Geometry lesson: Collapsible chair model

We live in a 3-dimensional world but why is it that we start teaching geometry we start from 2-dimensions to 3-dimensions? Shouldn’t it the other way around? Aren’t we supposed to start from students’ experiences? Making connections is not only about building connections between new concepts and previously learned concepts but most especially connecting it to students’ experiences in the world.

I had the chance to observe a public lesson in Japan when I attended a Lesson  Study conference. The lesson started with a classic problem in kinematics. Kinematics is a branch of classical mechanics that describes the motion of bodies (objects) and systems (groups of objects) without consideration of the forces that cause the motion. Put simply, kinematics is the study of how things move. I tweaked the lesson a bit by starting, instead of the history of the problem, I started with a video and photos where the concept that is introduced in the lesson is applied in real-world.  You can download the original Japanese lesson plan (yes, it’s in English). This lesson was given to Year 9 students. They’ve yet to learn analytic geometry.

Here’s my proposed lesson outline:

Introduction: For introduction and to catch students imagination you can show this video about space-saving furnitures. Then, follow it up by showing photos of collapsible furniture below.

You may use the following guide questions: What do you think is common to the chairs and gate below?  How do the parts move? What are the paths traced by the tips and intersections when they are folded?

Focusing on the parts that will be modeled:  Consider the movement of the back of the white chair. If the front legs is slid along the floor towards or away the back legs, how
will the back of the chair move? What about the path traced by the point intersection of the legs? (It’s best to have with you a similar folding chair so the students can experiment on the movement.) Make a 2-d drawing of the parts we are investigating.

Presenting the mathematical model: The most ideal of course is for students to be able to think of making the geometric representation and articulating the problem themselves. However, if they have not acquired that skill then you may present the mathematical problem. Problem: In the figure above, assuming that CG = GF = GA, if C is moved along CA, what is the path of G and F?

Going deep into the geometry: How can you be sure about the path (locus) of G? How about the path of F? Explain or write a proof. Click this link for the solution.