Unit IV: Evolution s3

Honors Geometry

Topic Outline

Course Description and Philosophy

The purpose of Honors Geometry is to develop the student’s abilities to formulate logical patterns of mathematical thought. The students will learn how to read mathematics, how to describe mathematical ides in rigorous language and how to use their creativity in the solutions of challenging problems. The aims and objectives of the course are to develop and show the value of the logic of deductive reasoning and to improve and increase the understanding and application of the terminology, the symbolism, and the structure of mathematics. The problems explored involve intense analytical thought and require strong algebraic skills. Non-Euclidean geometries such as fractal geometry and the Lenart sphere are investigated. Through the use of their algebraic skills and the knowledge developed relating to plane, solid, and analytic geometric concepts, the students will extend their foundation of mathematics for further mathematics courses and the sciences. The course has been updated to meet the 2010 Common Core Math Standards of High School Geometry. http://www.corestandards.org/Math/Content/HSG/GPE

Teacher Resources:

· Illustrative Mathematics Content Standards: High School http://www.illustrativemathematics.org/standards/hs

· The Teaching Channel (Common Core Math Channel) https://www.teachingchannel.org/videos?page=1&categories=subjects_math,topics_common-core&load=1

· Flipped Classroom resources including Khan Academy Geometry https://www.khanacademy.org/math/geometry and HippoCampus Geometry http://www.hippocampus.org/Algebra%20%26%20Geometry

· The Mathematics Common Core Toolbox (http://ccsstoolbox.agilemind.com/resources_samples.html) has both sample scope and sequence documents as well grades 4-12 PARRC assessment tasks.

Text Reference:

Written 2013

Unit I: Congruence, Proof, and Construction

Essential Questions: How do the fundamentals of geometry enhance inductive reasoning? How do rigid motion and formal constructions establish the triangle congruence conditions?

Objectives: Students will be able to:

· Make sense of problems and persevere in solving them.

o SLO 5 Plan a pathway to prove theorems about lines, angles, triangles, and parallelograms.

· Reason abstractly and quantitatively.

o SLO 4 Know and use properties of rigid transformations in proofs involving lines, angles, triangles, and parallelograms.

· Construct viable arguments and critique the reasoning of others.

o SLO 5 Build a logical progression of statements to prove conjectures about lines, angles, triangles, and parallelograms.

· Model with mathematics.

· Use appropriate tools strategically.

· Attend to precision.

o SLO 1 Use precise language in the definitions of angles, circles, parallel lines, perpendicular lines and line segments.

· Look for and make use of structure.

· Look for and express regularity in repeated reasoning

Unit I: Topic/Content Skills / Assessment / Resources / Instructional Method / Tech Infusion / CCS: Unit I
Topic 1: Use the undefined notion of a point, line, distance along a line and distance around a circular arc to develop definitions for angles, circles, parallel lines, perpendicular lines and line segments. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.1
Topic 2: Apply the definitions of angles, circles, parallel lines, perpendicular lines and line segments to describe rotations, reflections, and translations. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.1, G.CO.4
Topic 3: Develop and perform rigid transformations that include reflections, rotations, translations and dilations using geometric software, graph paper, tracing paper, and geometric tools and compare them to non-rigid transformations. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.2, G.CO.3, G.CO.4, G.CO.5
Topic 4: Use rigid transformations to determine, explain and prove congruence of geometric figures.
/ Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.6, G.CO.7, G.CO8
Topic 5: Create proofs of theorems involving lines, angles, triangles, and parallelograms.
(Please note G.CO.10 will be addressed again in unit2 and G.CO.11 will be addressed again in unit 4) / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.9, G.CO.10, G.CO.11
Topic 6: Generate formal constructions with paper folding, geometric software and geometric tools to include, but not limited to, the construction of regular polygons inscribed in a circle. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.12, G.CO.13

Differentiated Learning Activities

Strategic learner: Under different isometries, the students determine whether given statements are true or false. If they are true, the students give proofs using coordinate geometry. If they are false, the students give counterexamples.

Advanced learner: Students are introduced to the concept of runway bearings. They will complete a project where they research the runways of a major airport in the United States.

Ethical Decision Making/Character Education: Students are given four measures of a music staff. The students determine whether the combination of notes and/or rests form a frieze pattern. If so, they classify the frieze pattern. The students discuss the importance of music in schools, and the relationship between music and mathematics.

21st Century Skills: By discovering fresh insights and communicating them to others, students come to understand that mathematics is a creative endeavor that builds on previous knowledge. EXAMPLE: Working in teams, students study examples of Kolams from Southern India or Sona from Angola and Zaire and explore the mathematics in these complex geometric art forms. They explore Kolam designs based on Fibonacci numbers (http://vindhiya.com/Naranan/Fibonacci- Kolams/) and examine how an array of dots can give rise to a one-line sona (www.beloit.edu/ computerscience/faculty/chavey/sona/).Using the concepts they have learned, students try to create their own Kolams or Sona.

Unit II: Similarity and Proof

Essential Questions: How are proportions used to solve geometric problems? How will the understanding of dilations and proportional reasoning help to develop a formal understanding of similarity?

Objectives: Students will be able to:

· Make sense of problems and persevere in solving them.

· Reason abstractly and quantitatively.

o SLO 1 Proof of the similarity of specific circles used to reason about the similarity of all circles.

· Construct viable arguments and critique the reasoning of others.

o SLO 5 Construct proofs about triangles using assumptions, definitions, and previously established theorems.

· Model with mathematics.

· Use appropriate tools strategically.

· Attend to precision.

· Look for and make use of structure.

o SLO 3 Use the definition of rigid transformations to determine if two figures are similar.

· Look for and express regularity in repeated reasoning.

Unit II: Topic/Content Skills / Assessment / Resources / Instructional Method / Tech Infusion / CCS: Unit II
Topic 1: Generate proofs that demonstrate that all circles are similar. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.C.1
Topic 2: Justify the properties of dilations given by a center and a scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged (the dilation of a line segment is longer or shorter in the ratio given by the scale factor). / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.SRT.1
Topic 3: 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. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.SRT.2
Topic 4: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.SRT.3
Topic 5: Prove theorems about triangles. / Test/Quizzes
Homework
Class Participation
Projects / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.CO.10, G.SRT.4

Differentiated Learning Activities

Strategic learner: Students design a hole of a mini-golf course that requires the ball to hit one or two walls. The students draw at least one possible path to the hole using similar triangles that would result in a hole-in-one. Advanced learner: In groups, students read the well-known Lewis Carroll logic puzzle: “We are offered our choice of two clocks. Clock A loses 1 minute per day. Clock B doesn’t run at all. Which clock should we choose?” Students answer given questions and discuss the solution to the puzzle.

Ethical Decision Making/Character Education: Students find conditional statements in the school’s honor code and write the converse of each statement. The students will determine if the conditional statement and the converse are biconditional. The teacher will review the importance of the honor code and academic integrity.

21st Century Skills: Students articulate mathematical thoughts and ideas using oral and written communication skills. Using abstract and quantitative reasoning with attention to precision, they construct viable arguments and analyze others’ reasoning.

EXAMPLE: As a joint activity with a social studies class, students study the history of how seats have been allocated in the U.S. House of Representatives. They analyze the mathematics behind different plans and underlying socio-political issues, such as the effects on small versus large states or rural versus urban populations. Students then solve various “fair division” problems for their city council or state house of representatives using different plans. Finally, students hold a debate in which students advocate for different plans, considering both the mathematical and social issues that go into the allocation of seats.

Unit III: Trigonometry

Essential Questions: How does similarity apply to right triangles? What is right triangle trigonometry? How can we calculate the missing measures in all triangles – not just right triangles? How is trigonometry used to solve real-life problems?

Objectives: Students will be able to:

· Make sense of problems and persevere in solving them.

· Reason abstractly and quantitatively.

· Construct viable arguments and critique the reasoning of others.

o SLOs 3 Justify solutions to problems involving side lengths and angle measures using triangle congruence and similarity criteria.

· Model with mathematics.

· Use appropriate tools strategically.

· Attend to precision.

o SLO 4 Demonstrate the need for precision when deriving definitions.

· Look for and make use of structure.

o SLO 8 Look for hidden structures to prove and apply the law of Sines and Cosines.

· Look for and express regularity in repeated reasoning

Unit III: Topic/Content Skills / Assessment / Resources / Instructional Method / Tech Infusion / CCS: Unit 3
Topic 1: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.GPE.6
Topic 2: Prove theorems about triangles. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources: (e.g. Khan Academy) / G.SRT.4
Topic 3: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources / G.SRT.5
Topic 4: Derive the definitions for trigonometric ratios using similarity of right triangles. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources / G.SRT.6
Topic 5: Explain and use the relationship between the sine and cosine of complementary angles. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources / G.SRT.7
Topic 6: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources / G.SRT.8
Topic 7: Derive and use the formula for the area of an oblique triangle (A = 1/2 ab sin (C)). / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources / G.SRT.9
Topic 8: Prove and apply the Laws of Sines and Cosines to solve problems involving both right and oblique triangles. / Test/Quizzes
Homework
Class Participation / Text
Manipulatives / Lectures
Hands-On Activities
Lab Work
Group Collaboration / Geometer’s Sketchpad
Scientific Calculator
Links to “Flipped Classroom” resources / G.SRT.10, G.SRT.11

Differentiated Learning Activities