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Agriculture Two Weeks Geometry Lesson Plan Teacher: Grade: 8th Grade Math Teacher 8th Grade Lesson Title: Congruence and Right Triangles: Their Properties and How They Lend to Trigonometric Ratios and Real World Relevance STRANDS Similarity, Right Triangles, and Trigonometry Congruence LESSON OVERVIEW Summary of the task, challenge, investigation, career-related scenario, problem, or community link. The unit will start with an investigation of congruence and the real world relevance of using congruence in order to be time and cost efficient. Students will then investigate how different transformations affect the congruency of triangles. Students will also learn the minimum information needed in order to recreate a triangle that is congruent to the original. This will lead into student investigating right triangles and how their similarities make way into the trigonometric ratios that allow humans to solve large-scale measurement problems. Social Studies will be examining agricultural practices used in the past, how they have changed over time, and how agriculture has shaped the United States. Language Arts will be exploring agriculture through literature as well as practicing technical writing at the conclusion of the lab. As well as examining physical and chemical changes students will also develop a working definition for matter, gain an understanding of density, and investigate states of matter. This will culminate in a raised bed vegetable garden project that will integrate the finding from all subjects. Ben Hunter, with the Agriculture Extension Office, will come during the final 2 project days to assist and educate students on the construction and maintenance of the raised bed vegetable gardens. MOTIVATOR Hook for the week unit or supplemental resources used throughout the week. (PBL scenarios, video clips, websites, literature) Day 1 -"Modular Efficiency": This motivator will utilize the following video clip – “Modular Efficiency” (Appendix A). The students will then discuss the pros and cons of modular construction. What type of figures are these modules used to build (congruent figures), and how does having modules (congruent figures) make a company more efficient? The teacher will lead the discussion of how modular construction is utilized in STEM fields in order to be cost and time efficient. Day 4 -"Distance to the Stars": This motivator will utilize the following video clip – “Distance to the Stars” (Appendix F). The students will then discuss how trigonometry is used in today’s society after watching the informative video. Things will be brought up such as GPS and satellite location. The students will then debate Other types of STEM careers and technology that involve the use of triangles and trigonometry. Teacher will lead the discussion by offering how trigonometry is a leading component in structural analysis, as a measuring device for all things that are too large to physically measure, and sailing. DAY Objectives (I can….) 1 I can use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Materials & Resources Instructional Procedures “Modular Efficiency” (Appendix A) Essential Question(s): How can I use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent? “Verifying Triangle Congruence” (Appendix B) Ruler (or straight edge) Calculator “Need More Support” Verifying Triangle Congruence Task (Appendix C) “Need More Challenge” Triangle Congruence Task (Appendix D) Differentiated Assessment Instruction Remediation: Peer Tutoring Heterogeneous Grouping Set: Teacher will begin by showing the “Modular Efficiency” video clip, The students will then discuss the pros and cons of modular construction. What type of figures are these modules used to build (congruent figures), and how does having modules (congruent figures) make a company more efficient? The teacher will lead the discussion of how modular construction is utilized in STEM fields in order to be cost and time efficient. The teacher will then ask the students how they may prove that these figures are congruent (sides, angles.) Teaching Strategy: 1. Place students in pairs and hand out “Verifying Triangle Congruence” Task. Also have available the “Need More Support” Triangle Congruence Task and “Need More Challenge” Triangle Congruence Task for students in need of differentiated instruction. 2. Have the students work together to transform the figure, and determine if the new figure is congruent to the original figure. 3. Come back together as a class and conduct a class discussion using the following discussion questions: “How does a reflection, rotation, or translation affect the corresponding parts of a transformed figure?” “Is there a transformation that does not create a congruent figure?” “Using your prior knowledge of angles, side lengths, and triangles is there an alternative approach to verifying the triangles are congruent besides showing “Need More Support” Triangle Congruence Task. This gives the graphed figure with the angles and side measurements already completed. It also gives the students the different types of transformations they are investigating. Enrichment: Peer Tutoring Formative Assessment: Teacher Observations Performance Assessment: Exit Ticket Summative Assessment: Verifying Triangle Congruence Task graphs and Results that all of the corresponding sides lengths are congruent and all of the corresponding angles are congruent.” 2 I can explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Construction paper “Bulletin Board Congruence” Task (Appendix E) Rulers Protractor Markers Heterogeneous Grouping Summarizing Strategy: As an exit ticket, have students summarize their findings. What needs to be true in order for triangles to maintain congruency (sides and angles)? What is the only transformation that does not maintain congruency? “Need More Challenge” Triangle Congruence Task. This deepens students understanding of congruence by questioning what transformations do not conserve congruence, but do conserve similarity within the transformed figures. Essential Question(s): Remediation: Peer Tutoring How can I explain the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions? Set: Begin by asking students to jot down what it means for two or more triangles to be congruent to one another. Ask them to think about and write down how they would go about creating congruent triangles. Come together as a group and discuss answers from the students Teaching Strategy: 1. Divide the class into groups of 2-3. Provide the students with a copy of the “Bulletin Board Congruence” Task and a large piece of construction paper. The paper will be used as the size of the bulletin board. Allow students a few minutes to read over task and make any additional notes or jot down any questions. Heterogeneous Grouping Enrichment: Peer Tutoring Heterogeneous Grouping Formative Assessment: Teacher observations of methods used to create congruent triangles Performance Assessment: Ending discussion of methods used by students Summative Assessment: 2. Allow students time to work on the task and create solutions. Walk around and observe the students. Ask them about their techniques and how they are meeting the goal of the task. 3. Allow the class to complete a gallery walk to see the different solutions from around the classroom. Completed Bulletin Boards Exit Ticket Summarizing Strategy: As an exit ticket, have students summarize their findings. What is the smallest number of measurements needed (sides and/or angles) to recreate congruent triangles? 3 Project Day – See Unit Plan Feeding America: Exploring Raised Bed Gardening – Research and Planning 4 I can understand that by similarity, side ratios in right triangles are Printer Paper “Distance to the Stars” Video Clip (Appendix F) Essential Question(s): How can I 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? Remediation: Peer Tutoring Give students Formative Assessment: Teacher observations of properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. “Graphic Chart” (Appendix G) “Trigonometric Table Spread Sheet.” (Appendix H) Graphing Calculator Ruler Protractor Ruler “Pre-designed Right Triangle” (Appendix I) “Need More Challenge” Task (Appendix J) Set: Begin the class by showing the video clip: “Distance to the Stars.” The students will then discuss how trigonometry is used in today’s society after watching the informative video. Things will be brought up such as GPS and satellite location. The students will then debate Other types of STEM careers and technology that involve the use of triangles and trigonometry. Teacher will lead the discussion by offering how trigonometry is a leading component in structural analysis, as a measuring device for all things that are too large to physically measure, and sailing. Teaching Strategy: 1. Give each student a sheet of unlined paper a protractor and a ruler. Instruct students to draw, as accurately as possible, a right triangle with one of its acute angles measuring 30o. Ask students, in pairs, to compare triangles. Ask students to share their observations. (Students should recognize that, while some of the triangles may be approximately congruent, all the triangles in the room are approximately similar.) 2. Have students form trios by finding two partners that created triangles that are “different” than theirs. Ask, “What are the differences that made you choose your group members? What is still the same about your triangles?” 3. Have groups measure, as accurately as possible, the lengths of all three sides of their triangles and find the measure of the unknown angle. (They may find the angle by measuring, but some may use the fact that the sum of the measures of the angles of a triangle is 180o to find that the remaining angle measures 60 o.) Explain to students a system of identifying the sides of their triangles. (One side is the “hypotenuse” and will be denoted as “H.” The hypotenuse will be a previously learned concept. It may be necessary to review it in the context of the Pythagorean theorem. The legs can now be identified relative to one of the acute angles called the “reference angle.” The leg across from the reference angle is called the “opposite leg,” which will be denoted as “O,” and the leg which makes up one side of the reference angle is called the “adjacent leg” which will be denoted as “A.”) Instruct students to use the 30o angle as the reference angle. Have each group choose one pair of sides and create a ratio of those sides for each triangle in the group and find a decimal approximation of the ratio. (Groups should find that the ratio they find is equal for all three triangles.) Have groups report their results. Record the results on a board or projector (using the graphic organizer). Give each group the “Graphic Chart” to record the results for any of the ratios found by the class. If all six ratios have not been chosen, have the class find the remaining ratios in groups and report them to complete the table. Have groups add these labels to three of the ratios on the chart: O/H = sine 30o, A/H = cosine 30o, O/A = tangent 30o. Point out abbreviations “sin, cos, tan.” (Anticipate the possibility of questions as to why the other three ratios, the “Predesigned Right Triangle” with the 30o angle, hypotenuse, adjacent side and opposite side clearly identified. Enrichment: Peer Tutoring student conclusions Performance Assessment: Discussions throughout task Summative Assessment: Completed “Graphic Chart” Triangle Designs Give students the option of the “Need More Challenge” Task in which students deepen their understanding of the relationship between the acute angle, length of sides, and their trigonometric properties. Exit Ticket 4. 5. 6. cosecant or csc and secant or sec and cotangent or cot, are not being named at this point. In this case, point out that these ratios are merely reciprocals of the first three and not necessary for relating the sides at this point. Feel free to identify them by name to satisfy curiosity.) Ask, In any right triangle with a 30o angle, what do we now know about the ratio of the leg opposite the 30o reference angle to the hypotenuse (O/H)? (It is always the same and equal to .5.) In any right triangle with a 30o angle, what do we now know about the ratio of the leg adjacent to the 30o reference angle to the hypotenuse (A/H)? (It is always the same and equal to approximately .8660.) In any right triangle with a 30o angle, what do we now know about the ratio of the leg opposite to the 30o reference angle to the leg adjacent to the 30o reference angle (O/A)? (It is always the same and equal to approximately .5774.) What is the measure of the other acute angle? What would happen to the trig ratios if we used this angle as the reference angle? Ask, “What would happen to the trig ratios if we used a different angle? What, more specifically, do you think would happen to the sin and cos of the angle if we reshaped the triangle so that the reference angle was changed from 30o angle to 40o? Be prepared to explain why you think this will be true.” Have students discuss the accuracy of the answers. Elicit responses from groups who found specific answers gleaned from ratios. Note these answers as with 30 o before. Look for or make a suggestion that there are other ways than construction of a triangle to find the ratios. Make sure each group has a scientific or graphing calculator. (One calculator per student is preferred.) Ask them to explore and find the sin30o, cos 30o, sin 40o, and cos 40o on their calculators. Tell them they will know they have found it correctly, without using a constructed right triangle, when they get the same approximate answers as discovered earlier. Ask them to find tan of the two reference angles as well. Give each student a copy of the “Trigonometric Table Spread Sheet.” Ask students to find the sin, cos, and tan of 30o and 40o on the table and compare the answer to that on the calculator. (Note that the answers are approximate and that the decimals for the ratios non-terminating and non-repeating. Ask, “What generally happens to the sine as the reference angle gets bigger? What happens to the cosine and tangent?” Other than, looking at answers on the table or calculator, why do you think this is true? Summarizing Strategy: As an exit ticket, have students summarize their findings. How do acute angles lend to the trigonometric functions? How are these functions useful in STEM professions? 5 I can explain and use the relationship between the sine and cosine of complementary angles. “Trigonometric Table Spread Sheet.” (Appendix H) Ruler (or straight edge) Essential Question(s): How can I explain and use the relationship between the sine and cosine of complementary angles? Remediation: Peer Tutoring Set: Begin by asking students to jot down what it means for two angles to be complementary. Ask them to then write down what observations they can make about complementary angles in a triangle. Heterogeneous Grouping Teaching Strategy: 1. The goal of this lesson is to assist the learner in seeing that the sine and cosine of complementary angles are the same. There are a variety of ways to initially establish this. Some options include: a) Have students explore on the accompanying trigonometry table spreadsheet for answers that are the same. Ask them make a conjecture about what sorts of angles and trig functions produce equal trig ratio. b) Have students explore on a calculator for answers that are the same. Ask them make a conjecture about what sorts of angles and trig functions produce equal trig ratio. 2. Have students write a conjecture as to the equality of sine and cosine of complementary angles. 3. After a conjecture has been established of the equality of sine and cosine of complementary angles, have students work in groups and produce an explanation (or proof) of why these ratios would be equal. Have students draw a triangle and label parts. Note: Here, students will be asked to discover a rationale for the equality. These hints, given judiciously, will help the process along: a) Note that the two acute angles of a right triangle are complementary b) Note that the designations of the sides (opposite leg, adjacent leg) will change depending on the reference angle. c) On the right triangle, label the legs “a” and “b” and then find the sine and cosine relative to each reference angle. Enrichment: Peer Tutoring Heterogeneous Grouping Formative Assessment: Teacher observations of conclusions as to why sine and cosine of complementary angles is equal Performance Assessment: Proof by group work of the given conjecture Summative Assessment: Exit Ticket Summarizing Strategy: As an exit ticket, have students summarize their findings. What do we know about the sine and cosine of complementary angles? Why is this always true? 6 I can use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. I can understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles “Constructing a Clinometer” – Video Clip (Appendix K) “Checklist for the Presentation” (Appendix L) Protractors String Straws 3x5 Notecards Paper Clips Tape Pens iPad A basket for Each Pair Containing the Items for the Clinometer Measuring Tape Essential Question(s): 1. How can I use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems? 2. How can I understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles? Set: Ask students to guess the height of the school building, school flagpole, or the trees outside of your school building? Ask each of them to give justification to support their guess. After hearing their guesses, tell them they will learn to measure the height of these objects using their knowledge of trigonometric functions and a simply constructed tool known as a clinometer. Teaching Strategy: Place the students in pairs. Let them know that together they will determine the height of the school building using indirect measure and a clinometer that they will design and construct. They will organize their results into a presentation such as a poster, or power point, or iMovie. Be prepared to explain step-by-step how you determined your answer to your classmates. *Note: Students need to understand the indirect measurement and solving right triangles before completing this task. 1. View the video “Constructing a Clinometer.” 2. Allow students 8-10 minutes to construct their clinometer and determine how to use the instrument along with their mathematical knowledge of trigonometric functions to find the height of the objects 3. Come together as a class and discuss the length of time available for the measurements, as well exactly what data will need to be obtained to solve for the height of the object. 4. Develop a list of objects to measure around your school and create a tour of the school using the clinometers and indirect Remediation: Peer Tutoring Heterogeneous Grouping Enrichment: Peer Tutoring Heterogeneous Grouping Formative Assessment: Teacher observations of clinometer construction Teacher observation of techniques used for solving the problem Teacher observation of solutions to the problem. Performance Assessment: Discussion of data required in order to find the height of the object. Students using clinometers as a tool for measurement Summative Assessment: Exit ticket Calculators 5. 6. 7. 8. measurement to determine the height of objects. Take a walk around the school and obtain data that will help determine the height of the objects listed. Come back to the classroom and allow the students some time to calculate their solutions. Pass out a “Checklist for the Presentation” and discuss what will be expected for the presentation. Presentations could be posters, power points, iMovies, etc. Let the students know they are to bring in the supplies tomorrow to work on their presentations. Summarizing Strategy: As an exit ticket, ask students to summarize their findings. How did the clinometer work? What type of trigonometric function was used to solve for the missing side? Homework: Ask students to plan their presentation and bring in the needed supplies to complete the presentation 7 I can use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. I can understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles Laptops iPads Materials Students are Bringing for Their Presentation Essential Question(s): 1. How can I use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems? 2. How can I understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles? Set: Begin the class by discussing the various results from yesterday’s findings? Are the solutions found equal or approximately equal? What techniques were used? What steps did you take as a group to ensure the measurements are accurate? Teaching Strategy: 1. Have students get into their pairs and begin working on their presentations from yesterday’s findings. 2. Make sure each pair of students has the “Checklist for the Presentation.” 3. Observe the students as they are putting the presentation together. Make sure they are displaying correct information. Ask questions to Remediation: Peer Tutoring Heterogeneous Grouping Enrichment: Peer Tutoring Heterogeneous Grouping Formative Assessment: Teacher observations of justifications used in presentations for group solutions. Performance Assessment: Discussion of solutions found by various groups Summative Assessment: Final Presentation guide the students in the correct direction. 4. Ask the students to turn in their presentations with about ten minutes left in the class period. 5. Take a gallery walk of the presentations. Summarizing Strategy: As an exit ticket, ask students decide which presentation best explained how to use trigonometric ratios to solve the height problem? Why was this presentation method best? 8 Project Day – See Unit Plan Feeding America: Exploring Raised Bed Gardening – Building and Planting 9 Project Day – See Unit Plan Feeding America: Exploring Raised Bed Gardening – Building and Planting Exit ticket 10 I can derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. iPads Rulers Protractors Graph Paper Calculator “Superman’s Shopping Center Site Map and Scenario.” (Appendix M) “Triangles Resource” Sheet (Appendix N) Essential Question(s): How can I derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side? Remediation: Peer Tutoring Set: Tell students Superman’s Shopping Center wants to build a new store in the space bounded by three roads. This would mean they would have to develop on an area that was currently a forest. The Department of Natural Resources (DNR) has restrictions in this area that no more than 120,000 square feet of forest may be taken down during a single building project. You and your team have been hired by Superman’s Shopping Center to advise them on whether or not their building project will exceed these limits. Because much of the land has not been developed you only know certain distances. You do know the distance of Road A from points B to C measures 650 feet and the distance of Road B from points A to C measures 800 feet. You also know that Roads A and B intersect at a 35 degree angle. Using these measurements, determine if the area exceeds the 120,000 square feet. Heterogeneous Grouping Teaching Strategy: 1. Before beginning the task, access students’ prior knowledge of the proof of the area of a triangle. If students need a review of this topic, provide students with the virtual manipulates link below to explain why this formula works. http://math.kendallhunt.com/x19469.html 2. After accessing students’ prior knowledge of the area of a triangle, divide the class into small groups of 3-4 students. Then, provide them with the “Superman’s Shopping Center Site Map and Scenario.” Provide students with rulers, graph paper, and protractors in order to create a model of the store. 3. Allow students a few minutes to ask questions before they begin. Remind them after the questioning period is over; the teacher role will be to observe and not be to answer questions or help with the activity. *Note: This allows students time to try the problem on their own without interference and gives them appropriate time to struggle while encouraging them to use what they have previously learned. Students who are still struggling can use the “Need More Support” task. 4. As groups are finishing, have them summarize their solutions on their graph paper. Groups can then post their solutions with any necessary diagrams around the room. 5. Have students participate in a gallery walk, while thinking about the Enrichment: Peer Tutoring Heterogeneous Grouping Require students submit a scale model of the store. The scale must be included with the model and clearly visible. Formative Assessment: Teacher observations student prior knowledge of the Triangle Proof. Performance Assessment: Discussion about creating similar figures. Student transformed similar figures Summative Assessment: Exit Ticket following questions: a. How did other groups’ solve the problem? b. Which group had the most efficient method? c. Did everyone come to the same conclusion? Why or why not? Summarizing Strategy: As an exit ticket, ask students what technique their group used to solve the problem. Whose group had the most effective technique and why? STANDARDS Identify what you want to teach. Reference State, Common Core, ACT College Readiness Standards and/or State Competencies. G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.SRT.B.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.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.D.9 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).