Grade: - US-China Institute



Exponential Growth and Decay Models: Lesson 1

|Grade: Advanced Algebra |Section: 1 |

|Unit: Exponential Growth and Decay Models |Lesson #: 1 |

|Lesson: Intro to Exponential Growth and Decay |Date / Day: |

|T.S.W.B.A.T. |Warm-Up / Anticipatory Set: |

|Create exponential growth and decay models and use the models to solve |Compound interest problem on the board when students enter the class. |

|problems. While learning about some factors that lead to china’s | |

|exponential economic boost and its effects. | |

|Key Activities and Timing: |Resources Required: |

|Review of compound interest problem (5 min) |Overhead projector |

|Direct instruction: growth and decay models (20-30 min) |Ipads for each student |

|Students complete and submit in-class examples using ipad (20 min) |Geometer’s Sketchpad software installed on each ipad. |

| | |

|Old Homework: |Diversity / Differentiation Strategies: |

| |Direct instruction with Q&A. |

| |Individual Ipad-based work with individual teacher assistance. |

| |Cooperative learning as appropriate (students may discuss examples with |

|New Homework: |each other). |

|Exponential growth and decay worksheet 1 | |

| | |

| | |

|IL Learning Standards Addressed: |Lesson Critique / Student Notes: |

| | |

|SSE.B.5. Interpret the parameters in a …exponential function in terms of | |

|a context. | |

|IF. C. 7e. Graph exponential functions, showing intercepts and end | |

|behavior. | |

|SSE.B. 3c Use properties of exponents to transform expressions for | |

|exponential functions. | |

| | |

| | |

|Evaluation Procedures: | |

| | |

|Observe student performance during in-class examples. | |

|Homework assigned and checked. | |

|Individual project on growth and decay. | |

|Unit quiz on exponential growth and decay. | |

| | |

| | |

| | |

Set or “Hook”

Today we’re going to take exponential functions to the next level as we explore growth and decay functions, we will specifically look at China’s growth since its cultural revolution starting in the 1940’s. The Chinese economy grew at really fast pace going from being a very poor country to being a “super power” within 60 years. We will explore their “exponential growth.”

Lesson Body

Direct Instruction: Growth and Decay: 20 minutes

Yesterday we looked at what happens with an investment that grows at a given rate over a number of years.

Today we’re going to apply the concept to population growth. Projecting population has numerous applications including governmental (resource and services planning) and business (new product and marketing planning). Since China had such a rapid growth the government had to plan ahead using mathematical models.

Let’s look at an example:

The population of China in 1950 was 562,579,779 and was growing at 2.77% per year. We can create a formula for the population in any given year:

Population = 562,579,779 *(1.0277)^t where t is the number of years.

With this formula the Chinese government answer many useful questions:

1. When will the population reach 1,000,000,000?

2. What will be the population in 10 years?

Let’s discuss for a moment: what are the driving factors of population growth?

Ans: Birth rates

Death rates

Immigration in

Immigration out

There are also situations where a quantity is declining. We call this decay. Let’s look at an example:

Over-active beavers are destroying a forest at a rate of 50% every 5 years. We say that the half-life of the forest is 5 years. If we know the initial size of the forest (500,000 trees), we can create a formula for the size of the forest at any given time:

Number of Trees = 500,000*(1/2)^t/5 where t is the number of years.

With this equation we can answer many useful questions:

1. How many trees will be left in 50 years?

2. How many years will it take to the forest to be reduced to 10% of its original size?

No it’s your turn to work on examples!

(Introduce in-class exercise).

Close

Growth and decay are two very useful applications for exponential functions. Tomorrow will take a closer look at population growth!

Exponential Growth and Decay Models: Lesson 2

|Grade: Advanced Algebra |Section: 1 |

|Unit: Exponential Growth and Decay Models |Lesson #: 2 |

|Lesson: Exponential Growth: Limiting Factors |Date / Day: |

|T.S.W.B.A.T. |Warm-Up / Anticipatory Set: |

|Identify real-life factors that could affect exponential specifically in |The population of Phoenix, AZ is growing at a rate of 10% per year. At |

|China, what were some of the implications about One Child Policy in |this rate the city will be 10 times its current size in 25 years. Is |

|China. |this likely to happen? Why or why not? |

|Key Activities and Timing: |Resources Required: |

|Students discuss the warm-up question with a partner. Each pair shares a|Ipad for each student |

|perspective. (10 min) |Geometer’s Sketchpad and PowerPoint software installed on each Ipad |

|Class discussion: what factors can limit population growth? (5 min) | |

|Students receive an individual assignment and use in-class time to get | |

|started. (30 min) | |

| | |

|Old Homework: |Diversity / Differentiation Strategies: |

|Exponential growth and decay worksheet 1 |Cooperative learning (in pairs). |

| |Class discussions. |

| |Individual sketchpad problem solving. |

|New Homework: | |

|Individual project (assignment) | |

|IL Learning Standards Addressed: |Lesson Critique / Student Notes: |

| | |

|SSE.B.5. Interpret the parameters in a …exponential function in terms of | |

|a context. | |

|IF. C. 7e. Graph exponential functions, showing intercepts and end | |

|behavior. | |

|SSE.B. 3c Use properties of exponents to transform expressions for | |

|exponential functions. | |

|Evaluation Procedures: | |

| | |

|Observe student performance during in-class examples. | |

|Homework assigned and checked. | |

|Individual project on growth and decay. | |

|Unit quiz on exponential growth and decay. | |

| | |

Set or “Hook”

Mathematical models can lead to some wacky projections. Today we’re going to look at factors that can impact population growth, specifically at China’s One Child Policy.

Lesson Body

In response to the warm-up question and through class discussion, the following factors that can affect population growth may be identified:

Availability of resources (water, food, etc.)

Infrastructure (electricity, public transportation, roads, etc.)

Relative attractiveness of other cities

Technological innovations (air conditioning)

Government policies, China’s One Child Policy.

If we look at a range of or other factors they may include disease, medical innovations, climate changes, governmental actions, religious influences, etc.)

Today you will work on an individual assignment that looks at both the mathematical and sociological factors of population growth in China.

(Introduce the individual assignment)

Advanced Algebra

Section 1

Unit: Exponential Growth and Decay

Project

In this project you will compare the characteristics of one of the United States with the characteristics of China. In particular, you will explore the current population, the population growth trends, project future population, and discuss factors that may limit or accelerate population growth for the state/country.

You will be provided with the state/country and the necessary background information. Your project must include the following:

A two-page presentation prepared in PowerPoint:

Page 1:

1. Three defining characteristics for each state/country

2. Current population (in millions) and the projected growth rate between 1950 and 2010.

3. Three factors for each state/country that will either limit or accelerate population growth.

Page 2:

1. A graph prepared using Geometer’s Sketchpad that shows the projected population of each state/country over the next 100 years.

2. The exponential population growth function for each state/country.

3. The year and population at which each state’s/country’s population will double.

4. The year and population at which the state and country’s population will be equal (if applicable).

Your project will be graded according to the following criteria (total of 30 points):

1. Presentation is neat, well-organized, and easy to read: 10 points

2. Characteristics and factors for each state/country are insightful and accurate: 10 points

3. Growth functions and populations are accurate: 10 point

Project is due at the beginning of class on xxxxxxxx.

Close

An incredible amount of time and money is invested to study population growth. Your individual project gives you a taste of the complexity involved in projecting growth. I look forward to seeing your work tomorrow.

Exponential Growth and Decay Models: Lesson 3

|Grade: Advanced Algebra |Section: 1 |

|Unit: Exponential Growth and Decay Models |Lesson #: 3 |

|Lesson: Unit Quiz and Extension |Date / Day: |

|T.S.W.B.A.T. |Warm-Up / Anticipatory Set: |

|Show what they know. |Clear desks and prepare for the unit quiz. |

|Recall the basic concepts of amortization. | |

|Key Activities and Timing: |Resources Required: |

|Unit quiz (25 min) |None |

|Direct instruction: amortization (20 min) | |

| | |

|Old Homework: |Diversity / Differentiation Strategies: |

|Individual project (assignment) |Written quiz. |

| |Class discussion. |

|New Homework: | |

|Amortization problem (from the board) | |

|IL Learning Standards Addressed: |Lesson Critique / Student Notes: |

| | |

|SSE.B.5. Interpret the parameters in a …exponential function in terms of | |

|a context. | |

|IF. C. 7e. Graph exponential functions, showing intercepts and end | |

|behavior. | |

|SSE.B. 3c Use properties of exponents to transform expressions for | |

|exponential functions. | |

|Evaluation Procedures: | |

| | |

|Observe student performance during in-class examples. | |

|Homework assigned and checked. | |

|Individual project on growth and decay. | |

|Unit quiz on exponential growth and decay. | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

Set or “Hook”

It’s your turn to show what you’ve learned about exponential growth and decay.

Lesson Body

After completing quiz, begin lesson on amortization. Note: this is an extension. Not all students will fully grasp the concepts. The goal is to provide sufficient exposure so that students will have some recollection when they enter into an in-depth study of the topic in a later class.

Let’s think about compound interest. What would happen if you withdrew a fixed amount of money each month from the account? In other words, the account was growing (due to interest payments) and declining (due to with drawals) at the same time.

This is a common practice in the world of investments. It generates to relevant questions:

1. If I withdrew a fixed amount each month, how many months could I withdraw the amount before the account went to zero?

2. If I wanted to with draw a fixed amount for a given number of months, how much could I withdraw each months?

These questions and calculations are referred to as an amortization of the balance (or account).

Let’s look at the mathematical calculation:

i = fixed monthly interest rate

p = fixed monthly withdrawal

M = total number of months

B = account balance

We can solve for the amount that we could withdraw each month as follows:

P = B*i*(1+i)^M/[(1+i)^M – 1]

We could also solve for the balance after n months as follows:

Debt after n months = B*(1+i)^n – (p/i)*[(1+i)^n – 1]

Example:

You invest $10,000 at a monthly interest rate of 0.5%. How much can you withdraw each month if you want to be able to with draw the same amount for 60 months?

What is your balance in the account after 30 months?

Homework problem:

You are given $15,000 to cover your rent during the four years you are at college. You invest wisely, earning a fixed return of 1% per month. If you want to use the entire investment over the 4 years (48 months), what’s the maximum monthly rent that you can pay?

Close

You are now ready for the world of high finance (almost). These calculations are used for most every financing activity such as purchasing a car or a house.

Advanced Algebra

Section 1

Unit: Exponential Growth and Decay

Worksheet 1

1. Population Growth:

The population of a town in the year 1990 is 2500 and is increasing at a rate of 2.5% per year.

a. Find the population in 1992, 1995, and 1998.

b. Use a graphing calculator to graph the function for the years 1990 through 2015.

c. Use the graphing calculator to approximate the population in 2005 and 2010.

d. Verify your answers in part (c) algebraically.

2. Radioactive Decay:

Let Q (in grams) represent the mass of a quantity of radium 226, which has a half-life of 1620 years. The quantity of radium present after t years is

Q = 25(1/2)^(t/1620)

a. Determine the initial quantity (when t = 0).

b. Determine the quantity present after 1000 years.

c. Use a graphing utility to graph the function over the interval t = 0 to t = 5000.

d. When will the quantity of radium be 0 grams?

3. Data Analysis:

A cup of water at an initial temperature of 78 degrees Celsius is placed in a room at a constant temperature of 21 degrees Celsius. The temperature of the water is measured every 5 minutes for a period of ½ hour. The results are recorded in the table, where t is the time (in minutes) and T is the temperature (in degrees Celsius).

|t: (time in minutes) |0 |5 |10 |15 |20 |25 |30 |

|T: (degrees Celsius) |78.0 |66.0 |57.5 |51.2 |46.3 |42.5 |39.6 |

a. Use the regression capabilities of a graphing calculator to fit a line to the data. Use the graphing utility to plot the data points and the regression line in the same viewing window. Does the data appear to be linear? Explain.

b. Use the regression capabilities of a graphing calculator to fit a parabola to the data. Use the graphing utility to plot the data points and the regression parabola in the same viewing window. Does the data appear quadratic? Even though the quadratic model appears to be a good fit, explain why it may not be a good model for predicting the temperature of the water when t = 60.

c. The graph of the model should be asymptotic with the temperature of the room. Subtract the room temperature from each of the temperatures in the table. Use a graphing calculator to fit an exponential model to the revised data. Add the exponential model to the revised data. Add the room temperature to this regression model. Use a graphing utility to plot the original data points and the model in the same viewing window.

d. Explain why the procedure in part (c) was necessary for finding the exponential model.

Advanced Algebra

Section 1

Unit: Exponential Growth and Decay

Unit Quiz

Name: _________________________ Date: _________________________

Points Earned: ___________________ Percentage Earned: ______________

Points Available: 40 Letter Grade: _________

In 1980 the population of a town was 250,000. The town was projected to grow at a rate of 1.3 percent per year for the next 20 years. (20 points)

a. Write the formula for the population in any given year as a function of time. (8 points)

b. What is the projected population rate of this town if the government implemented a policy where couples could only have one child. Use that reate to calculate the population for 1990. (6 points)

c. If the growth rate stayed constant at 1.3 percent per year, when would the population reach 300,000? (6 points)

The typical male has 135,000 hairs on his head and loses half of his hair every two years beginning at age 50. (20 points)

a. Write the formula for the amount of hair that a man has in any given year as a function of time. (8 points)

b. What percentage of his hair will the typical male have left at age 60? (6 points)

c. At what age will a male have only 10 percent of his hair remaining? (6 points)

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