down, Flying Around to make sure students' understanding of operations involving directed numbers is secure. The article Adding and Subtracting Negative Numbers offers different models for helping students to make sense of negative numbers.

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