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.

Right triangles.

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.

[Updated 05/08/13]

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.

[Updated 05/11/13]

Look what the crazy and wonderful Desmos did (click on tweet below to see):

[06/28/14]

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**

**hillby wrote:**

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**

**fawnnguyen wrote:**

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**

**hillby wrote:**

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**

**fawnnguyen wrote:**

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.

## 22 Comments

When I first attempted to do what the task asked, I was somewhat frustrated as I wanted to get to solving for the answer. As I worked through the task, it became clear to me that the teacher had set the task to stimulate thinking. Every step was doable and it allowed for creative and flexible thinking, something we often leave out of math. The deeper I moved into the task, the more engaged I was. This made the task fun!

Thank you so much for sharing this, Tricia. I smiled as I read your comment because it’s not easy for teachers to “be less helpful.” We were programmed to show-show-show students what to do. :) Hooray to you and your kids!!

student ownership of problem is great – the key to it.

then the discussion and sharing of ideas underpins the development of understanding.

awesome problem – and approach!

“the discussion and sharing of ideas underpins the development of understanding” Yup and yup! Thanks, Lachie!

This was a fantastic lesson! I love how you put the ownership on the students to create and manipulate. This is the way that math should be.

Your lesson modification was awesome, but I think the size of the diagrams prevented students from thinking outside the box.

When teaching me to play pool, my partner once told me to aim for the reflection of the pocket to make a bank shot. When I constructed my miniature golf course, I didn’t use the entire sheet of paper. This gave me room to find H’, the reflection of the hole across the wall the ball will bank on. Drawing segment BH’ helped me to find the angle of incidence (and the angle of reflection). Finding B’ and drawing B’H helped me find the angle of reflection (which could also be accomplished by constructing segment CH).

Great lesson modification!

Nice, Robert, thank you! I didn’t even think of this; will definitely remember to leave some room around the box.

It was interesting to see how the students took ownership of the problem I will work on that in my class.

Great lesson! I see how having the students have more ownership of math problems causes them to think more and gives them a deeper understanding.

Great Idea! Having students own the problem is the same as empowering them. This is their own design of a miniature golf. Suddenly they must feel responsible for solving and being able to provide an answer.

This was so interesting, I found myself getting caught up in the math and trying to figure out what the students were doing/thinking.

I cringe thinking about stepping back and not being the safety net, so to speak, but I know it has to be done. As teachers this goes against everything we were taught to do with our students but it makes so much sense! I am now looking for ways to do this in the next lesson.

Wonderful job! As I was reading I was able to imagine students’ engagement, participation and learning. Of course, I will give it a try!!

Wow! I know that is where I want to take my students, but letting go of what I am use to is difficult to do. Your students were engaged and most likely learned more that day than any other normal math day.

Wow, what a great lesson. I loved the challenge you gave your students. Thank you for sharing!

Wow, what an incredible lesson. The different ways the students used to solve their problems was awesome. What was even more amazing was even though the problem was difficult, no one gave up

I like the way you formulated the math task so students could take ownership. I have acquired ideas from your lesson how I can do this in my class!

Thank you!!

Amazing modification on your part. I am interested in how the students who did not get the correct answer faired. Sounds like no one said ” I don’t get it” and that is awesome. I really like how the students took ownership of the problem ,and it makes me wonder how I can do the same with my sixth graders.

Hi Michael. Because each paper was unique, there was no one correct answer, but they all eventually came to the understanding of applying similar triangles. Kids just grabbed a different paper when theirs were not possible with one-bank shots. This was done with my 8th graders though. You could certainly do this with 6th graders as long as they already understand similar shapes and proportions. (Proportions has now been pushed back to grade 7 in Common Core, but that doesn’t mean kids don’t make this leap on their own.)

We are on block schedule because the Juniors are state testing in English classes, so I did this activity with my Honors Pre Calculus classes during the 2nd hour of the block.

Oh my goodness, these students have such a learned helplessness. Some had no idea how to start. Some came up with something (they drew a line from B to the side of the rectangle, and then just connected that point to H and thought they had solved the problem). I asked these students, “if you hit the ball and it bounces off the wall here, how do you know that the ball bounces along this path?” and I got a lot of “I don’t knows” and “because it just does” and “I just hope it does”. So when I asked them how they could figure this out, they had not clue. So when I nudged them to look it up on their phones, they were stumped about what question to even ask google.

I was trying to walk the fine line between productive struggle and frustration, but I’m still new to all this, so after about 20 minutes I pointed some groups in the right direction.

I almost think I am going to start the year off with this question… to start their training with this kind of task. I have to remember that when they get to me, they have had many years of training that I need to “untrain”.

PS After first period, one of my normally curious students came up and commented that today’s activity was awful… and why didn’t I just teach it to them… “basically, we had to teach ourselves, and aren’t we in here to learn?” So I countered with “don’t you learn the most when you’ve taught yourself something?” He didn’t look convinced and walked out the door. I worry about these kids cause they are 1-2 years away from being thrown in the big pond of college.

Hi Claire. Thank you so much for your two comments here. I think it has a lot, if not entirely, to do with the classroom culture. It’s something I’ve worked hard and honed starting on day 1 of school with my students. Productive struggle is the first sign of learning. I repeat this mantra, not in words alone but with rich tasks, over and over again as a way to HONOR and RESPECT their thinking, their struggles. The AHAs that eventually come out of that struggle are pure gold. But teaching is a fine art. We have to scaffold and keep a finger on the class’s pulse to know when and where to interject. Not enough instruction and scaffolding leave kids frustrated to a point of giving up is the last thing we want. There is much “learned helplessness” for sure, that’s why we need to break them of this lethargic state with small, measured, intentional doses of encouragement and empathy. Talk about how you struggle with a problem, share with students your personal strategies. This is why I go to Math Teachers’ Circle where we tackle fun and very challenging problems that kick my butt. But I get to go back into the classroom with that mindset and share with my students how much I appreciated that struggle, it’s a big dose of humility that always makes my mind more curious and my heart full of gratitude for such opportunities. Thank you again for dropping in, Claire.

There’s actually some pretty good (although now well buried) research from the 80s and 90s supporting this stuff.

It actually turns out that if you ask students to make up their own problems they’ll demonstrate advanced problem-solving skills, even if they’re not advanced problem-solvers.

I gave a talk recently in a different field (game design) that you might find relevant ( http://stfj.net/DesigningForProblem-Solvers/ ), or actually, more relevant might be this paper http://stfj.net/DesigningForProblem-Solvers/Winograd1995.pdf which sites some older and harder to find research on the benefits of asking students to make up and solve their own problems over asking them to just solve given problems.

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