Unit 4 Grade 9 and 10 Similar Triangles Regular and Honors GeometryLesson OutlineBIG PICTURE/LESSON ABSTRACTThe study of triangles includes proving many properties that students may already be familiar with. The angle sum of a triangle being 1800 and the relationship between an exterior angle the sum of the remote interior angles are familiar and connected ideas that can be proved. Discovering and applying combinations of sides and angles that are sufficient conditions for similarity or congruence of two triangles (for similarity: AA, SSS, SAS, and for congruence: SSS, SAS, ASA, AAS, HL) provides experience in making conjectures. The results of these relationships can be used to reason further about additional properties of triangles, isosceles triangles, and many quadrilaterals. Proofs related to triangles can again take many different forms including coordinate proofs. In addition to congruence relationships, similarity is an important area of study in triangles. In fact, it is reasonable to begin with the properties of similarity and then move to congruence properties as a special case of similarity. The properties of congruence and similarity should be used to solve problem situations.Focus Question:What are the similarities and differences between similar and congruent triangles?Common Core Essential State StandardsDomain: Congruence(G-CO) Similarity, Right Triangles and Trigonometry(G-SRT)Clusters: EXPERIMENT with transformations in the plane. UNDERSTANAD similarity in terms of similarity transformations. PROVE theorems involving similarity. Standards: G-CO.2 REPRESENT transformations in the plane using, e.g., transparencies and geometry software. DESCRIBE transformations as functions that take points in the plane as inputs and give other points as outputs. COMPARE transformations that PRESERVE distance and angle to those that do not (e.g., translation versus horizontal stretch). G-SRT.2 Given two figures, USE the definition of similarity in terms of similarity transformations to DECIDE if they are similar; EXPLAIN using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles, and the proportionality of all corresponding pairs of sides. G-SRT.5 USE congruence and similarity criteria for triangles to solve problems and to PROVE relationships in geometric figures. G-SRT.1 VERIFY experimentally the properties of dilations GIVEN by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a

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parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by

the scale factor. G-SRT.3 USE the properties of similarity transformations to ESTABLISH the AA criterion for two triangles to be similar. G-SRT.4 PROVE theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.Gaps from 8th Grade Common Core (after 2012-13, students will come to high school with the following):8.G.4 UNDERSTAND that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, DECSRIBE a sequence that EXHIBITS the similarity between them. 8.G.5 USE informal arguments to ESTABLISH interior and exterior angles CREATED when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. Standards for Mathematical Practices:2) Reason abstractly and quantitatively.3) Construct viable arguments and critique the reasoning of others.5) Use appropriate tools strategically.6) Look for and make use of structure.Intellectual Processes: Representation: Use representations of triangles to model and interpret physical, social, and mathematical phenomenas.Reasoning and Proof: Develop and evaluate mathematical arguments and proofs related to similar and congruent triangles.Problem Solving: Build new mathematical knowledge of triangles through problem solving. Key Concepts/Vocabulary:Similarity, composition, rigid motion, dilation, center, ratio, angle measure, side length/segment length, proportional, corresponding sides, corresponding angles, proof, Pythagorean Theorem, parallel, intersect, congruence, triangle similarity, distance, image, preimage, vertex angle, Triangle Congruency (SSS, SAS, AAS, ASA, *HL) Triangle Similarity (SSS, SAS, AA)Day

Title/Topic Learning Goal/Objectives Expectations

1 Who is Thales?

What is

Discover Thales theorems Investigate similar triangles Understand the Basic

Proportionality Theorem and its Converse. Investigate the properties of

G-SRT.2G-SRT.48.G.48.G.5

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Similarity? similar triangles2 It’s all just

similar to me. Investigate similar figures,

corresponding sides, and scale factors

Investigate the measures of angles in similar figures

G-SRT.1

3 to 4

Transformation

Dilation

Showing TriangleCongruence

Recall properties of transformation. Investigate dilation using

measurement Use inductive reasoning to

formulate reasonable conjectures and use deductive reasoning to justify formally or informally

G-SRT.2G-SRT.5G-SRT.4G-SRT.1G-CO.28.G.4,8.G.5

5 Showing Triangle Similarity

Identify and use AA, SAS and SSS similarity to solve a variety of problems.

Discover indirect measurement

G-SRT.3

6 How high? How far?

Using What You have Learned

Solve problems involving similar triangles using measurement data

Solve problems involving similar triangles from given situations.

G-SRT.2G-SRT.5

7 Assessment

Unit 4: Day 1: Similar Triangles: Who is Thales? What is SimilarityMinds On: 40 Min Action: 30 MinConsolidate/ Connection: 10 Min

Total = 80 Min

Learning GoalsStudents will apply proportionality in the context of parallel lines theorems.Prior Knolwedge: special angles formed by parallel lines, ratios, proportionsAnticipated Challenges: Students not reading on grade level.The sum of the triangles not congruent to 180 degrees due to human error.

Materials Thales story Worksheet Ruler Glue Calculator Computer

( Geogebra or TI -83 or Nspire)

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The student measurement verses a computer answer.

Assessment Opportunities

Minds On… Individual → Pretest Groups of 4 → Read: Thales Story Students will divide the reading assignment in their groups, take notes, and discuss their findings with their group members.Whole Class → Discussion Discuss the reading as a whole group and add to their individual notes as needed. If the students need help Facilitate a discussion by asking leading questions such as: What theorems did Thales discover that

we have discussed? How did he use ratios? Proportions?

Review the cooperative learning skills.

Encourage each groups to share, then next group to add what is new or unique and so on until all groups have shared.

Action! Groups of 4 → Guided Investigation Students will complete the activity.

Assess initiative learning skill, measuring using ruler, and protractor.

Consolidate/Connection

Whole Class → Guided Discussion Consider the results of the investigation. Share different solutions. Ask students to write a summary of what they learned during the investigations.

If needed say: “In order for two triangles to be considered similar, all three _____ ______ (corresponding angles) must be congruent and all three pairs of ________ _______(corresponding sides) must be ______proportional.”

Assess student understanding.

Extension/PREP/Hwk Students will complete pages 11 and 12 for homework. This will also help prepare students for the SAT and ACT exams.Accommodations/Special Needs: 1) Have students draw pictures by the Greeks to give them a better understanding about right triangles. *2) Given a triangle with an internal segment parallel to a side, ask students to give and justify three true proportions for the figure.* This can also be used for an opener of the next lesson or part of the closure if time permits.

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Teacher Reflection on Lesson: I really enjoyed presenting the history of Thales story at the beginning of this unit. Students realized that he is the father of Geometry. This motivated the students during the lesson. It was the foundation for the reminder of the unit. The students read about how he used indirect measurement to find the height of a pyramid. Looking Ahead: Ratio will become the scale factor in the world of similar figures, and proportions will be heavily utilized and manipulated in working with similar figures. Aspects that Worked.

Discussion of Thales was great. Communicating ideas. Sketching Thales theorem. Presenting the ratio and

proportional at the beginning of the lesson.

Students relied on prior knowledge from Algebra I in solving algebraic proportion.

Things to change for next lesson.Reserve the computer lab for all my classes. I would also use Geogebra to complete the investigation.

Unit 4: Day 2: Similar Triangles: It’s all just similar to me.Minds On: 20 MinAction: 25 MinConsolidate/Connection: 20 MinTotal =

Learning Goals: Investigate properties of similar triangles, corresponding angles are equal and corresponding sides are proportional using concrete materials.Prior KnowledgeMeasuring angles with a protractor and measuring lengths with a ruler.Anticipated Challenges: Fear of fractions and the numerical values.

MaterialsStudent handout scissorsProtractors rulers(cm) colored pencilsCalculatorFrayer graphic organizer

Assessment Opportunities

Minds On… Whole Class → Guided DiscussionConduct bell ringer. Assess how the

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Whole Class → Guided DiscussionStudents will begin the activity by cutting out the triangles and then grouping the triangles.

Whole Class → Guided DiscussionThe students will share with the class how they grouped the triangles.Next, ask students what similar triangles are: same shape, different size. Lastly, tell the students to group the similar triangles together.Whole Class → Guided InstructionsGuide the students through labeling the triangles in the following way: 1a, 1b, 1c, 2a, 2b, 2c, 3a,3b,3c with the number being the similar groups: 1- acute triangles, 2-right triangles, 3-obtuse triangles, and the letter being the size: a – smallest, b- middle size, c- largest.Have students take the groups of similar triangles and match them with the corresponding angles. So that they can see the corresponding angles of similar triangles are congruent.Help students to label the corresponding angles in groups of similar triangles. Students can use different colored pencils to mark the corresponding angles or they can mark the angles using arcs with one slash, two slashes, or three slashes.

different groups grouped the triangles.

Assess that the students are labeling correctly.

Action! Groups of 4 → Guided Investigation Students will continue with the activity in their groups.

Assess that students are labeling the triangles with the appropriate

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Answer question 2.Next, tell them to determine the measure of all other angles without measuring the angles. Then label the triangles appropriately and complete the chart. Now they will use a protractor to measure the angles of 1c, 2, and 3c. Using the discovery they made about angles in similar triangles, they will find all the other angles without measuring them.Lastly, students will discover what a scale factor is. Whole Class → Guided DiscussionDiscuss the concept of corresponding sides with the students. Have them label the corresponding sides of each set of triangles. They can use different colored pencils to mark the corresponding side or they can mark them using slashes.

measurement.

ConsolidateConnection

Individual → Practice Students will complete a Frayer model for similar triangles based on their learning. Optional: Discuss briefly the differences and similarities between similar shapes and congruent shapes.

Assess students understanding.

Extension/PREP/Hwk: Option 1)Write a summary of today’s lesson. Option 2) Find the missing information for pairs of similar triangles.Accommodations/Special Needs:

This lesson incorporates different techniques typically utilized for diverse learners (hands on manipulates and interactive online manipulatives). Another option is for students to work in pairs.

Teacher Reflection on Lesson: This lesson was a reinforcement lab to the previous activity. The students manipulated the triangles to visualize the

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parallel lines proportionality from the previous lessons. This was a great way to explore similar triangles. The students were able to think logically, using inductive reasoning to formulate reasonable conjectures.Aspects that Worked.

The hands on manipulative gave students an opportunity to visualize that angles are congruent and sides are proportional.

Use precise mathematical language and use symbolic notation.

This lab also served as a way for students to work cooperatively and independently to explore similar triangles.

Things to change for next lesson.I pondered eliminating this activity from my honors class, due to the high number of sophomores enrolled in Honors Geometry, I left it in as a challenge. This activity is a reinforcement lesson.

Unit 4: Day 3and 4: Similar Triangles: Transformation and Showing Triangle Congruence.Minds On: 30 to 50 MinAction: 60-90 MinConsolidate/Connection: 20 MinTotal = 1 to 2 days

Learning Goals Students will identify and

compare the three congruent transformations.

Apply the three congruence transformation to coordinates of the vertices of figures.

Identify and apply dilations. Students will verify congruent

and similar figures. Students will investigate, and

justify the conclusion for triangle congruence (SSS, SAS, ASA, and AAS)

You can use short cuts to determine if triangles are congruent.

Prior KnowledgeUnit 1 transformation, isometric and knowledge of rigid motion.Anticipated Challenges:

New students may not have the prior knowledge of transformation as needed.

Some of the measurements will vary due to the length

MaterialsWorksheet, protractor, ruler, straws, construction paperGraphic Organizer

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of the straws. How to use the straws to

measure the angles. Assessment Opportunities

Minds On… Groups of 4 → Guided Investigation Students will complete Activity OneIndividual → Practice 1) Discuss the ideas of transformations that occurred. 2) Which of the six the transformations were congruent or similar.Individual → Practice Students will write a summary of the activity.

Assess students understanding.

Assess students understanding and justifications for their reasoning.

Action! Groups of 4 → Guided Investigation Students will complete Activity 2-6.

Whole Class → Guided DiscussionConsider the results of the investigation. Facilitate a discussion about proving triangles congruent by SSS. This is a short cut. You can prove triangles are congruent if the three sides of the triangles are congruent. The students will also verify this by measuring the three angles.

Whole Class → Guided DiscussionStudents should be discussing triangles are congruent by SAS. Facilitate a discussion about proving triangles congruent by SAS. This is a short cut. You can

Assess students’ ability to use inductive, deductive, and analytical methods.

Assess students are making the correct notation for congruent sides and angles.

Assess students understanding and short cut of proving triangles are congruent by using three sides of a triangle.

Assess students understanding and the short cut of proving triangles are congruent by using two sides and the angle between the two

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prove triangles are congruent if the two sides of the triangles are congruent and the angle between those two sides is also congruent. The students will also verify this by measuring the remaining corresponding parts.

Discuss why some student’s third length varied from 8 to 9.5.

Whole Class → Guided DiscussionStudents should be discussing triangles are congruent by ASA. Facilitate a discussion about proving triangles congruent by ASA. This is a short cut. You can prove triangles are congruent if the two angles of the triangles are congruent and the side between those two angles is also congruent. The students will also verify this by measuring the remaining corresponding parts.

Whole Class → Guided DiscussionStudents should be discussing triangles are congruent by AAS. This is a short cut. You can prove triangles are congruent if the two angles of the triangles are congruent and the side not between those two angles is also congruent. The students will also verify this by measuring the remaining corresponding parts.

Whole Class → Guided DiscussionStudents should be discussing triangles are not congruent

sides.

Assess if students are correctly placing the straws on top of the protractor and then making markings to construct their angles.

Assess students understanding and the short cut of proving triangles are congruent by using two sides and the angle between the two sides

Assess students understanding and the short cut of proving triangles are congruent by using two angles and the side not between the two angles.

Assess students understanding of AAA and SSA are

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when they have three congruent angles. The triangles are similar but not congruent.

Whole Class → Guided DiscussionStudents will do a group summary for activities 2 -6.

not short cuts in proving triangles are congruent.

ConsolidateConnection

Whole Class → Guided DiscussionConsider the results of the investigation. Share results, and ask students to write any concerns of their findings in the investigations of the activities 2-6. Which of the following short cuts work and which did not and explain (SSS, SAS, ASA, AAS, AAA, and SSA)

Extension/PREP/Hwk: Briefly discuss why SSA and AAA do not work for triangle congruence.Assigned as needed due to conflict with honor roll celebrations. Students had an opportunity to debrief before the start of the following day about the activities.Accommodations/Special Needs: This lesson incorporates different techniques typically utilized for diverse learners (hands on manipulates and interactive online manipulatives.Another option is for students to complete the graphic organizer on triangle congruence in pairs.

Teacher Reflection on Lesson: I love the flow of the information. This activity allowed students to develop a step-by step plan for which they have prior experience. The students were able to grasp the concepts and proceeded with ease through the rest of the activities. The students did understand that you could use the short cuts to prove triangles congruent instead of verifying the measurements of the angles and the segment lengths for both triangles every time. My regular students grasp this concept and enjoyed using the short cut.Looking Ahead: This activity will help students to find which shortcuts may be used to prove triangles congruent. Aspects that Worked.The students were required to write individual summary statements for their first activity. Collaborative group summaries for activities 2 through 6 were

Things to change or modify for next lesson.Due to bench mark exams, I was not able to do this lab using Geogebra. I was only able to use

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require. This gave me an opportunity to assess the students understanding and grade them using a rubric model.

This also gave me an opportunity to talk about technology measurements verses human measurements. I gave the students this example: If I have new carpet installed in my house and the carpenter measured the perimeter incorrectly, it would cost the company and myself more money.

the computer lab with one Regular group of students.

The computer lab would have given the students an opportunity for the measurement to work every time verses human measurements.

Unit 4: Day 5 and 6: Similar Triangles: Showing Triangles Similarity and How High? How far? Using What you have Learned.Minds On: 10 MinAction: 60 MinConsolidate/Connection: 15 MinTotal = 85 Min

Learning Goals Students will be able to

identify and use AA, SAS, and SSS.

Similarity to solve a variety of problems including real world applications.

Prior knowledge Proving triangles

congruence, Using short cuts to prove

triangles congruent. Thales indirect

measurement of the pyramid.

Anticipated Challenges:In writing geometric statements, students tend to write the word “because” instead of using the symbol notation for therefore.

MaterialsDiscovery Activity SheetGraphic Organizer

Assessment Opportunities

Minds On… Individual → Complete the BellringerStudents will complete the bell ringer. Facilitate the students answers before beginning the Activity of Proving Triangles are Similar.

Action! Pairs → Guided Investigation : Pairs work through the Discovery Activity. Encourage

Assess students understanding of the short cuts of proving triangle

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students to show their work and present their solution in an organized manner.

Whole Class → Guided InvestigationStudents will discuss the short cuts of proving triangles similar by AA, SSS and SAS. Facilitate the discussion as needed.

Whole Class → Guided DiscussionDemonstrate to the class how to justify their reasoning. For example, write the statements using the correct notation.Given /A = /D, /ACD =/FCD, vertical angles are congruent( :.) ΔACD = ΔFCD.

Whole Class → Guided DiscussionFacilitate the discussion how tall is the wall activity and referring back to Thales indirect measurement of the pyramid.

Groups of 4 → Guided Investigation Facilitate by asking the students to sketch the drawing using two right triangles.

Whole Class → Guided DiscussionHave students to put their sketches and answers on the boards. Facilitate the discussion of the results.

similarity.

Assess students understanding of indirect measurement.

Assess students understanding and the students are drawing the triangles and labeling the information correctly.

Assess students are correctly using their notations and marking their vertical angles and reflexive segments.

Assess students are drawing the two right triangles and labeling the segments with the correct lengths.

Assess to make sure students are setting up the proportions correctly.

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ConsolidateConnection

Compare and contrast similarity and congruence. What makes two figures similar?

Extension/PREP/Hwk Students will complete the following: When are two triangles similar? Give examples of situations in which similar triangles occur. Compare and contrast SSS similarity and SSS congruence.Review for assessment.Use the Similar Postulate/Theorems worksheets for students to practice using notation and justify why the triangles are similar.Accommodations/Special Needs: This lesson incorporates different techniques typically utilized for diverse learners (hands on manipulates and interactive online manipulatives.Students will complete the graphic organizer for proving triangles similar.

Teacher Reflection on Lesson: After this lesson my students were able to answer the focus question: What are the similarities and differences between similar and congruent triangles? As I review this lesson I think the format of the groups played an essential role in this unit. The groups of four were selected by the students and my self. The students were ask to write down one person they did want to work with and two people they preferred not to work with. I sorted the groups based on their request, and work ethics. Looking Ahead: A classic proof of the Pythagorean Theorem and the use of the geometric means, the similar triangles created when the altitude to the hypothesis is drawn. The study of indirect measurement will continue to be used in our right triangle unit. Similarity is also key to theorems in circle geometry.Aspects that Worked.

The lab worked very well. Once the students had completed the activities, I returned their pretest, and individually they completed a review.

Changing the desk to diagonal rows facing the door was a great idea for the indirect measurement activity. I did not want the students to think of my classroom as a traditional room or depend on their classmates or myself for completion activity. My rationale was the importance of the indirect measurement. The concepts of problem solving will become more and more evident when we start our trigonometry unit. Students will be

Things to change for next lesson.

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faced with similar activities of problem solving and I wanted them to be prepared to work outside of their groups individually. Unit 4: Day 7: Similar Triangles: AssessmentThe pretest is used as an informal assessment. It provided me with the following rationales: what to teach, in what order, to provide appropriate activities to meet the needs of all students, and to include concrete and technology activities filled with continuous assessment opportunities:

I start with the history of Thales and his contributions to geometry, which laid the foundation for them to discover the relevant relationship. This activity also promoted mathematical thinking on part of the students.

Next, to use the Proportionality Theorem for the following purposes: 1) Give students prior knowledge of parallel lines and the algebraic portion would be separated from the indirect measurement. It would give students the opportunity to practice with solving proportions before having to solve proportions and set up proportions based on applications.

Review of Transformation would also be used as a prior knowledge and allow the students to see geometry at work as a cohesive subject.

Congruence and Similarity of Triangle will give the students an opportunity to make comparison.

Direct measurement

The activities during the lessons served as informal assessments, and I was able to make adjustments quickly. The informal assessments served as a check of how well the students were grasping the concepts. The activities were also used informally to assess the mathematical communication that occurred between students.

The Post Assessment for this unit consists of 20 questions. The questions format included five True/False, ten multiple choice, and 5 open ended questions. The pre assessment and the post assessment was a common exam that all Geometry teachers at my school used. We will meet next week to discuss this assessment.

Based on the post assessment data my students did learn the material and the instructional goals were met. (Please see graph below pg 18.)

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As I review the post assessment, I should continue to work on indirect measurement, a few

of the students set up the proportion correctly. However, they just made simple mathematical mistakes.

A few students did not mark their triangles with the correct notations to prove the angles were congruent, which sides were congruent and therefore did not choose the correct answer. To address this issue I will have students to use colored pencils to mark the drawings based on given information.

Pretest Vs. Protest

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G-CO.2

G-SRT.1

G-SRT.2

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