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: 1 = 6.1; = −1 + .085 Explicit: = 6.1 + .085( − 1) After 50 years, the population will be approximately 10.27 billion people. Geometric Recursive: 1 = 6.1; = −1 ∙ 1.014 Explicit: = 6.1(1.014) −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: = 33 + 4 Recursive: a1 = 37 and an=an-1 + 4 where 1≤≤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
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Lesson Title: Recursive and Explicit Formulas Course: Algebra II, Unit 2
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
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.
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
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.
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
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?
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
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.
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
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.
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