Unit 4 Grade 9 and 10
Similar Triangles Regular and Honors Geometry
Lesson Outline
BIG 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 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. 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.
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 Similarity? / · Discover Thales theorems
· Investigate similar triangles
· Understand the Basic Proportionality
Theorem and its Converse.
· Investigate the properties of similar triangles / G-SRT.2
G-SRT.4
8.G.4
8.G.5
2 / 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 Triangle
Congruence / · 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.2
G-SRT.5
G-SRT.4
G-SRT.1
G-CO.2
8.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.2
G-SRT.5
7 / · Assessment
Unit 4: Day 1: Similar Triangles: Who is Thales? What is Similarity
Minds On: 40 Min Action: 30 Min
Consolidate/
Connection: 10 Min
Total = 80 Min / Learning Goals
Students will apply proportionality in the context of parallel lines theorems.
Prior Knolwedge: special angles formed by parallel lines, ratios, proportions
Anticipated Challenges:
Students not reading on grade level.
The sum of the triangles not congruent to 180 degrees due to human error.
The student measurement verses a computer answer. / Materials
§ Thales story
§ Worksheet
§ Ruler
§ Glue
§ Calculator
§ Computer ( Geogebra or TI -83 or Nspire)
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.
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 Min
Action: 25 Min
Consolidate/Connection: 20 Min
Total = / Learning Goals: Investigate properties of similar triangles, corresponding angles are equal and corresponding sides are proportional using concrete materials.
Prior Knowledge
Measuring angles with a protractor and measuring lengths with a ruler.
Anticipated Challenges:
Fear of fractions and the numerical values. / Materials
Student handout scissors
Protractors rulers(cm) colored pencils
Calculator
Frayer graphic organizer
Assessment
Opportunities
Minds On… / Whole Class → Guided Discussion
Conduct bell ringer.
Whole Class → Guided Discussion
Students will begin the activity by cutting out the triangles and then grouping the 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 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. / Assess how the 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.
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 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. / Assess that students are labeling the triangles with the appropriate measurement.
Consolidate
Connection / 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 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 Min
Action: 60-90 Min
Consolidate/Connection: 20 Min
Total = 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 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. / Materials
Worksheet, protractor, ruler, straws, construction paper
Graphic Organizer
Assessment
Opportunities
Minds On… / Groups of 4→ Guided Investigation
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.
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.