TAXICAB GEOMETRY TWSSP Tuesday

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

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

Taxicab and Euclidean distances Under what conditions on points A and B does d T (A, B) = d E (A, B)? For any two points, how do the taxicab and Euclidean distances between the two points compare?

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?

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?

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?

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

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?

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?

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.

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

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.

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