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TaxiCab Geometry

Mario Ruiz, Mar 20, 2017

This book is design to introduce Taxicab geometry to a high school class.This book has a series of 8 mini lessons. Lesson 1 - introducing the concept of Taxicab geometry to students Lesson 2 - Euclidian geometry Lesson 3 - Taxicab vs. Euclidian geometry Lesson 4 - Taxicab distance Lesson 5 - Introducing Taxicab circles Lesson 6 - Is there a Taxicab Pi ? Lesson 7 -Are all circles, squares? and are all squares circles? Lesson 8 -Similar triangles The Common Core Standards that we cover are : CCSS.MATH.CONTENT.HSG.MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* CCSS.MATH.CONTENT.HSG.C.A.1 Prove that all circles are similar. CCSS.MATH.CONTENT.HSG.GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. CCSS.MATH.CONTENT.HSG.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). CCSS.MATH.CONTENT.HSG.GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

1.

Chapter 1 - Distance

It introduces Taxicab to the students. It has then compare Taxicab distance with Euclidian distance.

1. Lesson1
2. Lesson2
3. Lesson3
4. Lesson 4

1.1.

Lesson1

Imagine a city where the city blocks were a perfect grid. If you wanted to take a taxi from Point A to Point B. How many blocks is that?

[color=#ff0000][b][size=100]Draw in a few routes. For example draw the shortest route or the longest route.[/size][/b][/color]

How many different routes could you take?

Assuming you are on a taxi and you can only stay on the street. What is the shortest distance from point A to point B?

[color=#ff0000][size=200][size=150][b]This is called TaxiCab geometry because we are assuming you have to stay on the roads to get from one place to another.[/b][/size][/size][/color]

1.2.

1.3.

1.4.

2.

Chapter 2 - Taxicab Circles

It introduces the concept of a Taxicab circle to students and it leads them to discover the Taxicab Pi.

1. Lesson5
2. Lesson 6

2.1.

Lesson5

You have seen a Euclidian circle. A circle is the set of all points that have the same distance from a fixed given point known as the center point.

This is a circle with a radius 3, meaning all the points on the circle have a distance of 3 from the center C.

[color=#ff0000][b]How does a TaxiCab circle look like?[/b][/color][br][br]I want you to plot all the points that are 3 units away from the center given. This would be a TaxiCab circle with a radius of 3.[br][br]After you have plot all the points, connect the points.

When you connected the points what kind of shape does it make?

In Euclidian Geometry we define [math]\pi[/math] as the ratio of its circumference compare to its diameter.[br][math]\pi=\frac{C}{d}[/math][br][br][br]What is the TaxiCab equivalent of Pi?[br][br]Find out what the Circumference of your TaxiCab circle is and divide it by the diameter of the circle to find this ratio.

What was the ratio you found?

2.2.

3.

Chapter 3 - Things are not what they seem.

We ask the question, are circles also squares? and are squares also circles? We also ask students to look at congruent triangles with the side-side-side theorem.

1. Lesson 7
2. Lesson 8

3.1.

Lesson 7

Circle or Square which is it?

In Taxicab this is a circle with a radius of two. [br][br]It is also a square with side length equal to square root of 8. [br][br][br][br]

We see that this is a square with side lengths equal to 2.[br][size=150][color=#ff0000][b][br]But is it also a circle?[/b][/color][/size][br][br]Point E is the center but not all points on the square are equidistant from the center. [br][br]This is not a circle in Taxicab geometry. [br][size=150][color=#0000ff][b][u][br]So we conclude that all circles are squares but not all squares are circles.[/u][/b][/color][/size]

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