Memory

Lenth of unit: 9 days Unit Title: Similar Polygons and Fractals**
 * Lesson Plan: Geometry Honors (8th/9th graders):

__Stage 1 Desired Results__ Standard 4.2.12 B4 Generate and analyze iterative geometric patterns · Fractals · Patterns in areas and perimeters of self-similar figures(impact on dilation) Standard 4.2.12E Use techniques of indirect measurement to represent and solve problems involving similar triangles. Standard 4.2.8 A4 Understand and apply the concept of similarity. · Using proportions to find missing measures · Scale Factors(ratio) Students will understand that… · Geometry is a part of the everyday life and is very useful and can be seen in art, architecture, and construction. · The golden rectangle can be seen not only in architecture but in nature · Fractal Geometry is currently being used in technologies such as cell phone antennas, Bluetooth as well as aiding in researching cures to heart disease and cancer. · Scale drawings such as a house’s blueprints (survey) and projections such as movie projectors are examples of applications of dilation and similarity. · A relationship exists between the scale factor of a similar polygon(a:b) and the area of similar polygons(a ^2:b^2 ). _ (1) Can geometry be used in life outside of the classroom(for students)? (2) What significance does similarity play in our world and where is it being used and for what purpose? (3) Can fractal geometry be used to understand biological structures better? (4) Do you agree that the golden rectangle is the most beautiful rectangle that exists? _ **Students will know…** 1. How to work with ratios (including probability) and solve proportions. 2. How to apply the AA, SAS, and SSS similarity theorems to a variety of proofs and applications. 3. How to apply the Triangle Proportionality theorem and angle bisector theorem. 4. What the Golden Ratio is and be able to provide examples of its prevalence in nature. (1) Apply concepts of area ratios, when triangles are similar or when they share a common side of height. (2) Prove triangles are similar, and use corresponding parts to find values. (3) How to apply similarity concepts to art, architecture, and construction (4) Apply the concept of scale to a wide variety of real world problems, especially indirect measurement problems. (5) Use Geometry Sketchpad to create dilations and investigate fractals. (6) Apply similarity to the Minimum Distance problem and to billiard ball geometry. **__ Stage 2 – Assessment Evidence __** Students will have the option of completing one of the following assignments. (1) Write a one/two page analysis of one part of the video that intrigued you. (2) Complete the geometry sketchpad assignment to create your own Koch Snowflake. (3) Use what was taught from the video and sketchpad lab activity to create your own original fractal using sketchpad. ** Option 2: ** Student is to complete minimum distance Project. (1) Complete Geometry sketchpad lab minimum distance dilemma. (2) Use what was taught to solve minimum distance dilemma problem without sketchpad and write up work. (3) Explain what you have learned from doing this project. How does this problem relate to the path a pool ball takes when it bounces off a cushion and the path a ray of light takes when it bounces off a mirror. You may wish to type up your response. ** Other Evidence: ** (1) Completion and quality of the work (Answers to questions) of geometry sketchpad labs. (Similar polygons, Golden rectangle, Proportions with area, Hat curve fractal). (2) Quality of the completion of homework handouts. Coordinate geometry applications to similarity (2 worksheets) Similar Triangles Worksheets (4 worksheets) (3) Enrichment Handout: Minimum distance and its relation to the path a pool ball takes as it bounces off a cushion (Problems related to concept.) (4) Student Presentations on Minimum Distance/Fractal Project. (5) One day test to be given at end of unit. ** Stage 3 – Learning Plan ** (1) Streaming Video on the Golden Rectangle (+quick group activity related to golden rectangle) and my collection of Fractals (Geometry sketchpad demo) to introduce unit. Introduce the minimum tent problem and provide students with project options. (2) Geometry sketchpad labs: (A) Similar polygons (B) Proportions with area (C) Hat curve fractal (D) Golden Rectangle (3) 6 Worksheets: (A) 2 Coordinate geometry applications to similarity 4 Similar Triangles Worksheets · To be completed for h/w and reviewed in groups of 2-4 students. Handouts will investigate relationships of similar polygons, relationships between scale factor and area of similar polygons, real world examples of its application in terms of construction and architecture, as well as to assess prior mastery of the Triangle Proportionality theorem and angle bisector theorem and provide students with new applications of the theorems that were not previously investigated. · To investigate information needed to prove triangles are similar. (4) Student Project Presentations(Minimum Distance Project/Fractal Discovery) · To investigate how similarity is related to architecture, art, and construction(minimum distance project) · To investigate what a fractal is and their significance in biology, nature, and the real world.(Fractal video and project) (5) Pool Ball enrichment Problem (Time Permitting – To be done in pairs in class)
 * Established Goals: **
 * Understandings: **
 * Essential Questions: **
 * Students will be able to… **
 * Performance Tasks: **
 * Option 1:** Students may watch the following PBS special on fractal geometry and complete the following assignment. []
 * Learning Activities: **

The activity was not as difficult as I had first imagined it would be. My biggest difficulty was figuring out the essential questions and stage 1 of the process. I am used to planning my assessment before planning my lessons so part 2 of the process was not too difficult. My biggest question is whether I did the project correctly to ensure that I understand the UBD concept. I am still a little shaky on stage 1 but I understand it a lot better having practiced on my upcoming unit in geometry. Overall, I do like the UBD design. I know that our school is moving toward this change in curriculum writing and certain subjects have already revamped the curriculum in UBD format. I do feel that the process helped me to organize my thoughts and unit better. It forced me to plan a more cohesive unit that will make it easier to follow. It took a lot of initial effort organizing and planning but I think that it will be worth it as I teach the unit. Jim, I think you did a really nice job on your lesson plan. I like how you used videos in your lesson to get the point across (i am going to check them out). Giving the students two options for an assessment is a great way to differentiate your assessment. I see that both of the assessments deal with fractals and I am assuming they are assessing the similarity theorems as well? Shannon

Jim, Nice job with he structure and the EQ. For the assessment projects, what would understanding look like in term sof the presentations? How do you get students to move beyond spitting back your notes and sharing a transfer of the knowledge about fractals into a different format? BB

Jim, I really liked your essential questions, especially the one that integrates science with math. I think that having inter-disciplianry lessons are very important for students to make connections. One suggestion I would make is to possibly remove the test as part of the assessments. The students are doing a lot that could display their knowledge of the content you are trying to teach. Sometimes using a final project (like you listed) is enough. This would give them a break from a formal test. Very nice job. Frank