SQUARES AROUND A TRIANGLE

by Gary Flewelling

Introduction

What follows is just the beginning of a large investigation that is described in a some detail in an article entitled “Sense Making: Changing the Game Played in the Typical Classroom” . This article  appears both in the AMT: The Australian Mathematics Teacher Journal, 2002,  and as an appendix in  A Handbook on Rich Learning Tasks (2nd revision, 2002) by  Flewelling with Higginson, published (in CD format only) by the Centre for Mathematics, Science and Technology Education, A233 McArthur Hall, Faculty of Education, Queen’s University, Kingston, Ontario, Canada, K7L 3N6, Tel: 613-533-6221, E-mail: 

Welcome to Squares Around a Triangle. 

You have just landed on the shores of a small unexplored geometric world. At first all you see is a random triangle with squares drawn on each of its sides. You also notice that the centers of the three squares have been connected.

SQUARE1.GSP
 
Some of you “explorers” might want to push off on their own after you have taken the first few steps on this “voyage of discovery”.

Start your voyage of discovery.

1. Open the sketch  SQUARE1.GSP 

2. Drag a vertex of the triangle.  Notice some of the things that change as you drag this vertex.

3. When do the segments joining centers of the squares pass through the triangle’s
 interior? exterior? a vertex?

Three outer triangles make an appearance.

SQUARE4.GSP

4. Open the sketch SQUARE4.GSP .

5. Drag a corner of the inner triangle.  Investigate  properties and relationships of the four triangles. Record your findings.

Two more triangles enter the picture.

SQUARE5.GSP

6. Open the sketch SQUARE5.GSP .

7. Drag a corner of the inner triangle. Investigate the properties and relationships in this sketch. Record your discoveries.

Continuing the Investigation

8. Reader, take the investigation in a direction of your choosing.  If you prefer to be given some suggestions, consult “Sense Making: Changing the Game Played in the Typical Classroom” referred to in the Introduction.