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?
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.
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?
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?
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.
The same approach will work with 3 and 7. Here you can get 3, 6, 7, 9, 10, and everything from 12 onwards.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/3-and-5-stamps at 8:56pm on the 26th February 2024