Teachers Name: Mr. VanBogaert

Name of Unit: Congruency Proofs and everything in between

Subject Area: Geometry

Grade Level: 9th-10th grade

Instruction Time: 80 minutes

Lesson Summary: This lesson is all about exploring and deriving the theorems that work for

proving triangles congruent. This lesson, which is a lab based instruction, is the single most

important lesson that will take place throughout this unit. The concepts and ideas that are

determined and learned in this lesson will help guide the rest of the unit, as the entire unit deals

with being able to prove congruency or similarity. The other big idea of this lesson, triangles, is

also extremely important, because as we work towards our next unit and proving quadrilaterals at

the end of this unit, triangles and being able to manipulate them will become the biggest theme.

This lesson is part of the introduction into proofs and understanding/ analyzing theorems that

work and do not work when it comes to proving both triangles and the corresponding parts of

those triangles congruent. The lesson what will come after this one is an activity where we

examine the different types of proofs and how we set them up, and what they tell us. This

opening lesson, is one that will help when it comes to this idea, as it lays out step by step what

we are going to try and achieve (builds the proof without the structure of the proof). Because this

is the first lesson, there is no lesson that precedes it, rather, the writing and deciphering of class

definitions of what congruent and similar mean, so that we can have a foundation for the start of

the unit.

This lesson works by having the students be the ones that do the exploration. This exploration is

completed either using chicken wire, or angel-hair pasta, depending on what the school will and

will not allow (NOTE: chicken wire works better, however, it could be deemed dangerous or

unsafe). Once the students have the proper supplies the activity is laid out for them step by step,

starting with the SSS case than the SAS case, and finally the last case on the lab is one where

instead of proving congruency, we prove similarity, using AAA on equilateral triangles of

different sizes. Although this lesson does not prove the ASA case directly, the SAS case can be

altered so that both the ASA and the SAS case work. This lesson will take proximately 50 to 60

minutes to complete, by leaving 10 minutes, we can come together as a class and discuss through

what we found as well as determining if other cases work or do not work. These cases include:

SSA, AAS, AAA, SAS, SSS and ASA (four will work; one will show similarity and the other

one SSA will not work).

Standards:

1. Use the definition of congruency in terms of rigid motions to show that two triangles are

congruent

Corresponding pairs of sides and corresponding pairs of angles are congruent.

2. Prove theorems about lines and angles.

Congruency relationships

Enduring Understandings:

The biggest idea that students need to take away from this unit is the idea of congruent versus the

idea of similarity. In order to do this, the students will have to understand the qualifications for

both and be able to explain why those qualifications prove one or the other. They will also have

to be able to make sure that they understand that we can use similarity to prove congruency.

This is really the only enduring understanding for this lesson, because the lesson is so specific to

one task.

Objectives:

By the end of the lesson, students will be able to correctly notate triangles, labeling all parts

correctly, with zero error.

By the end of the second case, students will be able to correctly notate congruency between

triangles, with no mistakes.

After the beginning of class, students will be able to define congruent and similar.

By the end of the lesson, students will be able to analyze and explain congruency in triangles.

Differentiation: The biggest thing that would need to be differentiated is allowing the students

the opportunity to either work with partners or work alone. Working partners has an advantage of

being able to bounce questions and ideas off of one another, however, at the same time; I

understand that some students when they work in partner groups end up doing all the work

anyway. That is why I will leave the option open for the students whether or not they would like

to work with a partner or if they would rather work on the concepts and ideas alone. If they do

chose to work solo on this lab, than they will lose the advantage of being able to use a peer for

help and questioning, although this is something that most students would not be ok with, there

are some students that will feel more comfortable and more relaxed, if they can just bounce

questions off of me or sort through their questions or concerns on their own. Not every student

always likes to be social, there are times when people want to be more reserved and more

independent, and I understand that, which is why I am leaving the option open.

The next thing that needs to be differentiated is the amount of time the students get to do the

activity. Seventy minutes may seem like a long time do a lesson, however, when the students get

going and start working, at least a fourth of that time will probably be spent talking with friends

or socializing about something other than the task at hand. The students will also notice the time

start to flyby once they are working, because they are engrossed in the material and they are

keeping busy. On the other hand, I will also have students that will be done with the lab in a

super quick fashion, because it just naturally makes sense to them. These are the students that I

am really worried about differentiating for. When a student is way ahead of where everyone else,

it can be extremely hard for them to remain engaged in the lesson or remain active in the

classroom. In order to help ensure that this does not happen during this lesson, I have two ideas.

The first idea is allow these students to work on the analysis discussion that we will have at the

end of class, the second idea, is to have them write up proofs for each other, or if they do not

wish to do that, than I will allow them to help other students who might be struggling. The idea

behind letting them work with struggling students is because peer-to-peer interaction is just as

helpful when it comes to learning as student teacher interaction (social learning theory). By

having the students help other students, it will also allow for more thorough questions from the

struggling students. This approach also helps me out, because I cannot be in more than one spot

at a time; which is very important in something like this, because there is going to be more

questions by the students then I can answer all at the same time, without disrupting the

discussions and work that is taking place.

The other differentiation that I must account for is the students, who are really struggling and

instead of just needing more time, may also need a more concrete way of thinking about the

material as well as a different approach to deriving the material so that they can learn it. The

easiest thing that I can do to help these types of students is pretty simple, change the wording of

the questions and the wording of the ideas that we are trying to explore. For example the first

thing that I can do is instead of having them abstractly explore the ASA case, I can have the

option for the students to see and work with a visual representation for this case, as well as give

them the actual question that I am asking so that they have an idea of what they should be

looking for. Because how the lab is currently written, the ASA case does not actually appear at

all, instead we only are proving two cases and a similar case. The next thing that I can do is make

sure that I spend more of my time with the students that are having more issues and are frustrated

more. By deviating my time to these students, I can clearly answer any questions or concerns that

the students have, as well as make sure that any misconceptions that arise are taking care of. That

way I know exactly what the students know and do not know.

Materials:

1. Flat surface area

2. Chicken Wire or pasta noodles

3. Ruler

4. Protractor

5. Lab work sheet

6. Pencil or pen

7. Scissors

Lab Handout--

Worksheet lay out:

Answers expected:

SSS

1. Cut off six pieces of chicken wire, so that two lengths are 6 cm long, two are 8 cm

long, and two are 10 cm long.

2. Build a triangle use on 6 cm piece of chicken wire, one 8 cm piece of chicken

wire and one 10 cm piece of chicken wire.

The triangle that is constructed should be a right triangle, because of the 3-4-5

triple property. (See the picture above)

3. Build another triangle using the remaining pieces of chicken wire.

4. Compare the two triangles. -Are the two triangles congruent? Is there any way to build a second

triangle that is different than the first triangle?

Yes they are congruent, because the sides are the same length no matter what,

because there are only three different sides. There is no way to build a non-

8 cm

10 cm

6 cm

10 cm

8 cm

6 cm

congruent triangle, because we are restricted to the three lengths, and only using

those lengths in both triangles. Therefore, we will always have equal sides.

SAS

1. Using the one 8cm and one 10 cm piece of chicken wire, construct an angle. One

partner create an angle that is 30* and the other partner create one that is 60*.

2. Record the measurement of the angle here. ___

The angle measures will either be 30* or 60*

3. Find a partner and record their angle measure.

4. Record partners angle here. ___

The two angles will either be the same angle, or they will be a different angle, it is

ok, because the two angles will produce a 90* angle once the triangle is

completed.

5. Complete your triangle and measure the remaining angles.

-Are the two triangles congruent? If so why?

Yes the two triangles are congruent, because the two side lengths are the same

and the angle between those sides is the same (8 cm, angle X, and 6 cm side).

This also proves the ASA theorem as well, because the two angle measures

were shown to be congruent.

8 cm

10 cm 10 cm

8 cm

30* 60*

8 cm 8 cm

10 cm 10 cm

6 cm 6 cm

X X

AAA

1. Create a triangle using three 6 cm pieces of chicken wire.

Equilateral triangle

2. Create another triangle using three 10 cm pieces of chicken wire.

Bigger equilateral triangle

3. Measure the angles in the 6 cm triangle.

4. Record measurement here: __

60* (all angles in an equilateral triangle are the same)

5. Measure the angles in the 10 cm triangle.

6. Record measurement here: __

60*

7. Compare the two triangles are determine if they are congruent. -If they are not congruent, what would they be?

The two triangles are not congruent; they are similar, because although the angle

measures are all the same, the two triangles are completely different lengths.

Therefore the AAA postulate does not prove congruency, it proves similarity.

Student Assessment: In order to assess the students on how well they understand the material as

well as their mastery of it, I am going to do two things. First, we are going to have a ticket out.

This ticket out will have five questions on it. These questions will analyze if the students can tell

the difference between congruency and similarity, as well as test how well they can explain the

theorems and how they work together to prove triangle parts congruent. The other thing that I am

going to do is assign a little homework. This homework will be an essay question in which the

only thing that needs to be addressed is explain and describe the four ways that prove

All sides are

6 cm long All sides are

10 cm long

congruency. This question is the single most important question that will be asked throughout the

entire unit. If the students can answer this question correctly or accurately then the rest of the

unit will flow smoothly, however, if we have to do a lot of refreshing and reviewing, this unit is

going to be difficult and one that takes longer than expected.

In order to fairly and justly grade how well the students know the material, more emphasis will

be placed on the essay question and how well they respond and elaborate on it. The things that I

will be looking for are as follows: all four theorems are stated and described; the theorems are

explained correctly, each section has detail and evidence to support why it works, and each

section includes proper vocabulary and mathematical diction where it is needed.

Before handing out the assignment I will make sure to explain to the students what is expected

how they should set up their answers and how they will be graded. I think that is the only way to

make sure that the assessment is fair and graded accurately and thoroughly.

Instructional Strategies: The biggest strategy that I will use is a peer to peer cooperative

learning strategy. The reason that I am choosing to use this strategy is because not only does it fit

perfectly with the partnering up element of the lesson, it also allows other students to bounce

ideas off of each other in order to not just arrive at an answer but also to reason through their

answers. The other strategy that I am using in this lesson is the discussion/ experimentation

strategy. This strategy is actually the one that is used more throughout the course of the lesson,

because of the breakdown of how we go about proving and seeing the different theorems as well

as when we come together as a whole class to verify our results and discuss why certain cases

work and why others do not. Both of these strategies are direct strategies and both of them

involve me being able to ask the right questions as well as making sure that when a student ask a

question, I do not simply just answer the question but make sure that the other students have a

chance to offer a solution.

Sequence:

1. Pass out the materials and the lab sheets. (2 minutes)

2. Write the words congruent and similar on the board and as a class come up with a

definition for each one. This is extremely important, because it will be the focal point

for the lesson once we start proving congruency theorems. (3 minutes)

3. Begin the exploration by cutting each length of wire to the specifications asked. This

could take some time as the wire can be a little tricky to cut right away. (5 minutes)

4. Once everything is cut to the correct lengths, the first thing the students will need to

do is begin working on the SSS case. This case should not take that long as it is the

easiest one to show. (10 minutes).

5. After the Students have all finished the case, or just about wrapped the case up, bring

the whole class together for a few moments and have a discussion about why the case

works or why the case does not work. (2 minutes)

6. Work on proving the next case, which is the SAS case. This case will take a little

longer, because of what the students have to do with measuring angles and comparing

triangles. (15 minutes).

7. Bring the whole class together again for a discussion. Make sure to ask the question

can I use this idea of SAS with every triangle regardless of the size of the first angle?

The answer is no, because an obtuse angle and a right angle would violate the 180*

angle sum principle. This is an extremely important point and should be talked about

in detail. (10 minutes)

8. Prove the last case, which is AAA. This case will not show congruency, however, it

will show similarity, which means that the students class definitions will come into

play during this case. Also note that while the students are trying to prove and show

this case, when they measure the angles, they are not permitted to use a protractor,

because the triangle is a special case triangle. (15 minutes)

9. Bring the class together for one last discussion about how AAA shows similarity and

how we could adapt this case to show congruency, by implementing a side or equality

in between two of the angles. This case is known as the ASA case. (10 minutes)

10. Allow the students to ask any questions they have about any of the cases and be sure

to address those questions. (5 minutes)

11. Hand out the analysis worksheet. We can either go over this sheet as a class, or the

students can choose to work individually on their own. Note: if the students want to

work on their own, every 2 minutes, stop and have a student explain what they are

thinking in relation to the case and how they would label the triangle. (10 minutes)

12. Have the students do the ticket out and pick up the essay question on the way out the

door. (3 minutes)

Reflection: Although we do not have to do the lessons in this plan as part of the assignment, I

actually conducted this lesson during my practicum. When I first started the lesson out the

students seemed a little bit confused, I think was because I did not initially start with a whole

class definition of what congruent and similar meant, instead I just took for granted/ assumed

that the students had seen the words before and knew what they met. This was a huge mistake

because not only did it lead to massive confusion on what the purpose of the lesson was, it also

lead to a loss of time, when we had to stop the lesson and go back and readdress what each one

of the two words meant.

The biggest change that I made during the lesson was on the second day of teaching it. The very

first thing that I did was to make sure that we had a class wide definition of what each term

meant and how each term was going to be used throughout the course of the lesson. By doing

this, it really made clear the goal and purpose of the lesson and helped the students to put out the

key details and components to each theorem that we were trying to prove. I feel that if I was able

to do this with the first class, rather than having to wait and fine tune the lesson for the second

class, I think the first class would have been much more attainable and the lesson would not have

had such as huge issue like it did.

Teachers Name: Mr. VanBogaert

Unit: Congruency Proofs and everything in Between

Subject Area: Geometry

Grade Level: 9th-10th

Instruction Time: 70 minutes

Lesson Summary: This lesson has one focus and one focus only, being able to set up and write

proofs correctly. This can be done in three different ways. The first way is with a paragraph

proof. In this style of proof the mathematics is incorporated which the paragraph of logic. What

this means why this is significant it because the paragraph is simply just what the students are

thinking. Instead of just having them explain it out loud or give an answer, they are instead being

asked to write down their thoughts and ideas. These types of proofs are structured where the

paragraph consist of both words and symbols. For example, the congruency symbol can be used

instead of writing out the word congruent; same for the similarity symbol. The second type of

proof is the one that most people are familiar with. The two column proof is composed of

making a T-chart and then labeling one side STATEMENTS and the other side REASONS. The

way that these proofs work is the statement is writing and then the reason that statement is true is

than given. For example, if we were doing the proof of SAS, than we would say that triangle

ABC and triangle DEF are congruent. The reason for this would be because of SSS, since all the

sides are the same length as a corresponding side on the other triangle. In this type of proof

symbols are used more frequently than written expressions, simply because they take up less

room and are more easily read. The last type of proof is the outline proof. This style of proof can

get really messy and because it is just bullet points, it can be difficult to justify everything that

the problem requires. In this type of proof, the student will use almost all symbols and then give

brief rationale for each statement (the reason is no more than one sentence). This proof style will

be taught very briefly, because we want the students to try and avoid using this method when

they are writing their proofs. However, the reason that we teach it is because this style can be

really helpful when trying to figure out what needs to be proven first and what needs to be

proven second.

The lesson that comes before this one is the lesson on congruency theorems and the relation

between congruent and similar. The reason that this lesson is done after the previous one, is

because in this lesson, we are taking the proofs that we have shown and actually writing them out

and articulating them into words and phrases. If we introduced how to write and set up proofs

before having anything to prove, than the lesson would not be as fluid and would have a serious

gap in the content it is teaching.

This lesson is a construction lesson. The way this works, is the students will have a packet of

proofs that needs to be completed. They will also have an envelope filled with the reasons and

statements that are going to be important for this lesson. All the proofs that they will be doing

can be done using a certain combination of the statements and reasons provided. The

construction aspect of this lesson comes in when the students have to work through each proof

and lay out the steps in the correct order, before being able to write them down on the paper. The

reason that I am choosing to have the students blow up the statements and reasons is because by

having the visual and having to put them in a certain order, it will teach the students how to

correctly lay out a proof and what works and what does not work when it comes to proving

something. For example, if the student is proving triangles congruent, they cannot say the

triangles are congruent, until they have proven the parts necessary to prove the triangles

congruent first.

Standards:

1. Give formal and informal arguments about triangles and quadrilaterals to prove

congruency.

2. Write proofs about shapes

Congruent

Similarity

CPCTC (corresponding parts of congruent triangles are congruent.

Enduring Understandings: The single most important thing that the students need to take away

from this lesson is how proofs are written and how they logically flow. In order to do this the

students will need to be able to analyze proofs and how they are written and be able to explain to

themselves (because I already know the material) why the proof makes sense. The students will

also need to be able to correct proofs that are written incorrectly; by having the students do this,

it really demonstrates mastery and understanding of the material. Because proofs are the entire

unit, the only way we can move on at all, is if the students are able to grasp this lesson and the

concepts being taught behind it.

The second thing that students must understand for this lesson to be successful, is that they can

write a single proof in a variety of ways. That means that one way is not always the best,

however, one way is often more appropriate than another in the context of the problem, or what

is being asked. Usually when the problem ask for justification a paragraph proof is more

beneficial; however, if the problem just says prove, than a two column proof is the easiest to read

and is the most appropriate type of proof for the problem. The third proof style, outline proofs,

should only be used as a reference when trying to write a paragraph proof or a two column proof.

The outline proof should never be turned in or used as the means to solve the proof. The reason

for this is because the style does not require any detail or explanation. Therefore the style is

incomplete and the proof lacks merit.

Objectives:

By the end of the week, given a problem, students will be able to correctly write the proof.

Students will be skilled at identifying what type of proof is more logical and apply this style of

proof to a certain problem.

Given the congruency theorems that we proved the other day, Students will be able to write up and

articulate the justification for why each theorem works or does not work. (This is a proof by

contradiction or a direct proof).

By the end of the lesson students will be able to logically plan out the steps for constructing and

writing a proof.

Differentiation: There is not a whole lot that can be differentiated in this lesson, because the

lesson is so laid out for the students. However, there are a few small things that I can do to help

ensure that the students stay engaged and are really working on understanding the material. The

first thing that I can do is not require all the problems be completed. This has two benefits. The

first benefit is for the student who is way ahead of everyone else and is just bored and viewing

this assignment as busy work. By not requiring them to do all the problems this type of student

can avoid having to sit there and not challenge themselves. Instead of having them complete the

work sheet, I would instead give them proofs to solve. These proofs would involve using the

ideas that we previously learned and being able to apply them in the structure of proving

properties and notions about quadrilaterals. Although we have not covered this idea of a

quadrilateral, we are going to be working on it once we finish proof styles and triangles.

Therefore I am not just giving the students something that they could never actually accomplish

or understand, I am instead giving them a segway into the next lesson and the next big idea that

we will be looking at and working with. The other benefit that not requiring all the problems be

completed is with the student who might be struggling or stressing about the amount of work. By

reducing the number of problems it can help to alleviate some of this stress as well as not over

whelm the students. This can be a big factor in being able to keep students engaged in the lesson

as well as helping the students work at their own pace.

The next thing that I can do to differentiate the lesson is instead of making them use the outline

proof style; I could just get rid of it all together. This style is not a huge component to the rest of

the unit, we as the other two styles are going to be the focal point for proving everything from

this point on. That means that although I am showing the style and than having the students write

up the style, I could instead just have them write up two styles instead of an outline for their

thoughts. If it is the case that the students really like this and think it is helpful, I can leave the

option for them to use the style at their own discretion as a means to help them gather and collect

their thoughts and ideas, I would just than have to explain to them, that this style will not be

graded or allowed on a test/quiz/homework assignment.

Either way I choose to differentiate the lesson would work. I could even choose to do both ideas

that way for the students that are having a major struggle with this lesson, they can take a deep

breath and regain their composure and feel less stressed. Because alleviating the stress and the

pressure that time and work load pile onto a student is something that is really easy to correct and

fix, it is also a great way to lose the engagement of the students, therefore it is essential that the

work load is attainable and manageable, not ridiculous. That is why I could choose to do both

time and work load, rather than just choosing one.

Resources:

1. Statement and Reasons T-table

2. Book

3. Envelope with statements and reasons

4. Problems to be proved worksheet

5. Pencil

6. Paper

Worksheet:

Answers Expected:

Unpacking Proofs

This lesson is designed to help you unpack the parts of a proof and build the skills necessary to

formally write your own proofs. The following proofs will help you to build the structure of a

proof. It is important to remember that in order to successfully build a proof; the proof must

consist of all parts required. Do the first two in two column style, the next one in paragraph style

and you can choose the style for the final one.

1. Prove ABC (congruent) DBC

Given: quad ABCD is a square,

A C

B D

Line BC bisects

Given: ABCD is a square

DG, BE, CF, AH are midpoints

(Note this is a very complex long proof, this proof is designed to get you thinking about what

steps work and what steps do not work when solving a proof).

In order to solve this problem, students need to show that all large triangles are congruent, by using

SAS; they then need to show that all the smaller triangles are congruent to each other using ASA. Once

they have showed both of these, they want to use that information to show that all the sides of WXYZ

are all equal in length. This is done by using the subtraction property and the equality postulate. The

students than need to prove parallel sides and relating them to alternate interior angles as well as the

definition of right angles and supplementary angles. Finally the students need to prove that all interior

angles of quad WXYZ are congruent; this can be accomplished through the use of vertical angels. Once

they have shown all these parts, they can than make the claim that WXYZ is a square.

Assessments: The most effective way for me to assess the students would be to have them

complete a ticket out. This ticket out, or exit slip would consist of three questions. These

questions would be based on determining if the students can successfully sequence a proof or if

they are unable to do so. The first question would be what is the very first thing that must be

shown when writing a proof? The answer is the givens, however, I think that the students might

have a tendency to write down that the first thing that needs to be shown is what we are proving.

If I end up getting a lot of these responses then at the beginning of class tomorrow, we will have

a quick bell ringer that clarifies the parts of a proof and the order in which they must appear. The

second question will be does proof order matter? The answer to this question is yes, and any no

responses have to be immediate red flags. That would mean that the students do not understand

the structure at all and we need to spend a lot more time going over the notes and the styles

again. The final question will be what are the three styles of proofs? This is a straight forward

A

D

B

C E

G

F H

W X

Y Z

answer, paragraph, column, or outline. Anything but these three answers and I will have to talk

to the student directly to see if they are having trouble, or if they just did not pay particular

attention in class that day.

The other thing that I am going to do to assess the students understanding of proofs and their

structure is I am going to assign the three problems that they had to do during class, for

homework as well. However, they have to write their proof in the other style than the previous

one. For example in problem one they have to prove using two column format, that means for the

homework they have to prove using paragraph. The last problem on the sheet, problem number

four, is not going to be required that the students complete the proof, however, the students will

need to look up the properties of the square that we wrote down as a class, while we were saying

what would be needed to solve the problem. The reason I am choosing to add this to the

homework is not because it directly assess their ability to write a proof, however, it does make

them think about the proof and if it really does show what they want it to show. This will be

important come the unit test as well as when we get into the properties of quadrilaterals later in

the unit. The properties that the students need to look up are: parallel sides, all angles are right

angles, the diagonals bisect each other, and all sides are the same length. These properties must

appear under the picture and an explanation of definitions and ideas must also be written. This is

a great way to not only test if the students are just following the template or if they are logically

thinking about what goes where and how they all relate to each other; it also works as a great

segway into the properties of quadrilaterals.

Sequence:

1. Pass out the statement and reasons envelope (2 minutes)

2. Draw a T-chart on the white board and have the students copy it while I finish passing

out the envelopes. (1 minute)

3. Ask the question what does it mean to make a statement? Then wait for an answer (2

minutes).

4. Repeat the same question with a reason; once again wait for the students to provide an

answer. (3 minutes)

Note: The answer may not be exactly correctly, however, getting the ideas flowing and

getting them thinking about math will be more beneficial, if the two questions go

incorrectly answered.

5. Discuss and write the components of a two column proof. Be sure to include, givens,

proof statement, the body and the result. (5 minutes).

6. Do a simple example of how to write this style of proof. The example is nothing

complicated or difficult; it is simply just a reference for the students. (3 minutes)

7. As a class configure the proof that we have just created into written words. (5 minutes)

8. Once we have the paragraph constructed, pick on random students to go up and circle or

underline the important components of the proof. (5 minutes)

9. After everything is labeled ask the students what they notice about the two styles of

proofs. (3 minutes)

Note: they should come to the conclusion that the two proof styles can be interchanged

with one another. This will be important when it comes to being able to do simple proof

elements out loud or in a class teaching situation.

10. Answer any questions or concerns that the students have before going into the worksheet

activity. (3 minutes)

11. Have the students work with a partner on problem one. (10 minutes)

12. Add one person to each group so that now there are small groups of three and have these

groups work on problem two (10 minutes)

13. Once the students are done with problem two, have them complete problem three. (5

minutes)

Note: This problem is a lot smaller and easier to prove; therefore it should not take as

long as the previous problem did.

14. Explain what is expected of the students on the last problem. (5 minutes)

15. Answer questions about any of the proofs and have the students complete the ticket out.

(8 minutes)

Reflection: I got really luck with this unit. I was allowed to help teach or teach the entire unit. This was

one of the lessons that were taught and during this lesson I found out that a lot of the students really

struggled with being able to articulate the proof into words. They had no problem with the two column

statement and reasons they just had difficulty going from the column style and putting that into

sentences. This lesson actually took us two days to complete, because we could not move on until the

students had a firm grasp on how to write a proof. This lesson also took two days, because the problems

required the students to check with my cooperating teacher or myself before being able to move on to

the next problem. We found that by making the students check their answers, we were able to identify

weak points in the proofs and we used that to help sum up the lesson on the second day. Some of the

weaknesses we saw were: using the right reasoning at the right time, completing the proof, using the

right theorem, labeling the parts of the figure that we are trying to prove, and explaining why the proof

works.

The second day of this lesson, we spent finishing up the proofs and began the class by making sure that

we clarified and explained that when writing a proof there is a checklist of things that need to be

completed before you finalize the proof. Each student was asked to copy this checklist down and we

made them use this list on day two. The students better understood the material and by the end of the

lesson, only a handful of students (the ones who sit and do nothing) were unable to explain the steps

and why they logically work.

If I had to do this lesson all over again I would make two changes. The first change would be that I would

spend more time on making sure that the logical thinking aspect of a proof was well known and

demonstrated by everyone. The reason I would focus on this is because most of what we learn and do in

Geometry is driven by logic. If the students are unable to process and think logically than they will have

major issues for the rest of the unit. Similar to how important it was for them to master the triangle

theorems on the first day, it is equally as important that they master logic. I went into the lesson

thinking that they had already accomplished this; however, I quickly learned that this was not the case. I

think by spending more time on logic in the beginning, it would help to make the time of the lesson

more efficient and more valuable to the students.

The second thing I would do is I would require that more paragraph proofs were written as opposed to

the amount of two column proofs. This might not seem like a huge thing; however, the better the

students can write down and articulate the mathematics the better they truly understand it. Two

column proofs are organized and easy to read and follow; the only problem is that these proofs have a

format and a structure, making it difficult to truly assess just how well the students know the

mathematics or how well they can plug pieces in the right spot. If it was up to me I would have made the

students write a paragraph proof for every single problem. Now, in order to do this, they would have

had to write the paragraph before they created the two column proof, because otherwise it would be

just converting from a bullet to a sentence; this would take more time and might lead to more frustrated

students, however, the articulation is important for the student, and that is what every lesson needs to

focus on, the student. That is why the paragraphs are even more important.

Name: Mr. VanBogaert

Unit: Congruency Proofs and Everything in Between

Subject: Geometry

Grade Level: 9th-10th grade

Instruction Time: 70 minutes

Lesson Summary: This lesson focuses on being able to manipulate different quadrilaterals in

order to change them into different shapes, for example, converting a parallelogram into a

rectangle, converting a trapezoid into a rectangle, and converting a rectangle into a right triangle.

All of these conversions are important because each one of them focuses on how the area of one

shape relates to the area of anther shape as well as it helps when trying to prove triangles within

different shapes.

How this lesson works is students will be given different shapes and they will be asked to change

the shapes into one of the shapes above. In order to be able to do this, we must first go over how

the different shapes can be formed, by writing and using some definitions. The main definitions

that we will use are midpoint, midline, and perpendicular. These three definitions will be

extremely important in creating our manipulated shapes. In order to direct the lesson so that the

students are able to learn these different definitions, I will start the class off by having five

different quadrilaterals written on the board and as a class we will spend about twenty minutes

making sure that we fully understand each definition and are able to use them in the correct

format.

Standards:

1. Experiment with transformations in the plane

2. Make geometric constructions

Essential Understandings: There are two things that must be taken away from this unit and that

students must be able to understand by the end of it. The first thing is students must understand

the concept of a midline and a midpoint. These two things are essential in being able to create

and manipulate different shapes. When a student can manipulate one figure and turn it into

another figure that shows that they can think abstractly about the material and that will help them

when it comes to proving quadrilaterals are indeed quadrilaterals and it will also help to prove

the :truss idea which will be taught in the next lesson.

The second thing that the students must learn is the concept/ idea of being able to use

manipulatives in order to change from shape to shape. This idea is known as a transformation

and it will be a key to the rest of this unit as well as when it comes to surface area and volume in

the next unit.

Objectives:

1. Students will be able to correctly transform quadrilaterals into triangles

2. Students will be able to find the midpoint and midline of a triangle or rectangle.

3. Students will be able to explain the cuts and lines that they create for each figure in order

to transform them.

Differentiation: There is not a whole lot that can be differentiated for this lesson. The only thing

that I can really do is give the students who are struggling; the chance to have the explanations

in bullet format so that they can see what is expected. By doing this it will help them to

understand how the objects should be created as well as how the manipulations should work.

This lesson is one that will be teacher directed more than any other lesson that is contained in

this unit. Because of that the differentiation will not be as needed, because the whole class will

be working together. However, with that being said, it is also extremely important that the

assessment is differentiated as well, if the students need it. There is not a whole lot that can be

differentiated in terms of this lesson; however, the amount of work can be differentiated as well

as the explanations that I expect. When the students are doing the cutting and configuring they

are also expected to explain what each cut is and how they got from one shape to the other. This

is a crucial component of the lesson, because this will lead directly into the assessment.

Materials:

1. Pen or pencil

2. Vocabulary terms that we go over in class

3. Ruler

4. Shapes

5. Scissors

6. Notebook

Problems on the Board:

Answers expected:

1. Convert the parallelogram to a rectangle.

Draw a line perpendicular from one corner of the parallelogram to the base.

Shift the triangle to the right, and then rotate into place.

2. Convert the Rectangle into a right triangle.

Draw a line from the upper left vertex of the rectangle to the midpoint of the opposite side.

Rotate the triangle 180* and then slide into place.

3. Convert the trapezoid to a rectangle.

Cut a perpendicular from the top left corner to the base. Rotate the triangle 180* and then flip it over

and into place.

Student Assessment: There is not real formative or written assessment for this lesson. There

will not be a ticket out and there is no homework. The only thing that will be used to assess if the

students learned the material or not, is when they put the shapes into their notebooks, they have

to come up with the explanations in their own words. The reason I am choosing to do this, is

because the students will remember and learn them better if they have the chance to put it in

terms that they understand. This is different from my other plans, however, I think a break every

now and then from homework is a good thing, as well as it gives them the opportunity to reflect

on what we have done so far for our next upcoming quiz.

Instructional Strategies: This lesson is lecture based lesson with a lot of teacher direction and a

small amount of student explanation towards the end. The reason that I am choosing to direct this

lesson so much, is because there are critical definitions and terms that the students must

understand in order to do any of the scissor congruent transformations. These terms include, flip,

rotate, perpendicular, midpoint, midline, slide, and reflect. I feel more comfortable directing this

lesson; so that I can make sure that each student truly understands the definitions and that each

student knows how each transformation works and is used.

The student aspect of this lesson comes into play when they have to take the shapes above and do

the explanations for them in their own words, however, they should be similar to the words that I

used in the sample answers. This is the part when I will stop lecturing and give the students time

to configure each transformation into their words. By doing this, I put the learning in the students

hands, however, I am also able to make sure that they understand each transformation by

collecting and evaluating their work.

Sequence:

1. Have students get out their notebooks (1 minute)

2. Write the terms on the board: flip, rotate, slide, perpendicular, midpoint, midline (3

minutes)

3. As a class define each of the terms in the students notes. (15 minutes)

4. Draw the pictures on the board of the parallelogram, rectangle, and trapezoid. (2

minutes)

5. Have the students name each shape. (2 minutes)

6. Draw the other three shapes that they need to transform to: the rectangle, triangle, and

rectangle. (5 minutes)

7. As a class work through each one of these transformations, by having the students

decide what to do and then going step by step through the process. (30 minutes)

8. Allow the students to ask questions and return materials. (2 minutes)

9. Give the students the chance to convert each transformation into their own words (5

minutes)

10. Check the answers for understanding and coherency. (5 minutes)

Reflection: This lesson when I actually taught it went the smoothest of any lesson. The students

really liked being able to cut and manipulate shapes as well as learning about how each

transformation worked. The goals of this lesson were to show relationships between areas of

different shapes as well as be able to make geometric transformations. Both of these goals were

met, because every shape was changed from one shape to the other. The students had the

opportunity to work with each shape and work through how to get from one to the other. They

also were able to see how area formulas relate to that of a rectangle, because that is the shape that

is most used and most well known by everyone. The lesson also allowed the students to ask a lot

of questions and during the thirty minutes of teacher directed discussion; we got into some deep

conversations about the mathematics which really showed when the students took their test the

following week.

There is not a whole lot that I would really change about this expect maybe the amount of shapes

that the students had to do. I feel like three was a little unhelpful, because there are more shapes

then just rectangles that need to be transformed. However, because this is really not that involved

of a lesson in terms of the rest of the unit, it was ok that we did not do more than just three

shapes. The main focus was so that the students knew the terms and how to do each

transformation, and that is what they learned during this lesson. If I had to do it again, I would do

it exactly how we did, with the addition of a different problem for the students who were way

ahead. These students got kind of board and we actually had them help the students around them

that were struggling. That is the only thing I would do differently, I would make sure that the

advanced students stayed engaged and challenged, not assist the teacher.