Max Ray-RiekThe Math Forum / NCTM

Deidra BakerMid-Prairie High School

Wellman Iowa

Student Work for Triangle congruence ARC

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Student Work for the Triangle Congruence ARC

Students work for journal and assessment questions follow.

The first examples of student work show students who are able to

articulate understanding of the material. The final example will be

used to show a student who is not yet able to articulate a

concept, opening opportunities for discussion and follow up

ideas.

ARC Assessment

1. In the diagram below, B is the center of the circle and

ED and GF are diameters of the circle. Prove that EG

= FD. Can you prove this using more than one

method?

Answer: Assessment

Question 1

(understanding)

GB and EB are radius of

the circle and ½ the

diameter. Same for DB

and ED (FB). They are

both radiuses meaning

they are half, so they are

equal. Also, Angle DBF

congruent to angle GBF

because they are vertical

angles. So we have SAS

(side angle side)

post(ulate). And if we

patty papered them on

top of one another EG

and FD (segments) would

line up.

Arrow- bottom triangle if

you reflect over point B

you will see segments

EG and FD line up.

Answer: Assessment Question 1 (understanding)B is center

of circle:

given

Segments

BE, BG

congruent,

and BF and

Bd by def of

radius

Angles EBG

and GBE

(BFD)

congruent

by vertical

angles, so

triangles

EBF and

FBD are

cpngruent

by SAS and

segments

FD and Eg

are

congruent

by

correspondi

ng parts of

congruent

triangles.

This can be proved with more than 1 method. Like we could’ve done segments EB

and DB because of def of midpoint. Radii leave mulitple options.

Answer: Assessment Question 1 (opens more questions for discussion)

The definition of diameter is

the width of a circle and

diameters for one circle. They

are both the same.

B is the center - given

Segment BG is a radius- def of

radius

Segments, EB, FB, & DB are

radiuses - def of radius

The segments EB, BG, FB,

and DB are all congruent - all

radiuses are the same for

same circle

Segments ED and GF are

congruent- segment addition

Some possible questions for discussion:

Which segments did you add?

You have marked sides of the triangles

in the picture as congruent. How can use

this information? When we talked about

proving triangles congruent how many

parts did we need? - If you don’t have

enough parts are there any more parts

you can show are congruent- which

ones? How could this information help

you?

Spongebob and Patrick are given a construction challenge by

Squidworth. Squidworth says, “Draw a circle with

center Q and then draw a point P anywhere you want. Draw a line

segment from center Q to point P. That will be one side of

both triangles you’re going to draw. Now draw a radius from Q to a

point R anywhere on the circle. Make triangle QPR. Reflect Ray

PR over line QP. Finish drawing triangle QPS by making point S

where Ray PR’ (the reflected ray) intersects the circle.”

Squidworth claims that Patrick and Spongebob should both have

drawn two congruent triangles, QPR and QPS. Patrick’s triangles

are congruent, but Spongebob’s aren’t.

Make your own drawing. Can you make it so your triangles are

congruent? So they aren’t? Did Spongebob make a mistake? Is

Squidworth wrong?

Answer: Assessment Question 2 (understanding)

The triangles that we

made are congruent.

There is no way to

make them so they

aren’t congruent

because the side

lengths will always be

the same. (SSS)

Spongebob made a

mistake while drawing

because there is no

counterexample for

this.

Answer: Assessment Question 2 (understanding)

SSS because they

share a side. They

reflect a side and

reflections are

congruent. The

distance from the

points would be

the same because

the lines are the

same.

This work

demonstrates student

understanding that

reflections images are

the congruent to the

pre-image, and that

connecting points that

reflections would

create congruent

segments as well. This

student did not answer

the second part of the

question but seems

confident that

congruent triangle are

created.

Yes but only if all the

angles in the center

are 90 degrees.

Spongebob didn’t

necessarily do it wrong

he just needed to

specify that the middle

had to be 90 degress in

all 4 angles to be

congruent.

“

Some possible questions for discussion:

Why do the angles have to be right angles? Why all 4 angles or which 4 angles?

Did you try other cases? Are there any other ways to show the triangles are

congruent?

Assessment Question 2 (opens more questions for discussion)

After each lesson, students were asked some reflection

questions. After the first lesson students were asked to

explain or reflect on how they found the congruent halves.

Congruence worksheet reflections - Day 1 questions

1) Explain what you and your partner discussed as you

decided how to split the figures, and then how to get

them to line up with each other, what ideas did you try on

the different shapes, was the patty paper helpful- if so

how?

1) You wrote directions to get from shape A to shape B, how

would those be different if another pair went from B to A?

1) Your advice/ concern/ reflections on this lesson for the

authors?

Some responses for the first question:

What direction the shape was compared to the other one.

We turned the paper to see if another shape was there .

Turning the paper and using the patty paper, so it was very helpful - Kaylee & Alex

We found area and how to split it, we just looked at the shapes and we also used patty paper.

For the more simple shapes we just looked at them by for the more complicated shapes we

brought in other aspects.

Yes because we could see if we were correct or not. - Kessa and Sophia

Most student responses for the second question indicated that

going from the yellow shape to the green shape would be the

opposite or the reverse of going from the the green to the

yellow shape. - The only difference is that the order of transformations would be in the

opposite order. Jaiden

For shapes like this one, we had great class discussions about point of

rotation, and the order of transformations. Some students wanted to

rotate about the tip only, other students argued that you would have to

translate before or after the rotation.

Students have some advice for the authors of this lesson:

I like how this sheet has focuses on different kinds of ways to

get the congruent halves and doesn't include just one of the

many types. -Keaton

I like puzzles and all, but if this kind of stuff were to be homework I

would give plenty of time to finish it, because it is pretty difficult. -

Nolan

On day 2 students investigated whether triangles

with given side lengths would have to be congruent.

They could use patty paper, rulers, protractors or

compasses to decide if the triangles were congruent.

The next slide contains some reflection questions

about this lesson.

Some Day 2 reflection questions about shortcuts

What did you learn about triangle side lengths and congruence?

If you get the same 3 lengths in different orders, can you ALWAYS

find a way to map one triangle onto the other triangle? Why or why

not?

What other key ideas did you learn today?

Do you think that there are other combinations of 3 parts (besides

sides) that might result in triangle congruence using

transformations? What combinations do you predict might work?

If two triangles have the same side lengths, they are

congruent.- Cassidy

(If all sides are equal then the triangle is congruent)

If two triangles have congruence on each each side

with the matching side of the other triangle then the

triangles are congruent. - Abby

Yes; because no matter if the same 3 lengths are in different orders

the triangles will still be congruent. You would have to use

transformations to move one triangle on top of the other one.

-Jaiden

You can rotate them, flip them, or whatever they are just going to

match -Nenad

This depends on the shapes themselves because some triangles

you can simply rotate one or flip one to match the other but I feel

that some triangles are not always congruent to each other. -Adam

If two triangles have congruence on each each side with

the matching side of the other triangle then the triangles

are congruent. -Adam

(that you can use more than just side length to see if triangles are congruent.

That aaa does not work -Josh

Yes SAS, ASA - Matt

Yes; Flipping would work, also rotating and sliding would also

work. -Jaiden

Answers like Jaiden’s gave me an opening to talk with the class about

triangle parts and theorems about those parts, and transformations. There

were some vocabulary misunderstandings.

Students then had the opportunity to explore other ways to decide if triangles

are congruent.

Classes generated possibilities to try with measurements to try. We

investigated with paper-pencil tools, and online tools.

Students were able to test SAS, ASA, AAA, AAS, and SSA. Some students

were easily convinced which options would always work, and which would

not while other students remained unsure what would always work to prove

triangles congruent.

Lesson 3 Other Triangle congruences reflection questions

1. Right now, I can show triangles are congruent by

_____________________________ OR

_____________________________. For me, right now,

_____________________________ is the easier method because

_____________________________. I can image that if a problem were

like _____________________________, it might be easier to do the

other method.”

2. Patrick says that if you have two triangles and you know any two pairs

of corresponding sides are congruent and any pair of corresponding

angles are congruent, then the triangles have to be congruent. Sponge

Bob says he doesn’t think that’s true but he’s having a hard time

explaining just when it is true. Please help Sponge Bob, using what

you explored today.

1. Agree or Disagree: Looking at examples in the app proves that these

shortcut sets of conditions guarantee triangle congruence. Why do you

agree or disagree?

__reflexive_____ OR ____rotating_____. For me, right now,

___reflexive_____ is the easier method because __its basically

flipped____________. I can image that if a problem were like __a shape is

turned________, it might be easier to do the other method.” -Jillian

This lesson reflection enabled me to discuss many misconceptions that

students revealed in their reflections.

Right now, I can show triangles are congruent by determining what the

method is OR showing whether it's vertical or not For me, right now,

reflecting is the easier method because i know it is just a line that is put

on the other side. I can image that if a problem were reflecting, it might be

easier to do the other method.”- Jacob

_side side side __ OR _side angle side _. For me, right now, _side side

side___ is the easier method because _it is a lot neater and easier to

write out_. I can image that if a problem were like _side side side_, it

might be easier to do the other method.”- Abby

You need to have three points of information for it to be even close to

correct. -Kessa

If your only proof is set up SSA then the triangles dont have to be congruent.

-Matt

The sides will be the same and the other angle is the same so one of the

triangles may need to be turned after the triangles are made. -Ryan

This lesson was a wonderful way for students to explore triangle

congruence in a hands on way, and the reflections confirmed my

opinion that students really struggle with this concept. Having them

write gave me a way to address misconceptions in an anonymous

way.

I agree because if the sides and angles are congruent and in these orders

then the others are set in place. -Matthew

After this series of lessons students continued to study triangle

congruence, and worked on more traditional proofs, although

using a 3-column format- demonstrating which three parts were

congruent, in order of part (SAS) and how they could prove those

parts were congruent. This series of lessons was a terrific way to

introduce the need for more formal proof- students did not want

to continue testing cases forever, they wanted to know which

cases worked. As students worked on proof, they incorporated

some strategies from the lessons like color coding or drawing a

few cases to make sure they knew which parts were involved.

Students still struggled with formal proofs, but has a much more

intuitive idea about using transformations to decide if shapes

would be congruent.

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