**Body Double**
Introductory Activities
Learning Activities
Culminating Activity
Cross-curricular Extensions
Community Connections
**Introductory Activities
**
Distribute the

"Penny pincher" handout for students and 100 pennies to each student. ("Penny pincher"

answers are also provided.) Ask students to determine how many days it will take to consume all the pennies if five pennies are saved everyday. (

*Students should quickly identify that it will take 20 days to consume all the pennies in 5-penny increments*.)

Ask students to determine how many days it will take to consume the pennies if they double the number of pennies each day starting with one penny on day one, two pennies on day two, four pennies on day three and so on. (

*Students will run out of pennies by day 7-having only 37 of the required 64*.)

Assign students to 2-person working groups. Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to consider how doubling growth, as done in the previous exercise, compares to linear growth by 5-penny increments; which growth pattern will consume the pennies first?

CUE Cyberchase episode to the point at which you see Hacker and Cybersquad racing to the top of the mountain. Hacker will be standing on top of a large blue block; the Cybersquad will be standing on top of a narrow red block. PLAY tape. PAUSE tape when Cybersquad reaches the top of the mountain. Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to explain why the Cybersquad beat Hacker to the top of the mountain? (

*Even though they started with only one block, because they were doubling each growth cycle, it only took four growth cycles to reach the top. The same pattern could be observed with the pennies*.) REWIND tape if students need to review the explanation.

**Learning Activities
**
A function (by definition) is any relationship that links a domain to specific values in a range. Two important types of function are linear functions and exponential functions. The relationships in linear functions are direct relationships-as one variable increases, the other variable increases; as an independent variable increases (or decreases), the dependent variable follows the same trend. On the chalkboard, create a data table similar to the one shown here. Explain to students that this is a helpful way of organizing information when being collected. Ask students to create a data table that could coincide with their "penny pincher" stacks.

Explain to students that x and y are called variables because they represent numbers. If the value of one variable depends on the other variable, we have a function. The terms domain and range are also useful for describing x (the variable representing the domain) and y (the variable representing range).

Log onto the Cyberchase companion Web site game page. Choose the

"Double the Donuts" game. Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to observe what happens as the number of donuts is doubled with each click and determine how many donuts will the dragon have to eat at the end of 10 growth cycles. (

*Students will notice that the value increases very quickly. At the end of 10 cycles/clicks, the dragon will have 1024 donuts! *)

Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to observe the relationship between the number of clicks and the number of donuts. Is the function additive (related to addition) or multiplicative (related to multiplication)? (

*The number of donuts is multiplicative*.)

Cover the screen of the computer with a blank sheet of paper. Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to predict how many donuts the dragon will have after 13 clicks. (8192)

**Culminating Activity
**
Distribute two sheets of graph paper to each student. Ask students to graph the following functions:

- y = x
- y = x + 2
- y = x + 5
- y = 2x
- y = 2x

Graph functions A, B and C on one set of axes; graph functions C and D on a separate axis system.

Ask students to compare their results to a neighbor's results. Distribute the

"Growth patterns" handout. Ask student to compare their results to those generated using graphing software (Microsoft Excel). Explain to students that graphs are a meaningful way of organizing data just like tables are useful for organizing data. Remind students that tables are useful for showing the relationship between two variables-what function operates between two variables.

The equations written above are statements of the functions; they will help predict "future" values. Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to consider how to make predictions using graphical data when there is no equation. Log onto the

Population growth Web site. Ask students to determine the population of the Southern Region of the United States in 1962. (

*56 million is a reasonable estimate from the graph. This strategy is called interpolation because the student used information available on the graph to determine a value*.)

Provide students with a FOCUS FOR MEDIA INTERACTION by asking them to predict in what year the Southern U.S. regional population would reach 90 000 000. (

*1993 would be a reasonable estimate from the graph. This strategy is called extrapolation because the student must predict a value that extends beyond the information provided by the graph*).

**Cross-Curricular Extensions**
**Science**
Patterns are ubiquitous in nature. Explore the interconnectedness of math with science by studying the Fibonacci Series. Use the Fibonacci numbers and Golden Section in nature Web site as an interactive reference site for both.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/Rabbits (Quicktime player helpful)

**Technology-graphing**
Use spreadsheet creation software or a graphing calculator to create graphs of various functions. Use linear functions and exponential functions as the basis for graph development. Use methods of interpolation and extrapolation to predict values not part of your original domain/data set.

**Physics-parabolic pathways and falling**
Use the "Monkey jump" interactive web page at

http://www.explorescience.com/activities/Activity_page.cfm?ActivityID=32 to predict the velocity that will be required to catch a ball tossed to a falling monkey at different distances. Determine the minimum velocity with which the ball must be thrown in order to reach the monkey. Consider what happens when the velocity is halved or doubled.

**Community Connections**

**Internet safety/media analysis**

Log onto http://www.census.gov/prod/2002pubs/censr-4.pdf to obtain published census data. Go to page 156 (of 222) and compare the information provided by the United States Department of Commerce to that provided by the Population growth web page. Determine which data set is more reliable. Make sure to find out from where the "population growth" data were obtained. What is the probable explanation for the disparity in data?