Today we proceed with the next lesson in the text. Lesson 6-4 of the U of Chicago text is all about applying reflections to games such as miniature golf and billiards. I don't need to make any changes to the lesson, so I can just keep what I wrote last year for this lesson almost intact:
One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired almost exactly 24 years ago, was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."
Lesson 6-4 of the U of Chicago text discusses miniature golf and billiards. Just as Bart learns in this above video, one can use geometry to determine where to aim.
The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:
"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce off
We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:
Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG
1. H refl. over line AB is H' 1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG 4. Transitive Property of Equality
Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.
Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:
"Pictured [in text] is a table with cushions w, x, y, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]
"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"
In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.
In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.
For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page, so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.
Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.
Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.
Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right after the week-long Thanksgiving break.