4c and 7c Stamps
Describe what it means for a sequence to carry on infinitely;
Look for patterns using linear combinations of 4 and 7.
This problem involves the use of sums and multiples of 4 and 7 (linear combinations of 4 and 7). It also involves understanding that if you have 4 consecutive numbers then you can produce all subsequent numbers by adding multiples of 4 to those numbers. For example: starting with 23, 24, 25, 26 and adding 4 to each gives 27, 28, 29, 30, and then adding 4 to these gives 31, 32, 33, 34, and so on.
This problem is the second of a series of six problems that develop from a specific stamp problem to a quite general one. The other problems in this series are 3c and 5c Stamps, Number, Level 4, 5c and 9c Stamps, Number, Level 4, What is s?, Algebra, Level 6, What is t?, Algebra, Level 6, and What is s and t?, Algebra, Level 6. The earlier problems look at two more specific problems following on the theme of the current problem. While we hint at generalisations in the earlier problems we don’t follow these through until Level 6. If you plan to use more than one of these problems, it is probably a good idea to do them in order.
The Otehaihai Post Office only sells 4c and 7c stamps. What amounts of postage can be made up from these denominations? (The Post Office has an inexhaustible supply.)
- Pose the problem to the class.
- As a class list some of the values (4, 7, 4 + 7 etc).
- Let the children work on the problem with a partner. Ask them to look out for conjectures (guesses).
- The children may become stuck. For instance they may say that “the answer is all combinations of 4 and 7, so what is there to do?” In this case, you will need to tell them that they are looking for some simple way of telling if a number can be made from 4 and 7 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 18 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 4 to a single number or a set of numbers.
- Share solutions.
- Encourage them to go on to try the Extension problem.
- Ask them to write up what they have done. This should include a justification of what they have found.
Extension to the problem
How would things change if the stamps were 4c and 9c?
Can you guess a general result with two denominations of stamps where one denomination is 4c?
The solution method that we give here follows the same pattern as that of 3c and 5c Stamps, Level 3.
A good way to start here is to experiment. For instance, make a table showing the numbers 1 to 30 and put a tick against those that can be made and a cross against those that can’t. What amounts seem to be working are 4, 7, 8, 11, 12, 14, 15, 16, and everything from 18 onwards.
Now that makes a nice conjecture but how can it be justified? Can you make 18, 19, 20, and 21? Yes. Fine, then add 4 to each of these and you’ll get 22, 23, 24, and 25. But then add 4 to all of these and you’ll get 26, 27, 28, and 29. Can you see now that eventually you will get any number you want that is bigger than 18, simply by adding enough fours?
(Alternately, though a little longer, show that 18, 19, 20, 21, 22, 23, 24 can be done and add sevens to get to any number above 18 that you want.)
Solution to the extension
The same approach will work with 4 and 9. Here you can get 4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, and everything from 24 onwards.
Now 4 and t is a different kettle of fish. We suggest you get the students to try various values of t so that they can look for patterns and produce a conjecture. Forget about the small values you can get. They might come up with the conjecture that, from 3(t – 1) onwards, you can get all numbers. That is on the track but doesn’t work if t = 8. In fact you need 4 and t to have no factors in common in order to get 3(t – 1).
Proving this last conjecture requires a bit of algebra that may be beyond Level 5. But if you have a particularly bright group you might like to try it out. We give the full proof in What is t?, Level 6.