From our textbook:
Stuff like this makes my heart sink. (I actually wrote that it makes me fart — but that’s very unladylike. And I’m trying to write better.)
There is essentially nothing left in this problem for students to explore and figure out on their own. If anything, all those labels with numbers and variables conspire to turn kids off to math. Ironically even when the problem tells kids what to do (use similar triangles), the first thing kids say when they see a problem like this is, “I don’t get it.”
They say they don’t get it because they never got to own the problem.
I wiped out the entire question and gave each student this mostly blank piece of paper and the following verbal instructions:
- Make sure you have a sharpened pencil. Write your name and date.
- Inside this large rectangular border, draw a blob — yes, blob — with an area that’s approximately 1/5 of the rectangle’s area. No one will die if it’s not quite 1/5.
- Next, draw a dot anywhere inside the rectangle but outside the blob. Label this dot H.
- Now, draw another dot — but listen carefully! — so that there’s no direct path from this dot to the first dot H. Label this second dot B.
I asked the class if they knew what they just drew. After a few silly guesses, I told them it was a miniature golf course: blob = water, point B = golf ball, point H = hole location.
The challenge then was to get the ball into the hole. Since you can’t putt the ball directly into the hole due to the water hazard, you need to make a bank shot.
(Some students may have drawn the blob and points in such a way that this was not really possible, at least not in one-bank shot. I let them just randomly pull from the stack of copies to pick a different one. I made a copy of their sketches first before they started their work.)
The discussions began as they started drawing in the paths. One student drew hers in quickly and asked, “Is this right?” I replied, “I’m not sure, but that’s my challenge to you. You need to convince me and your classmates that the ball hitting the edge right there will bounce out and travel straight into the hole. Does it? What can you draw? What calculations are involved?”
What I heard:
The angle that the ball hits the border and bounces back out must be the same.
Because we’re talking about angles, something about triangles.
This is like shooting pool.
Similar right triangles.
Do we need to consider the velocity of the ball?
This is hard.
I can’t figure out how to use the right triangles.
Similar right triangles because that’ll make things easier.
Even though it’s more than one bounce off the edges, I’m still just hitting the ball one time.
I think I got this.
I have an idea.
Wish my golfer is Happy Gilmore.
BIG struggles, so I was happy and tried not to be too helpful. (I struggled big time too on some of their papers! And I think this made them happy.)
Lauren explained in this 55-second video how she found the paths for the ball to travel. I also had her explain to the whole class later at the document camera.
Jack took a different approach. Instead of measuring the sides and finding proportions to find more sides to create similar triangles like Lauren did, he started with an angle that he thought might work [via eyeballing] and kept having the ball bounce off the borders at paired angle until it went into the hole. (His calculation was off — or his protractor use was inaccurate — as he had angles of 90, 33, and 63. Or maybe if he had a better teacher, he’d know the sum of the interior angles of a triangle was 180.)
Gabe was quieter than usual today. When he finally shared, his classmates realized he was the only one to solve the entire problem using just constructions with a straightedge and compass. He walked us through his series of constructions until he found point C on the bottom border where the ball needed to bank off and end up in hole H.
Imagine none of this thinking and sharing would have occurred if I had given them problem #24 in the book.
Half of my kids were still struggling and working to find the correct bank shot(s), but they were giventhe chance to struggle. And none of them said, “I don’t get it.”
The cutest thing also happened while we were doing all this math. Yesterday (Monday) I bragged to the kids — and I’m doing it again right now — about the Rolling Stones concert that we went to on Friday. I am still over the moon ecstatic that we got escorted into the Pit from our way-in-back-floor-seats!!!!
Anyway, a kid today started humming to the tune of (I Can’t Get No) Satisfaction and quickly others joined in with THESE LYRICS:
I can’t get no similar triangles
I can’t get no similar triangles
‘Cause I try and I try and I try and I try
I can’t get no, I can’t get no
When I’m drawing in my lines
This lesson leaves me so full and proud. Their singing to the Stones while struggling in math makes me crazy in love with them.
Just so you know, I swooned shamelessly in front of my students over a 70-year-old rock star’s butt.
Today I had the kids work on someone else’s paper (remember I made copies of their papers before they worked on them) and find similar triangles to make the bank shots. Because I purposely told the kids to draw in the blobs and the 2 points without any mention of where exactly to place them, it was then by chance that these papers below allowed for one-bank shots to get the ball into the hole.
The ones below, however, are some of the ones that would not work with just one-bank shots, but I had the kids create similar triangles on them anyway because that was the learning goal of the lesson.
Look what the crazy and wonderful Desmos did (click on tweet below to see):
There were over 90 comments left for this post on the old site, but I’d like to feature this thread of comments between me and hillby as it involves us sharing some geometric constructions.
May 8, 2013 7:06 AM
Awesome lesson, excellent job of breaking the problem down, increasing cognitive demand and also getting students to share their thought process.
It took me a while to figure out how Gabe was able to find the point exactly with just some lines and a compass. I stumbled upon it, but I haven’t figured out why it worked. Did Gabe figure out that this approach would work through reasoning, or trial and error like me? I guess I’m basically asking if he added a proof, or did he check by measurement?
May 8, 2013 10:20 PM
Thank you, Chris!
I’m really glad you questioned Gabe’s constructions. I wrote down his steps and re-created it on GSP so you could see:
B is ball. H is hole.
Construct BA and HD, both perpendicular to horizontal bottom line. Both have the same measurements as what he wrote on his paper.
Draw in HA, forming angle(AHD).
Copy angle(AHD) over to angle(GBA).
Now this is his “just a hunch” step: construct the midpoint of AD, label this E.
Construct the midpoint of AG, label this F.
Construct the midpoint of FE, label this C.
Draw in BC and HC, forming the yellow and green triangles.
He just checked by measurement. What do you think?
May 9, 2013 7:30 PM
Oh, how INTERESTING!! I did something similar based on the picture in the post, but it wasn’t quite the same. On the other hand, I got a perfect match. Picking up from BA & HD,
Draw in HA
Draw in BD
Draw in a line perpendicular to AD through the intersection of BD & HA
The intersection of the perpendicular and line AD will be your exact point of reflection.
I think Gabe’s method is similar to the Newtonian method of finding zeros – he’s basically iterating closer and closer to that exact point of reflection.
May 9, 2013 7:54 PM
And look how beautiful yours looks!
I will share your construction with Gabe. I love how Gabe persevered on this problem and appreciate his “hunch” too — it’s a risk I want more kids to take! You can see his tedious work of constructing those midpoints. Any other kid would have just eyeballed it or used a ruler.
Thanks again, Chris.