Découverte : Le cercle unitaire

This one takes 2 days!  It takes some monitoring, some very clear instructions, glue and scissors, but the result is worth it.

I have been struggling for years with a way to help students discover/understand the unit circle.  Why do we care?  Why is it there?  Typical answers have always been :

• Because there are angles larger than 180 degrees, so we can just stick to triangles
• Because there are rotational angles so we need to talk about them
• Because it helps us organize sine, cosine and tangent for any angle, especially the remarquable ones
• Because it helps us model trig identities

Lots of great answers but I think the students always memorized it, accepted it as true and went on with their lives.  

L’année passée, je me suis assis avec mon collègue pour discuter d’une façon de démontrer le cercle unitaire en utilisant les connaissances antérieures des élèves.  J’ai vu d’excellentes idées sur Pinterest et celle-ci m’a sauté aux yeux.

Pourquoi ne pas construire des triangles et utiliser les triangles semblables pour faire un lien entre le cercle unitaire et les rapports trigonométriques de base?

Voici le plan de ma leçon et je tenterai de le résumer ici suivi par un diaporama de ce que l’élève créera :

Day 1

1. I separate the class into groups, making sure that I have sets of 4 groups.  This is because each group will create 1/4 of the unit circle.
2. Students construct right angle triangles of given hypotenuse using 3 different colours of paper.  They create congruent triangles with the same paper colour.
3. Students make sure to take note of all angles and dimensions of their triangles which they will use later.
4. I give each group a different corner of a white piece of paper to glue their triangles.  This is where I can talk to them about where the rotational angle should be and that the right angle is opposite of the corner.

Ce diaporama nécessite JavaScript.

Day 2

1. Each group now has a 1/4 of the circle.  I tell them to get together.  This is where I will hear the following : Oh!  I think we are supposed to be a circle!  If you put our pieces together, the triangles form a circle!
2. Once they have their circle figured out, this takes some time because they need to think about which angle joins in the middle (Geolegs help them model this), I have them write down the angle that is in the center of the circle for each of their triangles.
3. They realize at this point that the triangles in the other quadrants of the circle can’t be between 0 and 90, so they start to use trig identities and find all their angles.
4. I then visit each group and talk to them about models – math models should help us represent other situations/other circles.  Would you be able to use your model to quickly tell me the measurements of a circle with a larger radius?  a radius of 51cm?
5. WHAT KIND OF CIRCLE COULD HELP US REPRESENT ANY OTHER CIRCLE EASILY?  This is where I get the unit circle out of them.  They communicate to me that if our circle was of a radius of 1, we could just multiply all the dimensions of the triangles to represent any other circle.
6. They get to work using similar triangles and find the values of (x,y) for every point on their circle of radius 1.
7. BIG IDEA: CHECK OUT THOSE COORDINATES, YOU HAVE SEEN THOSE NUMBERS SOMEWHERE BEFORE!

OMG!  Madame, the values of (x,y) of our circle of radius 1 are the same values in our trig table.  So, the sine and cosine of the angle are the (x,y) of the circle!

All supporting documents can be found in the LESSON PLAN.