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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