This problem solving activity has a number focus.
The Otehaihai Post Shop only sells $3 and $5 stamps for larger letters and for parcels.
What amounts of postage can be made up from these denominations?
- Describe what it means for a sequence to carry on infinitely.
- Look for patterns in combinations of numbers.
This problem involves the use of sums and multiples of 3 and 5. It also involves understanding that if you have 3 consecutive numbers, then you can produce subsequent numbers by adding multiples of 3 to those numbers. For example: starting with 13, 14, 15 and adding 3 to each gives 16, 17, 18 and then adding 3 to these gives 19, 20, 21 and so on.
This problem is the first of a series of six problems that appear at other levels. There is also a series of three Table problems students might explore before solving this problem. These are Jim’s Table, Algebra, Level 1, Jo’s Table, Algebra, Level 2, Sara’s Table, Algebra, Level 3.
- Copymaster of the problem (Māori)
- Stamps/ envelopes to pose the problem
- Copymaster of the problem (English)
The Problem
The Otehaihai Post Shop only sells $3 and $5 stamps for larger letters and for parcels. What amounts of postage can be made up from these denominations?
Teaching Sequence
- Discuss mail. Ask:
How often do you or your family post letters?
Why have postage stamps become more expensive?
Do you think more or less people post letters now than they did 5 years ago? Why do you think that?
Who thinks there will still be letter mail in 5 years? Why do you think that?
How else do you communicate with people in far-away places? - Pose the problem to the class.
- As a class list some of the values important to the problem (3, 5, 3 + 5 etc).
- Have students work on the problem, making it clear that they are looking for a simple way of telling if a number can be made from a combination of 3 and 5 or not. You might suggest that they start at 1, then 2 and so on to see which values they can make.
- If they can see that it looks like everything from 8 onwards can be made, ask them if they can justify this. You might need to encourage them to think about what happens when you keep adding 3 to a single number or a set of numbers.
- Share solutions including giving a justification for what they have found.
- Make the extension problem available.
Extension
How would things change if the stamps were $3 and $7?
Can you investigate and suggest a general result with two denominations of stamps where one denomination is $3?
Solution
One approach is to make a table showing the numbers 1 to 20 and put a tick against those that can be made and a cross against those that can’t. From this it seems that 3, 5, 6, and everything from 8 onwards can be made. See also: Sara’s Table, Algebra, Level 3.
This is a conjecture that should be justified: You can make 8, 9, and 10. Add 3 to each of these and the sums are 11, 12, and 13. Add 3 to each of these and the sums are 14, 15, and 16. In this way you will get any number you want that is bigger than 8, simply by adding enough threes.
Alternately, though a little longer, show that 8, 9, 10, 11, 12 can be made, and fives can be added to get to any number above.
Solution to the Extension
The same approach will work with 3 and 7. Here you can get 3, 6, 7, 9, 10, and everything from 12 onwards.