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Recursive Routines with Repeated Multiplication

On this web page, you will investigate the patterns created by recursive routines that use repeated multiplication. This exploration will reinforce the concepts in Lesson 7.1 on pages 367–373 of Discovering Algebra: An Investigative Approach.

Sketch

You can study a recursive routine with repeated multiplication using the sketch below. Adjust the starting value by dragging point S on the green slider. Adjust the percent increase by dragging point P on the blue slider. You can adjust the scale on the y-axis by dragging point B. Click Step to show the effect of the percent increase after one time interval. Click Start Over to reset everything. (You may have to click Start Over twice.)

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Investigate

  1. Imagine that a bug population has invaded your classroom. One day you notice 16 bugs. Every day, new bugs hatch, increasing the population by 50% each week. How many bugs will there be after one week? After two weeks?
  2. In the sketch above, adjust the scale, set the starting value and the percent increase, and then see how many bugs there are after one week, two weeks, and so on. How does the slope change from point to point? Does the rate of change show a linear pattern? Why or why not?
  3. How does this growth pattern differ from the pattern of growth created from repeated addition?
  4. If the bug population increases by 50% each week, what is the constant multiplier for the bug population?
  5. Write a recursive routine that shows the growing numbers of bugs.
  6. What other situations are there in which growth takes place in this way?
  7. See what happens for different percents of increase and different starting values. (Try dragging S or P after you have several points plotted.) What characterizes this kind of growth?
  8. What do you think would happen if the percent increase were more than 100%?

Sketch

The next sketch shows another recursive routine that uses repeated multiplication, but this one shows a decreasing pattern. Drag sliders S and P to adjust the starting value and the percent decrease. You can adjust the scale on the y-axis by dragging point B.

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Investigate

  1. Click Step several times to plot points, then drag S and P. How is this pattern different from the one for a percent increase? How are the patterns similar?
  2. Many decreasing patterns, like cooling liquids and decay of substances, can also be described by repeated multiplication. If the next stage is less than the present stage, what must be true of the number that you multiply the first stage by in order to get the next stage?
  3. If you multiply by 0.7 each time to get the next number in the sequence, what percent decrease is that? Use the sketch to check your answer.
  4. For any starting value S and any percent decrease P, explain how you could write a recursive routine for the sequence of of values.
  5. What happens for most starting values and most percent decreases in the long run?
  6. Drag slider P past 100%. What happens when the decay rate is more than 100%? Would this pattern make sense in a real-world situation? Explain why or why not.

Sketch

Use the sketch below to compare exponential growth (from repeated multiplication) and linear growth (from repeated addition). Adjust the starting value by dragging point S on the green slider. Adjust the percent increase for the exponential growth by dragging point P, and adjust the amount added for the linear growth by dragging point A. You can adjust the scale on the y-axis by dragging point B. Click Step to show both patterns after one time interval.

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Investigate

  1. Compare exponential growth and linear growth using different starting values, different percents of increase, and different amounts added. What similarities do you notice? What differences do you notice?
  2. When is linear growth faster than exponential growth? When is exponential growth faster than linear growth?
  3. How do the percent increase and the amount added affect which kind of growth is faster?
  4. Do you think exponential growth will always be able to outgrow linear growth? Why or why not?