Geometry
Chapter 1 Answers
Visual Thinking and Mathematical Models
When you look at any model of the earth, it is important to be able to precisely describe any location on its surface. To do this, cartographers (map-makers) devised a system of longitude and latitude lines.
Part A, Geometric Models
1. Go to the University of Tennessee's Science bytes site to find an explanation of latitude and longitude.
a. Draw a picture of the Earth with lines of latitude and longitude. Label the lines.
Lines of latitude.
Lines of longitude.
b. How many degrees of latitude are there? [180]
c. What is the 0 degrees line of latitude called?
[The equator]d. How many degrees of longitude are there? [360]
e. What is the 0 degree line of longitude called?
[The prime meridian]
Part B, Algebraic Models
2. Use the List of cities in the world to find the positions of some cities on the globe. (Note: Negative latitude means south and negative longitude means west.)
a. Find the latitude and longitude of the city or town in which you live.
b. Find the latitude and longitude of Sydney, Australia. [33.91660 degrees south, 151.28330 degrees east]
c. Find the latitude and longitude of Shannon, Ireland. [52.70000 degrees north, 8.91700 degrees west]
Part C, Making Connections
3. Which are always the same distance apart, lines of latitude or lines
of longitude? Explain.
[Lines of latitude are always the same distance
apart. Lines of longitude get closer together near the poles as the distance
around the earth gets smaller.]
Logic and reasoning are important skills in mathematics and in life. Knowing
how to reason will help you find to the answers to many questions, even
if you don't have all the information you need to figure out the answer
directly. Learn how scientists use logic and reasoning on the Internet.
Part A, Inductive Reasoning
1. Learn how bacteria grow at the Cells alive site.
a. According to the site, how long does it take for one cell to split into two cells? [about 20 minutes]
b. Start with two cells. Make a table showing the number of cells you will have after 0, 20, 40, 60, 80, 100, 120, and 140 minutes.
Time period
0
1
2
3
4
5
6
7
Number of minutes
0
20
40
60
80
100
120
140 Number of cells
n2
4
8
16
32
64
128
256
c. Use inductive reasoning to determine the number of cells after 4 hours. [81
d. Find a formula that relates n, the number of cells, to t, the number of 20-minute time periods that have passed.
[n = 2^ (t +1)]e. Do you think this growth can continue indefinitely? Explain why or give some potential problems with it.
[The population cannot grow indefinitely. Eventually the bacteria will use up their food supply and the population will level off or decrease.]
Part B, The Language of Logic
2. Learn how penicillin kills bacteria at the Cells alive site.
a. Complete the statement below:
All bacteria are killed by __________. [penicillin.]b. Describe a counter example to your statement in a.
[A counter example would be a statement about one form of bacteria that is not killed by penicillin.]c. For the statement "All X are Y," if you find only one X that is not Y, have you proved the statement to be false? [yes]
d. If not all bacteria were killed by penicillin, how could you rewrite the statement in a to make it true?
[Possible answers include: all X are not Y.]
3. Read about some of the different places that bacteria and other organisms have been found in the Why Files. Use the information you find here to write three statements using the terms all, some, and none.
Geometry is everywhere! Learn how to measure segments and angles and to categorize and find costs to mail letters and postcards.
Part A, Measuring Segments
1. Read the size restrictions that the U.S. Postal Service has on the postcards it will accept.
a. What are the maximum acceptable length and height of a postcard?
[The maximum acceptable size is 6 inches long, 4.25 inches high, and 0.016 inch thick.]b. Draw a postcard that is the maximum acceptable size. [Check students' work.]
c. Measure the length of the diagonal.
[The length should be approximately 7.3 inches.]
Part B, Measuring Angles
2. Use your protractor to help you find the measure of each angle of
the postcard you drew in Chapters 1-3, Part A, 1b.
[90 degrees, 55 degrees, 35 degrees]
Symmetry is found in the art of many different cultures.
Explore some of the different symmetries you can find in art.
Part A, Symmetry
1. Different types of line symmetry frequently appear in quilts.
a. Look at pictures of two Amish quilts. Sketch the quilts and draw the lines of symmetry in both.
Answer:
b. Look at a picture of a Late 18th century quilt. Does this quilt have line symmetry? If so, sketch the quilt with its lines of symmetry. If not, explain why it does not have line symmetry.
[No. There is repetition of a pattern, but no line symmetry.]c. Look at a Picture of a quilt. (Ignore the applique cover in the bottom right.) Does this quilt have line symmetry? If so, sketch the quilt with its lines of symmetry. If not, explain why it does not have line symmetry.
Answer:
Part B, Reflections
2. M.C. Escher is famous for the way he incorporated geometry into his art.
a. Look at Escher's Drawing Hands . What kind of symmetry does this picture exhibit? Explain.
[Rotational symmetry. If you rotate this picture 180° it will look the same.]b. Look at the picture at the Escher's Snakes Web site. What kind of symmetry does this picture exhibit? Explain. [Rotational symmetry. If you rotate this picture 120° or 240° it will look the same.]
Part C, Properties of Reflections
3. Mirrors provide another interesting way to look at reflections.
a. Describe what is happening in Escher's Magic Mirror.
[The image on one side of the mirror is essentially a reflection of what is happening on the other side of the mirror. However, some of the animals appear to be split in half by the mirror.]
b. The ball on the right side of the picture appears to be about 3 tiles away from the mirror. If the image on the left of the mirror is a reflection of the image on the right, how many tiles is it from the mirror? [three]
c. How did you get your answer to b?
[Since the image is a reflection, the same point is the same distance from the mirror on both sides.]