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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

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taxicab geometry

Taxicab Geometry

TWSSP Tuesday

welcome
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
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
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?
what do we know
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
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
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
Taxicab Circles
  • I notice/ I wonder about taxicab circles (at least 2 of each)
taxicab congruence
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
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
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
Cognitive Demand Framework
  • Low Level Tasks
    • Memorization
    • Procedures without connections
  • High Level Tasks
    • Procedures with connections
    • Doing mathematics
low level vs high level
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
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