Unit 2: Recursion

Algebra IIDay 1: Recursive Notation and Definitions

Objectives - I can write recursive formulas for generating sequences using proper notation given a list of numbers or a real-world scenario. - I can use a recursive formula to generate a sequence. - I can distinguish between arithmetic, geometric, and shifted geometric sequences given a formula, list of numbers, or a graph.

Materials

Anticipatory Set take a look at the following picture:this is recursion. Adding on each time a similar amountstart with row 1next row: what changes?next row: how does it relate

How does the figure increase layer by layer?

We are going to be talking about recursive sequences and by the end of the chapter you should be able to look at sequences and be able to create a recursive formula that accurately fits.

Procedures

What is recursion?- process in which each step of a pattern is dependent on the step or steps that come before it

patterns create sequences, or ordered list of numbers

Recursive formula- formula that defines a sequence, must specify one or more starting terms and a recursive rule that defines the nth term (any term/number) in relation to the previous term(s)

For example, if your sequence is 4, 6, 8, 10, 12 we can create a formula that worksour first term, 4, would be labeled as:

f(1)= 4f(1)= f(n-1) + 2 where n ≥ 2

Unit 2: Recursion

What does this all mean?the first term is 4 and each subsequent term is equal to the previous term plus 2.

** f(n) is defined in relation to the previous term, f(n-1). *so the 10th term would be f(n) = f(10-1) + 2 = f(9) + 2

**good idea to organize the numbers into a table

use book for explanation of scenariodraw picture of rows and label

Rows 1 2 3 4 …Seats 59 63 67 ????? …

If we want to find the number of seats in the 4th row, we can find an equation first

1) Need a starting terma. This starting term is 59, or f(1) = 59

2) need to look at common difference between successive termsa. 63 is 4 more than 59; 67 is 4 more than 63

3) Write recursive rule using the informationa. f(1) = 59 b. f(n) = f(n-1) + 4 where n ≥ 2

4) Plug in termsa. Can use the formula to then solve for any rows number of seatsb. Just continue the sequence by adding 4 to the previous rows

i. Ex: find # seats in 8th row

Arithmetic sequences- sequences that increase/decrease by a common difference (the constant of the sequence)- let d be the common difference- form: f (n)=f (n−1)+d

---need to recognize the common difference to know it’s an arithmetic sequence-try subtracting consecutive terms- Take the 2nd term and subtract the 1st term if it is constant for each pair of terms, then it must

be arithmetic.

Geometric sequences- when each term is equal to the previous term multiplied by a constant, or common ratio- let r be the common ratio-form: f (n)=r∗f (n−1)

Example: 1, 2, 4, 8, 16…what is the common ratio in the sequence?how can you find that ratio?(2nd # / 1st #)write recursive formulafind f(7)=?

80, 20, 5, 1.25, …formula: f(1) = 80 f(n)= 0.25 f(n-1)

Unit 2: Recursion

Multiplying by 4 between them, but you are really dividing because your sequence is decreasing

Example:

- Count the number of triangles in each stage of the series- what does it start at? 1

- So, f (0)=1- f (n)=3∗f (n−1) where n ≥ 1

** can identify by dividing the consecutive terms

- If f (n)

f (n−1) is constant for each pair of terms, then it is geometric

Shifted geometric sequences

-same as geometric sequences, but also has a y-intercept includedform of f (n)=r∗f (n−1) + a

In class: pg 34 # 1a,b, 4, 5a together if time allows before closing and assigning homework

Closure

When trying to determine whether or not it is geometric or arithmetic

1) Determine the differences between each number in the sequencea. Look for patterns

2) What are the patterns between each sequential number?a. Just adding to the previous number? Arithmeticb. Multiplying by some number between each terms? Geometric

3) Multiplying by each term and then adding? Shifted geometric

briefly explain how recursion can be helpful and that these formulas help simplify the problems. Tomorrow we’ll be learning how to use our graphing calculators to learn how to evaluate recursive sequences.

Assessment in class problems and homework assigned

homework: Recursively Defined Sequences WS1

Unit 2: Recursion

Algebra IIDay 2: Recursive Formulas and Calculators

Objectives I can use technology to simulate arithmetic and geometric sequences.

Materials graphing calculators

Anticipatory Set check to make sure homework is done before asking:

Who can tell me what we talked about yesterday in class? What were the general formulas used?

Procedures now we are going to take what we know about algebraic and geometric sequences and use our graphing calculators to evaluate (this is how we are going to check the answers on the homework!)

Calculator steps/notation:

1) Change mode to seq on calculator2) Press “y=” button at the top and set…

a. nMin = 1b. \u(n)= u(n-1)+## this is the recursive formula used

i. Get the u by pressing [2nd] + [7]ii. the u represents “f” in f(n)

c. U(nMin) = {# that f(1) equals} brackets, not parentheses3) Press “table” ([2nd] + [graph]) in order to scroll through

a. OR mainscreen: [2nd] + [7] = u(some desired #) = ?? to get u(n)

Closure/Assessment see how they did on their homework from the previous day and that they are understanding how to use their calculators to do before handing out the worksheet for additional practice

homework: finish “recursive sequence-calculator WS2” sheet

Unit 2: Recursion

Algebra IIDay 3: Growth/Decay

Objectives I can use geometric sequences to model growth and decay.

Materials calculators

Anticipatory Set go over previous day’s homework and answer any questionsyesterday you learned how to use your calculators to evaluate for geometric and arithmetic sequences by creating/using formulas. Today, we are going to learn how to use the geometric sequences to model growth and decay.

Who can tell me what the equation for a geometric sequence is?

f (n)=r∗f (n−1)

Let’s start out with an example right away on growth:

You go to the bank and put in $2000. The bank gives you 7% annual interest each year.what do you think our variables will be?What is our starting amount? 2000= f(0).What is our rate of change? 7% = .07 such that = (1+.07)****

*** because we think of r, the growth or decay, as a common ration as a whole number, 1, plus or minus the percent of change, which would we use?

- positive change (growth) r = (1+ p)-negative change (decay) r = (1-p)

so, f (n)=(1+ .07)∗f (n−1) where n ≥ 1**use your calculators like we did yesterday to evaluate the balances at the end of each yearHow much money will you have at the 11th year? ($4209.70)

How do you think we can use this to evaluate for growth and decay?

As I said before…

When it comes to growth and decay, we think of r as a common ration as a whole number, 1, plus or minus the percent of change

- positive change (growth) r = (1+ p)-negative change (decay) r = (1-p)

why 1 ± p?? 1 represents 100% plus or minus the %-it’d be more than 100% if you are increasing because you end up with more than what

you started-it’d be less than 100% if you are decreasing (Decaying) because you end up with less

than you started

Unit 2: Recursion

For an example of decay, we need something decreasing in value.

Let’s say you buy washing machine for $500. Every year it’s value decreases by 1/5. How much will it be worth in 5 years?what are our variables?initial cost = 500 = f(0)are we going to be increasing or decreasing??our rate of decay is 1/5. So r = (1-0.2)so our formula is f (n)=(1−.2)∗f (n−1) where n ≥ 1

in 5 years our washing machine will be worth… $163.84

Closure go over the key steps again1) find you variables2) are you increasing (growing) or decreasing (decaying)?3) plug in for f (n)=(1± p)∗f (n−1)

Assessment

homework: p41 #1-3, 6, 9, 14 and Modeling Growth and Decay wkst 3

Unit 2: Recursion

Algebra IIDay 4: Modeling Growth/Decay

Objectives I can use geometric sequences to model growth and decay.

Materials graphing calculator; stations set up

Anticipatory Set go over homework from previous day and how to set up for a growth/decay formula

Procedures questions from worksheet 4 will be individually placed at various places in the room such that there are 2 stations for each question. Students will be put into groups and rotate to each station throughout the class period, lasting 9 min per station. They will work as a small group to go through the problem and help each other understand it

Closure bring class together to see what the various groups were getting for answers in the station work

Assessment see how students are understanding the station work

Unit 2: Recursion

Algebra IIDay 5: More Modeling Growth/Decay

Objectives I can use geometric sequences to model growth and decay.

Materials

Anticipatory Set continue going over problems from yesterday and recap growth/decay formula

Procedures students will be put into small groups and will become experts on an assigned problem, but must complete the whole handout the students will be expected to present their assigned problem to the class the following day to go over

1) Find ratio2) Find recursive formula

f(?)=f(n)=

3) Answer question—be prepared to present

Closure emphasize making sure they know their assigned problem thoroughly and are able to present it to the class as a group the next day and that they are still expected to finish the rest of the problems

Assessment homework: finish any problems not finished during class

Unit 2: Recursion

Algebra IIDay 5 (Part 2): More Modeling Growth/Decay

Objectives I can use geometric sequences to model growth and decay.

Materials

Anticipatory Set continue going over problems from yesterday and recap growth/decay formula

students were put into small groups and became experts on an assigned problem Today, the students will be expected to present their assigned problem to the class.

Procedures students present their problems, emphasizing what they got and how they got their answers for: Find ratioFind recursive formulaf(?)=f(n)=Answer question asked

Today we are going to work on a growth/decay lab relating to these limits and shifts. Do the first problem as a class – walk through slowly, giving them time to work on each

question a bit before asking as a wholeAnalyze results of plotting the points and what it means in terms of the question-find the rate, starting value and calculate the table-graph data to see what’s happening-ask what they notice about the points—almost make a line by leveling off

Closure Explain: sequences of numbers of a system sometimes reach a limit; being able to predict the limits are important for analyzing situations. The long-run value helps you estimate the limits.

Explain difference between arithmetic sequences (creates linear graph) vs geometric sequences creates nonlinear graphs, curved)

Tomorrow we will spend more time working on graphing the growth/decay and what long run limits are.

Assessment no homework: assessed on presentation of questions from previous day and their understanding

Unit 2: Recursion

Algebra IIDay 6: Shifted Geometric Sequences (Limits)

Objectives I can determine long run values of the geometric and shifted geometric sequences.

Materials graphing calculator

Anticipatory Set students will present their problems that they became experts on the previous class.

*recap growth/decay lab (front side was finished in class last time) -know how to create the recursive formula-can use our calculators to get the data in a table- plotted our data to show that the medicine is approaching, but never is completely out of

our system

*start side 2 of growth/decay lab sheet as class- read the problem

-point that they take 16ml each day on top of the decrease-have students help set up the problem excluding the daily dosage-ask: what will taking 16ml of medicine each day do to our recursive formula? How

will that affect the graph of the data?

we call this a geometric shift

Procedures shifted geometric sequence- includes an added term in the recursive rule

-for example, if you are doing a lab experiment that has some growth or decay that occurs each day, but you area also adding more to the experiment each day, you’d have your regular growth/decay formula, but also the shifted sequence for what you add additionally each day.

Finish working through problem on backside of lab 1.3

sequences of numbers of a system sometimes reach a limit; being able to predict the limits are important for analyzing situations. The long-run value helps you estimate the limits.

Long run value = limit that the equation reaches without surpassing; Limit that a sequence or a function approaches.

-think of as a “cut off” value

Unit 2: Recursion

Example 1:

Evaluate: f(0) = 16f(n) = f(n-1) * (1-.050 + 16)long run value = ?? (answer = 320.00 at u(400))

**how can you easily find the long run value and make sure? Go to [TBLSET] and punch in random numbers further along to see what the table show—usually 50 or 60 works**type in u(#) on home screen goes faster than table when looking that far ahead

**don’t always have to add for the geometric shift, we can subtract

Example 2:evaluate: f(0) = 50

f(n) = f(n-1) * (1-.5) -2long run value = ?? (answer = -4)

** it has to be (1- #) because it has to be a decay to reach a limit, but this doesn’t mean that the graph will always be a decay overall, it can be a growth as in example 2 on the worksheet in class (dependent upon the geometric shift)

Closure recap:in example 2, what was the percent growth/decay?is it going to be an increase or decrease? (hint: must use your table!)What makes a sequence arithmetic vs geometric?How do you know there is a geometric shift?What does the shift represent?

Assessment

Homework: limits worksheet 6

Arithmetic Sequences (Increase/decrease by a common difference ‘d’)

f ( n )=f ( n−1 )+d

Geometric Sequence (Increase/decrease with a common ratio ‘r’)

f ( n )=(1± r ) f (n−1)

Shifted Geometric Sequence (Increase/decrease with a common ratio ‘r’ and common difference ‘d’)

f ( n )=(1±r ) f (n−1 )+d

Unit 2: Recursion

Algebra IIDay 7: Applications of Shifted Geometric Sequences (limits)

Objectives I understand and can apply the concept of a “limit.” I can determine the long run value of a geometric sequence. I can model real-world situations using a shifted geometric sequence.

Materials graphing calculators

Anticipatory Set

Go over homework

Who can tell me what’s different in a shifted geometric series vs a regular geometric series?

Procedures

Go over another example of a shifted geometric sequence

Example:You are a farmer and your chickens produce 24 eggs the first week. Each week you take 60% of the eggs to cook for food, but your chickens also produce an additional 30 eggs per week.

a) write the recursive formulaf(0)= 24 f(n) = f(n-1)*(1+.6)+30

b) list the first five terms of this sequence of balances, starting with the initial investment.f(0)=24 f(1) =40; 46; 48; 49; 50 (round up for egg #)

c) what is the meaning of the value of f(4)?you have 49 eggs in the coup after four weeks

d) what is the balance a year later? What about 10 years later? What is the long-run value?f(1) = 40 f(10) = 49.997 = 50 eggs long-run=50

Closure Why would we want to use a shifted geometric equation? How did it help in this case?

-in this situation, it helps us map what we’ll have in our account over time

-can use it to see how accruing interest on bills is impacted with various payments

What are some other situations that this might be helpful?

Assessment Homework: WS7 #1-4 (not 5)

Unit 2: Recursion

Algebra IIDay 8: Finite Geometric Sequences

Objectives I can find the sum of a finite geometric series

Materials

Anticipatory Set

Go over the homework

What if we wanted to add up all the numbers we get in a geometric sequence?

For example, we know how to set up a geometric equation for number of people that attended a fair and knew how it increased each day. What if we wanted to know the total number of people that attended that weekend? How would we add it up quickly instead of individually counting out the numbers?

Procedures

Walk through proof for sum of geometric sequenceswhat does finite mean? What does infinite mean?

(using the problem presented in the powerpoint…)

1) Define s, r (%), n, a2) S = a + a(1.03) + a(1.03)(1.03) +…+a*rn-1 **student don’t need to write this line

S = a + a*r + a*r2+a*r3+…+a*rn-1

3) Multiply each side by r and distribute(r)S = a(r) + a(r)2+ a(r)3+ a(r)4+ …+ a(r)n-1+ a(r)n

-multiplying by n, add another n such that (n-1)+n = n

4) Subtract S-(r)SS-(r)S =

5) Simplify:S-(r)S = a - a(r)n

6) Factor out S and aS(1-r) = a(1-rn)

Unit 2: Recursion

7) Get S by itself:

S=a (1−r n)(1−r)

Example 1) f(0) = 10 f(n) = f(n-1) * (1-.8)

Find the sum of the first 15 termsa=0r=.2n=15S= 12.5

Ex 2) f(0) = 20f(n) = f(n-1) * (1+.55)

Find the sum of the first 11 terms, rounding to the nearest decimala= 20r=1.55n=11S=4475.1

Closure

****this is the formula that you need to know to find the sum of a geometric sequence instead of counting up the tables in the calculator!

r = common ratio (%)

a = starting value

n = # terms

S = sum

Do problem number 3 from ws 8 together

Assessment new ws8

Unit 2: Recursion

Algebra IIDay 9: Loans and Investments with Compound Interest (No Payments)

Objectives I can use a recursive formula to model a loan. I can use a recursive formula to model an investment.

Materials

Anticipatory Set go over the new ws8 together as a class, answering any questions that remain.

Shifted geometric sequences can be applied to a lot of day-to-day things but we mostly see them where money is involved, such as loans, bank accounts, farming investments, etc. (anywhere money is involved, really).

Today we are going to talk about loans and investments, in particular, ones with compounded interest.

Procedures

First, we need to go over a few terms:Loan- take money out to pay back later; equation has increase in r but decrease in d (because paying back loan so amount overall should decrease)

Investment- putting money into something; equation has increase in r as well as increase in d (Because continually contributing to the cause)

Interest- percentage of money added to your account

Compounded- the paying of interest on the accrued interest as well as on the principal

Principal- amount owed, invested, or the face value of a debt

Annually- yearlysemi-annually- twice a year

Quarterly- divide by 4 (so every 3 month period)

Monthly- divide by 12

Weekly- divide by 52 weeks (assumes consecutive weeks in a year)

Days- divide by 360 and NOT 365

Unit 2: Recursion

Example) Find the balance after 4 years if $1000 is deposited into an account with an annual interest rate of 1.25%, compounded monthly.

1) What’s the rate?

2) Start value?

3) Write the recursive formula

4) How much money would be in your account after 4 years? (answer the question)

Example) You take out a loan for $30000 in order to buy a house. Each month you accrue 7.9% interest quarterly.

1) What’s the rate? What does quarterly imply?

2) Start value?

3) Write the recursive formula.

4) How much do you owe on the loan after 3 months? 5) 6 months?

Closure interest is something that we see everywhere. Sometimes it works in our favor: bank accounts, and sometimes it works against us: loans. Either way, we can calculate to see how it will impact us over time.

Assessment WS9: loans and investments / compound interest

Unit 2: Recursion

Algebra IIDay 10: Loans and Investments with Compound interest and payments

Objectives I can use a recursive formula to model a loan. I can use a recursive formula to model an investment.

Materials

Anticipatory Set go over the homework from yesterday and ask for questions

Yesterday we went over loans and investments that had interest on them. However, now we are going to talk about those loans and investments with payments included because we don’t want to take out a loan and let the interest accrue without paying it back!

In general, if we set up a geometric sequenced recursive formula for a loan or investment, how would we set up our equation if we are also making payments? (hint: shifted geom. sequence)

Procedures

Example 1) Find the first month’s interest on a $32000 loan at 4.9% annual interest rate and paying $1000 a month.-principal = $32,000-f(0)=32000-f(n)=f(n-1)(1+.049/12) – 1000how much do you owe after a year? F(12)= $21330.50two years? f(24)= $10126.28

Example 2) given f(0)=6000 and f(n)=f(n-1)*(1+.08/4) + 200loan or investment? Investmentdeposit or payment? Deposit

how much? 200% interest rate? 8%frequency compounded? Quarterly

Example 3) find amount of interest given a $15,000 loan at 10% daily compounded interest at1 day: 15000(.1/360) = $4.171 week: [15000(.1/360)]7 = $29.17

Unit 2: Recursion

Example 4) How much owed after 3 years if you pay $300 a month on a $25000 loan with 3.9% interest compounded annually?a) principal: 25000b) formulaf(0) = 25000 f(n)= f(n-1)*(1-.039)-300c) answer: f(3)= $21,322

Assessment informal assessment by walking around the room and helping students as well as seeing where they are at

Homework: WS10: loans and investments with compound interest and payments

Unit 2: Recursion

Algebra IIDay 11: Loans and Finding the Payments

Objectives I can use a recursive formula to model a loan. I can use a recursive formula to model an investment.

Materials

Anticipatory Set go over yesterday’s homework, answer questions

Recap: loans and investments with payments require a shifted geometric recursive formula

How can we use our knowledge of these equations and how to set them up in order to find the amount we would need to pay in order to pay off loans?

Procedures

So let’s say that we took out a $10,000 loan that has 3.5% interest rate compounded monthly and we want to see how much we’d need to pay each month in order to get it paid off in 4 years.

1) Set up the formula:

f(0) = 10000; f(n) = f(n-1)*(1+.035)+ ??

the blank is because we want to see how much we’d have to pay each month in order to get it paid off in 4 years

2) Look at how long we want it paid off in: 4 years = 12*4 = f(48) we want to look at

3) Plug into equation various numbers until we get f(48) to be less than $0

a. f(n) = f(n-1)*(1+.035)+ 435=195.17 at 47 and -233 at 48

b. f(n) = f(n-1)*(1+.035)+ 434 = 310.52 remains at 47 and -112.60 at 48

4) what do the negative numbers at 48 imply??

implies that you paid more than you needed to on the final month; your payment for the final month is not going to have to be as much as the previous 47 months

-hand out the making payments investigation

Unit 2: Recursion

Walk through part 1with classstudents work on part 2 and 3 slowly, coming together as a class to go over them

Closure Why would figuring this out be important?

Because you have a credit score and you want it to be good, which means you need to pay things according to plan without getting too far in debt.

The longer it takes for you to pay off loans and credit cards, the more you end up paying overall.

Assessment informal assessment by walking around the room and helping students as well as seeing where they are at

Homework: WS11: loans & investments w/ payments