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Teaching Notes for Postage Stamp Arithmetic

Phase 1

Ask your students to read the original problem statement.  Better still, with the help of a student or aid you could act it out!  Let them think about the problem for a few minutes, but explain that you'll start with some smaller numbers and return the 33 and  20-cent case later.

Have students work in pairs or triplets on the warm-up problems 1, 2, and 3.  

When it seems everyone has completed #1, ask people to pause in their work and share some solutions.  Most likely there will be a couple of different solutions.  If everyone did the same thing, you may want to present an alternate solution yourself.    Here is one solution based on the fact that two equals seven times four plus thirteen times negative two..  "Dump" means pour as much as possible, keeping what's left.

Gallons in the seven gallon pail Gallons in the 13 gallon pail
Fill 7 and dump into 13 0 7
Fill 7 and dump into 13 1 13
Empty 13 1 0
Dump 7 into 13 0 1
FIll 7 and dump into 13 0 8
Fill 7 and dump into 13 2 13

Ask students to continue with the stamp warm up problems.  

These problems will probably take the rest of the class time -- and then some!   The progress they make will depend a lot on what your students already know, and on how comfortable they are working on open-ended tasks.  If students aren't sure how to approach the problem, you can suggest making a table of numbers from 1 to 30 and writing "yes" or "no" depending on if they can make the postage or not.  Better still is a table like this:

yes     3   5 6   8 9 10 11
no 1 2   4     7        

Patterns emerge pretty quickly, and students develop techniques like "can I add 3 or 5 to an earlier "yes" to get this number?"

These problems may take more than one day of class time depending on your students and the length of your class.  If some groups are working more quickly than others, encourage them to test other cases, looking for a relationship between the values of the stamps  and the largest amount of postage you can't make.

When everyone has completed the first three warmup problems, discuss the results.  Was there always a point beyond which you could make all values?  What was different about 3 and 6?  What does "relatively prime" mean?  What would happen with 8 and 10?   (With 3 and 6 some kids might thing that divisiblity of 3 into 6 is the problem that motivates the "relatively prime" condition)

The discussion should end with a conjecture like this one:  "If a and b are relatively prime, then there's some point beyond which you can make all values of postage using just a-cent and b-cent stamps.   Explain that the next goal is to figure out where that point is and how it depends on a and b.    Also discuss the differnce between the bucket and stamp problems -- you can't have negative stamps but you can "subtract" with buckets.

Phase 2

Discuss the solutions to the warm-up problems.  

Give each student or team of students a pair of relatively prime denominations, each between 3 and 11 cents.   Suggest that they identify which amounts can't be made and make any other observation that emerges.

Here is a random list of these pairs that you can use as the sign-up sheet that appears below.  Two spaces are provided in case you want to assign two students to each number pair.   You can also copy this sheet, print it on a transparency sheet,  and use it to collect results.

{4, 5} __________________________ __________________________
{9, 11} __________________________ __________________________
{7, 8} __________________________ __________________________
{9, 10} __________________________ __________________________
{7, 10} __________________________ __________________________
{7, 9} __________________________ __________________________
{3, 10} __________________________ __________________________
{3, 8} __________________________ __________________________
{5, 11} __________________________ __________________________
{4, 9} __________________________ __________________________
{4, 7} __________________________ __________________________
{4, 11} __________________________ __________________________
{5, 8} __________________________ __________________________
{7, 11} __________________________ __________________________
{5, 6} __________________________ __________________________
{5, 9} __________________________ __________________________
{10, 11} __________________________ __________________________
{3, 11} __________________________ __________________________
{6, 11} __________________________ __________________________
{8, 11} __________________________ __________________________
{6, 7} __________________________ __________________________
{8, 9} __________________________ __________________________
{5, 7} __________________________ __________________________

Phase 3

Collect the students' work.  You should anticipate that some of the results will be wrong.  This is a good point at which to discuss error-checking.  Pose the question  "How can you be really sure that a certain amount is the largest one that can't be obtained?"   Point out that the most general answer to this question is our ultimate goal and so you won't resolve it yet, but make sure that this question is always in the background.

Discuss observations that might be made based on the results.  If nothing general emerges, give the students time to reflect.   Suggest that they concentrate on both the number of unattainable amounts and what those amounts are.   If that doesn't generate some discussion, suggest that they concentrate only on results where some specific denomination, like 3.  Then put the results into a table like this:

Denominations three and four three and five three and seven three and nine three and ten ...
Largest amount that cannot
be made

Give the class a chance to prove any conjectures that might be suggested.  While the proposers of the conjectures are attempting proofs, you might want to spend time with some of the others verifying that the conjectures are true for warm-up problems 2 and 3.   

Assignment:  Have students continue working on proving their conjectures.   Have them also come up with a problem that is related to the postage stamp problem.  

Phase 4

Allow time for students to report on their success in proving their conjectures from the previous phase.

Depending on whether you want to leave this problem as a long term project or want to reach some closure, you'll need to decide what to do if the class hasn't figured out the formula for the largest amount that can't be made in terms of a and b.  You might want to give them further hints.

Allow enough time at the end of the class to discuss the new problems the students came up with for the assignment in the previous phase.  How much time you spend on these extensions will depend on whether you complete the original problem and/or how engaged the students are in solving the original problem.

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