How Much Can We Spend?
Why do this problem?
This problem offers students opportunities to explore fundamental ideas about number theory in a simple context. They are encouraged to explore, conjecture, generalise and justify.
There are opportunities for older students who are familiar with algebraic manipulation or modulo arithmetic to produce rigorous proofs.
Possible approach
This printable worksheet may be useful: How Much Can We Spend?
Introduce the context, just two coins 3z and 5z.
- 7 is the largest total that cannot be made
- 8 and all numbers above can be made
'Could we have predicted this?'
Are there any conjectures?
'What about 3z and 7z? 3z and 8z? 3z and 9z?......'
Allow some time for the students to test and refine their conjectures.
Bring the students together to share their findings and display their results, focussing on the largest total that cannot be made and the number that follows.
Were any of the original conjectures correct? Which?
Can they now use the combined results to make some generalisations?
Can they use their generalisations to predict what will happen for any pairing of 3z with another coin?
With older students who are familiar with algebraic manipulation, there is an opportunity here to express the findings algebraically in different ways and convince themselves that they are equivalent.
Ask students to consider now what would happen when the pairings contain:
7z and another coin...
any prime number value and another coin.
Key questions
How do you know you can make all the totals after a certain total?
Does it make a difference if the other coin is 1 more or 2 more than a multiple of 3z? (or 5z or 7z...)
Possible support
The whole class introductory activity as described above should provide the necessary support for all students to access this problem.
Possible extension
Ask students to consider what happens if neither coin is a prime number.
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