Using postage stamps of ONLY denomination 5 cents and 7 cents, carefully complete the table below to find all the possible postages.

Confirm that 23 is not possible.

Then: Find 24, 25, 26, 27 and 28 in the table and explain why all greater amounts are possible.

Extension: Look for pairs of consecutive amounts of postage in the chart (such as 34 and 35).To get from 34 to 35, you can either

add 3 7’s and take away 4 5’s or you can add three 5’s and take away 2 7’s.

Write about why there are at most two ways to add one to a number on the chart.

Then explain why it is impossible to add one to 22.

Circle all the numbers in the table that do not allow you to add 1 in either way.

Explain why 23 cents is the greatest impossible postage.

Further extension: Do this for two other stamp values.

Greatest challenge: Find a pattern that relates the two stamp values to the greatest impossible postage.

Enjoy!