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Leadership Two Weeks Geometry Lesson Plan Teacher: Grade: 8th Grade Math Teacher 8th Grade Lesson Title: Experimenting with Perimeter, Area, Transformations, Congruency and Similarity for Future STEM Career Leaders. STRANDS Similarity, Right Triangles, and Trigonometry Expressing Geometric Properties with Equations Congruence LESSON OVERVIEW Summary of the task, challenge, investigation, career-related scenario, problem, or community link. The unit will start with a STEM career professional speaking to students about their competitive advantage entering the workplace given their STEM education. Students will take information from their readings in English class and write an essay with central ideas on the changing face of education since the 1900s to STEM in the workplace. Students will review finding perimeter and area of polygons in the different forms in which they can be represented (coordinates, figures, and equations) by using various methods such as the distance formula and Pythagorean’s Theorem. Students will then design a floor plan of their dream home and backyard, scale the drawing, and estimate the costs of putting up fencing and laying down sod given a price list for each of those items. This not only reinforces important Common Core Standards, but also prepares students in planning for their future homes and showing how to estimate costs before purchasing items. This will also prepare students for their scale drawing that will be required with the catapult project. The video clip about the Mercedes Bens Logo and it’s design will lead into the lessons on transformations and tie into the final math component for the State of Stem culminating presentation while tying in the history of mathematics used in business. Students will then experiment with transformations in the plane. Students have had experience with transformations: translations, reflections, rotations, and dilations from Algebra I and Pre-Algebra, and they will be reviewed. Comparisons of transformations will provide the foundation for understanding similarity and congruence, which will also play a part in the culminating presentation requirements. Students will then perform various transformations as requested. Similarity and Congruence will then be visited, first with definitions and following up with discussion questions. These questions are intended to formatively assess prior knowledge and to begin student discussion of similarity versus congruence. Students should be encouraged to answer in their own words and to critique each other’s assessments. This allows students to practice communication of knowledge using language rather than algebraic expressions to demonstrate definitions and the importance of math concepts. Hook for the week unit or supplemental resources used throughout the week. (PBL scenarios, video clips, websites, literature) MOTIVATOR Day 1 "Home Quick Planner": This motivator will utilize the following video clip – “Home Quick Planner” (Appendix A). The students will then discuss what formulas are used to calculate distance, and how these could be used in real world applications. Day 3 "Motion Geometry": This motivator will utilize the following video clip – “Motion Geometry” (Appendix F). The students will then discuss how transformations are used in today’s society after watching the informative video. Things will be brought up such as games and digital animation. The students will then debate what kinds of skills and education would be necessary to be a video game designer or creator of digital animation. Teacher will lead the discussion by offering how mathematics is a leading component in programming and animation. Day 6 “The History of the Mercedes Logo”: This motivator will utilize the following video clip – “The History of the Mercedes Logo” (Appendix I). The students will discuss pre-image, and how the pre-image of the Mercedes Logo would compare to a rotated image of the same figure. Since they would appear the same, the teacher would explain this gives the Mercedes Logo the property of rotational symmetry. The students will need this knowledge to design their own logo with rotational symmetry for the “State of STEM Presentation” as well as lead into further investigation of transformed figures and their pre-images. DAY Objectives (I can….) 1 I can use coordinates to compute perimeters of polygons and areas of triangles and rectangles. I can know precise definitions of angle, perpendicular line, parallel line, and line segment, Materials & Resources Instructional Procedures “Home Quick Planner” (Appendix A) Essential Question(s): 1. What are polygons? 2. How to find the area and perimeter and area of a polygon? 3. How to define angle, perpendicular line, parallel line and line segment based on undefined notions of point, line and distance along a line? Set: Teacher will begin by showing the “Home Quick Planner” video clip, then asking students to write down the distance formula, Pythagorean’s Theorem, and name one situation where you may use each formula. How are they similar and can they be used for the same purpose? Teacher will activate discussion regarding distance formulas and their real world applications such as floor plans for homes, determining how to estimate supplies needed for construction, and how knowing perimeter and area of a given location could change the approach to a design. Graph paper Station Work (Appendix B) Ruler (or straight edge) iPad Calculator “Need More Teaching Strategy: Differentiated Assessment Instruction Remediation: Peer Tutoring Heterogeneous Grouping “Need More Support” Stations Enrichment: Peer Tutoring Heterogeneous Grouping “Need More Challenge” Stations Formative Assessment: Opening Writing Assignment Teacher Observations Performance Assessment: Exit Ticket Summative Assessment: Work Station calculations with graphs where based on the undefined notions of point, line, and distance along a line. 2 I can use coordinates to compute perimeters of polygons and areas of triangles and rectangles. Support” Station Work (Appendix C) “Need More Challenge” Station Work (Appendix D) Graph paper “Dream Home Design” Handout (Appendix E) Ruler (or straight edge) I can use geometric shapes, Protractor their measures, and their iPad properties to Calculator describe objects. I can apply geometric methods to solve design problems. 1. Teacher will assign students to heterogeneous groups of 2-4. 2. Each group will travel to Stations 1-4 (Appendix B) and complete the directions given at each station. All stations are concerning perimeter and area of differently shaped polygons presented in different forms. Groups can pick from three folders at the station: Assigned Work (Appendix B), Need More Challenge, (Appendix D) and Need More Support (Appendix C). Teacher will direct students to the folders that are appropriate given the group ability level. 3. Have the students complete the stations and regroup for a discussion. Were any of the stations harder than the others? What technique was most effective for calculating the length of each polygon’s side? Would the order in which you complete the stations make a difference? Summarizing Strategy: As an exit ticket, have students summarize their findings. Ask what they think their strengths and weaknesses are for finding perimeter and area for given polygons. Essential Question(s): 1. How to use geometric shapes, their measurements, and their properties to describe objects? 2. How to apply geometric methods to solve design problems? Set: Begin by asking students about their dream home floor plan. How many bedrooms and bathrooms would you like? Explain that they get to design their own one floor, dream home design. All of this must be put into a plan first, and perimeter and area must be calculated to estimate the total costs in order to estimate the amount of materials to buy and what they may cost. Teaching Strategy: 1. Have students design their floor plan design on a coordinate plane. Students should use each of the quadrants and make sure that all vertices are plotted on integer coordinates. Students may have to modify their design slightly in order to transfer the diagram successfully. Allow students access to graph paper, rulers, and protractors for this task. 2. Ask students to calculate the length of each of the walls and the total perimeter their home’s floor plan design. Have students keep each of these measurements on their “Dream Home Design” handout. appropriate Remediation: Have students create floor plans with only 90º angles. Enrichment: Have students create at least six angles that are not 90º angles. Formative Assessment: Teacher observations of methods used to find area and perimeter. Performance Assessment: Ending discussion of methods used by students Summative Assessment: Floor Plan drawing with perimeters and areas, along with cost 3. Within the design, have students add a backyard that includes a pool (with straight sides) behind their dream home design. Have students calculate the perimeter of the backyard, the perimeter of their dream pool, and the area of their backyard. Using the area, have students calculate the number of square feet of sod that would need to be purchased. 4. Ask students to use the provided prices of fence, stone and sod to determine the total cost of all the landscaping materials on the “Dream Home Design” handout. estimates for sod, brick and fencing. Summarizing Strategy: Ask the class to compare costs, and see what caused some floor plans to be more or less expensive than others. Ask students to share their techniques for finding area and perimeter of the spaces they created, and let the class discuss which would be easiest to use and why. Adapted from: Leinwand, S. (2009). Accessible Mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann. 3 I can use geometric descriptions of rigid motions to predict the effect of a given rigid motion on a given figure. Video “Motion Geometry” (Appendix F) Essential Question(s): How can I use geometric descriptions of rigid motions to predict the effect of a given rigid motion on a given figure? ½ Project Day – See Unit Plan The Catapult Project - Writing Set: Teacher will begin by showing “Motion Geometry” a video clip on use of transformations in video animation. Teaching Strategy: Students and teacher will discuss how transformations are used in today’s society after watching the informative video. Things will be brought up such as games and digital animation. Ask students if they realized that it is all transformations that are programmed that create the look of movement on Remediation: None Enrichment: none Formative Assessment: Teacher observations of current knowledge on the subject. Summative Assessment: Exit ticket. screen. Spark a debate on what kinds of skills and education would be necessary to be a video game designer or creator of digital animation. Lead the discussion by offering how mathematics is a leading component in programming and animation.” Summarizing Strategy: Tell students that after the following project days, they will begin diving into transformations and their properties. As an exit ticket, have them write down 3 things they know about transformations and 2 things they would like to learn. 4 Project Day – See Unit Plan The Catapult Project – Traditional Design and STEM Design 5 Project Day – See Unit Plan The Catapult Project – STEM Design 6 I can articulate the definitions of the transformations: reflection, rotation, and translation. I can recognize the difference between the examples and non-examples of reflections, rotations, and translations “Transformation Examples and Non-Examples” (Appendix G) “Transformation Identification” (Appendix H) “The History of the Mercedes Logo” video clip (Appendix I) Graph Paper Calculator Essential Question(s): 1.What are the definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments? 2.How to recognize the difference between examples and non-examples of reflections, rotations, and translations? Set: The teacher will ask students to make a list of the different types of transformations that can occur in a plane. The teacher will then show “The History of the Mercedes Logo” (Appendix I) video clip. The students will discuss pre-image, and how the pre-image (the original figure before the transformation) of the Mercedes Logo would compare to a rotated image of the same figure. Since they would appear the same, the teacher would explain this gives the Mercedes Logo the property of rotational symmetry. The students will need this knowledge to design their own logo with rotational symmetry for the “State of STEM Presentation” (See Unit Plan) as well as lead into further investigation of transformed figures and their pre-images. Teaching Strategy: Determining the definitions for transformations activity Create three transformation stations: reflection, rotation, and translation, from the Transformation Examples and Non-examples. 1. Divide the class into heterogeneous groups of 3-4 and have students look at different examples provided for rotation, reflections, and translations and write their own definitions for those types of translations. 2. Go through each group and have them provide their definition for each. As a group discuss the definitions, and decide as a class on the Remediation: - Peer Tutoring Heterogeneous Grouping Enrichment: - Peer Tutoring Heterogeneous Grouping Formative Assessment: Teacher observations of definitions developed by students for transformations Performance Assessment: -Ending discussion of determining the best definitions - Homework: Ask students to create their own examples of a rotation, reflection, and translation using the definitions they developed in class Summative Assessment: -Exit Ticket nest one. Use this as an opportunity to clear up any misconceptions students may have. Identifying transformations activity 1. Have student access the “Transformation Identification” activity. 2. Have the groups go through each of the transformations and classify them into the appropriate group. Some of the transformations may be classified as more than one transformation. 3. After completing the “Transformation Identification” activity have students reexamine their definitions. Discuss as a group if the definition is well written or needs to be changed to fit their new findings. 4. Discuss with students each group’s definitions. Decide which are the best and why. Summarizing Strategy: As an exit ticket, ask students how does a reflection, rotation, or translation affect the lines and angles of a transformed figure? Where do you find examples of reflection, rotation, and translation in the real world? Adapted from: Howard County Public Schools (HCPSS) Secondary Mathematics Office (v2.1) 7 I can accurately identify and verify images that are the product of dilation. Graph paper I can create representations that preserve angle measure and double all lengths. Protractors Rulers Compass Calculator Essential Question(s): 1. How to accurately identify and verify images that are the product of dilation? 2. Can I create representations that preserve angle measure and double all lengths? Set: Ask the students to write down the types of transformations they learned about in the previous lesson. Ask students to identify the only transformation they did not define yesterday (dilation), and have them attempt to define this once they have identified it on their paper. Explain to the class that today’s lesson will be investigating the properties of dilation. Teaching Strategy: Divide the class into groups of 3-4. Groups will work together to verify the properties of dilation. Have the groups follow the steps below for the verification. Remediation: Peer Tutoring Heterogeneous Grouping Rather than recreating the dilated figure for investigation, give students in need of support the already dilated figure, and have complete step 10 from the Formative Assessment: Teacher observations of students verifying dilation Performance Assessment: Ending discussion of determining how numbers less than one would effect the dilated figure Dilated figures Using graph paper, rulers and protractors: 1. Draw a line segment. 2. Select a point not on the line for the center of dilation. 3. Extend a ray from the center of dilation through each endpoint of the segment. 4. Choose a scale factor that will double the size of the segment (k = 2). 5. Determine the lengths of the segments from the center to each endpoint along each ray. 6. Multiply the lengths by the scale factor to determine the dilated distance. 7. Measure the dilated distance along the appropriate ray from the center to the new endpoint. 8. Connect the dilated endpoints. 9. Determine the length of the original and dilated segments. (Measure the corresponding angles formed by the intersection of the ray, original segment, and dilated segment.” 10. Verify the following properties associated with similar figures: a. The dilated distance is twice as large as the original distance. b. Corresponding angles are congruent. c. The distance along the ray from the center to the dilated endpoint is twice as large as the distance from the center to the original endpoint. 11. Without dilating the figure, describe how the properties of dilation would affect a size change with a scale factor less than 1 (k = 0.75). Discuss as a class how this would affect the figure. Summarizing Strategy: As an exit ticket, ask the students to write down 3 careers that use dilation while working. How does dilation relate to the scale drawing of the floor plan they created in the previous lesson? Homework: Fill in a dilation table and determine whether or not figures have been dilated teaching strategy. Enrichment: Peer Tutoring Heterogeneous Grouping For students in need of a challenge, have them actually create the figure discussed in Step 11 of the teaching strategy. Summative Assessment: Exit Ticket 8 Project Day – See Unit Plan The State of STEM – Research 9 I can determine if figures are similar by comparing their sides and angles. Rulers I can create similar figures by maintaining angle measure and creating proportional sides. Calculator I can find lengths of sides of proportional figures given enough information about their proportionality. Protractors Graph Paper “Transformed Figure Handout” (Appendix J) Essential Question(s): 1. What are the similarities in angles and sides in congruence? 2. How to create similar figures by maintaining angles and creating proportional sides? 3. How to find lengths of proportional figures given enough information about their proportionality? Set: Ask what needs to be true for proportions (ratios) to be equal. Have students draw a geometric figure in which they think they could create a similar and proportional figure. Recommend a triangle or quadrilateral. Remediation: Peer Tutoring Heterogeneous Grouping Rather than recreating the similar figure for investigation, give students in need of support Teaching Strategy: the 1. Place students in groups of 3-4. Have students exchange their drawings “Transformed with another group of students. Figure Handout” 2. Have each group of students draw a “similar” version of the figure they and have them were given from the other group, either twice or half as large (the option complete step 5 will avoid some problems with size limitations). from the 3. Once they have completed their new figures, have students discuss and teaching share how they created their similar figure with the entire class. During whole group sharing, highlight the necessity for angles to be the same and strategy. all three pairs of sides to be proportional by 2:1 or 1:2. Enrichment: 4. Ask, “What must be the same and what does not for the figures to be Peer Tutoring similar?” Once it is established that the angles must be equal in measure Formative Assessment: Teacher observations of student conjectures and use of those conjectures. Performance Assessment: Discussion about creating similar figures. Student transformed similar figures Summative Assessment: Exit Ticket (start using “congruent”.) 5. Have them measure with a ruler and record the lengths of the sides they have drawn. Have students discuss the criteria for the corresponding sides. Have students write their conjecture and explain how the conjecture is verified. Refer to the opening set for guidance, if necessary. Summarizing Strategy: As an exit ticket, ask students what happens to the angles when transformations are applied to create similar figures? What happens to the sides when transformations are applied? Do these things always happen? 10 Presentation Day – See Unit Plan State of STEM Heterogeneous Grouping For students in need of a challenge, rather than having them complete step 5, have them measure one side of the transformed figure and calculate the lengths of the remaining sides of the figure given the information received from the first side. STANDARDS Identify what you want to teach. Reference State, Common Core, ACT College Readiness Standards and/or State Competencies. G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.A.2 Represent transformations in the plane using 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.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.SRT.A.1.b Verify experimentally the properties of dilations given by a center and a scale factor. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.A.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.