# Exploring Reflections with Desmos Activity Builder

Geometric transformations take up a good chunk of the first quarter of Geometry (at least the way that I had it sequenced). The tricky part of teaching transformations in Geometry is the delicate balance between the non-algebraic techniques and understandings and their algebraic counterparts.

For example, consider the two following statements.

Two images are reflections if they are congruent, equidistant from a single line of reflection and oriented perpendicularly from said line of reflection.

The reflection of A(x, y) is A'(-x, y) if the line of reflection is the y-axis and A'(x, y’) if the line of reflection is the x-axis.

As a geometry teacher, am I to prefer one over the other? In my experience, they both present challenges.Students are often a little more enthusiastic about the first (which is why I start there), but can often be more precise with the second. The second requires the figure to have vertices with coordinates and the line of reflection be an axis. The first requires the figure be drawn (and fairly accurately, at that.)

So, in my attempt to learn how to use Desmos Activity Builder, I wanted to produce something useful. So, I made a Desmos Activity to bridge the transition from a visual, non-algebraic understanding of reflections to an algebraic one.

Full disclosure: This activity presupposes that students are familiar with reflections in a visual sense. It isn’t intended to be an introduction to reflections for students who are brand new to the topic.

Feedback, questions… all welcome.

# Fun with The Magic Octagon

So, we are wrapping up our unit in Geometry on rigid transformations, which means it is that wonderful time of year when I show the students Dan Meyer’s The Magic Octagon!

Seriously… have you seen this? (Go through it like a student. Pause it to make your first prediction.)

Isn’t that cool? Not sure whether you predicted correctly or not (I did not the first time), but I’ve used this video with 6 or so geometry classes and the results are somewhat predictable. 85-90% of the class guesses 10:00. Most of the remaining voters choose 7ish:00 and get called crazy.

Then they see the answer and they JUST… CAN’T… BELIEVE IT!

“Wait… wait. How does one side go clockwise and the other side go counterclockwise?

“No. Run the video again. What did I miss?

That would be a 10-out-of-10 on the perplexity scale, when, like 85-90% of the class gets the math problem wrong and that suddenly becomes a motivator!

Then as they start trying to figure it out, they start making lots of hand gestures, which is surprisingly helpful to them and

Then they don’t want you to move on. They want two more minutes to talk about it. Then a classmate starts explaining it. Not all of them get it the first time, but some of the demand to have it explained again.

Then they move on to the second rotation and they feel so confident. You ask for explanations. They give them… quickly. Quickly because they can’t wait to see the answer. And then they did.

And those who got it right cheered! Quite loudly.

Then a boy stopped us and offered a sequel.

“If the front side arrow is pointed at 5:00, would the other arrow point at 5:00, too?”

He turned the tables enough close to the end of the hour that we left with that question unanswered.

And I fully expect a couple students to have something to say about it tomorrow.

# Perplexing the students… by accident.

It seems like in undergrad, the line sounds kind of like this: “Just pick a nice open-ended question and have the students discuss it.” It sounds really good, too.

Except sometimes the students aren’t in the mood to talk. Or they would rather talk about a different part of the problem than you intended. Or their skill set isn’t strong enough in the right areas to engage the discussion. Or the loudest voice in the room shuts the conversation down. Or… something else happens. It’s a fact of teaching. Open-ended questions don’t always lead to discussions. And even if they did, class discussions aren’t always the yellow-brick road lead to the magical land of learning.

But sometimes they are. Today, it was. And today, it certainly wasn’t from the open-ended question that I was expecting. We are in the early stages of our unit on similarity. I had given a handout that included this picture

… and I had asked them to pair up the corresponding parts.

The first thing that happened is that half-ish of the students determined that in figures that are connected by a scale factor (we hadn’t defined “similar” yet), each angle in one image has a congruent match in the other image. This is a nice observation. They didn’t flat out say that they had made that assumption, but they behaved as though they did and I suspect it is because angles are easier to measure on a larger image. So, being high school students and wanted to save a bit of time, they measured the angles in the bigger quadrilateral and then simply filled in the matching angles in the other.

Here’s were the fun begins. You see, each quadrilateral has two acute angles. Those angles have measures that are NOT that different (depending on the person wielding the protractor, maybe only 10 degrees difference). And it worked perfectly that about half of the class paired up one set of angles and the other half disagreed. And they both cared that they were right. It was the perfect storm!

Not wanting to remeasure, they ran through all sorts of different explanations to how they were right, which led us to eventually try labeling side lengths to try to sides to identify included angles for the sake of matching up corresponding sides. But, that line of thought wasn’t clear to everyone, which offered growth potential there as well.

By the time we came to an agreement on where the angle measures go, the class pretty much agreed that:

1. Matching corresponding parts in similar polygons is not nearly as easy as it is in congruent polygons.

2. Similar polygons have congruent corresponding angles.

3. The longest side in the big shape will correspond to the longest side in the small shape. The second longest side in the big, will correspond to the second longest side in the small. The third… and so on.

4. Corresponding angles will be “located” in the same spot relative to the side lengths. For example, the angle included by the longest side and the second longest side in the big polygon will correspond to the angle included by the longest side and the second longest side in the small polygon. (This was a tricky idea for a few of them, but they were trying to get it.)

5. Not knowing which polygon is a “pre-image” (so to speak) means that we have to be prepared to discuss two different scale factors, which are reciprocals of each other. (To be fair, this is a point that has come up prior to this class discussion, but it settled in for a few of them today.)

I’d say that’s a pretty good set of statements for a class discussion I never saw coming.

# “Right and Wrong” vs. “Good, Better, Best”

So, I’m finding that I am becoming hooked on showing student work back to the students for them to explore as an opportunity to develop a deeper understanding of mathematical topics. But I’m learning there are a couple ways to handle this situation.

I could show this picture…

… and ask if the reflection is right or wrong. Which would activate a certain type of thinking. But it is a short and stifling line of thinking, especially when you consider that most hand-drawn student work isn’t perfect so “wrong” is the most likely answer. Then it risks becoming an annoying knit-picking session, which might have a negative effect on engagement.

A different approach would be to acknowledge that which we all know (that nobody’s perfect) and ask the question differently. Suppose I show the follow four photos together…

… and ask “Which of the reflections is the best?”

That question opens up a lot of potential thought-trails to wander down. As I did this activity with students today, the class settled on three criteria for rating these reflection attempts. The first is that the image and pre-image should be congruent. Attempt A won that battle (with B at a close second). The next thought was that the image and pre-image should be the same distance away from the line of reflection. Attempt B was the closest (with D pretty good, too.) Finally, the students thought that the segment connecting the image/pre-image pairs would should be just about perpendicular to the line of reflection. Attempt B took that contest quite comfortably.

The students concluded that attempt B was the best reflection and I was able to confirm that by showing them this photo, which seems to agree quite strongly with that conclusion.

If you want to try this activity, go ahead. I’d love to hear some feedback on how it went in your class. I’m still dealing with some quality control issues with some of my multi-media projects, so I apologize for that. I didn’t notice it to be too distracting when I was showing it to the students, but there is still room for improvement.

Best Reflection – Act I

Best Reflection – Act II (reflection distances)

Best Reflection – Act II (reflection angles)

Best Reflection – Act II (segment lengths of image and preimage)

Best Reflection – Act III

# Bankshot (I give 3-Act Lessons another try…)

Feel free to read this post, but the video I present at the bottom has since been revised. After you read this post and watch the video, I’d encourage you to read the comments. Then head over to Bankshot 3-Act Revised to check out how I interpreted the feedback.

I recently attended a workshop led by Dan Meyer (@ddmeyer) which I found to be incredibly valuable. In the workshop, Mr. Meyer broke down the 3-Act lesson design model that he describes on his blog. He demonstrated it and showed a few of his own examples as well as some of Andrew Stadel’s (@mr_stadel)I have been trying to make sense of the 3-Act model for a while now. I had continually felt confused by a handful of the aspects. I tried to do some activities but found that I was often either giving the students too much information or not nearly enough.

Bottom line, I was confused about the basic point of the 3-Act Model. It is designed in a way that maximizes engagement and allows you to raise the bar while being more inclusive (which is tricky business). The first act, you set the students up to be curious about the situation. You prepare a scenario where a handful of outcomes seem likely and ask the students to choose which of the outcomes they suspect will happen. This is a short amount of time. You get the students curious, you have them choose a camp and then move on.

In act 2, you start leading the students through the mathematical processes that will allow the students to rule out focus on the outcome(s) that seem to be the most supported by the math. This is where the students begin to explore the variables of the situations, determine the appropriate modeling mechanisms and choose which tools they are going to use. This might also be where you lecture them a bit if they are trying to use models that they are unfamiliar or uncomfortable with.

In act 3, you reveal the answer and the allow the students to make sense of any differences that “real life” has with mathematical modeling.

In an earlier post (the value of face-to-face) I commented on how powerful I find in-person, face-to-face interactions and how the #MTBoS, as powerful as it is, is unable to accommodate for this particular shortcoming. That is only more solidified in my mind now, after seeing how much better I understand the overall approach and value of the 3-Act model having gotten to interact with Mr. Meyer face-to-face, ask him questions, and hear his responses.

So… I decided to try again.

For my first Act I, I decided to go with a geometry/physics topic. Also, the Act II is still in production. So, it probably isn’t quite ready for implementation yet, but I want to see if I am actually making progress in being able to understand, deliver, and (hopefully) create 3-Act lessons.

Enjoy…