# Series Formative Assessment Practices to Support Student Learning: Formative Assessment: Understanding Congruence

Math.HSG.CO.B.6

Common core State Standards

• Math:  Math
• HSG:  High School: Geometry
• CO:  Congruence
• B:  Understand congruence in terms of rigid motions
• 6:
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

|
Math.HSG.CO.B.7

Common core State Standards

• Math:  Math
• HSG:  High School: Geometry
• CO:  Congruence
• B:  Understand congruence in terms of rigid motions
• 7:
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

|
Math.HSG.CO.B.8

Common core State Standards

• Math:  Math
• HSG:  High School: Geometry
• CO:  Congruence
• B:  Understand congruence in terms of rigid motions
• 8:
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

## Formative Assessment: Understanding Congruence

Lesson Objective: Formatively assess understanding of congruence in terms of rigid motions
Grades 9-12 / Math / Geometry
Math.HSG.CO.B.6 | Math.HSG.CO.B.7 | Math.HSG.CO.B.8

## Discussion and Supporting Materials

### Thought starters

1. Why does Ms. Pforts begin her class by asking students for real-world examples of congruent figures?
2. Why does Ms. Pforts give the students graph paper and shapes in addition to the geometry software?
3. How does Ms. Pforts know when students are ready to work on the extension activity?

The reason why Mrs. Pforts asks for real-world examples of congruent figures is because when students identify shapes around them they understand them better. They can then apply that understanding to a more formal definition of congruent shapes and rigid motions. The use of graph paper and triangles along with geometry software allows students to access problems in different modes and make connections between the abstract and the concrete. Formative assessment allows Mrs. Pforts to know when her students are ready to move on to the extension activity.

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Ms. Pforts helped her students act on their evidence by first using a mathematical software program. The students could translate, rotate, and reflect the rigid triangles.  When she could see that her students were successfully moving the triangles using multiple combinations of translations, rotations, and reflections, she knew they were ready to act on their evidence by applying what they knew to the real world.  Her students jumped right in and found examples of bridges, buildings, and other structures that used translations, rotations, and reflections.  The conversations within the groups were extremely helpful. The students built on their knowledge and question their conclusions.

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I love the way Ms. Pforts introduces real-world examples  of congruent figures to her class because students can see and touch the shapes. The students no longer has to imagine what the shape is because she made learning easire and possible for all learners.

Recommended (0)

Giving the students graph paper and shapes in addition to the geomety software allowed the students to access the shapes at different and especially good for students who are hands-on learners. They were able to see the  shape instead of trying to imagine it.

Recommended (0)

Ms. Pforts requires the students to think of real-world examples of congruent figures so that they can make connections between school and life. This requires them to pull from their background knowledge in order to be successful in the classroom. Giving the students graph paper and shapes in addition to the learning software was a great idea for students that need that hands-on, tactile input to facilitate learning. Then the software is a bonus to what they have done for themselves. She knew they were ready for the extension activity when their thinking went on to the next level.

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### Transcripts

• Formative Assessment: Understanding Congruence Transcript

Dalton: There's a lot of everyday things that are-- it's going to be the same part

Formative Assessment: Understanding Congruence Transcript

Dalton: There's a lot of everyday things that are-- it's going to be the same part over and over again, which means every single part is going to be congruent to each other.
April Pforts: All right, so good morning, ladies and gentlemen. So today we're going to talk about high school geometry and congruence and what that looks like for us.
Lower Third:
April Pforts
9-12th Math Teacher
Mount Pleasant High School, Mount Pleasant, IA
April Pforts: When I clarify intended learning with the students, I like to explicitly make sure that they know what's the hot ticket items basically.

Card:
STUDENT LEARNING GOAL
I understand that congruent figures remain congruent through the rigid motions of translations, rotations, and reflections.

April Pforts: One, I understand that congruent figures remain congruent through the rigid motions of translations, rotations, and reflections.
Card:
STUDENT LEARNING GOAL
I understand that justifying my conclusions, communicating with others, comparing plausible arguments and asking useful questions help to clarify mathematical reasoning.
April Pforts: I understand that justifying my conclusions, communicating with others, comparing plausible arguments and asking useful questions help to clarify mathematical reasoning.

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STUDENT LEARNING GOAL
I understand that using clear and precise definitions helps to simplify and strengthen the mathematical reasoning process.
April Pforts: And our third learning goal is that I understand that using clear and precise definitions helps to simplify and strengthen the mathematical reasoning process.

April Pforts: The learning goals today were from the common core and we were talking about rigid motion to show that triangles are congruent and how to make mathematical arguments.
April Pforts: You do not have to write every single word down, just the bold words as we go through it. And by the end of this lesson, we want to really have a deep understanding of what those bold words mean.

Card:
STUDENT SUCCESS CRITERIA
I can use a series of rigid motions to show that two triangles are congruent.
April Pforts: I can use a series of rigid motions to show that two triangles are congruent, so series.

Card:
STUDENT SUCCESS CRITERIA
I can justify that there is more than one series of rigid motions to show two triangles congruence.
April Pforts: I can justify that there is more than one series. So now, we're going to build on what a series is and can there be more than one.

Card:
STUDENT SUCCESS CRITERIA
I can define congruence in terms of rigid motion to construct arguments explaining why two triangles are congruent.

April Pforts: Our third success criteria today: I can define congruence in terms of rigid motion to construct arguments explaining why two triangles are congruent.
April Pforts: I wanted them to focus on really a series and rigid emotion, how are they going to argue that and how does look in the real world and the properties of rigid motion.

Lower Third:

Ali
Mount Pleasant High School, Mount Pleasant, IA

Ali: Having the success criteria and the learning goals just kind of-- it clarifies okay this is what I need to do. And it does add pressure but it's kind of-- it's good to know what you're doing.
April Pforts: Where might we find congruent figures in our practical world? Think about everyday objects. So let's just take a couple of minutes in your pairs and then in your small groups. So brainstorm with each other.

Student: The manufacturing aspect of things like how a press would stamp everything exactly the same.
April Pforts: What might be some other congruent shapes that are around us? Specifically, look around.
Student: The cabinets.
April Pforts: The cabinets. Honeycomb.
Student: Like in-- like the octagon or the hexagon.
April Pforts: Why would that need to be the same for this to be…
Dalton: Well, it'll fall apart if they don't all fit together.

Lower Third:

Dalton
Mountain Pleasant High School, Mount Pleasant, IA

Dalton: It's really nice for her to put in that real world thing because if you're walking down the street, you're really not going to think, oh, those are congruent. But when she points it out and then we go searching, like one thing we used was the Epcot ball at Disney World and it's all made of triangles, like little triangles. But then the more you put it together, it makes bigger triangles and you see the reflections and the rotations. And it makes everything like a lot more interesting.

April Pforts: When the students identify congruent shapes around them, I felt that they understood that and could then apply that to a more formal definition of rigid motion.
April Pforts: So previously, we've learned about rigid motions. What are some rigid motions that you've learned about. Dale?
Dale: Translation.
April Pforts: Translation. Yes.
Student: Rotation.
April Pforts: Rotation. What's another way that we could describe rotation? Adam?
April Pforts: As a turn.

April Pforts: I want them to be thinkers. I want them to be problem-solvers and not just contrived problem-solvers but solvers of problems that we don't know the answers to.
April Pforts: Today, we will be exploring how we can use rigid motion to prove that two triangles are congruent. So we're going to use that reflection, that rotation, that translation to prove that two triangles are congruent, same size, same shape.

Card:

Student: To show F, to show the point of the rotation.
April Pforts: I would share that with like-- that's a really great question.
Student: When you reflect it, lay on-- right on that triangle.
Student: Yeah, opposite.
Student: Mm-hm.
April Pforts: In the first part of our exploration today, we will be exploring how we can use rigid motion.
Lower Third:
April Pforts
9-12th Math Teacher
Mount Pleasant High School, Mount Pleasant, IA

April Pforts: I transition the students from the launch activity where they brainstormed where they would see congruent shapes into the Explorer activity where they were going to look at the shapes in the geometry software to be able to apply that to congruent shapes.
April Pforts: Your task this morning is to use a series of rigid motions to prove that the two triangles shown are congruent and compare and contrast your work with your peers. And then when you think you have something, what your series is going to be, then share it with another pair in your group.

April Pforts: When I elicit evidence from students, I'm listening. I'm looking. I'm looking for vocabulary. I'm looking, do they understand that task at hand. Do they under-- does their question seem to match what we're doing.
April Pforts: So you could rotate it- so you could rotate it around the origin but then you might have to also take another step. Okay, how can I move this triangle over to this triangle to be this triangle?
Student: Reflect it.

April Pforts: Okay, so you could reflect it maybe. Okay. Is reflection the only thing you could do to it?
Student: You could like…
April Pforts: What if my shape was like this?
Student: Then you could like reflect it and rotate it.
April Pforts: Then you could reflect it and rotate it, good.
Logan: Now, we've got to get this one to that one by rotating it. Translations.

April Pforts: What transformations are you going to apply to that? Are you going to slide it or are you-- you got it, Logan?
Logan: Mm-hm.

Lower Third:

Dalton
Mountain Pleasant High School, Mount Pleasant, IA

Dalton: We use the terms that Mrs. Pforts gave us: translation, reflection and rotation. And we used it on the Geogebra program and also on our own with the paper and the plastic triangles to show that what she was saying was true. When you rotated it and you slid it and you did all these things, no matter what, it would still be the same.
Student: Oh, wait, no. Like here, you just slide it. It's still over here.
Student: Yeah, so we could just slide it.

April Pforts: I figure out how they can move on through asking them those different questions. And if they can explain it to me or they have another deeper question, then I'm like okay, you're on the right track. But if they're like-- sometimes they'll just look at me and say, "Yeah, I don't get it." And so, then we go with that. Okay, what don't you get? Where are you at?
April Pforts: Is it possible for all of those angles to be the same?

April Pforts: As I walked around, I observed and I listened to their conversations. And I heard students using the vocabulary for rigid motion such as the reflection and translation. I also heard students to say, "Well, this would be congruent because this angle would be the same over here."
April Pforts: If I know one of them is going to be 9.7 degrees.
Student: Yeah-- no.
Student: No.
April Pforts: No, why?
Student: One eighty.
April Pforts: One eighty, okay, so the most they can be is 180. I've already got almost 80 degrees here, right? Okay.

April Pforts: As I'm assessing my student's understanding, I try not to give them the answers because I want them thinking about the mathematics. I want them kind of justifying let's see, this does this because of why.
April Pforts: Oh, the graph, sorry.

Lower Third:

Ali
Mount Pleasant High School, Mount Pleasant, IA

Ali: She just kind of gives us hints and say, "Okay, what do you need to do to figure out how to do this?" And like she'll come around and assist us. But the key learning goals, I think she wants us to figure out on our own and not have-- spoon feed, I guess.
Student: The exact side and that side. Those two. IH and then HJ.
Student: Yeah.
Student: A2, B2 and C2. Our final.

Student: There we go, 180.
Student: We have a triangle.

Card:

April Pforts: Try to come up with as many different series as you can.
Student: Use reflection at the center of the figure.
Student: Yeah, let's try it.
Student: Side angle side.
Lower Third:
April Pforts
9-12th Math Teacher
Mount Pleasant High School, Mount Pleasant, IA

April Pforts: During the interpreting evidence, that's more of an internal dialogue. That's the connection for me between eliciting the evidence and acting on the evidence because unless I understand where the students are at for the information that they've provided to me, I really don't know what to do with it.
April Pforts: In second part of our explanation. Today, we will create our own pair of congruent triangles to share with another group. Label the sides and the angles, trade your construction with another group. So you can do it on the software, but be sure to sketch it out on your graph paper because you're going to trade it.

April Pforts: In the one activity where the students had to draw the shapes and then come up with their own series of rigid motion, I noticed as I walked around some of the students were like, "Well, I could just reflect it." I was like, "Well, that's one way." So as I interpreted that, I wanted them to not-- I was like well, they understand simple rigid motion, but I wanted them to understand rigid motion on a more complex level.
April Pforts: If you reflect it first and then translate it or translate it and then reflect it.
Student: Yeah, is that two different series?

April Pforts: My question to you would be would those give me the same thing. So maybe what you want to do is on your paper, okay, draw it out and then try it both ways and see if you get the same triangles, the same congruent triangles. Does that make sense?
Student: Mm-hm.
April Pforts: So you really have to think about what does this mean, what is the student saying, how are they able to make the connection if they're here and I need them to be over here. What does that mean?
April Pforts: What could you do to get it down there?
Student: We can rotate it.

April Pforts: You could rotate it. Okay, what else could you do?
Student: Reflect it twice.
April Pforts: You could reflect it twice. So you could rotate it.
Student: We could reflect it over here and translate it down.
Student: Yeah.
April Pforts: Does that make more sense? Okay.
April Pforts: When I interpret evidence from students, I really think about how is that making sense to them, or why are they thinking that, or why do they have that question.
April Pforts: Okay, so you guys can go ahead and trade and see if you come up with the same answer. And if you did, remember take it to the next level and try to find even another series. Okay?

Student: Like there's arrows.
Student: Side length side.
Student: You guys took this right here, reflected it over the y-axis. So then it'd be right there. And then you translated it.

Lower Third:

Dalton
Mountain Pleasant High School, Mount Pleasant, IA

Dalton: If you're asking questions and having us kind of have like that light bulb moment, I think that really is crucial to learning.
Student: Translate it up two and-- up three and to the right two.
Student: And this one rotated…
Student: Ninety degrees counterclockwise.
Student: But they all have the same angle measurements and side measurements.
Student: The same with yours over here, which means they're congruent still.
April Pforts: I feel the students were getting it because they were able to come up with multiple ways. They were having a conversation.

Student: This is what we did for our triangle and then this is another way they figured out how to get there.
April Pforts: Okay.
Student: We found like three ways.
Student: Two.
Student: Two or three.

April Pforts: How many different ways do you think there are? That would be a good question to explore, huh?

Card:

April Pforts: It seems like your group is ready to move on. And this next part, I want you to do an Internet search. So go out to the Internet and find a design or structure that has congruent shapes, okay, and then use your rigid motions to prove that those are congruent.
Lower Third:
April Pforts
9-12th Math Teacher
Mount Pleasant High School, Mount Pleasant, IA

April Pforts: Acting on evidence is all about knowing when to move the students after they've-- you've interpreted, okay, they have it and say, okay, now what am I going to have them do, what am I going to do to have them take it to the next level.
April Pforts: Using your found design, use rigid motion to approve the parts are congruent. And I want to use that part for our summary. Why am I asking you to do that, use rigid motion to prove the parts are congruent? What do I have to remember about rigid motion that helps me to prove they're congruent?

Student: Well, rigid motions never change the side lengths or angle measures.
April Pforts: Right. So same shape, same size, so my angle measures, my angle sides are going-- or my side lengths are going to be the same.
April Pforts: I knew the students were ready to move on to the real world activity part of the lesson because they were able to come up with multiple rigid motions for problems that they had created themselves.
April Pforts: Okay, so tell me that again. So they started with one triangle and then how did they get the next triangle?

Dalton: They reflect it here and they just keep doing that over and over in the lines. And then, they take this triangle and they rotate it along this point, right here.
April Pforts: Once they make a bigger triangle, then they start-- could they look at these three triangles together and are-- so they reflect that one down? Okay, so each time that shape is going to get bigger and bigger. So it doesn't always have to be just that original thing.
Dalton: Because right here, you can see lines.

April Pforts: Right. So once they made that pattern, and then they just started reflecting that over and over again. That's really, really cool. So that…
April Pforts: The students were getting very excited because they were able to see congruent shapes in the world around them and how it shapes their world. So, for example, they were looking at bridges. They were looking at some of their different athletic equipment, such as soccer balls. They were looking at playground equipment. They were talking cars.

April Pforts: Giving them open-ended problems that they don't have the solutions to and not contrived problems and problems where they really have to think about it and say what does this mean in the world around me because when they go out into the world, there's no hard and fast solution for their problems. They're going to have to think and take all those considerations in and apply the properties that they knew about things to come up with those solutions.
Student: We're doing the jungle-- the jungle gym.
Student: That you can climb.
Student: We're doing that.

April Pforts: I know formative assessment works because students get excited about the math. And they're allowed to wonder. It helps me to know where they're at, and it helps them to know where they're at. And so, they get excited. They feel successful about it. They're like, "Hey, I can do this. What else can I do?"

Dalton: This one shows it pretty good because it has big triangles-- a big triangle and it just keeps it multiplying to make it.

Card:

### School Details

Mount Pleasant High School
2104 South Grand Avenue
Mount Pleasant IA 52641
Population: 651

Data Provided By:

April Pforts

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