Lesson 6.1 Recursive Routines In this lesson you will ● explore patterns involving repeated multiplication ● write recursive routines for situations involving repeated multiplication ● look at tables and graphs for situations involving repeated multiplication

Lesson 6.1 Recursive Routines Essential Questions ●Are all rates of changes constant? ●Can we write recursive routines for repeated multiplication? ●What do the tables and graphs for situations involving repeated multiplication look like?

Recursive Routines Investigation: Bugs, Bugs, Everywhere Bugs 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. So, in the first week the population increases by 8 bugs.

Recursive Routines Investigation: Bugs, Bugs, Everywhere Bugs Step 1: In a table like this one that you have drawn in your notes, record the total number of bugs at the end of each week for four weeks. (Complete Column 2) Remember: We started with 16 bugs, and increase by 50% each week Weeks elapsed Total number of bugs Increase in number of bugs (rate of change per week) Ratio of this week’s total to last week’s total Start (0) 1 2 3 4 16 24 8 36 12 54 18 81 27

Step 2: The increase in the number of bugs each week is the population’s rate of change per week. Calculate each rate of change and record it in your table. Does the rate of change show linear pattern? Why or Why not? Complete step 2 (column 3 in your table), then below the table you have completed, Label step 2 and answer the questions in complete sentences that restate the question as part of the answer. Example: Step 2: The rate of change (does/does not)show a linear pattern. Now justify your answer.

Step 3: 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 and describe how the slope changes from point to point. ● ● ● ● ●

Step 4: Calculate the ratio of the number of bugs each week to the number of bugs the previous week, and record it in the table in column 4. Divide the population of each week by the previous week. Example: Week 1 /Week 0 yields 16/24 = 1.5 Repeat the process to complete your table. Now label step 4 below step 3 and answer the questions how do the these ratios compare? Explain what the ratios tell you about the bug population growth.

Step 5: What is the Constant Multiplier for the bug population Step 5: What is the Constant Multiplier for the bug population? (Again, write the step number and restate the question when you answer in a complete sentence). Example: The Constant Multiplier or number each term is multiplied by to get the next term is _________. How can you use this number to calculate the population when 5 months have elapsed? Step 6: Model the population growth by writing a recursive routine that shows the growing number of bugs. (Calculator Note 3A) Describe what each part of this calculator command does.

Step 7: By pressing a few times, check that your recursive routine gives the sequence of values in your table (in the column “Total number of bugs”). Use the routine to find the bug population at the end of weeks 5 to 8. Step 8: What is the bug population after 20 weeks have lapsed? After 30 weeks have lapsed? What happens in the long run? ENTER

In the investigation you found that repeated multiplication is the key to growth of the bug population. Populations of people, animals, and even bacteria show similar growth patterns. Many decreasing patterns, like cooling liquids and decay of substances, can also be described with repeated multiplication. Example A: Maria has saved $10,000 and wants to invest it for her daughter’s college tuition. She is considering two 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?

+ interest (balance X interest rate) Plan A Plan B Year Current balance + return New balance 1 10,000 550 10,550 2 3 4 Year Current balance + interest (balance X interest rate) New balance 1 10,000 10,000 X .05 10,500 2 3 4 Write a recursive routine to do this on your calculator for Plan A. Write a recursive routine to do this on your calculator for Plan B. Note: (10,000 + 10,000 x .05) can be written in factored form as 10,000(1+.05)

+ interest (balance X interest rate) Plan A Plan B Year Current balance + return New balance 1 10,000 550 10,550 2 11,100 3 11,650 4 12,200 Year Current balance + interest (balance X interest rate) New balance 1 10,000 10,000 X .05 10,500 2 10,500 X.05 11,025 3 11,025 X.05 11,576 4 11,576 X.05 12,155 Plan A. {0, 10000} {Ans(1)+1, Ans(2)+550} , ,….. Plan B. {0, 10000} {Ans(1)+1, Ans(2) •(1+.05)} ENTER ENTER ENTER ENTER ENTER ENTER ENTER ENTER

Graph the results for plan A and B on a graph like the one below.

Example B: Birdbaths at the Feathered Friends store are marked down 35%. What is the cost of a bird bath that was originally priced $34.99? What is the cost if the birdbath is marked down 35% a second time.

Examples 1. Give the starting value and constant multiplier for each sequence, then find the 8th term. 3, -6, 12, -24, 48, …. 2. Use a recursive routine to find the first five terms of a sequence that starts with 32 and has a constant multiplier of .25, then find the 10th term. 3. Write a 5% increase percent of change as a ratio comparing the result to the original quantity, then write it as a constant multiplier.

Practice, Your Turn 1. Give the starting value and constant multiplier for each sequence, then find the 8th term. 192, 96, 48, 24, 12,…. 2. Use a recursive routine to find the first five terms of a sequence that starts with 10 and has a constant multiplier of 2.2, then find the 10th term. 3. Write a 5% decrease percent of change as a ratio comparing the result to the original quantity, then write it as a constant multiplier.