Ohio Resource Center

Treasure Map

Estimated Time
Two half-periods, one for drawing and one for the proof.
Prerequisites
• Facility with drawing equipment
• Proofs involving congruent triangles
Materials Needed
• Rulers, protractors
• Worksheets
• Transparencies
Ohio Standards Alignment

Topics

Problem solving heuristics, constructions, formal proof with SAS & SAA

Overview

A treasure map has bizarre directions that seem a little vague but, indeed, locate a treasure accurately. Students construct examples on a worksheet to help them form a conjecture as to what is going on. They then undertake a formal proof of the conjecture. It is quite a surprise as to what is ultimately behind these vague directions.

The "Hook"

It is not quite clear how it happened, but here you are on Treasure Island, holding an old, weatherbeaten, but still legible map. The map tells you to find a lone pine tree at the edge of the beach, and a large pinkish boulder nearby. You worry about whether they are still there, but you manage to find both the tree and the boulder, and you continue reading the map.

It says: "Call the tree P, and call the boulder B. Put yourself on the line PB, somewhere between the two, and walk out on the beach toward the water. Stop wherever you like, and mark the point W where you stopped. Now, carefully locate two points X and Y on the inland side of PB, as follows:  PX = PW and angle WPX is a right angle.  BY = BW and angle WBY is a right angle. Dig at the midpoint of XY and you'll find the treasure there."

These directions are not so hard to follow, but wait a minute: just what is going on here?

Critical Question
Isn't the bit about picking any point W a little vague?
Well, you try the directions anyway, you find that midpoint, and what do you know, there is a small, elegant, wooden chest. Locked. You will have to wait until you have something to open it with, but in the meantime, since -- for all you know -- you will be stuck on this island for a while, you decide to figure out how these directions work.

You start over, and walk out to a different point W, follow the directions, and amazingly, you wind up right back at the treasure chest.

Critical Question
What is going on? How can this be?
Critical Question
Since you can stop anywhere you want, where might be some clever places to put the point W so you can get some ideas how this works?

The Investigation

Read the problem with the class (give them a copy, along with the worksheet Figure 1).

Critical Question
How is this map different from the typical treasure map?
Pass out drawing equipment and give the students plenty of time to work out the directions with at least two different placements of the point W. If the students work carefully, they will see that they do, in fact, arrive at the same place to dig each time (see Figure 2).  If two students stack their papers and hold them up to the light they will get even more corroboration -- again, if they have worked carefully.

Once they see that the directions work, start encouraging them to think of clever places to put W -- for instance, at the midpoint of the segment PB, or right on P, or right on B. When that is done, it should become clear that there is a large square, with side PB, and with the treasure in the center (see Figure 3).

Teaching Tips

• The mental leap to the "aha" moment of envisioning that square may be a big one. This will be the trickiest part of the teaching -- to lead students to the conjecture that the treasure is in the center of that square without actually giving it away.
• You may want to prepare transparencies ahead of time with key placements of W.
• Once the news is on the table, the task is to prove it. There is a worksheet (see Figure 3) with a good diagram for the proof. It will be up to you and the class to see how much they can do by themselves.

Citation

From the teaching files of Rudd Crawford.