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1 Plainfield Public Schools Mathematics Rigorous Curriculum Design Unit Planning Organizer Grade/Course Unit of Study Pacing Geometry Trigonometry 7 week , Standards for Mathematical Practice ● MP1. MP2. MP3. MP4. MP5. MP6. MP7. MP8. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. ● Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 2 UNIT STANDARDS G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 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.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.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 3 “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels Focus Standard: G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Find Line segment 1 ratio “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels Focus Standard: 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. Prove Triangle 4 “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels Focus Standard : G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures Use congruence and similarity 2 Solve problem 3 Prove relationship 4 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 4 “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels Focus Standard: 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. Understand Right triangle 1 Trigonometric ratios “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Explain and use sine and cosine of complementary 2 angles. “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ Use trigonometric ratios and the Pythagorean Theorem 2 “Unwrapped” Skills “Unwrapped” Concepts DOK (students need to be able to do) (students need to know) Levels G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 4 Derive area of a triangle Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 5 “Unwrapped” Focus Standards Concepts Laws of Cosine Laws of Sines Area of Triangle Cosecant Secant Cotangent Angle of elevation Angel of depression Pythagorean Theorem Trigonometric Ratios Perpendicular Lines Right Angles Congruent Similarity Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 6 Essential Questions When a tv picture is forced to fill a screen why may the picture be distorted? Corresponding Big Ideas Proportions are used to identify similar polygons How can a coordinate grid be used to model Properties of similar polygons may be used and describe trigonometric ratios? to solve problems. How do the angles of two similar triangles compare? You may determine triangles similar using AA, SSS and SAS. Proportional relationships of corresponding angle bisectors, altitudes and medians in similar triangles may be used to solve problems.Scale factors are used to interpret and solve problems How do you find a side length or angle measure in a right triangle? Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height How are the Pythagorean Theorem and special right triangle rules used to find the lengths of the sides of right triangles? The Pythagorean Theorem and special right triangle rules are used to find the lengths of the sides of right triangles. How do trigonometric ratios relate to similar There are three trigonometric functions that are used to find the lengths and angle right triangles? measures of right triangles. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 7 Engaging the Learner The Hopewell people were Native Americans whose culture flourished in the central Ohio Valley about 2000 years ago. They constructed earthworks using right triangles. The bestsurviving features of the Hopewell Tradition era are mounds built for uncertain purposes. Great geometric earthworks are one of the most impressive Native American monuments throughout American prehistory. Eastern Woodlands mounds have various geometric shapes and rise to impressive heights. The function of the mounds is still under debate. Due to considerable evidence and surveys, plus the good survival condition of the largest mounds, more information can be obtained. Earthworks were built for various purposes by heaping earth in piles and shaping them. Some of the earthworks were large burial mounds, others served as platforms for structures such as temples, and still others served as defensive walls. Mounds were usually cone-shaped, oval, or formed into the shape of an animal." Which standard(s) (priority/supporting) will the task address? 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.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.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems What essential Question(s) and corresponding Big Idea(s) will this task target? How do the angles of two similar triangles compare? Proportional relationships of corresponding angle bisectors, altitudes and medians in similar triangles may be used to solve problems. Scale factors are used to interpret and solve problems. You may determine triangles similar using AA, SSS and SAS. How do you find a side length or angle measure in a right triangle? The angles of elevation and depression are the acute angles of right triangles formed by a horizontal distance and a vertical height. Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem How are the Pythagorean Theorem and special right triangle rules used to find the lengths of the sides of right triangles? The Pythagorean Theorem and special right triangle rules are used to find the lengths of the sides of right triangles. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 8 How do trigonometric ratios relate to similar right triangles? There are three trigonometric functions that are used to find the lengths and angle measures of right triangles. Proportional relationships of corresponding angle bisectors, altitudes and medians in similar triangles may be used to solve problems. Scale factors are used to interpret and solve problems Which “unwrapped specific concepts and skills will this task target? Prove theorems about triangles Use congruence and similarity Solve problems and prove relationships in geometric figures Use the relationship between the sine and cosine of complementary angles Use trigonometric ratios and the Pythagorean Theorem How will the students apply the concepts and skills? What will they do and/or produce? Students will study the history of the Hopewell people( Native Americans) and their contribution to mathematics Apply Pythagorean Theorem to solve for length of the hypotenuse Proportional relationships of angle bisectors, altitudes and medians in similar triangles and scale factors to interpret and solve problems Use trigonometric functions to solve problems What resources, instruction, and information will students need in order to complete the task? Appendix 1 : Hopewell Geometry Task What evidence of learning will I look for to show that I know all of my students have conceptually learned the concepts and skills – the standard(s)? Completion of tasks with data table and analysis. Appendix 2 : Hopewell Geometry Task Rubric How can I differentiate the application and/or evidence to meet the varying needs of my students? Give a personal cue to begin work Give work in smaller units Provide immediate reinforces and feedback Make sure the appropriate books and materials are available Introduce the assignment in sequential steps Check for student understanding of instructions Check on progress often in the first few minutes of work Provide time suggestions for each task Provide a checklist for long detailed tasks Use technological resource Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 9 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 10 Advanced or Exemplary All “Goal” criteria plus: Goal ( 7 – 10 points) All answers are correct Use Pythagorean Theorem Use trigonometric functions Apply and use scale factor to justify reasoning Gives a correct explanation such as finds the lengths of all three sides and shows lengths don’t satisfy the Pythagorean Rule. Progressing Meets _3-4__ of the “Goal” criteria Provides partially correct explanation Beginning Meets fewer than 4 of the “Goal” criteria Provides an incorrect explanation Task to be repeated after re-teaching Comments: Interdisciplinary Connections and Related Focus Standards Specific to Task CCSS.ELA-Literacy.RST.9-10.7 Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words CCSS.ELA-Literacy.RH.9-10.4 Determine the meaning of words and phrases as they are used in a text, including vocabulary describing political, social, or economic aspects of history/social science 21st Century Learning Skills Specific to Task #1 those that apply for each task: ❑ Teamwork and Collaboration ❑ Initiative and Leadership ❑ Curiosity and Imagination ❑ Innovation and Creativity ❑ Critical thinking and Problem Solving ❑ Flexibility and Adaptability ❑ Effective Oral and Written Communication ❑ Accessing and Analyzing Information ❑ Other Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 11 Instructional Strategies/Resources Task /Activities that solidifies mathematical concepts Use questioning techniques to facilitate learning Reinforcing Effort, Providing Recognition Practice , reinforce and connect to other ideas within mathematics Promotes linguistic and nonlinguistic representations Cooperative Learning Setting Objectives, Providing Feedback Varied opportunities for students to communicate mathematically Use technological and /or physical tools Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. 12 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved.