Content Standards
Algebra, Functions

Process Standards
Communication, Reasoning, Connections

School: Capuchino High School; 964 students
Location: San Bruno, California
Teacher: Chicha Lynch
Years Teaching: 15
Students in Classroom: 32
Grade: 10

Alice to the Moon

Video Overview

In groups, students discuss their homework on using large numbers to solve four problems. Different groups then present their work for solving each homework problem. Ms. Lynch asks students to write about how large number skills fit the metaphor of Alice growing in Wonderland and if scientific notation would be important for Alice. Students share their writing in groups and then groups present their writing to the class. At the end of the video, Ms. Lynch presents homework, which involves exponents that are not integers. Students must figure out what power of 10 will get Alice to the moon, which is 239,000 miles from Earth.

The Class Challenge

Students review two problems from the previous night's homework:

1. How many inches to a light year?
   
2. How many Earths put together equal the mass of the sun?

Students then apply this work to the Alice in Wonderland metaphor.

Course and Curriculum

This required course, which covers algebra, trigonometry, and discrete mathematics, is the second-year course in the four-year Integrated Mathematics Program (IMP) core curriculum. This lesson focuses on exponential notation and is part of an extended lesson in the exponents unit based on Lewis Carroll's Alice in Wonderland.

In this unit, students learn basic principles of working with positive, negative, zero, and fractional exponents; logarithms; scientific notation; and manipulating numbers written in scientific notation. Prior to this unit, students studied the Pythagorean theorem, trigonometry, area, perimeter, and volume. Following the lesson, students worked with logarithms for computational purposes, and used the TI-82 graphing calculator log key and a "guess and refine" method for finding logs on a calculator. In the next unit, they studied graphing, systems of linear inequalities, and solving systems of linear equations.

In addition to emphasizing communication, writing, and cooperative learning, Ms. Lynch uses various assessment strategies and teaches students to use a variety of approaches to solve problems. To promote student collaboration, she places students in heterogeneous groups, tells them to ask one another questions before they ask her, and asks the groups to present their work to the class.

---

A Pre–Viewing Exploration for Teacher Workshops

Eruptions in the World of Mathematics
Objective: To calculate and estimate answers to application problems using numbers in scientific notation.
Materials: One scientific calculator for each teacher.

Activity
Solve this problem individually.
The fallout rate of each volcano varies. The table below gives the maximum eruption rate (cubic meters of fallout per second) of six volcanoes that erupted in the last century.

Volcano and date of eruption	Maximum eruption rate (cubic meters per second)
Santa Maria, 1902	4.0 x 104
Hekla, 1947	2.0 x 104
Bezymianny, 1956	2.0 x 105
Agung, 1963	3.0 x 104
Hekla, 1970	4.0 x 103
Ngauruhoe, 1975	2.0 x 103
Mount St. Helens, 1980	2.0 x 104

1. Use the information in the table to compare the total fallout if Mount St. Helens erupted at maximum rate for three hours and Bezymianny erupted at maximum rate for 1.5 hours. Which volcano had the greater total fallout?
2. What is the average rate for the two eruptions of the Hekla volcano?
3. Which volcano has the greatest eruption rate? Which has the smallest?
4. What is the ratio of the largest to the smallest eruption rate?
5. If a particle weighs 2.0 x 10^–5 grams, estimate the number of particles in a metric ton. (A metric ton = 2.0 x 10 ³ kilograms.)

Exploring Content

1. Discuss two ways to solve #5.
2. Create a scoring rubric for assessing student responses to questions 1 – 5. How would you weigh demonstration of conceptual understanding? The use of appropriate procedures? The expression of the results in verbal form?

Exploring Teaching Issues

1. What impediments do students encounter when they work with scientific notation? How can these impediments be minimized?
2. In what situations would you accept a student's approximation in lieu of an exact answer? When would you not accept an approximation?
3. What is the difference between "raising something to the power of 10" and "raising 10 to a power?" How would you help students understand this difference?

---

Exploration Answers (Pre–Viewing)

Activity

1. One hour = 3.6 x 10³ seconds. Mount St. Helens is 2.16 x 10⁸ seconds and Bezymianny is 1.08 x 10⁹ seconds, so Bezymianny has a greater total fallout.

2. 1.2 x 10⁴ cubic meters per second.

3. Bezymianny has the greatest eruption; Ngauruhoe has the smallest eruption.

4. 100:1.

5. A metric ton = 2.0 x 10³ kilograms. If 2.0 x 10^–5 grams = 2.0 x 10^–8 kilograms, then (2.0 x 10³)/(2.0 x 10^–8) = 10¹¹.

Activity

1. Answers will vary. One possible method is presented in the answer to #5. another possible method is to translate the scientific notation into expanded form, then multiply and divide decimal numbers.

2. Answers will vary depending on the teacher's assessment style and conceptual emphasis in the lesson presentation. Some possible answers are: Did the student interpret in scientific notation, or did she expand the numbers to find the answer? Did the student understand how to manipulate expressions in scientific notation or expanded form? Could the student describe verbally the process she used to solve the problem?

---

A Post–Viewing Exploration for Teacher Workshops

Growing Out of this World
Objective: To explore the relationship between exponential form and logarithmic form.
Materials: One scientific calculator and graph paper for each teacher.

Activity
Solve this problem individually.
Ms. Lynch tells her students that each time Alice eats an ounce of cake, her height increases by a power of 10. Then Ms. Lynch asks how much "base 10 cake" Alice must eat in order to stretch to the moon, which is 239,000 miles from Earth. How can you use this problem to help students understand the relationship between powers of 10 and logarithms in base 10? How can you find a number that will yield 239,000? (Assume Alice's height is represented by 10 to the zero power which equals 1.)

1. Write a mathematical expression that states the problem.
2. Draw another mathematical model of the problem on graph paper by graphing the function, h = 10^x where x is the number of ounces and h is the height, with ounces on the x–axis and height on the y–axis. (The actual function to represent Alice's growth is y = A * 10^x where A = Alice's original height. In this case we have assumed Alice's original height is 1.) Use a scale on the y–axis of 10,000 miles per unit, and one ounce per unit on the x–axis. Draw the graph for ounces = 1, 2, 3, 4, 5, 6. It will become clear that the exponent is going to be a nonintegral value between 5 and 6.
3. Use your calculator to determine a value of ounces between 5 and 6 to the nearest hundredth that provides a value closest to 239,000 miles.
4. Enter: LOG (239,000) in to your calculator. Here LOG refers to logarithms base 10. What is the result?
5. Now find the number of ounces of cake necessary for Alice to stretch to the planet Pluto, which is about 3.6 billion miles from Earth.

Exploring Content

1. What do you think is the relationship between the log and the exponent? Write a logarithmic equation that represents this problem.
2. How would you describe the growth of the function?
3. Explain the importance of the graphic representation of a function, such as 10^x = 239,000, to the definition of logarithm.

Exploring Teaching Issues

1. What examples could you give students to describe exponential growth?
2. Could your students do this same activity? Explain how you would structure the activity.
3. What are some appropriate questions for students which would facilitate calculator use to develop conceptual understanding and support skill development?
   

---

Exploration Answers (Post–Viewing)

Activity

1. What power of 10 equals 239,000?
2. 10^x = 239,000.
3. (5.1,125892), (5.2,158489), (5.3,199,526), (5.4,251188) (5.5,234422), (5.38,239883). Approximately 5.38 ounces of cake.
4. 5.37839701.
5. 9.5 ounces of cake.

Exploring Content

1. Log (239,000) = x (where x is the number of ounces).
2. The growth is not linear. With each successive power of ounces, the height grows at a greater rate.
3. It allows you to see that the logarithm is an exponent and has a unique value.

---

Topics for Discussion

Discuss other examples of exponential growth that might capture students' interest.

• What does "doubling in size" mean? How did the model Ms. Lynch used influence the students' understanding of thd concept?
• How could the models of doubling and increasing by 10 in each period be explicitly represented and distinguished from each other within the lesson?
• What examples of large numbers, such as the circumference of the earth, could help students comprehend the large numbers in the lesson? What other examples of exponential growth (or decay) might capture the interest of your students?
• How could this lesson be centered more around students' interests and questions?
• How could exponential growth be taught—using such devices as doubling, powers of 2, and scientific notation—to minimize stufent confusion?

What is the value of free–focus writing in this lesson?

• How does the free–focus writing assignment help to prepare students for the next lesson on logarithms?
• What do you think is the effect of telling students to write non–stop for five minutes and that the writing is not for publication?
• Ms. Lynch has an opportunity to learn from her students when they do free–focus writing, share their writing in groups, and report their observations in groups. How would you facilitate each of these settings in your classroom to assess your students?
• How would you use the student reporting of free–focus writing to make instructional decisions for this class and for individual students?
• Should a mathematics teacher teach and assess writing skills? Should written expression of mathematical concepts be expected of students in mathematics classes? What kind of support or training would you need to make written expression part of your mathematics class?

Why did Ms. Lynch make groups responsible for a group response?

• How did Ms. Lynch make the lesson accessible to students' different learning styles?
• Cite ways you might further promote cooperative work within the groups.
• How is the lesson structured for students to discover the rules of exponents?
• When is it appropriate to use formal and informal mathematical language? What are the risks and benefits of using each?

Home | Catalog | About Us | Search | Contact Us | Site Map © 1997-2009 Annenberg Media. All rights reserved. Legal Policy