Stage One- Desired Results

Established Goals Transfer

(Common core Math Standards) Students will be able to independently use their learning to…

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.

Identify the properties that describe the congruent relationships between triangles, in terms of

how they relate to the ideas of similarity, congruency, in a well written and articulated proof.

Prove theorems about triangles

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

SUBSTANDARDS Meaning

Quadrilaterals and their properties

Writing proofs UNDERSTANDINGS Essential Questions

Congruency Relationships

1) 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.

2) The second thing that students need to understand is the idea of triangle congruency and how we

prove triangles congruent. (SSS, ASA, SAS).

3) The different types of quadrilaterals and how they relate to triangles and proving triangles congruent.

4) Students will understand the proper way to correctly right a proof and be able to explain what a proof

is saying if they are given one to look at or examine. The students will understand what the first part of a

proof is (givens) what the last step of a proof is (the result) and how the pieces in between correlate/

relate to the first part and the last part (they are the theorems used to reach the result).

How do the concepts of SSS, SAS, and ASA relate to the concepts of AAS and AAA similarity?

How could these concepts be applied to quadrilaterals?

Why are proofs important? What is their purpose?

Students will know… Students will be skilled at…

1) Students will know the difference between similarity and congruent.

2) Students will know how to prove triangles congruent using one of three theorems.

3) Students will know the difference between different types of quadrilaterals and their properties.

4) Students will know how to set up and correctly write a paragraph proof, two column proofs, and an

outline proof.

5) CPCTC (Corresponding parts of congruent triangles are congruent)

1) Students will be able to explain why the three triangle congruency theorems work.

2) Students can explain why AAA (angle-angle-angle) is not a good indication of congruency.

3) Students can explain the properties of quadrilaterals and be able to prove those properties.

4) Students will be able to determine the different parts of a proof.

5) Students will be able to answer the question: “what is the fourth way I can prove triangles congruent?

How come we do not use this method very often?”

Stage Two- Evidence

Evaluative Criteria:

Theorem Application Create a new way to determine how to configure the different congruency

theorems in terms of how they are applied to one another. Through this plan be able to explain how you

know that each theorem holds as well as be able to teach the concept to another student through the

use of peer mediation (examples include: real life analysis of the proofs, such as ski slopes or the

curvature of trees, pyramids, mountain landscape, and housing architecture).

Accuracy of the proof Draft a well written and well-articulated proof that explains the differences

between congruent and similar. Within the proof students must have the correct use of each theorem

as well as the definitions of both congruent and similar.

Compare and contrast different types of triangles, including scalene, isosceles, and equilateral. Be able

to show and explain which ones are similar and which ones are congruent. Also be able to differentiate

between the different types of triangles and be able to evaluate what each type of triangle tells us and

how that relates to being able to solve for congruency or similarity.

Evaluation of similar triangles Write a proof using all three methods that have been taught: paragraph,

two column, and outline.

Solve a proof using all three methods Use the properties of quadrilaterals and the concepts of

congruent triangles to prove different aspects of quadrilaterals (including but not limited to: proving the

shape is actually that shape, proving one shape is similar to another shape, and proving the specific

properties of each shape).

Other Evidence:

Vocabulary articulated correctly and well argued in explanation Students must be able to demonstrate

full knowledge of the content by using proper vocabulary in the above assessments whether that be in

the form of an essay, a constructive response, or a selected response. The more vocabulary that is

properly used the more mastery the student will demonstrate (vocab includes: ASA, SSS, SAS, congruent,

similar, scalene, isosceles, equilateral, and definition of proof).

Stage Three- Learning Plan

Summary of Key Learning Events and Instruction:

The key to the transfer goal is that the students’ needs to be first guided through the process of each

theorem and procedure in order to determine congruency or similarity. The aim for this unit is that by

the end of it the students will be able to independently create and articulate a proof of the theorems

and postulates that are the desired transfer and understanding/skill goals.

Pre-assessment: Use a pre-test with five questions. These questions are: name the three styles of

proofs, name the four ways to prove triangles congruent, what is the difference between similar and

congruent, and then two proofs that will have to be solved. (The students are expected to really struggle

with this because most of them would have never had any experience in any of these areas).

Begin with an introduction into the definitions of congruent and similar. From this, we will start the

launch of the activity by taking these two propositions and applying them to just basic shapes. For

example we will begin with a simple right triangle and show that they can either be congruent or similar

or both (congruent implies similarity, while similarity helps to prove congruency).

Include text handouts that help to explain and reiterate the ideas above. These include a handout on the

different theorems for congruency as well as a worksheet that allows the students to begin to explore

their general understanding of the concepts. For students who are ahead of the game or seem to have a

firm grasp on these basic concepts, we will add more sources that expand on the ways that the ideas

relate and work together or against each other. If a student is struggling, we will add a more hands on

approach to the two concepts, such as a small guided instruction or a quick little drawing out of the two

ideas.

The lab is a critical part of the unit. The lab will encompass the main exploration of the four ways to

prove congruency and one way to show similarity (AAA). This lab is a full day exploration for the day and

it might even require more than one days’ time to complete. After each subsection of the lab we will

have a large class discussion in which we go over what the students are thinking about if the theorem

works for each case or if they only show similarity. This type of questioning will take a good chunk of

time, as it really gets the students thinking about the mathematics and forces them to think about how

they are thinking (a great precursor into an assessment without actually assessing them).

In order to prepare students for acquisition supply them with the following essay based question. When

is the appropriate time to use AAS congruency? The answer to this question must contain the follow in

the answer: SSS, SAS, and ASA must be addressed somewhere in the answer as well as that AAS will

show similarity not congruency, if one of the other three fails. If the students do not have the proper

requirements for the answer than we will review the concepts again, only this time we will not explore

them in as great as detail as we did the previous day. If it is the case that they do in fact have the proper

answer than we will move on to the different types of proofs and then get into the structure of a proof.

Teacher supplies the graphic organizer that helps to set up the different stages of each proof as well as

helps to build the structure for a proof. NOTE: this organizer will only be given out at the end of the

lesson, as to make sure that the students are engaged and focused during the actual lesson. This lesson

might not take as long, which means that there will be more time for questions (students are

encouraged to use this time wisely).

Another day of exploration, only this time, we are not proving anything, we are using the different proof

types to help facilitate our lesson. We will be focusing on the difference between a statement and

reason as well as how to lay a proof out in a correct manor (e.g.: you cannot prove one thing before you

have proven all the parts required for it). NOTE: This type of logic is very common and frequent when it

comes to solving and articulating proofs. This is another lesson that if it is a carry over, that is ok. In fact,

it is probably better that we spend more time on this lesson than try and rush through it in order to

reach the assessment.

After we have dealt with triangles and the students are able to build and construct proofs correctly, we

will get into the area of quadrilaterals and how they can be proven based on the previous triangle ideas.

This entire concept is focused in one three to four day lesson, in which we explore and examine the

properties and triangle relations in different quadrilaterals (I.E: parallelogram, kite, rectangle, square,

rhombus, etc.).

During the final few days (maybe three) we will focus on the volume and proving volume. We are not

going to use traditional shapes, rather, because this is just an introduction and we will cover this more in

the next unit, we are instead going to find the volume of objects that the students bring into the

classroom for their everyday lives. This is a fun and easy way to get the students engaged early and help

them have something to build on once we really get into the heart and soul of the volume unit. NOTE:

This is a small Segway, not the overall product.

Monday Tuesday Wednesday Thursday Friday

1) Start the

Congruency and

proofs unit.

Labeling triangles

will be the first

lesson

2) Introduce the

triangle types again

and review the

properties of the

three triangle

types.

5) Triangle

congruency

theorems (1st

lesson)

6) The idea of

parallel lines and

alternate interior

angles

7) Introduction to

proof styles and

how to write them.

(2nd lesson)

8) Discussion on the

previous days

problems

9) QUIZ on proof

styles and triangle

theorems.

12) Introduction

into quadrilaterals

and discussion of

what the students

found with the

square

13) Proving

properties about

quadrilaterals (the

main six)

14) QUIZ over the

six quads and being

able to tell me all

the properites of

each.

15) Introduction

into truss and

midline for a

quadrilateral or

triangle

16) Scissors

congruent webinar

and hand out

project. (3rd

lesson)

19) Scissors

congruent day two

20) QUIZ over

scissors congruent

21) THANKSGIVING 22)THANKSGIVING 23)THANKSGIVING

26) Review midline

and truss (relation

to base)

27) Truss

application proofs

28) PROJECTS DUE 29) REVIEW FOR

TEST

30) TEST

Calendar Of Events

Month= November

Key: Yellow= start of Unit Green= Quizes Blue= Project days Orange= Break Red= Test Days