## Recursive and Exponential Rules

In this lesson make connections between exponential functions and recursive rules. Students will use tables to create graphs, define recursive rules and find exponential formulas.

To begin the lesson, you can use the Recursive Rules Overhead to ask students to predict the next term in each sequence and write a recursive rule.

- 5, 15, 45, 135, …
- 4, 12, 36, 108, …
- 1/7, 3/7, 9/7, 27/7, …
- 3, 6, 12, 24, …
- 5, 10, 20, 40, 80, …
- 2.4, 4.8, 9.6, 19.2, …
- 4, -12, 36, -108, …
- -7, 14, -28, 56, …
- 7, -14, 28, -56, …

To trigger students' memories, you may want to remind them that a recursive rule starts with *A _{n}*. Assess students by circulating the room and monitoring student progress. Look for students finding different recursion rules.

- Questions 1-3 can be expressed with any of the following:
*A*= 2_{n}*A*_{n}_{ – 1}+ 3*A*_{n}_{ – 2}*A*=_{n}*A*_{n}_{ – 1}+ 6*A*_{n}_{ – 2}*A*= 3_{n}*A*_{n}_{ – 1}

- Questions 4-6 all have the same recursive rule of either of the following:
*A*=_{n}*A*_{n}_{ – 1}+ 2*A*_{n}_{ – 2}*A*= 2_{n}*A*_{n}_{ – 1}

- Questions 8-9 can be expressed by either of the following:
*A*=_{n}*A*_{n}_{ – 1}– 2*A*_{n}_{ – 2}*A*= -2_{n}*A*_{n}_{ – 1}

Once students have finished, ask them to write down and explain any patterns they notice or any conjectures they have. Give them a few minutes for this. Provide prompts as needed, such as, "I noticed that…" or "One interesting thing I saw was…" Have students share their findings with the class.

Students may notice the even though the patterns in questions 1‑3 start with different values, they have the same recursive rule (similarly, so do questions 4‑6 as do questions 8‑9). If students do not see this, be sure to bring it to their attention, and ask them if they see this happening anywhere besides questions 1‑3. Students may also notice the common ratio for each series. Unlike problems from the previous lessons, the common ratio is exact (rather than approximate) in these problems.

Explain to students that in question 1, to get 15, you do 5 ⋅ 3, and to get 45, you do 15 ⋅ 3, which is the same as 5 ⋅ 3 ⋅ 3.

Ask them to show two or three ways to get 135. One possibility
that should emerge is 5 ⋅ 3 ⋅ 3 ⋅ 3. Show that the terms from
question 1 can be written in exponential form as 5⋅3^{0}, 5⋅3^{1}, 5⋅3^{2}, 5⋅3^{3}, ….

You may need to remind students that multiplying by 3^{0}
is the same as multiplying by 1. A way to help students grasp this is
to say: How many times did you multiply 5 by 3 to get 45? [2.] Then
ask, how many times did you multiply 5 by 3 to get 15? [1.] Finally
ask, how many times did you multiply 5 by 3 to get 5? [0, as in 5⋅3^{0}.]

Ask students to find the exponential form for the other
examples. If time is an issue, students can skip questions 3, 6, and 9.
*Be sure students use parentheses on questions 7 and 8*. Again,
circulate while students do this to monitor their progress. Answer
questions and help them with any misconception they may have. When
finished, student should be able to represent an exponential series
using a recursive rule and using exponents.

Ask students how the recursive rules relates to the terms with exponents.

To check for understanding, present the following to students:

- For the sequence 3, 12, 48, 192, …, ask students to:
- Find the next term.
- Write a recursive rule.
- Express each term with exponents.

- Give the recursive rule
*A*= -5_{n}*A*_{n}_{ – 1}, where the first term is 3. Ask students to:- Write out the first 4 terms in the sequence.
- Express each term using exponents.

- If students are still having trouble, ask them what their questions are and model an additional example like each of the previous two.

Student should now have the tools they need to write exponential functions. To get students to attempt this without direct instruction, display the overhead below and ask students to complete the table individually.

Recursive and Exponential Rules Overhead

One possible answer is **Number of Ways = 2 ⋅ 6 ^{n – 1}**.

The calculator gives **Number of Ways = 1/3 ⋅ 6 ^{n}**. Students should recognize that

^{1}/

_{3}is the same as

^{2}/

_{6}. Further, students should realize that

^{2}/

_{6}⋅ 6

^{n}is equivalent to 2⋅6

^{n – 1}.

You may need to convince your students of this with the following example:

^{2}/_{7}⋅ 7^{5}=^{2}/_{7}⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7,which is the same as 2⋅7⋅7⋅7⋅7, since one of the 7s was divided by 7 to give 1.

Thus,

^{2}/_{7}⋅ 7^{5}= 2 ⋅ 7^{4}.Do this with

^{2}/_{6}⋅6^{5}if students need further convincing of reducing the exponent by 1.

Ask student to write rules using exponents to find the 5th and *n*th terms in questions 1‑9 from the Recursive Rules Overhead.

### References

- Benjamin, A. T. and J. J. Quinn. 2003. Proofs That Really Count: The Art of Combinatorial Proof, by Dolciani Mathematical Expositions, Volume 27. Mathematical Association of America.
- Wilf, H. S. 2006 Generatingfunctionology. A. K. Peters, Ltd. http://www.math.upenn.edu/~wilf/DownldGF.html

**Assessment Options**

To assess students give the following:

- Complete the table:
Train Length (n) 1 2 3 4 5 6 Number of Ways 4 20 100 Using Exponents: - Write the number of way to make a train of length 7 using exponents. Do the same for a train of length 8. Extend your table from #1 to check you answer.
- How would you find the number of ways to make a train of length 100? Length 2007
- Write a rule for the number of ways to make a train of length n.
- What rule do you get on the calculator? How does this compare to your answers in #3?

**Extensions**

Have students experiment with different starting values for the recursive rule from lesson 1: (A_{n} = A_{n-1}+A_{n-2}).

- They should pick any two numbers that they wish to start with and find the next eight terms in the recursion.
- Repeat these several times with numbers of their own choosing.
- Ask them to record any conjectures they have.
- When finished, specifically ask them if they notice anything about the common ratio. Does it behave similar to numbers 1-3, 4-6 and 8-9 from this lesson? What happens to the common ratio when different starting values are used in lesson 2?

**Questions for Students**

1. Can different sequences have the same recursive rule?

[Yes, but they may have different initial values.]

2. What is a number raised to the 0th power?

[1.]

3. How can you write a rule using exponents for the nth number in a sequence?

[If the sequence is geometric, then the nth term involves multiplication by some constant n times.]

4. How are the recursive rule and the exponential rule related?

[An exponential rule has the form An = a ⋅ bn – 1. When written recursively, the same sequence can be described as A1 = a and An = b ⋅ An – 1. For example, the sequence 3, 12, 48, …, can be described by the exponential rule An = 3 ⋅ 4n – 1 and by the recursive rule An = 4 ⋅ An – 1, where A1 = 3.]

**Teacher Reflection**

- What did you learn by preparing for this lesson? Did you have any new mathematical or pedagogical insights?
- What did students know already about recursion and exponential growth?
- How does this lesson relate to exponential growth?
- Did the students exceed your expectations in some areas and not meet them in others?
- What were some of the ways that the students illustrated that they were actively engaged in the learning process?
- Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?

### Trains, Fibonacci, and Recursive Patterns

### Recursive and Exponential Rules

### Learning Objectives

Students will:

- Represent data using tables, graphs and rules.
- Investigate patterns and make conjectures.
- Explain their reasoning when making conjectures.
- Describe and interpret exponential functions that fit the patterns.

### NCTM Standards and Expectations

- Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

- Relate and compare different forms of representation for a relationship.

- Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

- Generalize patterns using explicitly defined and recursively defined functions.

- Use symbolic algebra to represent and explain mathematical relationships.

- Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations.

- Draw reasonable conclusions about a situation being modeled.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP8

Look for and express regularity in repeated reasoning.