4.1+Making+Smaller+Ballots


 * __

JSM __** **Notes from Class: Vocabulary Exponential Decay -** A pattern of decrease in which each value is found by multipying the previous value by a constant factor greater than 0 and less than 1. **4.1 Making Smaller Ballots ** **A. The sheet of paper Alejandro started with had an area of 64 in². Copy and complete the table below to show the area of a ballot after each of the first 10 cuts.** a.
 * Big Idea: //The rate at which many things grow or decay can often be described mathematically.// **
 * Essential Question: //How can I show a problem of repeated division?// **
 * Decay Factor -** The constant factor that each value in an exponantial decay pattern is multipied by to get the next value. 
 * Investigation 4: Exponential Decay**
 *  __Problem 4.1__ **
 * Cut a sheet of paper as Alejandro did, and look for a pattern in the area of the ballots after each cut. Use what you discover to help you complete this problem. **
 * = **Cuts** ||= **Area (in^2)** ||
 * = **0** ||= **64** ||
 * = **1** ||= **32** ||
 * = **2** ||= **16** ||
 * = **3** ||= **8** ||
 * = **4** ||= **4** ||
 * = **5** ||= **2** ||
 * = **6** ||= **1** ||
 * = **7** ||= **1/2** ||
 * = **8** ||= **1/4** ||
 * = **9** ||= **1/8** ||
 * = **10** ||= **1/16** ||

**b. The area of a ballot decreases by half with each cut. c. The pattern of change in area is different from the exponential growth patterns, because the pattern decreases with each cut. The pattern of change in area is similar from the exponential growth patterns, because they have constant factor that the value is multiplied to each time. **uno.
 * B. How does the area of a ballot change with each cut?
 * C. How is the pattern of change in the area different from the exponential growth patters you have seen in this unit? How is it similar? **
 * __Problem 4.1 Follow-Up__ **
 * 1. Copy and complete the table below to show the area of a ballot at each stage.
 * = Stage ||= Area (cm^2) ||
 * = 0 ||= 27 ||
 * = 1 ||= 9 ||
 * = 2 ||= 3 ||
 * = 3 ||= 1 ||
 * = 4 ||= 1/3 ||
 * = 5 ||= 1/9 ||
 * = 6 ||= 1/27 ||

**dos. The area of a ballot is multiplied by 1/3 with each stage. tres. A=27(1/3^//n//) or A=27(1/3)n quatro. x=stage, y= area, title: Area of a Ballot After Each Stage
 * 2. How does the area of a ballot change at each stage?
 * 3. Write an equation for the area, A, of a ballot at any stage, n.**
 * 4. Make a graph of the (stage, area) data from your table.**