Foundations of Algebra and Geometry
Chapter 3 Answers
Drawings and Patterns
Symmetry is very pleasing to the eye. It represents beauty as well as order. Use the Internet to investigate symmetry in art, architecture, and biology.
Part A, Symmetry
1. Examine and print the pictures at the following sites:
a. Draw the lines of symmetry on each picture. If the picture has more than one line of symmetry, draw all the lines.
b. How many lines of symmetry does each picture have? [2, 1, 1]
2. What is the most number of lines of symmetry a picture could have?
[A picture, such as a circle, could have infinitely
many lines of symmetry.]
3. How do you know where to draw a line of symmetry?
[Answers may vary.]
Part B, Tessellations
4. People have used tessellations in art, architecture, clothing, and pottery for hundreds of years. Explore some uses of tessellations in architecture.
a. Tour the Alhambra in southern Spain. Click on the camera icons to see the pictures.
b. Which places show examples of symmetry and tessellations?
[All but the gardens have symmetry. Patio de los Leones and Sala de las dos Hermanas have tessellated regions.]c. Explore geometrical patterns in Islamic architecture and design. Click on each picture and describe the geometrical principles being used.
[Most of the pieces are symmetric, many of them have tessellations.]
Part C, Making Connections
5. Review tessellations at the MathForum
math site. Name 3 polygons that tessellate.
[Triangle, square, hexagon]
Part A, Slides, Flips, and Turns
1. View at least three Tessellations made by other students. Print out a tessellation you like and describe it. What kind of transformations did the student use?
2. Create your own tessellation at the Tessellation tutorial site
a. Make your tessellation by cutting out a shape and tracing it.
b. Explain how you made your tessellation. Show the shape you started with, the shape you used to tessellate (if it is different), and what transformations you used.
Part B, Translations
3. Look at M.C. Escher's Fish
and Boats. Explain how the art uses translations.
[All the fish are the same. The boats are all the
same. Each fish and each boat is a slide of one original image.]
Part C, Reflections
4. Examine M.C. Escher's Horsemen
tessellation. Explain how the art work uses transformations.
[There are two transformations in the picture. The horsemen are going both
toward the right and the left. This indicates reflection. There is also
a translation. The dark horses are shifted horizontally in front of each
other going to the right. The light horses are placed diagonally downward
from the dark horses and are shifted horizontally in front of each other
going to the left.]
5. Review translations at the Swarthmore site . Categorize all capital letters in one of four categories:
Category 1: letters that look the same when reflected across a vertical line
Category 2: letters that look the same when reflected across a horizontal line
Category 3: letters that look the same when reflected across both a vertical line and when reflected across a horizontal line
Category 4: letters that do not look the same after they are reflected across either a vertical line or across a horizontal line
[Category 1 -- A, M, T, U, V, W, Y; Category 2 -- B, C, D, E, K; Category 3 -- H, I, O, X; Category 4 -- F, G, J, L, N, P, Q, R, S, Z]
Part E, Making Connections
6. Explore the life and works of the Dutch Painter, M.C. Escher. Print out 3 pictures you like and explain the symmetry, transformation, geometry, etc. used in each piece.
Part A, Finding Patterns
1. Examine the windows on the side of the TransAmerica building facing you.
a. Make a table that shows the number of windows on each floor for the top 7 rows.
Floor Number of Windows
1 8
2 8
3 9
4 9
5 10
6 10
7 11
b. What is the pattern?
[Two rows are the same then increase by one for the next two rows.]c. Assuming this pattern continues, fill in the table for the next 20 rows.
Floor Number of Windows
8 11
9 12
10 12
11 13
12 13
13 14
14 14
15 15
16 15
17 16
18 16
19 17
20 17
21 18
22 18
23 19
24 19
25 20
26 20
27 21
Part C, Making Connections
2. Look at the arrangement of cells in a Bee hive.
a. Bees make honey combs from a pattern of cells shaped like hexagons. Start at any cell and count the number of cells around it. Let that be the first "circle." For the second circle, count the number of cells about the first circle. Make a chart showing the number of cells in the first 10 circles.
Circle Number of Cells
1 6
2 12
3 18
4 24
5 30
6 36
7 42
8 48
9 54
10 60
b. What is the pattern?
[Multiples of six]c. What is the rule for the nth circle?
[6n]