EXPLODING NUMBERS
Grades 9 - 12

This lesson will explore the concept of exponential growth. The concept is introduced using a paper folding activity, and developed using the models of population and money growth. Patterns are explored and a definition for exponential growth is derived. Various doubling times are discussed. Graphs of exponential growth situations will be introduced in the video and expanded on graphing calculators. Extensions will explore the rule of 72, comparisons of linear and exponential growth situations, and the writing of a story using exponential growth as its basis.

BILL NYE THE SCIENCE GUY: Populations #207
MATH TALKS: Soaring Sequences, Episode #108

Students should be able to:

• understand and explain how quickly numbers increase when exponential growth is occurring.
• use a graphing calculator to compare exponential and linear growth.
• compare various growth rates, and watch the speed with which populations grow numerically.
• compare the various ways to model exponential growth on 2 types of calculators, scientific and graphing.
• interpret a growth situation, determine an equation for the situation, and draw a graph of it.

• newspaper - one large sheet per student
• graph paper- 8 1/2 by 14 works quite well
• graphing calculators
• scientific calculators
• rulers-one per student
• lesson worksheet-one per student

Exponential - an algebraic expression containing an exponent.
Exponent - the small raised number in an exponential, shows how many times to use the base as a factor.
Base - the large number in an exponential, used as a factor.
Linear equation - an equation with a degree of one, usually in two variables having the form y = mx + b, where y is the total, m is the rate of change and b is the starting quantity.
Exponential equation - an equation having two variables and a degree of more than one, having the form y = bgx, where y is the total, b is the starting quantity, g the growth rate, and x the number of times it grows.

Distribute one large sheet of newspaper and the paper folding worksheet to each student. Tell them that they are going to fold the paper in half 50 times. Ask them to write down their estimate for the thickness of the newspaper after these folds. Instruct them to begin folding the newspaper, while recording the results as they go. It will take a few folds before they can get an approximate measurement of the thickness. Find a value the class agrees upon, and have all students use this number as the beginning of the thickness column. Also, after about 7 folds, they will be unable to fold it anymore. Once they realize this, have them complete the chart by discovering the patterns and continuing them. Have them answer the questions at the bottom of the chart. Discuss the findings of the activity, the patterns that they discovered, and determine an estimate for the original question of the thickness after 50 folds.

Distribute "Notes From Videos" worksheet. To give students a specific responsibility while viewing, explain that they will watch 4 video clips that are examples of rapid number growth. Tell the students that each clip will focus on a different idea, and that they will be taking notes on their worksheets to identify these.

Begin with the Bill Nye video at the "consider the following" segment, dealing with penny populations. This is a few minutes into the video following the weed problem. Tell the students to watch the first section to determine how long it takes the human population to double, and be able to give an estimate the population of the world in 20 years. START the video, and watch until the penny population demonstration is completed.

STOP the video. Discuss the rapid growth of the examples in Bill Nye's penny population. Have students complete questions for Clip #1 on the worksheet. EJECT the video.

INSERT the Math Talks video which has been set at the beginning of the talk show segment. Tell students to listen for a mathematical term that defines the rapid growth in this clip. START the video.

PAUSE when the musician Paul says "exponential growth", and ask students for a definition of the term. Help them formulate a definition for exponential growth that includes a base and an exponent, in this case the base is 2, and the exponent changes or varies and is thus named x. Have students get their scientific calculators ready and RESUME the video.

PAUSE when David Numberman finishes the calculator demonstration of the waitress' salary. Instruct the students to repeat the steps as a class on their calculators. They could even count the exponent out loud as it increases. Emphasize that the exponent changes for each day, and the base remains at 2, which is the doubling rate for this situation.

FAST FORWARD past the "Infinity Song" and the cartoon commentators to Fax Headfull. Instruct the students to watch this clip for a graph showing the growing population of the earth. RESUME the video.

PAUSE after Fax explains how long it would take to count all the people in the world, and have students answer the question on the worksheet. RESUME the video.

PAUSE the video when Fax begins the bar graph of the population, after the second bar goes up at 10.6 billion people. Use an overhead marker and draw a line on the TV screen joining the two points at the top of each bar. Comment that the line is straight, and ask the students if they think that this trend will continue. RESUME the video.

PAUSE when the graph is complete at 42.4 billion. Extend the graph on the screen, and ask what differences they see in this line as opposed to the linear graphs they are familiar with. Have students notice the scale at the side of the graph, and determine the doubling time for the human population. RESUME and watch to the end of this segment as students finish part 3 of the worksheet. STOP the video and EJECT.

RE-INSERT the Bill Nye video, at the place where it was previously stopped. Have the students focus on the doubling time of the bacteria in their stomachs. START the video. STOP when Bill offers the audience some of his yogurt cone. Ask what the doubling time for bacteria is, and how much bacteria the students have living in their stomachs. Instruct students to answer the questions on the worksheet.

To tie all aspects of the four clips together, pass out the graphing calculators, and instruct students to complete Part III of the worksheet. (This could be completed on graph paper if graphing calculators are not available.) Have the students get the calculators ready to graph equations by clearing all previous equations and setting the windows to a standard window. Have them complete the next section of the worksheet as the teacher demonstrates on the overhead, or alone if students have quite a bit of experience with the graphing calculators. This section will explore varying growth rates.

Have students contact the local Chamber of Commerce and obtain data relating to the population growth of their community. This could go back 100 years or so. First, have them graph the data and try to determine if the growth is linear or exponential. Then, have them find the current growth rate, and have them graph the population growth until the population doubles.

Computer chip sizes are shrinking at an incredible rate. Invite a computer engineer to come and discuss the processes involved in the manufacture and engineering of computer chips. Have them explain why this trend cannot continue for much longer. This is an example of exponential decay.

Obtain a pair of fruit flies from a biological supply company for your classroom. Allow them to reproduce, and watch as a saturation point is reached when most die. The process will then repeat again.

Invite a biologist to come to class and discuss the reasons why some species are endangered. He could address specific species in the region that are endangered and what local efforts are being made to remedy the problem.

Complete an investigation into how long it would take to double an investment using various growth rates. Good rates to use are factors of 72; 2, 3, 4, 6, 8, 12, 18 and 36. Begin with an investment of $100 for example. All on the same graph, (a 8 1/2 x 14 size works very well), plot the investment for each growth rate. Once they have found the length of time it takes at each specific rate to double, lead them to discover the 'Rule of 72' by drawing a table relating growth rate to doubling time. This rule used in the financial world states that when the investment rate and the investment time have a product of 72, the investment has doubled. Read The Rajah's Rice to the students. This is a story similar to the waitress example in the video, except the location is ancient, and rice is the commodity being doubled. Students could then write a story using their own characters in a situation where doubling occurs.

Have the students compare exponential and linear growth in various settings. This can be completed on a graphing calculator, or on graph paper. (See the accompanying worksheet.)

Can be taped off air. Contact your local PBS station for broadcast schedule.
Lesson plan developed by Kit Parker, South Junior High School, Boise, Idaho