Unit 4 - Winston-Salem/Forsyth County Schools



Unit 4 Grade 9 and 10

Similar Triangles Regular and Honors Geometry

Lesson Outline

|BIG PICTURE/LESSON ABSTRACT |

|The 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 Standards |

|Domain: 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 dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center |

|unchanged. |

|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? |Discover Thales theorems |G-SRT.2 |

| | |Investigate similar triangles |G-SRT.4 |

| | |Understand the Basic Proportionality |8.G.4 |

| | |Theorem and its Converse. |8.G.5 |

| |What is Similarity? |Investigate the properties of similar triangles | |

|2 |It’s all just similar to me. |Investigate similar figures, corresponding sides, and scale factors |G-SRT.1 |

| | |Investigate the measures of angles in similar figures | |

|3 to 4 |Transformation |Recall properties of transformation. |G-SRT.2 |

| | |Investigate dilation using measurement |G-SRT.5 |

| |Dilation |Use inductive reasoning to formulate reasonable conjectures and use deductive|G-SRT.4 |

| | |reasoning to justify formally or informally |G-SRT.1 |

| |Showing Triangle | |G-CO.2 |

| |Congruence | |8.G.4,8.G.5 |

|5 |Showing Triangle Similarity |Identify and use AA, SAS and |G-SRT.3 |

| | |SSS similarity to solve a variety | |

| | |of problems. | |

| | |Discover indirect measurement | |

|6 |How high? How far? |Solve problems involving similar triangles using measurement data |G-SRT.2 |

| | |Solve problems involving similar triangles from given situations. |G-SRT.5 |

| |Using What You have Learned | | |

|7 | |Assessment | |

|Unit 4: Day 1: Similar Triangles: Who is Thales? What is Similarity | |

|Minds On: 40 Min Action: 30 Min |Learning Goals |Materials |

|Consolidate/ |Students will apply proportionality in the context of parallel lines |Thales story |

|Connection: 10 Min |theorems. |Worksheet |

| |Prior Knolwedge: special angles formed by parallel lines, ratios, |Ruler |

|Total = 80 Min |proportions |Glue |

| |Anticipated Challenges: |Calculator |

| |Students not reading on grade level. |Computer ( Geogebra or TI -83 or |

| |The sum of the triangles not congruent to 180 degrees due to human |Nspire) |

| |error. | |

| |The student measurement verses a computer answer. | |

| Assessment Opportunities |

|Minds On… |Individual → Pretest | |

| |Groups of 4 → Read: Thales Story |Review the cooperative |

| |Students will divide the reading assignment in their groups, take notes, and |learning skills. |

| |discuss their findings with their group members. | |

| |Whole Class → Discussion |Encourage each groups |

| |Discuss the reading as a whole group and add to their individual notes as |to share, then next |

| |needed. If the students need help Facilitate a discussion by asking leading |group to add what is |

| |questions such as: |new or unique and so on|

| |What theorems did Thales discover that we have discussed? |until all groups have |

| |How did he use ratios? Proportions? |shared. |

|Action! |Groups of 4→ Guided Investigation |Assess initiative |

| |Students will complete the activity. |learning skill, |

| | |measuring using ruler, |

| | |and protractor. |

|Consolidate/ |Whole Class → Guided Discussion | |

|Connection |Consider the results of the investigation. Share different solutions. | |

| |Ask students to write a summary of what they learned during the investigations. |Assess student |

| | |understanding. |

| |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.” | |

|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. | |

|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. |Things to change for next lesson. |

|Discussion of Thales was great. |Reserve the computer lab for all my classes. I would also use Geogebra |

|Communicating ideas. |to complete the investigation. |

|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. | |

| | |

|Unit 4: Day 2: Similar Triangles: It’s all just similar to me. | |

|Minds On: 20 Min |Learning Goals: Investigate properties of similar triangles, |Materials |

|Action: 25 Min |corresponding angles are equal and corresponding sides are |Student handout |

|Consolidate/Connection: 20 Min |proportional using concrete materials. |scissors |

|Total = |Prior Knowledge |Protractors rulers(cm)|

| |Measuring angles with a protractor and measuring lengths with a |colored pencils |

| |ruler. |Calculator |

| |Anticipated Challenges: |Frayer graphic |

| |Fear of fractions and the numerical values. |organizer |

| Assessment |

|Opportunities |

|Minds On… |Whole Class → Guided Discussion | | |

| |Conduct bell ringer. | | |

| | | | |

| |Whole Class → Guided Discussion | |Assess how the different |

| |Students will begin the activity by cutting out the triangles | |groups grouped the |

| |and then grouping the triangles. | |triangles. |

| | | | |

| |Whole Class → Guided Discussion | | |

| |The 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 Instructions | | |

| |Guide the students through labeling the triangles in the | | |

| |following way: 1a, 1b, 1c, 2a, 2b, 2c, 3a,3b,3c with the number| |Assess that the students |

| |being the similar groups: 1- acute triangles, 2-right triangles,| |are labeling correctly. |

| |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. | | |

|Action! |Groups of 4→ Guided Investigation | | Assess that students are |

| |Students will continue with the activity in their groups. | |labeling the triangles with|

| |Answer question 2. | |the appropriate |

| |Next, tell them to determine the measure of all other angles | |measurement. |

| |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 Discussion | | |

| |Discuss 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. | | |

|Consolidate |Individual → Practice | |Assess students |

|Connection |Students will complete a Frayer model for similar triangles | |understanding. |

| |based on their learning. | | |

| |Optional: Discuss briefly the differences and similarities | | |

| |between similar shapes and congruent shapes. | | |

|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 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. |Things to change for next lesson. |

|The hands on manipulative gave students an opportunity to visualize |I pondered eliminating this activity from my honors class, due |

|that angles are congruent and sides are proportional. |to the high number of sophomores enrolled in Honors Geometry, I |

|Use precise mathematical language and use symbolic notation. |left it in as a challenge. This activity is a reinforcement |

|This lab also served as a way for students to work cooperatively and |lesson. |

|independently to explore similar triangles. | |

|Unit 4: Day 3and 4: Similar Triangles: Transformation and | |

|Showing Triangle Congruence. | |

|Minds On: 30 to 50 Min |Learning Goals |Materials |

|Action: 60-90 Min |Students will identify and compare the three congruent |Worksheet, |

|Consolidate/Connection: 20 Min |transformations. |protractor, ruler, |

|Total = 1 to 2 days |Apply the three congruence transformation to coordinates of the|straws, construction|

| |vertices of figures. |paper |

| |Identify and apply dilations. |Graphic Organizer |

| |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 Knowledge | |

| |Unit 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 of the | |

| |straws. | |

| |How to use the straws to measure the angles. | |

| Assessment |

|Opportunities |

|Minds On… | Groups of 4→ Guided Investigation | |Assess students understanding.|

| |Students will complete Activity One | | |

| |Individual → Practice | | |

| |1) Discuss the ideas of transformations that occurred. 2) | | |

| |Which of the six the transformations were congruent or | | |

| |similar. | |Assess students understanding |

| |Individual → Practice | |and justifications for their |

| |Students will write a summary of the activity. | |reasoning. |

|Action! | Groups of 4→ Guided Investigation | |Assess students’ ability to |

| |Students will complete Activity 2-6. | |use inductive, deductive, and |

| | | |analytical methods. |

| |Whole Class → Guided Discussion | | |

| |Consider the results of the investigation. Facilitate a | |Assess students are making the|

| |discussion about proving triangles congruent by SSS. This is | |correct notation for congruent|

| |a short cut. You can prove triangles are congruent if the | |sides and angles. |

| |three sides of the triangles are congruent. The students will| | |

| |also verify this by measuring the three angles. | |Assess students understanding |

| | | |and short cut of proving |

| | | |triangles are congruent by |

| | | |using three sides of a |

| | | |triangle. |

| |Whole Class → Guided Discussion | | |

| |Students should be discussing triangles are congruent by SAS. | | |

| |Facilitate a discussion about proving triangles congruent by | | |

| |SAS. This is a short cut. You can prove triangles are | | |

| |congruent if the two sides of the triangles are congruent and | |Assess students understanding |

| |the angle between those two sides is also congruent. The | |and the short cut of proving |

| |students will also verify this by measuring the remaining | |triangles are congruent by |

| |corresponding parts. | |using two sides and the angle |

| | | |between the two sides. |

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

| | | | |

| | | | |

| |Whole Class → Guided Discussion | | |

| |Students should be discussing triangles are congruent by ASA. | | |

| |Facilitate a discussion about proving triangles congruent by | |Assess if students are |

| |ASA. This is a short cut. You can prove triangles are | |correctly placing the straws |

| |congruent if the two angles of the triangles are congruent and| |on top of the protractor and |

| |the side between those two angles is also congruent. The | |then making markings to |

| |students will also verify this by measuring the remaining | |construct their angles. |

| |corresponding parts. | | |

| | | | |

| |Whole Class → Guided Discussion | |Assess students understanding |

| |Students should be discussing triangles are congruent by AAS. | |and the short cut of proving |

| |This is a short cut. You can prove triangles are congruent if | |triangles are congruent by |

| |the two angles of the triangles are congruent and the side not| |using two sides and the angle |

| |between those two angles is also congruent. The students will| |between the two sides |

| |also verify this by measuring the remaining corresponding | | |

| |parts. | | |

| | | | |

| |Whole Class → Guided Discussion | | |

| |Students should be discussing triangles are not congruent when| | |

| |they have three congruent angles. The triangles are similar | | |

| |but not congruent. | | |

| | | | |

| | | | |

| |Whole Class → Guided Discussion | |Assess students understanding |

| |Students will do a group summary for activities 2 -6. | |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 not short |

| | | |cuts in proving triangles are |

| | | |congruent. |

|Consolidate |Whole Class → Guided Discussion | | |

|Connection |Consider 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. |Things to change or modify for next lesson. |

|The students were required to write individual summary statements |Due to bench mark exams, I was not able to do this lab using |

|for their first activity. Collaborative group summaries for |Geogebra. I was only able to use the computer lab with one |

|activities 2 through 6 were require. This gave me an opportunity to|Regular group of students. |

|assess the students understanding and grade them using a rubric | |

|model. |The computer lab would have given the students an opportunity |

| |for the measurement to work every time verses human |

|This also gave me an opportunity to talk about technology |measurements. |

|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. | |

|Unit 4: Day 5 and 6: Similar Triangles: Showing Triangles Similarity and How High? How far? Using What you have | |

|Learned. | |

|Minds On: 10 Min |Learning Goals |Materials |

|Action: 60 Min |Students will be able to identify and use AA, SAS, and SSS. |Discovery Activity |

|Consolidate/Connection: 15 Min |Similarity to solve a variety of problems including real world |Sheet |

|Total = 85 Min |applications. |Graphic Organizer |

| |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. | |

| Assessment |

|Opportunities |

|Minds On… | Individual → Complete the Bellringer | | |

| |Students will complete the bell ringer. Facilitate the | | |

| |students answers before beginning the Activity of Proving | | |

| |Triangles are Similar. | | |

|Action! |Pairs → Guided Investigation : | |Assess students understanding |

| |Pairs work through the Discovery Activity. Encourage students| |of the short cuts of proving |

| |to show their work and present their solution in an organized | |triangle similarity. |

| |manner. | | |

| | | | |

| | | | |

| |Whole Class → Guided Investigation | | |

| |Students will discuss the short cuts of proving triangles | | |

| |similar by AA, SSS and SAS. Facilitate the discussion as | | |

| |needed. | | |

| | | | |

| |Whole Class → Guided Discussion | |Assess students understanding |

| |Demonstrate to the class how to justify their reasoning. For | |of indirect measurement. |

| |example, write the statements using the correct notation. | | |

| |Given /A = /D, /ACD =/FCD, vertical angles are congruent( :.) | | |

| |ΔACD = ΔFCD. | |Assess students understanding |

| | | |and the students are drawing |

| |Whole Class → Guided Discussion | |the triangles and labeling the|

| |Facilitate the discussion how tall is the wall activity and | |information correctly. |

| |referring back to Thales indirect measurement of the pyramid. | | |

| | | | |

| | | |Assess students are correctly |

| | | |using their notations and |

| | | |marking their vertical angles |

| | | |and reflexive segments. |

| |Groups of 4→ Guided Investigation | | |

| |Facilitate by asking the students to sketch the drawing using | | |

| |two right triangles. | |Assess students are drawing |

| | | |the two right triangles and |

| | | |labeling the segments with the|

| |Whole Class → Guided Discussion | |correct lengths. |

| |Have students to put their sketches and answers on the boards.| | |

| |Facilitate the discussion of the results. | |Assess to make sure students |

| | | |are setting up the proportions|

| | | |correctly. |

| | | | |

|Consolidate |Compare and contrast similarity and congruence. What makes | | |

|Connection |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. |Things to change for next lesson. |

|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 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: Assessment | |

|The 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.) | |

| | |

|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. | |

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