lesson title: recursive and explicit formulas course ...€¦ · lesson title: recursive and...

Lesson Title: Recursive and Explicit Formulas Course: Algebra II, Unit 2

Date: _____________ Teacher(s): ____________________ Start/end times: _________________________

HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann.

Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?

F.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the

integers.

MP2: Reason abstractly and quantitatively.

MP3: Construct viable arguments and critique the reasoning of others.

MP4: Model with mathematics.

Algebra II, Unit 2

Lesson Launch Notes: Exactly how will you use the

first five minutes of the lesson?

You were asked to study the birth patterns of local

fish. There are currently 2 fish housed in a tank at the

nearest fishery. If the population should grow

exponentially by a factor of 1.5 each year, how many

would there be in the fishery after 11 years?

Lesson Closure Notes: Exactly what summary activity,

questions, and discussion will close the lesson and provide

a foreshadowing of tomorrow? List the questions.

The Population Reference Bureau has claimed, “In 2000,

the world had 6.1 billion human inhabitants. This number

could rise to more than 9 billion in the next 50 years.” Will

the 2000 growth rate of 1.4%, when applied to the world's

6.1 billion population, yielding an annual increase of about

85 million people, support this claim?

Create either an arithmetic or geometric model for this

problem. Write the explicit and recursive forms for the

model you choose.

Arithmetic

Recursive: 𝑎1 = 6.1; 𝑎𝑛 = 𝑎𝑛−1 + .085

Explicit: 𝑎𝑛 = 6.1 + .085(𝑛 − 1)

After 50 years, the population will be approximately 10.27

billion people.

Geometric

Recursive: 𝑎1 = 6.1; 𝑎𝑛 = 𝑎𝑛−1 ∙ 1.014

Explicit: 𝑎𝑛 = 6.1(1.014)𝑛−1

After 50 years, the population will be approximately 12.06

billion people.

Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,

problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic

connections to appropriate mathematical practices. Part 1:

1. Review recursive and explicit forms for arithmetic sequences. (This was introduced in Algebra 1, but it is

important to determine where the students are and what they remember.) If needed, provide an example to help

them think. For example: A company needs to increase its output of chocolate bars by 4 per hour. At the end of

the first hour there will be 37 bars. How many bars will they have made after 8 hours?

Hour 1 2 3 4 5 6 7 8

Bars 37 41 45 49 53 57 61 65

Explicit: 𝑎𝑛 = 33 + 4𝑛

Recursive: a1 = 37 and an=an-1 + 4

where 1 ≤ 𝑛 ≤ 8

Allow students to work in groups of 3 or 4 so that they can discuss the concepts learned in Algebra I in order to




link prior knowledge to this scenario.

Students should be given about 2 to 3 minutes to generate a either a table or graph to show their solution.

Reintroduce the concept of explicit and recursive formulas from Algebra I.

Give the students 4 to 5 minutes to use their data to generate the two forms for this problem. (Look for

evidence of MP2 and MP3.)

2. After reviewing the arithmetic example, refer back to the warm-up. (Most students should have determined a

solution but may not have made the connection to recursive and explicit forms.) Allow students to work in

groups of 3 or 4 to determine each form for the geometric example. Make sure to circulate around the room

providing leading questions like, “If arithmetic sequences use addition and subtraction, what do you think

geometric sequences will use?”, “Is the pattern getting larger or smaller?”, “How can you represent

multiplication of the same number differently?” “How can we represent division in terms of multiplication?”

Year 1 2 3 4 5 6 7 8 9 10 11

Fish 2 3 4.5 6.75 10.1

3

15.

19

22.

78

34.

17

51.

26

76.

89

11

5.3

3

Approximately 115 fish will be in the pond after 11 years.

Explicit:

𝑎𝑛 = 2(1.5)𝑛−1

Recursive:

𝑎1 = 2 ; 𝑎𝑛 = 1.5𝑎𝑛−1

Where 1 ≤ 𝑛 ≤ 11

Review the different forms with them as a class to allow them to share out their success or difficulty in

remembering them. With the review this should take about 10 to 12 minutes. Students can graph their values, use

tables, and other tools to show their data points.(Look for evidence of MP2 and MP3.)

3. Use the websites below for example problems. As a class, identify each scenario as arithmetic or geometric. In

their groups, students should determine the explicit and recursive formulas for each scenario.

http://www.mathwarehouse.com/exponential-growth/exponential-models-in-real-world.php

http://www.austincc.edu/powens/+Topics/HTML/10-2/10-2.htm

(If students need additional practice, you can use the attached list of word problems for students that are

struggling to create a model.)

In groups, have students create their own examples of arithmetic and geometric sequences.

Each group will be provided a poster to present their work to the class.

On each poster they should list the forms and graph to show the differences visually between arithmetic

and geometric models. Have students graph their values, use tables, and any other tools to show their

data points.(This should take the majority of the class time to complete.)

Monitor the time and collect work during the last 6 to 8 minutes. (Look for evidence of MP4.)

4. While collecting hand out the exit ticket and have students complete it. Collect it to determine concept

attainment. (Look for evidence of MP2.)

Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I

measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside

observer) that your students have developed a deepened (and conceptual) understanding.

Students should be able to write an explicit and recursive formula given an arithmetic or geometric model. They

should be able to recognize the limitation of the recursive formula and the connections between the two formulas.

Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.

http://www.mathwarehouse.com/exponential-growth/exponential-models-in-real-world.php

http://www.austincc.edu/powens/+Topics/HTML/10-2/10-2.htm




Sequence notation can often be a source of student struggles; make sure students understand 𝑎𝑛 represents the nth

term in a given sequence, 𝑎𝑛−1 is the previous term and 𝑎𝑛+1 is the subsequent term.

Sequences can be defined either explicitly or recursively. An arithmetic sequence has a constant rate of change

while a geometric sequence has a constant multiplier. An explicit formula allows the user to find a sequence term by

simply substituting the term-number into the formula and evaluating. A recursive formula relates each term of a

sequence to the term before. In order to find a sequence term, the user must find all prior terms in chronological

order.

Take the time to look over the different groups work to determine if more time will be needed the next day to

complete the activity.

Resources: What materials or resources are essential for students to

successfully complete the lesson tasks or activities?

Posters

Graph paper

Graphing Calculator

Computer

Extra Example sheet (if needed)

http://www.otherwise.com/population/exponent.html

http://www.pbs.org/teacherline/courses/math125/session3/session3.htm

http://mathbench.umd.edu/applets/applets-catalog.html

Homework: Exactly what follow-up

homework tasks, problems, and/or

exercises will be assigned upon the

completion of the lesson?

Find or generate a real-life data set that has

either an arithmetic or geometric

relationship and include at least 5 data

points. Write the explicit and recursive

formulas for the data sets.

Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use to

reflect on the effectiveness of this lesson?

Can students distinguish between explicit and recursive formulas?

Can students write explicit and recursive formulas given an arithmetic or geometric model?

Can students determine and justify the appropriate formula to be used for different input values?

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this

product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

http://www.otherwise.com/population/exponent.html

http://www.pbs.org/teacherline/courses/math125/session3/session3.htm

http://mathbench.umd.edu/applets/applets-catalog.html

http://creativecommons.org/licenses/by-nc-nd/3.0/

Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Unit 2: Exponential Functions

Recursive and Explicit Formulas

For each scenario below complete the following:

a. answer the question

b. classify the scenario as an arithmetic or geometric sequence

c. write an explicit formula

d. write a recursive formula

A zombie bacteria strain will double in size every 30 minutes. If a person is infected with 20

bacteria and it takes about 2,621,440 for a person to become a zombie, how long will it take for a

person to mutate into a zombie?

A sequence of equilateral triangles is constructed. The first triangle has sides 14m in length. To

get the second triangle, midpoints of the sides of the original triangle are connected. What is the

length of each side of the 4th triangle?

An infectious bacteria has leaked from a government facility. You discover that you are infected

with 15 bacteria. You then discover that it doubles every 30 minutes. After 6 hours you will

become contagious. How much of the strain of bacteria will you have?

During a free fall, a skydiver falls 16 feet in the first second, 48 feet in the 2nd second, and 80

feet in the third second. If she continues to fall at this rate, how many feet will she fall during

the 8th second?

If you have $145 in your account at the beginning of week 13 and $205 at the beginning of week

18, how much are you depositing weekly?



You decide to open a savings account. Each week you are going to deposit $3 more than the

previous week. The first week, you deposit $10. How much will you deposit during the 25th

week?

A culture of bacteria doubles every 30 minutes. There are currently 75 bacteria, how many

would there be after 4 hours?

Iodine-131 is a radioactive element used to study the thyroid gland. It takes approximately 8

days for half of a sample of Iodine-131 to decay into another element. How much of an 80-mg

sample would be left after 32 days?

A population of fruit flies is growing in such a way that each generation is 1.5 times as large as

the last. If there were 100 flies in the 1st generation, how many would be in the 4th generation?

George’s current salary is $40,000. His annual pay raise is a percentage of his current salary.

What would his salary be if he got four consecutive 4% pay raises?

A one-ton ice sculpture is melting so that it loses 1/5 of its weight per hour. How much will be

left after 5 hours? Write your answer in pounds.



The final step in processing a black-and-white photographic print is to immerse the print in a

chemical fixer. The print is then washed in running water. Under certain conditions 98% of the

fixer is removed after 15 minutes of washing. How much of the original fixer is left on the

photograph after 2 hours?

A colony of bacteria began with 200. It grows at a rate of 40% every 20 minutes. How many will

there be in 3 hours?



Unit 2: Exponential Functions

Recursive and Explicit Formulas

For each scenario below complete the following:

a. answer the question

b. classify the scenario as an arithmetic or geometric sequence

c. write an explicit formula

d. write a recursive formula

A zombie bacteria strain will double in size every 30 minutes. If a person is infected with 20

bacteria and it takes about 2,621,440 for a person to become a zombie, how long will it take for a

person to mutate into a zombie?

Write a formula to represent this.

a. 8.5 hours

b. Geometric

c. 𝑎𝑛 = 20(2)𝑛−1

d. 𝑎1 = 20; 𝑎𝑛 = 2𝑎𝑛−1;

A sequence of equilateral triangles is constructed. The first triangle has sides 14m in length. To

get the second triangle, midpoints of the sides of the original triangle are connected. What is the

length of each side of the 4th triangle?

a. 1.75 m

b. Geometric

c. 𝑎𝑛 = 14(1

2)𝑛−1

d. 𝑎1 = 14; 𝑎𝑛 =1

2𝑎𝑛−1; 1 ≤ 𝑛 ≤ 4

An infectious bacteria has leaked from a government facility. You discover that you are infected

with 15 bacteria. You then discover that it doubles every 30 minutes. After 6 hours you will

become contagious. How much of the strain of bacteria will you have?

a. 61,440 bacteria at 6 hours

b. Geometric

c. 𝑎𝑛 = 15(2)2𝑛

d. 𝑎1 = 15; 𝑎𝑛 = 2𝑎𝑛−1

During a free fall, a skydiver falls 16 feet in the first second, 48 feet in the 2nd second, and 80

feet in the third second. If she continues to fall at this rate, how many feet will she fall during

the 8th second?

a. 272 feet

b. Arithmetic

c. 𝑎𝑛 = 16 + 32(𝑛 − 1)

d. 𝑎1 = 16; 𝑎𝑛 = 𝑎𝑛−1 + 32

If you have $145 in your account at the beginning of week 13 and $205 at the beginning of week

18, how much are you depositing weekly?



a. $12 per week

b. Arithmetic

c. 𝑎𝑛 = 12 + 12(𝑛 − 1)

d. 𝑎1 = 12; 𝑎𝑛 = 𝑎𝑛−1 + 12

You decide to open a savings account. Each week you are going to deposit $3 more than the

previous week. The first week, you deposit $10. How much will you deposit during the 25th

week?

a. $72 on the 25th week

b. Arithmetic

c. 𝑎𝑛 = 10 + 3(𝑛 − 1)

d. 𝑎1 = 10; 𝑎𝑛 = 𝑎𝑛−1 + 3

A culture of bacteria doubles every 30 minutes. There are currently 75 bacteria, how many

would there be after 4 hours?

a. 19,200 bacteria at 4 hours

b. Geometric

c. 𝑎𝑛 = 75(2)𝑛−1

d. 𝑎1 = 75; 𝑎𝑛 = (2)𝑎𝑛−1

Iodine-131 is a radioactive element used to study the thyroid gland. It takes approximately 8

days for half of a sample of Iodine-131 to decay into another element. How much of an 80-mg

sample would be left after 32 days?

a. 5 mg left after 32 days

b. Geometric

c. 𝑎𝑛 = 80 (1

2)

𝑛−1

d. 𝑎1 = 75; 𝑎𝑛 = (1

2)𝑎𝑛−1

A population of fruit flies is growing in such a way that each generation is 1.5 times as large as

the last. If there were 100 flies in the 1st generation, how many would be in the 4th generation?

a. 337 fruit flies in the 4th generation

b. Geometric

c. 𝑎𝑛 = 100(1.5)𝑛−1

d. 𝑎1 = 100; 𝑎𝑛 = (1.5)𝑎𝑛−1

George’s current salary is $40,000. His annual pay raise is a percentage of his current salary.

What would his salary be if he got four consecutive 4% pay raises?

a. $46,794.34 after 4 raises

b. Geometric

c. 𝑎𝑛 = 40,000(1.04)𝑛−1

d. 𝑎1 = 40,000; 𝑎𝑛 = (1.04)𝑎𝑛−1

A one-ton ice sculpture is melting so that it loses 1/5 of its weight per hour. How much will be

left after 5 hours? Write your answer in pounds.



a. 655.36 after 5 hours

b. Geometric

c. 𝑎𝑛 = 2000 (1

5)

𝑛−1

d. 𝑎1 = 2000; 𝑎𝑛 = (1

5)𝑎𝑛−1

The final step in processing a black-and-white photographic print is to immerse the print in a

chemical fixer. The print is then washed in running water. Under certain conditions 98% of the

fixer is removed after 15 minutes of washing. How much of the original fixer is left on the

photograph after 2 hours?

a. 2.56 × 10−14 percent left after 2 hrs

b. Geometric

c. 𝑎𝑛 = 𝑎1(. 02)𝑛−1

d. 𝑎1 = 𝑎1; 𝑎𝑛 = (.02)𝑎𝑛−1

A colony of bacteria began with 200. It grows at a rate of 40% every 20 minutes. How many will

there be in 3 hours?

a. 4,132 bacteria after 3 hours

b. Geometric

c. 𝑎𝑛 = 200(1.40)𝑛−1

d. 𝑎1 = 200; 𝑎𝑛 = (1.40)𝑎𝑛−1


Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product

under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Recursive and Explicit Formulas Homework

For each of the following problems:

a. Identify the formula as arithmetic or geometric.

b. Identify the formula as explicit or recursive.

c. If the formula is explicit, determine the recursive formula. If the formula is recursive, determine

the explicit formula.

d. Determine the first 5 terms of the sequence.

e. Determine the 14th term of the sequence.

1. 𝑎1 = −2; 𝑎𝑛 = −3𝑎𝑛−1

2. 𝑎𝑛 = −3 + 5𝑛

3. 𝑎1 = 1.5; 𝑎𝑛 = 𝑎𝑛−1 − 2.5

4. 𝑎𝑛 = 3(1.2)𝑛−1

5. Find or generate a real-life data set that has either an arithmetic or geometric relationship and include

at least 5 data points. Write the explicit and recursive formula for the data sets.




6. Marcy is training for the bicycle portion of a triathlon. She plans to ride every day, increasing her

distance by 8 km each subsequent week. During week 5, she rides 56 km.

a. How far is Marcy riding during the second week of training?

b. Write both a recursive and explicit formula to represent this scenario.

c. In what week will Marcy be riding 192 km?

7. A new home theater was purchased for $3500 in 2012. Unfortunately, it depreciates in value by 25.4%

each year.

a. How much will the computer be worth in 2018?


c. You decide that once the value of the system falls below $500, you will need to upgrade. In what

year will you be purchasing a new system?




Recursive and Explicit Formulas Homework

For each of the following problems:

a. Identify the formula as arithmetic or geometric.

b. Identify the formula as explicit or recursive.

c. If the formula is explicit, determine the recursive formula. If the formula is recursive, determine

the explicit formula.

d. Determine the first 5 terms of the sequence.

e. Determine the 14th term of the sequence.

1. 𝑎1 = −2; 𝑎𝑛 = −3𝑎𝑛−1

a. geometric

b. recursive

c. 𝑎𝑛 = −2(−3)𝑛−1

d. −2, 6, −18, 54, −162

e. −9,565,938

2. 𝑎𝑛 = −3 + 5𝑛

a. arithmetic

b. explicit

c. 𝑎1 = 2, 𝑎𝑛 = 𝑎𝑛−1 + 5

d. 2, 7, 12, 17, 22

e. 67

3. 𝑎1 = 1.5; 𝑎𝑛 = 𝑎𝑛−1 − 2.5

a. arithmetic

b. formula

c. 𝑎𝑛 = 1.5 − 2.5(𝑛 − 1) = 4 − 2.5𝑛

d. 1.5, −1, −3.5, −6, −7.5

e. −31

4. 𝑎𝑛 = 3(1.2)𝑛−1

a. geometric

b. explicit

c. 𝑎1 = 3; 𝑎𝑛 = 𝑎𝑛−1 ∙ 1.2

d. 3, 3.6, 4.32, 5.184, 6.2208

e. 32.098

5. Find or generate a real-life data set that has either an arithmetic or geometric relationship and include

at least 5 data points. Write the explicit and recursive formula for the data sets.

Answers will vary

6. Marcy is training for the bicycle portion of a triathlon. She plans to ride every day, increasing her

distance by 8 km each subsequent week. During week 5, she rides 56 km.

a. How far is Marcy riding during the second week of training?

32 km


Explicit: Recursive




𝑎𝑛 = 24 + 8(𝑛 − 1) = 16 + 8𝑛 𝑎1 = 24 and 𝑎𝑛 = 𝑎𝑛−1 + 8

c. In what week will Marcy be riding 192 km?

22nd week

7. A new home theater was purchased for $3500 in 2012. Unfortunately, it depreciates in value by 25.4%

each year.

a. How much will the computer be worth in 2018?

3500(1 - .254)6 = $603.26; please note that 2018 is the 7th year, which is why the exponent is 6.


Recursive: 𝑎1 = $3500 𝑎𝑛𝑑 𝑎𝑛 = 𝑎𝑛−1 ∙ .746

Explicit: 𝑎𝑛 = 3500(. 746)𝑛−1

c. You decide that once the value of the system falls below $500, you will need to upgrade. In what

year will you be purchasing a new system?

In between year 7 and 8, specifically July 21st, 2018


lesson title: recursive and explicit formulas course ...€¦ · lesson title: recursive and...

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