From circles to ellipses and functional transformations

The post is a wrap-up to the geometry class. I've run out of time and won't be able to have the class finish this, but I wanted to outline how the course would proceed to the end.

Students have just learned how to find the area of a circle's segment. This is exactly what is needed to find the area of overlap between two circles. The next step would be to give students some simple situations where the sector angle measurement and radii are known.

The next step would be to introduce the idea that all they know are the centers and radii of two circles. After considering this situation for a bit, the hope would be that they recognize they need to find the points of intersection of the two circles.

Because of the messiness of finding the points of intersection for two circles in the general case, I would focus on the situation where the centers fall on either the same vertical or horizontal line. These situations allow students to write out equations in standard form and then substitute expressions to obtain a quadratic equation in one variable. From here, using the quadratic formula or some other technique, they can find the two values for that variable. Back substituting into the original equations would provide the two values of the other variable.

When developing this approach, I discussed this idea with a colleague. He teaches pre-calc and I wanted to see if there were things they were doing in that class that I could bridge. Basically there wasn't anything to connect to directly. He did agree that the general case would be messy and that keeping the situations simpler would probably be best.

Knowing the points of intersection of the two circles now provides additional information to use. In this case, the radius is given and the base of the triangle has been calculated. These two pieces of information, along with an appropriate trigonometric ratio, can be used to find the central angle. This leads students to be able to calculate the segment area.

This investigation and work provides students many connections between their circle work, the ideas of area, and trigonometric ratios. I feel this is a good way to begin a review process without actually stopping everything to review for a final.

The last unit was to cover functional transformations. The idea was to do some algebra review as students start thinking about algebra 2 for the following year. My feeling was that this should be developed from the circle work rather than moving into linear or other functional forms. A colleague really wanted to move toward connections with ellipses.

With all the circle work, the idea of translations and how this affects the equations of circles was a central part of the circle unit. This was an introduction to functional transformations, but in a much more natural way.

The next step was going to be "normalizing" the equation of a circle. This would be accomplished by dividing all terms in the standard form of a circle equation by the square of the radius:

(x - h)² + (y - k)² = r² goes to (x - h)² / r² + (y - k)² / r² = 1

The "normalizing" is drawn from the concept in statistics of dividing through by a standardized value. I wanted a term to distinguish a standard form of the equation from this new form.

The next step would be to re-write the normalized equation as
((x - h) / r)²  + ((y - k) / r)²  = 1

A final modification in notation and we'll be ready to have some fun. Re-write the equation with subscripts for the radii as
((x - h) / r₁)²  + ((y - k) / r₂)²  = 1

The investigation would allow students to change any of the values for h, k, r₁ or r₂. What deformations of the circle can be made? How would a dilation that makes the figure bigger look? How would a dilation that makes the figure smaller look? What happens as one of the r values gets very large or very small.

This could be expanded into taking two ellipses and asking how you could tell if they overlap. You could look at approximating the overlap of the ellipses. You could ask students to re-write these in the form y = or to expand the equation out and ask what completing a square, as they did with circles, would look like. What is similar and what is different? How can they determine if the equation results in a circle or an ellipse?

That would be the concluding unit for geometry. If any one has the class time to go down this path, I'd love to hear how it worked for you. If you do want to try this, but are unclear about some of what I wrote, please feel free to contact me.

Next year I am back to teaching discrete math and statistics class only. I'll be blogging about changes to my discrete math content as we move from a single semester course to a full-year course.

I'll also be highlighting changes to my probability and statistics content as I add additional subjects into the course. I found myself short on content and having students doing too many projects during the second semester.

Have a great end to the school year!