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Exploring Area & Perimeter within City Blocks: an Application of Taxicab Geometry (TIU 11 Enrichment)

Lesson Plan

Exploring Area & Perimeter within City Blocks: an Application of Taxicab Geometry (TIU 11 Enrichment)

Grade Levels

10th Grade, 9th Grade

Course, Subject

Geometry, Mathematics
Related Academic Standards
G.2.2.2.1 G.2.2.2.1
Estimate area, perimeter or circumference of an irregular figure.
G.2.2.2.4 G.2.2.2.4
Develop and/or use strategies to estimate the area of a compound/composite figure.
CC.2.2.HS.C.1 CC.2.2.HS.C.1
Use the concept and notation of functions to interpret and apply them in terms of their context.
CC.2.3.HS.A.2 CC.2.3.HS.A.2
Apply rigid transformations to determine and explain congruence.
CC.2.3.HS.A.3 CC.2.3.HS.A.3
Verify and apply geometric theorems as they relate to geometric figures.
Expand
  • Big Ideas
    Mathematical statements can be justified through deductive and inductive reasoning and proof.
    Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
    Objects can be transformed in an infinite number of ways. Transformations can be described and analyzed mathematically.
    Patterns exhibit relationships that can be extended, described, and generalized.
    Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
    Similarity relationships between objects are a form of proportional relationships. Congruence describes a special similarity relationship between objects and is a form of equivalence.
    Some geometric relationships can be described and explored as functional relationships.
    Spatial reasoning and visualization are ways to orient thinking about the physical world.
    There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
    Geometric relationships can be described, analyzed, and classified based on spatial reasoning and/or visualization.
    Patterns exhibit relationships that can be extended, described, and generalized.
  • Concepts
    2- and 3-dimensional figures
    Analytic Geometry
    Geometric Relations: Congruence and Similarity
    Geometric Representations
    Reasoning and Proof
    Trigonometric Ratios
    Circles
    Measurement and Dimension
  • Competencies
    Define and describe types of geometrical reasoning and proof, using them to verify valid conjectures as they surface in the study of geometry; develop a counter example to refute an invalid conjecture.
    Define, describe, and analyze 2- and 3-dimensional figures, their properties and relationships, including how a change in one measurement will affect other measurements of that figure.
    Use concepts of congruence and similarity to relate and compare 2- and 3-dimensional figures, including trigonometric ratios.
    Use coordinates and algebraic techniques to interpret, represent, and verify geometric relationships.
    Apply geometric concepts in modeling situations.
    Describe how a change in the linear dimension can affect perimeter, circumference, area, surface area, and/or volume.
    Extend the concept of similarity to determine arc lengths and areas of sectors.
    Identify, determine, and/or use parts of circles and segments, lines, and angles associated with circles.
    Understand and apply theorems about circles.
    Use and/or compare measurements of angles.
    Use and/or develop procedures to determine, describe, or estimate measures of perimeter, circumference, area, surface area, and/or volume.
    Visualize the relation between two-and three-dimensional objects.

Rationale

The students will learn the rules of Taxicab Geometry, and use them to make conjectures and develop formulas (when possible)for finding distance, perimeter, and area, in a real-world Non-Euclidean Geometry. Often the rules/properties learned in Euclidean Geometry are not directly applicable to real life.

Vocabulary

Distance, Perimeter, Area, Plane, Euclidean Geometry, Conjecture, Postulate, Theorem

Objectives

1.  The students will accurately use the rules of Taxicab Geometry (on Graph Paper).

2.  The students will develop formulas for finding taxicab distance between two points, and compare Euclidean with Taxicab distances.

3.  The students will appoly their taxicab distance formula to explore taxicab perimeter.

4.  The students will copmlete an activity regarding taxicab area, and make conjectures about a possible formula for area of a triangle (and maye rectangle).

5.  The students will make generalizations for Taxicab Geometry, and list possible questions to explore in the future.

Lesson Essential Question(s)

How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real situations or solve problems?
How can geometric properties and theorems be used to describe, model, and analyze situations?
How can patterns be used to describe relationships in mathematical situations?
How can recognizing repetition or regularity assist in solving problems more efficiently?
How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?

1.  If we change any of the 5 basic rules of Euclidean Geometry, what else changes?

2.  Can we develop formulas for distance, perimeter, and area in a non-Euclidean Geometry (specifically Taxicab Geometry)?

Duration

One 1-hour Period

Materials

Each pair (or group) of students needs the following:

  • Red & Green Cup
  • Straight Edge
  • Several sheets of graph paper
  • 2 different colored pencils/markers
  • Set of Activity Sheets
  • Display Graph (either poster-size graph paper or grid board in classroom)

(If Diplay Graph is not available, consider overhead or Doc Cam)

Suggested Instructional Strategies

W:  Asking real-life questions and making real-life applications (in regard to city blocks) will allow students to see where the lesson is going and why.
H:  The Taxicab Geometry treasure hunt will hook the students and help to gain interest for the rest of the lesson.
E:  The powerpoint presentation will equip the students with any rules and/or techniques needed to successfully work through the rest of the lesson's activities.
R:  Think-Pair-Share will allow studeents to rethink, reflect, and revise.
E:  Students will utilize Red & Green cups to evaulate their own progress.  If they are progressing successfully through the lesson, the green cup will remain showing.  Students will put the red cup on top if they require teacher assistance.
T:  Pre-Test will be given to ensure that the lesson is tailored to meet the individual needs. (85% is required on pre-test to proceed with out intervention)
O:  Class Discussion & Collaboration will help to organize the lesson to ensure deep (not superficial) knowledge is acquired.  (The ticket-out-the-door will provide evidence of this).

Instructional Procedures

1.  Pre-Test.  If students cannot successfully answer 6 of the questions, have them independently use websites to fill in the answers.  Taxicab Geometry Pretest.docx

2.  Begin PowerPoint presentation to guide lesson. Taxicab Presentation.pptx

3.  Discuss Euclidean and Non-Euclidean geometries.

4.  Introduce Taxicab Geometry.

5.  Follow link (from PPT) to the Taxicab Treasure Hunt.  Have each group of students take a turn to play.  Each group of students should have a spokesperson, but the whole group should consult before making a move.

6.  Provide each group with some sort of graph paper to display (postersize, transparency, large grid board, etc.).  Give each member the Taxicab Geometry Activity 1.docx.  Activity should take 10 minutes.  Utilize "think, pair, share."  Representative from each group should share results with the class, illustrating the path taken to measure the Taxicab Distance.  (Provide Taxicab Geometry Graphic Organizer.docx to lower level learners, as necessary.  They should fill in the blanks as the lesson progresses.)

7.  Each student should get Taxicab Geometry Activity 2.docx.  Each group should present to the class, discussing the different results.  Lead discussion, using the following discussion points, if necessary.

  • Is it possible to get an area of zero?
  • Can we come up with a rule/definition for Taxicab Area?
  • Can this rule/definition be applied to all polygons?  (Consider convex vs. concave)
  • Extension question for advanced learners:  Can you convert other Euclidean definitions into Taxicab defitinitions?  (Consider the four conic sections: parabolas, ellipses, circles, and hyperbolas.)

8.  Provide advanced learners (or whole class if the entire class is gifted) with Taxicab Geometry Extension Activity.docx.

9.  Ticket out the door - Give each student an index card.  Answer the following question on the card (also can be found in the power point presentation):  Come up with a scenario when it would be better to use Taxicab Geometry, and a scenario when it would be better to use Euclidean Geometry.  Explain your responses. 

 

 

 

 

Formative Assessment

Taxicab Geometry Graphic Organizer.docx
Activity 1 Worksheet (Taxicab Geometry Activity 1.docx)

Activity 2 Worksheet (Taxicab Geometry Activity 2.docx)

Ticket out the Door

For Lower Level Learners: Taxicab Geometry Graphic Organizer.docx

For Advanced Learners: Taxicab Geometry Extension Activity.docx

Related Materials & Resources

Author

Rachel Caruso

Date Published

July 19, 2014
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