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Lesson 6.1: Recursive Routines

Lesson 6.1: Recursive Routines. To begin to investigate geometric sequences using recursive routines. To see examples of growth and decay that can be modeled recursively. Have you noticed that it doesn’t take very long for a cub of steaming hot chocolate to cool to sipping temperature?

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Lesson 6.1: Recursive Routines

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  1. Lesson 6.1: Recursive Routines To begin to investigate geometric sequences using recursive routines. To see examples of growth and decay that can be modeled recursively.

  2. Have you noticed that it doesn’t take very long for a cub of steaming hot chocolate to cool to sipping temperature? • Have you also noticed that the temperature seems to stay the same for a very long time?

  3. Have you ever left your lunch in your locker for several days, then suddenly after a few days some mold appears and a few days later it’s completely covered with mold.

  4. These situations describe the same principle. • These patterns are different from the linear patterns we saw in the rising elevator and the shortening ropes (constant addition and subtraction) • Now we will investigate patterns that increase rapidly.

  5. Bugs, Bugs, Everywhere Bugs • Imagine that a bug population has invaded your classroom. • One day you noticed 16 bugs. • Every day new bugs hatch, increasing the population by 50% each week. • In the first week the population increases by 8 bugs.

  6. In a table, record the total number of bugs at the end of each week for 4 weeks.

  7. The increase in the number of bugs each week is the population’s rate of change per week. Calculate each rate of change. What are the units? Does the rate of increase show a linear pattern? Why or why not?

  8. Let x represent the number of weeks elapsed and let y represent the total number of bugs. Graph the data using (0,16) for the first point. Connect the points with line segments. Describe how the slope changes from point to point.

  9. Slope = 27 bugs per week Slope = 18 bugs per week Slope = 12 bugs per week Slope = 8 bugs per week

  10. Calculate the ratio of the number of bugs each week to the number of bugs the previous week. Record it in the table. How do the ratios compare? Explain what the ratios tell you about the bug population growth.

  11. What is the constant multiplier for the bug population? How can you use this number to calculate the population when 5 weeks have elapsed?

  12. Write a recursive routine that models the populations growth for the growing number of bugs. Describe what each part of this calculator command does. By pressing ENTER a few times, check that your recursive routine gives the sequence of values in your table. Use the routine to find the bug population at the end of weeks 5 to 8.

  13. What is the population after 20 weeks? After 30 weeks? What happens in the long run?

  14. Comparing Growths • Maria has saved $10,000 and wants to invest it for her daughter’s college tuition. She is considering to options. • Plan A guarantees a payment, or return, of $550 each year. • Plan B grows by 5% each year. • With each plan, what would Maria’s new balance be after 5 years? After 10 years?

  15. Plan A Write a recursive routine to do this on your calculator.

  16. Create a chart • Record your totals on the chart for Plan A for the first 20 years.

  17. The factor 1+0.05 is called a constant multiplier Plan B Write a recursive routine to do this on your calculator.

  18. Create a chart • Record your totals on the chart for Plan B for the first 20 years.

  19. Create a graph of Plan A vs. Plan B • Enter the data from the chart in your calculator. • Place the year in L1 • Balance for Plan A in L2 • Balance for Plan B in L3 • Create a graph of • L1 vs. L2 - Linear Growth • L1 vs. L3 - Exponential Growth • Compare the graphs. What statements can you make about the two graphs?

  20. The factor 1-0.35 is called the constant multiplier. Example • Birdbaths at the Feathered Friends store are marked down 35%. What is the cost of a birdbath that was priced $34.99? • What is the cost if the birdbath is marked down 35% a second time?

  21. Example • Would the cost be the same if the birdbath had been reduced by 70% one time? Why or why not?

  22. Constant Multipliers • Constant multipliers can be either positive or negative. • These two sequences have the same starting value, but one has a multiplier that is 2 and the other has a multiplier of -2. • 3, 6, 12, 24, 48,… • 3, -6, 12, -24, 48, … • How does the negative multiplier affect the sequence?

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