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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.
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:
||$4 \times (+£2) + 1 \times (-£3)$
||$7 \times (+£2) + 3 \times (-£3)$
||$10 \times (+£2) + 5 \times (-£3)$
"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.
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?
Once students are confident at manipulating positive and negative numbers, Weights
offers an interesting investigation.
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.