# Day 136: Do The Really Intersect?

Its the day before Spring Break.  We finished the Midterm yesterday and they did brilliantly!  Not a great time to introduce new material.  An easy way out is to “show a movie” but to waste precious time with students with an experience that has no real academic purpose goes against everything I believe.

So, how do I keep my precalculus students engaged and interested?  I wanted something that reviewed an old, but not critical, learning but added an intellectual twist that will catch and keep their attention.  And I remembered a workshop I attended about 5 years ago (I’m sorry but I just can’t find the name of who presented and would love to give you credit…just comment below with your name and any other info you’d like to share) using the Nspire to look at the relationship between polar and rectangular equations in graph form.  The presenter shared 3 Nspire documents that had nice animations.

For this experience, though, I used Geogebra because the kids could manipulate easily with their iPads.

It was the 4th document which inspired me to create an investigation around a system of two polar equations.  Luckily the document was created, but I wanted my students to reflect more while using the Nspire document.

For instance, I had the students do this:

Move to 2.1. Without talking to your group, watch the animation alone!  You must watch carefully because you cannot re-graph again.  Try to observe how many times these graphs’ paths intersect.

1. What did YOU notice?
2. Once everyone has completed the task alone, talk with your group. What did other people in your group notice? Try to paraphrase your discussion.

Then later the real thinking started.  My students  used their Nspire to do the following:

Move to 4.1. You will see the results of graphing the two polar graphs in the rectangular plane.  Does this affect your answer?  Be specific!

1. Use the rectangular graphs to give the coordinates of the points (in polar form please) of intersection of the limacon, r1 = 3 + 2 cosθ, and the four-leaved rose, r2 = 5 sin(2θ). Label the intersection points on both the rectangular graph and the polar graph.  The first point is labeled P1 on both graphs.

1. Why could the apparent intersection point Q1 be called a “false” intersection point? What aspect(s) of a polar graph make it appear to be a point of intersection?  Label the other false intersection points on the polar graphs in the same manner.
2. Show on the rectangular graphs above that the second-quadrant angle θ for point Q1 corresponds to a point on the limacon but not to a point on the rose. What are the coordinates of point on the rose that correspond to the location Q1?  Is there a mathematical relationship between the point on the limacon related and the point on the rose which correspond to the location Q1?

I was so delighted with how engaged my students were…to see their heads down, fingers posed over their iPads and some conversations, too.  The Math Practices were everywhere today! In fact, they didn’t even realize class was over, and that’s a biggie for a 7th period just before Spring Break.

Happy Spring Break!  Happy Easter!  I’ll be back in a week.