Analyzing Triangle Congruence with AngLegs

I’ve tried to explain why AngLegs are a must have for high school geometry and should not only be considered a tool for younger students. Here is an example of how I find them indispensable in teaching triangle congruence. This lesson is adapted from MARS: Evaluating Conditions for Congruency.

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“Ok class, you are sitting in pairs and at each table is a bag of AngLegs. On the board I have written the question we are trying to answer for each of the scenarios I will present.”

“I made a triangle that includes a blue AngLeg. Can you make a different triangle that also has a blue AngLeg?”

“Make one. Hold it up.”

“How do you know that these triangles are different?”

“So, you are saying that keeping one side the same does not mean that the triangles must be congruent.”

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Next, lets look at Card 3:

 

“I made a triangle out of a blue, a purple and a yellow AngLeg. Can you make a different triangle using the same three AngLegs?”

“What if you put them in a different order? …or move the purple between the blue and the yellow? Are you sure they have to be the same? How can you tell?”

Student:  “The triangles still fit perfectly on top of each other”.

Student: “If the three sides on one triangle are the same as three sides of another triangle, then the triangles must be the same.”

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Look at Card 7:

“How can you tell if an angle is the same in two different triangles?”

Student: “They fit perfectly on top of each other!”

“Is there a way to make these triangles so that they are not congruent?”

Student: “No way. These have to be the same”

Student: “Wait! I made two different triangles with all three angles the same and one side the same.” Does this count? Look!”

Student:”If two angles are the same, then the third angle always has to be the same because they add up to 180 degrees!”

“So what is the conclusion for this one?”

Student:”The triangles can still be different sizes, but their angles are all the same.”

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“For the remainder of this class period, individually analyze card 5 and any other card, so 2 additional cards. Take good notes and write down your conclusions for tomorrow, where you will be randomly assigned a partner to complete the triangle activity. ”

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From here the lesson continues as described in the SHELL center teacher guide linked above and described further in a blog post here.

Removing the hurdle of constructions allows students to focus on the learning goal for this activity: determining the minimum information required to guarantee triangle congruence, and what congruence means. It also connects nicely to congruence proofs through transformations as students are physically checking of the triangles fit on top of each other.

Painless Proofs!

Last year I didn’t really teach proofs in geometry. It seemed so procedural and I knew it would take time away from more productive problem solving tasks.

I felt guilty for not including proofs in enough detail and I decided that this year I was going to teach it well.

Here is what I did:

1. Our unit began with the Shell Center task: Evaluating Conditions for Congruency, I don’t always follow these lessons exactly as described, but we focused on the Must the Two Triangles be Congruent? part of this lesson. I found that this activity is much smoother and more effective when using AngLegs instead of trying to draw each triangle (described here and here).
2. After this task we formalize our findings in our notebooks documenting which combinations guarantee congruent triangles [SAS, SSS, ASA, AAS] and which do not [SSA, AAA].
3. Next students practice determining if pairs of triangles must be congruent based on the information given. I am careful to include a few with overlapping triangles and triangles who share a side or contain vertical angles to generate observations and class discussions. At this point I address the reflexive property and vertical angles as reasons sides or angles may be congruent. I follow up any student observations of congruency with “How do you know?” or “Explain why you decided these two angles must be the same.”
4. Review definitions of midpoint, bisector, perpendicular, then a few images where I ask if you are given this information, what can you conclude is congruent. See the interactive notes here. This is the day students complete their first few **Really Basic** congruent triangle proofs.
5. Prior to this class I printed Proof Blocks on colored paper and laminated them, then I used masking tape to affix them to  whiteboards. Once class began I randomly assign student pairs and had them work on a wall mounted white board (Vertical non-permanent surface) with a set of Proof Blocks at each work space.

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Next I project a proof and students copied the image onto the whiteboard, then marked the givens, saw what else they could determine was congruent, and decided if they had enough information to conclude and prove that the triangles must be congruent.

In order to make sure all students got timely feedback and to hold them accountable, I gave each group an index card and when they finished a proof I would check it. Once it was done well, they got a stamp on their card. At the end of class they turned in their index card with their names & stamps on it, similar to the review activity I described earlier.

I returned to this format with the proof blocks and different pairs about once per week after learning and incorporating new skills. The second half of the slides above were from a day after learning and practicing angles formed by parallel lines & a traversal. I intend to create new “blocks” as we advance through the school year. Next week students will prove that all triangles have a sum of 180 degrees and that base angles of isosceles triangles are congruent using the same format.

With students standing at white boards, they can glance around and see their peers work easily, they are more likely to collaborate, and the whiteboard & Proof Blocks make it much easier to adjust their work when there are any corrections required.

I just graded students quizzes over this unit and it is the first time in my 12 year teaching history that they did very well on the first attempt!

Day 151 & 152: Similarity transformation proofs

I waited until the end of the school year for this because I wanted to encourage retention by requiring students to remember basic transformations in addition to applying dilatation which we just worked on. I searched for a good way to help students develop understanding of this. The best description of this standard that I found was Kate Nowak here. In this post she included a link to Khan Academy’s interactive modules. I am not a fan, but these really helped students to see how to prove similarity. I searched for other options and even asked Desmos, but I found this Khan Academy module to be my best option. I set up a “coach” account, then on the first day my students set up accounts, made me their coach, and accessed the module. It took a while for them to get the hang of how it worked. On the second day they will try to get 5 correct in a row, really I just want them to experiment with this some more and develop mastery.

Geometry: Planning the last 6 weeks

The last six weeks of geometry will focus on similarity, dilatation, indirect measurement, similarity transformation proofs, and circles:

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We will start with a review of real world ratio and proportion practice, interactive notes page & review, possible including activities such as New York Minute (NCTM Illuminations).

We will then head into applying the properties of proportion to geometric figures in discovering and defining similarity as described in the investigation activities in Discovering Geometry.We will complete an interactive notes page and practice. We will use similarity to measure tall objects using a mirror and measuring tape with indirect measurement. It is also an excuse to get outside, since it is getting warmer out and we are all developing cabin fever.

I am hoping we will have time to incorporate the Math Assessment Project‘s lesson: Solving Geometry Problems: Floodlights (MARS).

I am very excited to try to channel my inner Fawn by spending 2 days on Letting Them Own the Problem applying similar triangle properties.

We will explore & discover properties of dilatation’s with the Flip Family (page 13 of this pdf), followed by an interactive notes page and practice. I am just going to have to create a Dilatation Station Rotation because that has a fun name (and students will need some practice).

The scariest part for me will be attempting to teach similarity proofs through transformations as describes by Kate Nowak here. Hopefully, at the least we will all learn something.

Next we will discover & apply the relationship between similarity and proportions with area and volume through discovery activities, interactive notes and this little gem.

Then we are on to a short excursion into circles. We will begin with developing an understanding of radian measure, followed by a 3 act lesson by Mr Stadel. I plan to dedicate a few days on the Math Assessment Project’s Sector of Circles lesson. We will also do the 3 act lesson be Dan Meyer: Lucky Cow.

I plan to end the school year with a Modeling project adapted for my students to be more self directed: rolling cups.

Day 94: Coordinate geometry proofs-day 1

Day 53: Proofblocks part 1

We set up our INB’s for proof’s today. Clearly I need to emphasize correct spelling in the future!

We completed 1 proof, but it was a little rough. Tomorrow’s plan is to construct lotsa proofs on chart paper. just practice, no notes…It still feels a little soul crushing though, even with hot pink paper.

[update: next year laminate 1 class set to stay in the room so that I do not loose as much instructional time]

Day 52: Today was just OK

We did an estimation warm up, which everybody loves. Got the new learning target sheet & practiced identifying which congruence rule we were using depending on the “given’s” and the picture. In some classes we also did the second page from this made by Math Teacher Mambo. I plan to introduce proof blocks tomorrow. I don’t know why, maybe its the time change?, but I am just pooped. it is a real struggle to generate a creative plan despite drinking 3 cups of coffee. Just low energy. Hopefully tomorrow will be better!

Day 51: Triangle congruence notes: making it all official

This was not a highly engaging lesson. I found out that all other teachers were showing movies today, & I was the only one “making them learn”

Day 49: Intro to triangle proofs – better then expected!

I debated for a long time on the best way to teach triangle congruence proofs. I decided to start with the Math Assessment Project’s lesson Analyzing Congruency Proofs.

I broke it into a multi day lesson since we have 50 minutes classes, and I do not want to rush this! Some students got frustrated with pencil & paper constructions, since they are time consuming. I handed a few students who were struggling a set of AngLegs & a protractor. Boom! they understood what the task was.  Next year, I plan to modify this lesson to use the AngLegs instead of pencil & paper to increase student engagement. Once they draw conclusions, then write it down & explain. They even got frustrated and excited and debated about weather or not they could construct 2 different triangles meeting the specified requirements. The most hotly contested one was weather or not they could make 2 unique triangles with 2 sides the same & 1 angle. I thought about videotaping the debate, but I was scared that me pulling out a camera would disrupt the awesomeness. This was all individual work, but tomorrow they will work in small groups and compare their conclusions.

Day 44: Triangle Side-Angle Relationship

We analyzed the relationship between sides and angles within triangles using my fun AngLegs and developed conclusions as a class. I noticed this student has some errors in her notes & practice. We will be reviewing these conclusions and correcting any misconceptions tomorrow at the start of class.

Unplanned but awesome: some students noticed that some lengths did not create a triangle, which led to great discussions and an awesome lead into tomorrow. They are wondering when you get a triangle & when you don’t and why!