June 2007 Area and Tilings Slide 1

Area and TilingsA Lesson in the “Math + Fun!” Series

June 2007 Area and Tilings Slide 2

About This Presentation

Edition Released Revised Revised

First June 2007

This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during four school years (2003-07). “Math + Fun!” material can be used freely in teaching and other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami

June 2007 Area and Tilings Slide 3

Finding the Area of a Geometric Shape

The area of a triangle is half that of a rectangle that encloses it

Area = Width Height

Rectangle

Triangle

Area = Base Height / 2 Area = Base Height / 2

Triangle

Circle

Area = (/4) Diameter2

= Radius2

Square

Area = Side2

The area of a circle is about 80% of the square that encloses it

June 2007 Area and Tilings Slide 4

On a sturdy piece of cardboard, draw a 4” 6” rectangle

Activity 1: The Area of a Triangle

Place thumbtacks or pushpins into the two lower cornersPut a rubber band around the two thumbtacks or pushpins and stretch it so that it forms a triangle, with the top vertex at the upper left corner of the rectangle. Is it obvious that the area of the triangle is half the area of the rectangle?

Now, slowly move the rubber band so that the top vertex shifts to the right along the rectangle’s top edge. What happens to the area of the triangle as you move the top vertex?

June 2007 Area and Tilings Slide 5

Method 1: Approximate the irregular shape by a regular one

Measuring the Area of an Irregular Shape

Method 2: Cover with 1 1 tiles; count whole tiles and half of broken ones

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Draw a large irregular area on a piece of cardboard or construction paper

Activity 2: Tiling an Area with Square Tiles

15” or more

Draw a straight line through the middle of the area in any direction

Use square post-it notes as your tiles

Place tiles, one by one, on one side of the straight line that you have drawn, taking care that the tiles are aligned and there is no gap between them(real tilers actually leave a gap between tiles where they will pour the grout)Now, moving up and down from the row of tiles placed next to the line, finish tiling of the area, leaving spaces only where whole tiles would not fit; make sure the tile sides are perfectly aligned, with no gap between them

Cut tiles to appropriate shapes to fill the irregular areas at the edges

Taking your tiles to be 1’ 1’, estimate the area of the irregular shape in ft2

June 2007 Area and Tilings Slide 7

Any shape with right angles and side lengths that are integers can be tiled using 1 1 tiles.

Simple Tilings with Nonsquare Tiles

Some, but not all, shapes can be tiled using 1 2 tiles

To be completely covered with 1 2 tiles, a shape’s area must be even, but this is not enough

June 2007 Area and Tilings Slide 8

A chess board, or any rectangle with at least one even side, can be completely covered with 1 2 tiles

Covering a Chess Board with 1 2 Tiles

What if we remove two squares at opposite corners?

June 2007 Area and Tilings Slide 9

Tile a 4 6 rectangle using 1 2 tiles of two different colors. Try to find at least two tilings that look nice (have interesting color patterns)

Activity 3: Tiling with 1 2 Tiles

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Tile a 4 6 rectangle using L-shaped tiles that cover three squares. Is there more than one way to do this?

Activity 4: Tiling with L-Shaped Tiles

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Challenge: Try to come up with other ways of mixing 1 2 and 1 1 tiles

Some Possible 1 2 Tiling Patterns

Mixed with 1 x 1

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Some Irregular Tiling Patterns

Challenge: Try to come up with other interesting irregular tiling patterns

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Triangular, Hexagonal, and Other Patterns

These mixed hexagonal and pentagonal tiles don’t quite cover a flat surface area but . . .

June 2007 Area and Tilings Slide 14

Cut out a number of hexagonal and triangular tiles with sides of equal length (use paper of different colors) and use them to tile a square area

Activity 5: Mixed Triangular and Hexagonal Tiles

June 2007 Area and Tilings Slide 15

Two-Color Tiles

June 2007 Area and Tilings Slide 16

Multicolor and Patterned Tiles