Target Learner: High school and middle students who are qualified to take high school Geometry based on the Common Core State Standards.

Accelerated students are sometimes accepted in this class (Geometry) and it creates some challenges in classroom management.  

Prior Knowledge: Students need to have knowledge of theorems of parallelograms and congruent triangles (G.CO.11), make formal geometric constructions (G.CO.12), use coordinates to prove geometric theorems algebraically (G.GPE.4), and apply the geometric methods to solve problems (G.MG.3).

Students understand the concepts of congruent triangles and quadrilaterals before they enter this unit. 

Curriculum Standards:

G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.
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.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems.

The next unit is Right Triangle and Trigonometry and will focus on :

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.

Intended Learning Outcomes:

After learners finish this learning module, learners will be able to -

• Recognize dilations.
• Recognize similar polygons.
• Recognize AA. SSS. and SAS similarity and the parts of similar triangles.
• Apply parallel lines and proportional parts.
• Anticipated duration: Learners should be able to complete this learning module in 4 90-minute sessions.

Here are the learning objectives of this unit. 

Material Requirement:

Textbook: John A Carter, Gilbert J Cuevas, Roger Day, & Malloy, C. E. (2018). Glencoe Geometry (2018 ed.). USA: McGraw-Hill Education.
Apps: Desmos Graphing Calculator, Desmos Geometry and Geogebra.

Teachers' resources are available. Users' manuals of the applications are also available.

7.1 Dilation

Learning Objectives:

• Draw dilations.
• Draw dilations in the coordinate plane.

The learning objective focuses on 

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).

1. Warm-up: Work on Dilation Mini Golf by clicking on here or Go to student.desmos.com and type in: WK9PSV.

Multimodal Meaning & Recursive feedback: 

This warm-up focuses on presenting the conpcept of dilation with multimodal content. Students will experience the effect of dialation by playing the golf game. The recursive feedback happens when the student tries to hit the golf ball into the hole. The ball will fall into the hole if the dilation is performed successfully. Otherwise, the ball will not fall into the hole. We assume, after several attempts, students will be able to understand that dilation is the different sizes of a shape by the recursive feedback provided by the program. 

Teachers' guide is avaiable at https://teacher.desmos.com/activitybuilder/teacherguide/5dc15f09c2706737ce01664f

Dilation_20Mini_20Golf_20_E2_80_A2_20Teacher_20Guide.pdf

 

2. Comment: After finishing the warm-up, make a comment and discuss what you think about "scale factor". Use @ to talk to at least 3 of your peers.

Collaborative Intelligence & Active Knowledge Making: 

This comment encourages active knowledge making by asking students to figure out their own definitions of scale factor. The reading peers' comment and replying part encourages the collaborative intelligence. Students can reflect on the concept of "scale factor" by comparing their own thoughts with their peers' thoughts. 

3. Concept Presentation: How do we dilate a figure?

Multimodal Meaning & Ubiquitous learning

The concept "dilation" is presented in a multimodal display. Students learn the concept of dilation in Geometry by watching the video. The video allows students to forward, rewind and replay and this leads to the ubiquitous learning. With internet, students can access the learning material anytime and anywhere.

4. Practice: Khan Academy: Dilation

Differentiated learning & Recursive feedback

​The practice in Khan Academy is set up to issue questions based on students' previous response. It also provides differentiated feedback based on the students' answers. This realizes the differentiated learning by working out different levels of questions based on the previous response. At the same time, the students are allowed from the program to re-do and see the different kinds of responses based on the students' responses. This makes the recursove feedback happen in the classroom. 

5. Update: Make an update to share your strategy about the concept "Dilation." What's the difference between using origin (0,0) as the center of dilation and another point as the center of dilation. Respond to three of your peers. Your update should include one media.

Metacognition & Active knowledge making:

When students are asked to share their strategies about the concept of "dilation." They are thinking about their thinking and trying to know what they are knowing. When students respond to each other, they are actively making knowledge- the strategy of dilation and different centers of dilation- collaboratively. 

7.2 Similar Polygons

Learning Objectives:

• Use the definition of similarity to identify similar polygons.
• Solve problems by using the properties of similar polygons.

This section focuses on 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.

1. Warm-up: Play the Congruent or Similar Shape game by clicking on here or Go to https://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/SSCongruentSimilar.htm

Multimodal Meaning & Recursive feedback:

​The congruent and similar shape game displays the content with the multimodal meaning. Base on students' responses, the program will provide different feedback and students' can then modify their choice of the game base on the feedback from the program. Students can continue the cycle of sumit and modify and therefore realize the recursive feedback.

2. Comment: After finishing the warm-up, make a comment and discuss the difference between "Congruence" and "Similar". Use @ to talk to at least 3 of your peers.

Collaborative Intelligence & Active Knowledge Making:

​Through the process of discussing the difference between congruence and similar, the concepts of congruence and similar get clarified. Students are actively discovering and figuring out "congruence" and "similar" by themselves. This forms the collaborative intelligence moment. Through actively making the analysis by themselves, students understand the targeted concepts more by the recursive process. 

3. Concept Presentation: Similar Polygons

Multimodal Meaning & Ubiquitous learning

The concept of similar polygons are presented in a video, which is a multimodal display. The video allows students to pause and rewind which makes the ubiquitous learning happen. 

4. Practice: Kahoot Similiar Polygons

Please use the Kahoot app on your device to open this link and practice similar polygons!

(PIN: 0565617)

Differentiated learning & Recursive feedback

The Kahoot program gives recursive feedback based on students' input. With the different level of understanding, students can receive differentiated feedback based on their previous input. 

5. Update: Make an update to share a media (a picture or video) of a real-life example of similar polygons. Comment to at least three of your peers' update.

Metacognition & Active knowledge making:

By sharing a real-life example, students are immersed in a process of learning their learning. In the sharing and searching process, students are actively making the knowlege of similar polygons by the mathematics discourse with their peers. 

7.3 Similar Triangles: AA Similarity

Learning Objectives:

• Use the AA similarity criterion to prove triangles similar.
• Solve problems by using the properties of similar triangles.

This section focuses on : 

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

1. Warm-up: Play through the simulation in Geogebra by clicking on here.

Multimodal Meaning & Recursive feedback:

The simulation displays the concept of AA similarity in a multimodal way. Base on the previous input, students receive differentiated feedback. By continue to input the answers, students are able to modify their answers based on the feedback from the program and it forms a recursive feedback cycle. 

2. Comment: Notice and Wonder. After performing the simulation, make a comment about what do you notice and what do you wonder? Use @Name to talk to three of your peers.

Collaborative Intelligence & Active Knowledge Making

By sharing the notice and the wonder, students are actively making knowledge of what they have experience. Through reading and responding to peers' input, students are collaboratively building their concepts of AA similarity.

3. Content Presentation: AA Similarity

Multimodal Meaning & Ubiquitous learning

The concept of AA similarity is presented in a video with multimodal content. The video allows students to play, pause, rewind and forward. Students are able to access the instructional material anytime and anywhere with the access of the internet. 

4. Practice: Quizizz AA Similarity. Please click on here to practice.

Differentiated learning & Recursive feedback

Quizizz gives student instant feedback based on their answer. Students are allowed to take the quiz again to modify their answers based on their previous response. Teachers can curate questions by students' levels in the program.

5. Make an Update: Make 5 examples of pairs of triangles and have your peers identify whether it is AA similarity. Comment at least 3 of your peers to identify the AA similarity.

Metacognition & Active knowledge making:

By identifying the AA similarity, students are thinking what they have learned. By responding to their peers' , their concept of AA similarity gets clarified. 

7.4 Similar Triangles: SSS and SAS Similarity

Learning Objectives:

• Use the SSS similarity criterion to prove triangles are similar.
• Use the SAS similarity criterion to prove triangles are similar.

G.SRT2 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.4 Prove theorems about similarity.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

1. Warm-up: Play with this simulation and that simulation.

Multimodal Meaning & Recursive feedback:

The simulation is a multimodal display with curated feedback based on students' input. 

2. Comment: What do you Notice and Wonder? What's the same and what's the difference? Use @ to talk to 3 of your peers.

Collaborative Intelligence & Active Knowledge Making:

Students collectively produce the knowledge of SSS and SAS similairity by reading others' feedback and reflecting their own response. 

3. Concept Presentation:

Multimodal Meaning & Ubiquitous learning

The content knowledge is displyed with multimodal meaning and students are allowed to revisit with the access of the internet. 

4. Practice: ThatQuiz: Triangle Similarity Practice Mix

Differentiated learning & Recursive feedback

​Students received different responses based on their input. By the recursive feedback, students are able to re-anchor their answers to the core of the content knowledge. 

5. Make an Update: What is the difference between SSS triangle similarity and SSS triangle congruence? How about SAS triangle similarity and SAS triangle congruence? Reply to at least 3 of your peers to keep the discussion going.

Metacognition & Active knowledge making

Students are thinking what they are thinking by comparing and identifying the difference between SSS/SAS similarity/congruence. Knowledge is actively being created by reading and responding to peers' comments. 

7.5 Parallel Lines and Proportional Parts

Learning Objectives:

• Use proportional parts within triangles.
• Use proportional parts with parallel lines.

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.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.)

1. Warm-up: Play with the simulations: 1) Proportional Parts within Triangles 2) Proportional parts with parallel lines

Multimodal Meaning & Recursive feedback:

 

2. Comment: What is in common of proportional parts within triangles and proportional parts with parallel lines? What are the differences?

Collaborative Intelligence & Active Knowledge Making:

3. Concept Presentation:

1) Proportional Parts within Triangles

2) Proportional parts with parallel lines

Multimodal Meaning & Ubiquitous learning

4. Practice & Peer-evaluation/Discussion: Finish the practice in groups. Once you finish, turn it into your teacher. Then, your teacher will give you an answer from another group. After this, your teacher will lead a whole group discussion about the concepts - Proportional Parts in Triangles and Parallel Lines- and the strategy.

7.4 Practice- Proportional Parts in Triangles and Parallel Lines
7.5 Practice- Proportional Parts in Triangles and Parallel Lines

Differentiated learning & Recursive feedback

5. Make an Update: Application: Proportional parts are widely used in architecture, design, and art. Find a real-life example and share it with your peers. For example, a dollhouse. Comment to at least three of your peers' updates.

Metacognition & Active knowledge making:

7.6 Parts of Similar Triangles

Learning Objectives:

• Recognize and use proportional relationships of corresponding segments of similar triangles.
• Use the Triangle Angle Bisector Theorem.

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.
G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

1. Warm-Up: Play with the simulation here.

Multimodal Meaning & Recursive feedback:

2. Comment: Make a comment about what you notice and what you wonder. Use @ to talk to your friend.

Collaborative Intelligence & Active Knowledge Making:

3. Concept Presentation:

Parts of Similar Triangle

Multimodal Meaning & Ubiquitous learning

4. Practice: Use this practice from Khan Academy to practice.

Differentiated learning & Recursive feedback

5. Make an Update: The Angle Bisector Theorem. Watch and read the proof of the Angle Bisector Theorem. Make an update to describe the proof you saw about the Angle Bisector Theorem. Comment on at least three of your peers' updates.

Onlinemath4all.com: Angle Bisector Theorem Proof.

Metacognition & Active knowledge making:

Fractal: Fractal is a never-ending pattern composed of shapes that are similar to each other. To wrap up this Similarity unit, I would like you to experience the never-ending similarity.

This project supports the experienmental and application learning experience. 

Peer-reviewed Fractal design:

Make your own fractal design or choose one of the fractivities from the fractal foundation.
Turn in your fractal design and answer the following questions in a word file.
(1) Which fractivities did you choose? Why? (200 words)
(2) Describe the similarity in the pattern. (200 words)
(3) What do you learn by doing this activity? Analyze the relationship between the fractal and the similarity. (200 words)

Image by Barbara A Lane from Pixabay
A Sample ​List of Fractal Activities:

• Fractal Triangle
• Fractal Cutout Card
• Fractal Trees
• Watersheds and Rivers
• The Koch Curve and Coastlines
• Fractal Tetrahedrons
• Choose your own
• Peer-reviewed Process:

In the discussion class, present your fractal design and word file submission for 15 minutes.
Use the rubrics below to provide meaningful feedback to your peers.
Include 30 words of feedback to each peer-review feedback.
Rubrics for the fractal design: New Tech Network Oral Presentation Rubrics

Self-Evaluation: Please use this survey for your self-evaluation and reflection.

Analyze and reflect on learners' feedback to guide the future instruction.

Unit Test: A unit test will be given to the whole class.

Use this unit test for students. 

Similarity_20Unit_20Test.pdf

 

Scope of the test:

• Draw dilations.
• Draw dilations in the coordinate plane.
• Use the definition of similarity to identify similar polygons.
• Solve problems by using the properties of similar polygons.
• Use the AA similarity criterion to prove triangles similar.
• Solve problems by using the properties of similar triangles.
• Use the SSS similarity criterion to prove triangles are similar.
• Use the SAS similarity criterion to prove triangles are similar.
• Use proportional parts within triangles.
• Use proportional parts with parallel lines.
• Recognize and use proportional relationships of corresponding segments of similar triangles.
• Use the Triangle Angle Bisector Theorem.