Taxicab Geometry

1. Taxicab Geometry TWSSP Tuesday

2. Welcome • Grab a playing card and sit at the table with your card value • Determine your role based on your card’s suit.

3. Tuesday Agenda • Agenda • Refresh Euclidean definitions • Given constraints on relative distances, determine the taxicab sets of points satisfying those constraints • Establish and explore ideas of taxicab congruence • Question for today: How do common Euclidean definitions of figures and congruence compare to corresponding definitions in taxicab geometry? • Success criteria: I can … • Find sets of points taxicab and Euclidean – equidistant to two points • I can plot, define, generalize, and describe properties of a taxicab-circle • I can define, and make use of taxicab- congruence

4. Taxicab and Euclidean distances • 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?

5. What do we know? • Use the Think (5 min) – Go Around (5 min) – Discuss (10 min) protocol • What is the Euclidean definition of equidistant? • What is the Euclidean definition of a circle? • What is the Euclidean definition of an equilateral triangle? • What is the Euclidean definition of a square? • What does it mean for two line segments to be congruent? • What does it mean for two figures to be congruent?

6. Midsets • A midset is the set of all points equidistant between two points. • Find the taxicab and Euclidean midsets for the maps provided using a Think – Go Around – Discuss protocol • When are Euclidean and taxicab midsets different? Why?

7. Fixed distances • Complete the activities on the Fixed Distances sheet, using the Think – Go Around – Discuss protocol • How would you describe the boundaries of the regions you found?

8. Taxicab Circles • I notice/ I wonder about taxicab circles (at least 2 of each)

9. Taxicab Congruence • Think – Go around – discuss: What would need to be true in order for two line segments to be taxicab-congruent? • What would need to be true in order for two figures to be taxicab congruent?

10. Taxicab Triangles • It can be shown that taxicab geometry has many of the same properties as Euclidean geometry but does not satisfy the SAS triangle congruence postulate. • Find two noncongruent right triangles with two sides and the included right angle congruent • Explore taxicab equilateral triangles. What properties do they share with Euclidean equilateral triangles? How do they differ?

11. Exit Ticket (sort of…) • When will taxicab and Euclidean midsets between two points be the same? When will they be different? • Suppose you have a taxicab circle of radius 3 centered at (0, 0) and a Euclidean circle of radius 3 centered at (0, 0). List everything that the two circles will have in common, and everything that will be different about them. • If two triangles are Euclidean congruent, will they be taxicab congruent? Explain.

12. Cognitive Demand Framework • Low Level Tasks • Memorization • Procedures without connections • High Level Tasks • Procedures with connections • Doing mathematics

13. Low Level vs. High Level • Find the product of (x+2)(x+3) in simplest terms • Draw pictures to model the product (x+2)(x-3). Explain why and how your model works.

14. General Techniques for Modifying Tasks • Ask students to create real world stories for “naked number” problems • Use an additional representation and make connections between two (or more) representations • Solve an “algebrafied” version of the task • Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied • Eliminate components of the task that provide too much scaffolding