1. A Lesson in the “Math + Fun!” Series
Area and Tilings
A Lesson in the “Math + Fun!” Series
June 2007
Area and Tilings

2. About This Presentation
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 ( ). “Math + Fun!” material can be used freely in teaching and other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami
Edition
Released
Revised
First
June 2007
June 2007
Area and Tilings

3. Finding the Area of a Geometric Shape
Circle
Area = (p/4)  Diameter2
= p  Radius2
Square
Area = Side2
Area = Width  Height
Rectangle
Triangle
Area = Base  Height / 2
Area = Base  Height / 2
Triangle
The area of a circle is about 80% of the square that encloses it
The area of a triangle is half that of a rectangle that encloses it
June 2007
Area and Tilings

4. Activity 1: The Area of a Triangle
On a sturdy piece of cardboard, draw a 4”  6” rectangle
Place thumbtacks or pushpins into the two lower corners
Put 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

5. Measuring the Area of an Irregular Shape
Method 1: Approximate the irregular shape by a regular one
Method 2: Cover with 1  1 tiles; count whole tiles and half of broken ones
June 2007
Area and Tilings

6. Activity 2: Tiling an Area with Square Tiles
15” or more
Draw a large irregular area on a piece of cardboard or construction paper
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

7. Simple Tilings with Nonsquare Tiles
Any shape with right angles and side lengths that are integers can be tiled using 1  1 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

8. Covering a Chess Board with 1  2 Tiles
A chess board, or any rectangle with at least one even side, can be completely covered with 1  2 tiles
What if we remove two squares at opposite corners?
Because each tile covers one black and one white square regardless of how it is placed, the figure on the right cannot be tiled (it has 32 black squares and only 30 white squares).
June 2007
Area and Tilings

9. Activity 3: Tiling with 1  2 Tiles
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)
June 2007
Area and Tilings

10. Activity 4: Tiling with L-Shaped Tiles
Tile a 4  6 rectangle using L-shaped tiles that cover three squares.
Is there more than one way to do this?
June 2007
Area and Tilings

11. Some Possible 1  2 Tiling Patterns
Mixed with 1 x 1
Challenge: Try to come up with other ways of mixing 1  2 and 1  1 tiles
June 2007
Area and Tilings

12. Some Irregular Tiling Patterns
Challenge: Try to come up with other interesting irregular tiling patterns
June 2007
Area and Tilings

13. 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

14. Activity 5: Mixed Triangular and Hexagonal Tiles
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
June 2007
Area and Tilings

15. Two-Color Tiles
June 2007
Area and Tilings

16. Multicolor and Patterned Tiles
June 2007
Area and Tilings