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.

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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.)

The Magic Octagon from Dan Meyer on Vimeo.

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

2014-10-06 12.53.28

 

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

Similar Quads

… 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.

2014-02-11 13.51.01

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…

Geom #4-D

… 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…

Best Reflection

… 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.

Best Reflection - Act III

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…

Circles from Cedar Street

A sloppy bit of construction work makes for an interesting geometry question.

A sloppy bit of construction work makes for an interesting geometry question.

I was driving to the grocery store. This particular trip took me down Cedar Street. I drove past this manhole cover. It caught my eye in such a way that I decided to pull into a nearby parking lot and, when the traffic cleared, tiptoe out to the middle of the five lane road and snap a photo of it.

So, my mind instantly went straight to rotations. (Which is on my mind because transformations are Unit 1 of the Geometry Course that I start teaching in, like, two weeks.)

What if I started out rotations by showing this picture and simply asking the students how much of a rotation would fix the yellow lines.

My goals would be for the students to explore how to investigate an apparent rotation, learn how to visually represent a rotation, and struggle through the task of explaining it out loud to another person.

It would be okay with me if we made it to the convenience of using degrees as a descriptor of how much something is rotated. That would be up to them. I do suspect that after a short time of “about this far” and “about that much” they’ll like to find something to ease the trouble of explaining the transformation.

If you can think of a way to frame this learning opportunity better, please make me a suggestion. I feel like this is a good opportunity that I don’t want to waste.

Common Core Geometry: An update one semester in

End of first semester provides a chance to reflect

Photo Credit: Flickr User “Neil T.” Used under Creative Commons.

This is our first go-’round with the Common Core Geometry. Without a usable textbook, our local geometry team has been responsible for most of the content. So, where are we?

We completed three units: Unit 1 was an introduction to rigid transformations. Unit 2 used rigid transformations to develop the idea of congruence, specifically congruence of triangles. Unit 3 began to formalize the notion of proof by using angles pairs and rigid transformations to discuss parallel lines cut by transversals, isosceles and equilateral triangles and parallelograms. (We should have finished one more unit, but that always seems to be the case…)

So, how did the first semester go? Well, the algebra-writing relationship is an interesting one. In previous years, our first unit was dedicated to writing and solving equations based on geometric situations. To see if the students thought that those two angles are congruent, we would put an algebraic expression in each angle and see what the students did with it. It turned largely into Algebra 1.5 with the mysterious “proof” added in for good measure. It didn’t seem to make much sense.

We have moved away from this for two reasons:

First, proofs are about expressing relationships in writing. It seemed silly that students would learn most of the geometry concepts through an algebraic lens and then we had to change gears to try to develop proof. We thought to start with the writing to smooth out the transition. Then, we could add the algebra back in later. The students have been through three straight algebra-based math courses leading up to geometry, so that will come back much more easily.

Second, we thought that another “same-ol’-math-class” might be what was sapping the enthusiasm and engagement. So, we have gone heavy on writing, creation, and visual transformations. The students have responded well. We’ll see what happens when the algebra makes a comeback in the second semester.

But, improvement is definitely needed. We weren’t prepared for the shift in the needs of our students in a non-algebra class. Our students aren’t learning with a great deal of depth. They are having a tough time writing with technical vocabulary. They are having a hard time making connections across topics. These are problems we are going to have to address moving forward. We were algebra teachers. I know dozens little tricks for solving all sorts of different equations. I don’t know a single one to help a student remember the different properties of a parallelogram.

I think we have done a nice job creating activities that will engage the students, but now we need to make sure the content depth is appropriate as well. It might be as easy as changing the questions that I ask. I can think right now of an activity regarding parallelograms as one that I didn’t draw out nearly enough depth from them. I didn’t facilitate the student thought and discussion. It slipped back into teacher-led note-taking. They struggle with the parallelogram proofs on that test. No surprise.

Please. If you have ideas, resources, processes, thoughts, lessons, handouts, anecdotes, or other helpful offerings, I will be accepting them starting now. Load the comment section up and be prepared for follow-up questions.

 

Oh, and if you are interested in reading a semester’s worth of my previous reflections on our new common core geometry course, here they are:

From Jan 7 – Vocabulary: The Common Core Geometry’s First Real Hiccup

From Dec 12 – Why you let students explore and discuss – an example

From Dec 7 – When measuring is okay…

From Nov 16 – Proof: The logical next step

From Nov 15 – When open-ended goes awesome…

From Nov 2 – Improvement under the common core…