# Treasure Map

- Facility with drawing equipment
- Proofs involving congruent triangles

- Rulers, protractors
- Worksheets
- Transparencies

### 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?

**Isn't the bit about picking any point W a little vague?**

*Critical Question*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.

**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?**

*Critical Question*### The Investigation

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

**How is this map different from the typical treasure map?**

*Critical Question***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.