Superlesson
Project 10-1

Answers 10-1

Superlesson
Project 10-2

Answers 10-2

Algebra

Chapter 10, Answers
Functions and Structure of Algebra

Leonardo da Vinci's illustrations for the book De Divina Proportione by Luca Pacioli resemble the constructions we did in previous chapters. Let's explore!

Part A, Rational and Radical Expressions

1. Begin at the Golden Ratio Web site. Read this information and then select three of the sites highlighted in green. Working with your group, make notes on three key ideas from each site that you investigate.
[Answers will vary.]

2. Make a large poster to share your information with the class. Stress the mathematics of each site, including how the Golden Ratio is applicable.
[Answers will vary.]

Part C, Making Connections

3. Make a model showing the Golden Ratio. Begin by making a square with a length of 1 unit.

• Choose one side and extend its length to 2 units.
• Complete the rectangle with sides of length 1 and 2.
• Now complete a square on top of this rectangle with sides of length 2.
• Continue until you have constructed 7 squares.
  [Answers will vary.]

4. Begin with the inner most square. With a corner as a radius, construct a quarter of a circle in each square. Continue the spiral until you have a diagram of the nautilus spiral. How many Golden Rectangles can you find in your construction?
[Nine rectangles]

TOP

Growth and decay are two areas of science that depend on mathematics to make sense. Let's explore some additional functions!

Part A, Exponential Functions

1. Open the Median income Web site.

a. Make a chart showing the Age of the Householder and the Median Income at ages 24, 34, 44, 54, and 64.

AGE	MEDIAN INCOME ($)
24	20,979
34	34,701
44	43,465
54	48,058
64	38,077

b. Predict if the shape of a graph made with this information would be linear or exponential.
[Predictions will vary.]

c. Graph the information from above.
[We apologize for the inconvenience, but this graph is not yet available. Please check back soon.]

d. Is your graph linear or exponential? Explain.
[The graph is close to exponential. Population is an exponential function. At the age of 64, some people are dying. Therefore, the population is declining.]

Part B, Exponents and Logarithms

2. Open the Richter scale Web site.

a. Explain why this is a logarithmic function.
[The scale is logarithmic because each value on the scale is 10 times the value of the one before.]

b. Graph this function on your graphing calculator. Don't forget to change the scale of your graph. What do you notice?
[The graph goes up sharply...so sharply that it may appear to be a straight line. Explore by zooming in on the graph to see that it is not a linear function.]

Part C, Making Connections

3. Continue to use the information at the Richter scale site. Explain why an earthquake of Magnitude 8 is 100,000,000 times stronger than an earthquake of Magnitude 1.
[A Magnitude 8 is 10⁸ times stronger than an earthquake of Magnitude 1. Therefore, 10⁸ would require a one followed by eight zeros. Some students may solve this problem using logs and the formula M = log x divided by x₀.]

4. Would the principle paid on the monthly payment of a home mortgage be an example of linear or exponential growth? Explain.
[Home mortgage payments would be exponential. The payments on an individual loan will remain constant. However, the amount paid to interest each month will decrease, and the amount paid toward the principle will increase. This is reflected in the percent applied to the unpaid balance.]