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? 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

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

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

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

Monday Agenda Before lunch: After 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

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.

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?

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

Van Hiele Levels Level 0 – Visualization Level 1 – Analysis 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

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?

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

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

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

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)?

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

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?

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?