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It started when my colleague, Mari, showed me her moveable pentagon. At one point, the pentagon transformed into an angle which sparked an interest in angles – creating angles, moving angles, measuring angles. I reached out to both Desmos and my colleague, JJ Martinez, for help. Their equations gave me the foundation needed to move forward with the concept of a moveable angle.

Last year about this time, I worked with, Candace Jewel, on complementary and supplementary angles using Geogebra. This year, I was inspired to run with the moveable angle and rework those lessons using Desmos.

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## Complementary Angles Activity Builder

The Activity Builder: Angle Relationships – Complementary Angles

**Slide 1:** Secret Message

The first slide shows a secret message of “Complementary Angles” when the angles hit 90 degrees. The message bleeps for a second (if that). One can’t read it when the play button is on a fast speed.

**Slide 2**: Describe what happens in slide 1. SMP#3 states construct viable arguments and critique the reasoning of others. Writing about math isn’t an easy or natural process. Quick and short prompts embedded within an Activity Builder provide opportunities for students to practice using academic language to express their mathematical thoughts in written form.

**Student**: It moved.**Me:**What moved?**Student:**A line.**Me**: What’s the line a part of?**Studen**t: An angle.**Me**: Please refine your sentence.**Student**: The side of the angle moved.

**Student:**The numbers changed**Candace:**What did the numbers describe?**Student**: The angle.**Candace:**What about the angle changed**Student:**The size. The measurement.**Candace**: Add that to your description.

Over the 2 days I worked in Ms. Jewel’s room, a considerable amount of effort was spent helping students refine their mathematical writing. The refining process occurred in the nooks and crannies of the lesson. It doesn’t take a lot of time, just a quick question.

- What does “it” or “thingy” mean?
- You said, “all”. How many angles are you talking about?
- You mentioned equation, what was the equation you used?

The convenient aspect of Desmos … You can follow up on all your questions by scrolling through the teacher overlay.

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### The Perfect Segue

**Student**: Why does that message pop up when it pops up?**Ms. Jewel**: That’s exactly what we are going to explore today.

**Slide 3:** Figure out what makes two angles complementary.

**Slide 4**: Write about conclusions from slide 3.

**Slide 5**: Official stopping point. Honestly, I stop and have a class discussion when ever I feel it’s needed. Although, there are times when I don’t want students to move forward. Those are the times I use a stop sign.

**Slide 6 & 7:** I suggest finding an pair of complementary angles and inputting them into the x/y chart as a class. Then let the students find a few on their own. At first, students got lost toggling back and forth between the slides. Once we filled in one pair (point) on the x/y chart, students were good to go.

**Slide 8:** Stop. Get ready to share out your complementary angles. At this point, I show the class overlay. Then call on a number of students to share while I record responses on the white board. This is typically rapid fire.

The overlay gives the graphical pattern. The x/y chart on the board shows the algebraic pattern. You will notice, that someone shared 90 + 0 = 90. This was addressed. Yes, that example follows the patterns, but if one angle is 0 degrees then it’s really not an angle. Therefore the pair of 90 and 0 degrees doesn’t represent complementary angles.

This misconception led me to alter the Desmos activity builder. Now when a student creates a 90 degree angle and a 0 degree angle, the Complementary Sign does not pop up. See picture below

**Slide 9:** Turn the information in the x/y chart into a general equation (x + y = 90) Enter the equation back into slide 7. Describe the equation’s meaning.

For the rest of the period, Ms. Jewel and I circulated through groups questioning students on the equation. This was their exit card. Each member of each group had to describe a part of the equation, x + y = 90. This happened simultaneously as students were returning computers to the cart.

**Me:**What does the x represent?**Student:**The first angle**Me:**What does the y represent?**Student:**The second angle**Me:**What does the 90 represent?**Student**: The sum of the angles. The angles combined total etc…**Me**: If two angles have a sum of 90, what are they called?**Student:**Complementary angles

**End of day 1**

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## Day 2

**Slide 10:** Describe the animated picture. Candace and I set the expectation of writing a meaty description. We did not except: I see lines and points. We asked for details and encouraged our writers to show off their skills.

**Slide 11: **Slide 11 created a lot of excitement and conversation.

** **

**Slide 12**: Describe what happens in slide 11. The excitement reached a twitter worthy level. This sounds corny but I had one of those, “This is what teaching is about” moments. I heard rumblings of “Oh, I know. I know.” They wanted to share immediately, but we redirected them to slide 12 to transfer their thoughts to the computer.

Students of all levels were participating. A general sense of pride filled the room for they were gaining confidence in both their understanding of complementary angles as well as their writing ability.

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**The Next Day**

Slide 12 wraps up this activity builder. The next day, we moved onto discuss Supplementary Angles. The lesson plan contained 2 major activities: The opener followed by the Supplementary Activity Builder.

- The opening activity: Using Assessing & Advancing Questions
- The blog post and lesson guide: Desmos, Progressions & Supplementary Angles .

What a great lesson! I love the idea of a concept as simple as complementary/supplementary angles helping students take the time to think about how they are expressing their thoughts. While I work to embed the SMPs into the activities I ask students to do, this is the most explicit instruction/collaboration I’ve seen. Plus, anything with Desmos is sure to peak student interest, at least in my case. Thanks for sharing!

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I’ve heard about the Desmos Activity Builder before, but this is the first time I’ve seen it in action. Thank you, the practical application you’ve shared for your great lesson gives me the incentive to explore further. Thank you for sharing 🙂

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I really like how you tied in the equation and had students explain their representations. Nice lesson and thank you for sharing your activity builder talents with the world! 🙂

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Oh exciting! I love those moments in teaching! What a great lesson. I really need to dig into Desmos so much more. It’s such a new resource to me and I just love seeing what sort of things you’ve been able to come up with. Thanks for sharing!

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