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Taxicab Geometry

Taxicab Geometry. TWSSP Monday. Welcome. Grab a playing card and sit at the table with your card value Fill out a notecard with the following: Front: Name Back: School, Grade What do you like about teaching geometry? Learning geometry?

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Taxicab Geometry

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  1. Taxicab Geometry TWSSP Monday

  2. Welcome • Grab a playing card and sit at the table with your card value • Fill out a notecard with the following: • Front: Name • Back: School, Grade • What do you like about teaching geometry? Learning geometry? • What don’t you like about teaching geometry? Learning geometry? • Introduce the person to your left by name, school, grade, and share something that person will do this summer

  3. Week Overview • Focus on two non-Euclidean geometries, taxicab and spherical • What is a geometry? • The properties, definitions and measurements of points, lines, angles and figures • Often accompanied by a visual model consistent with the properties of the geometry – but the model is NOT the geometry

  4. Week Overview • Why non-Euclidean geometries? • When we contrast with Euclidean, our understanding will deepen • We will attend to the analogues of several CCSSM for Euclidean geometry • Put us in our students’ shoes • Van Heile level 4

  5. Week Overview • Content focused, but transparent in pedagogy • Dedicated time every day to consider whole group work in our content area • Purple cup time if desired

  6. Monday Agenda • Before lunch: • Community Agreements • Preassessment • After lunch: • Establish basic definition for and model of taxicab geometry • Explore properties and make conjectures • Question for today: How do we define distance differently in taxicab geometry, and what impact does that have on geometric objects and properties? • Success criteria: I can use a model for taxicab geometry to draw a point and a line, I can find the distance between two points in taxicab geometry, and I can find the set of points equidistant to two points

  7. A protocol we will use • Think - Go Around – Discuss • Private Think Time: Quietly and privately respond to questions. Respect the need for others to process quietly. • Go Around: Share your ideas, all ideas one person at a time. • Discuss: Come to agreement or consensus that can be shared out with the whole group. Make sure everyone in your group understands the ideas discussed.

  8. Community Agreements • What do you need from each other in order to be able to feel safe to explore mathematical ideas, share thinking, and build on and connect with others’ ideas? • What do you need to feel respected and valued as part of the mathematical community?

  9. Van Hiele Levels • Describes levels of understanding through which students progress in relation to geometry • Levels 0-4 (or 1-5) • Not dependent on child’s age or development level • Dependent on experiences and activities in which students engage • Levels are sequential – must pass through one to reach the next

  10. Van Hiele Levels • Level 0 – Visualization • Can identify a shape; can’t articulate its properties • Level 1 – Analysis • Can identify the properties of a shape; can’t articulate relationships • Level 2 – Informal Deduction • Can articulate relationships and informally justify conclusions; can’t construct formal proofs • Level 3 – Deduction • Can construct mathematically sound proofs of conjectures • Level 4 – Rigor • Understand geometry in the abstract, and that other geometries exist

  11. What do we know? • Use the Think (5 min) – Go Around (5 min) – Discuss (10 min) protocol • How would you define points and lines in Euclidean geometry? • How do we measure distances in Euclidean geometry? • How do we measure angles?

  12. Taxicab Geometry • Imagine a city set on a perfect east-west, north-south grid. • Taxicab geometry allows motion only along the grid, and measures distances accordingly. • A reasonable model for taxicab geometry is a grid or a Cartesian plane. • What are the points and lines in taxicab geometry? • Think – Go Around - Discuss

  13. Taxicab lines • What is a line in taxicab geometry? • How does a line in taxicab geometry compare to a line in Euclidean geometry?

  14. Proof • What constitutes a mathematical proof? • Think – Go around – discuss protocol

  15. Taxicab distances • Given two points A = (a1, a2) and B= (b1, b2), find the taxicab distance between those two points. • Notation – to make our life easier… • dT(A, B) := taxicab distance between points A and B • dE(A, B) := Euclidean distance between points A and B • If dT(A, B) = dT(C, D), must dE(A, B) = dE(C, D)? • If dE(A, B) = dE(C, D), must dT(A, B) = dT(C, D)?

  16. More taxicab distance thoughts • Under what conditions on points A and Bdoes dT(A, B) = dE(A, B)? • For any two points, how do the taxicab and Euclidean distances between the two points compare?

  17. Common Euclidean definitions • What are the definitions of parallel and perpendicular lines in Euclidean geometry? • What should those definitions be for taxicab geometry? • In how many points can two Euclidean lines intersect? • In how many points can two taxicab lines intersect?

  18. Exit Ticket (sort of…) • How are Euclidean and taxicab geometries similar? How are they different? • Given any two points, would you expect the distance between them to be larger in Euclidean geometry or in taxicab geometry? Why?

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