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Written by Joel Bezaire for University School of Nashville www.usn.org www.pre-algebra.info

University School of Nashville

Sixth Grade Math

Self-Guided Challenge Curriculum

Unit 5

Exponential Growth

This curriculum was written by Joel Bezaire for use at the University School of Nashville, funded by a grant from Quaker Hill in the Summer of 2011. We are making

it available for anyone to use. You may not alter, distribute, or disseminate this document except for use in your own personal classroom or home-school/tutoring

situation. You may not charge a fee for this document. Do not email this document or share it via USB drive or “the Cloud” – if you wish to share it with someone please

direct them to pre-algebra.info so that person can download it for themselves from the source website.

All rights reserved by the author and USN (www.usn.org)


About This Curriculum

What you hold in your hands is a special 6th grade curriculum, designed by the math teachers at USN to make sure that the best and brightest math students in the grade are

being challenged in the way they should be challenged.

There are a number of Units in this curriculum. They are very different in the way the activities are arranged: Some use computers, and some don’t. Some require a great deal

of reading, and some don’t. Some might require you to do a great deal of writing, and some won’t. Because that is the case you may enjoy some Units more than others,

depending on your preferred activities. Because of your particular strengths, you may be invited to do some Units and not invited to do others.

What they all have in common is as follows:

1) These are all designed to be self-studies. While you may need to receive some

help from a teacher from time to time, if you find yourself having to constantly receive assistance, you might be better served to be with the rest of your 6th grade

class during this unit.

2) This curriculum is designed to be difficult. Since these Units are designed to give a challenge above and beyond the regular 6th grade curriculum, you should expect to spend some time and effort completing the tasks. Sometimes you may need to set this booklet aside and just think. Sometimes you may need to walk away and work on something else while your brain stews on a difficult problem. All of that is OK. Only when you’re convinced that you are stuck should you seek help from

a teacher. Of course, the work in this booklet should represent the work of the student, not a parent, sibling, or classmate.

3) This curriculum assumes that you enjoy learning math. We don’t put a lot of

effort into “selling” mathematics to you, trying to convince you that mathematics can be fun and worthwhile. We assume that you already think so, and that’s why

you’re tackling this curriculum!

4) Since this is a self-guided curriculum, please let a teacher know if something sparks your interest! There might be a way to spend more time on a topic that

you find particularly interesting. You won’t know unless you ask…

Have fun!


"The greatest shortcoming of the human race is our inability to understand the exponential function." – Albert Bartlett

Introduction

It’s been argued that one of the most influential forces in the world is exponential

growth. Exponential growth occurs when the amount being added to a population is proportional to the amount already in the population: In other words, the bigger the population, the faster it grows.

Examples of exponential growth can be found in biology, chemistry, social

studies, physics, and economics. We’ll examine some of each of these applications in this Unit.

It’s important that you already have a little experience with exponents before

tackling this unit, including experience working with exponents of integers.


1. A Classic Problem: Rice on a Chessboard

This is a famous old problem that is often used as an introduction to the power of exponential growth. You may have even heard it before.

A peasant created a beautiful chessboard for his king. Upon being presented

this gift, the king was overwhelmed with gratefulness. “What would you like in return for this gift?’ the king asked the peasant.

“I would simply ask for a single grain of rice on the first square, two grains of rice on the second square, four grains of rice on the third square, and so on until the

chessboard is filled with rice.” The king readily agreed to such a reasonable request, and ordered rice to be

brought to the peasant until all 64 squares on the chessboard were filled with the rice as the peasant described.

a) How many grains of rice was the peasant given after the first 5 squares?

b) How many grains of rice was the peasant given after the first 10 squares?

c) How many grains of rice was the peasant given after the first 20 squares? d) Can you calculate how many grains of rice the peasant was given in total? If you

can, tell how much. If you can’t, explain why not.


e) Can you find a way to generalize how many grains of rice were on any particular

square? In other words, can you write a formula for how many grains of rice were supposed to placed on the xth square? (If you find this difficult, plug in values for x and see if you can start to create a formula that way).

f) Can you find a way to generalize how many grains of rice were on the chessboard

after a particular number of squares? In other words, can you write a formula for how many grains of rice were on the board after x squares? Your answer from part e) might be helpful.

g) Did the king make a good bargain with the peasant? Explain.


2. The Water Lilies There’s a school with a pond in its yard. The pond is full of living creatures (fish, frogs, turtles, and more). It is also filled with beautiful water lilies: The problem with the water lilies is that they double in number every day, and if left unchecked they will grow out of control and smother the pond in 30 days, killing all other living things. After the first day of observing the pond, the students notice that the water lilies are small and they decide that they will wait until the pond is half-covered with the water lilies before worrying about cutting them back.

a) On what day will the lilies cover half the pond? Explain your answer.

b) How many days does this give the students to save the pond? Explain your

answer.

c) When they finally do decide to cut back the water lilies the students decide to

only remove half of them, since they are so beautiful. How many days from then will the pond be smothered by the water lilies? Explain your answer.


d) The students decide that they like the water lilies so much (and hate the work of cutting them back), that they propose doubling the size of their pond so that they won’t have to cut them back as often. How many extra days does this give the students before having to cut back the water lilies? Explain your answer.

e) The students decide to purchase a fish that eats the water lilies. Each fish eats exactly one water lily per day. They hope that this helps control the growth of the lilies. It is on the 10th day of the month, and they want to purchase enough fish to maintain the population of water lilies at its current number. How many fish do they need to purchase? Explain your answer.


3. World Population

Chessboards and water lilies are fun problems to play around with a get a better sense of how the numbers work when we talk about exponential growth. But how about we try a real-life application?

Below is a list of dates when earth’s population reached one billion, two billion, and

so on. For the purposes of this activity, let’s say that the population of humans was “zero” at 100,000 BCE (that’s a generally-agreed-upon estimate of when Homo Sapiens started migrating around the earth, coming out of Africa)

Year Population (Billions)

100000 BCE 0 1804 CE 1 1927 2 1960 3 1974 4 1987 5 1999 6

a) Use the next page to construct a graph. Let the horizontal axis (x-axis) represent

years and let the vertical axis (y-axis) represent population in billions. The axes are drawn for you, but you’ll have to turn the paper to orient it properly. Decide on a scale that makes sense to represent this data.

b) Use this data to approximate what year the earth’s population will hit 7 billion people. Explain your answer.

c) Use this data to approximate what the earth’s population will be in the year 2075. Explain your answer.



d) How many years are there between the start of human history and the earth’s population reaching 1 billion people?

e) How many years are there between the earth’s population reaching 6 billion people and the estimation of when the earth’s population will reach 7 billion people?

f) What reasons can you think of for why there’s such a drastic difference between

your answers in part d) and e), when both represent a change of 1 billion people?

g) What long-term implications might there be for the current rate of population

growth? Are there any problems might we have to solve in the next 100 to 200 years that humans have never had to deal with before? What might they be?


4. Exponential Decay: Atmospheric Pressure

Related to exponential growth is exponential decay. As you might guess from the name, exponential decay happens when the amount being removed from the population is proportional to the amount already in the population. Atmospheric Pressure is an example of exponential decay. Atmospheric pressure essentially describes the mass of air that exerts pressure on a surface. For example, if you travel to a very high altitude the atmospheric pressure is much less than at a low altitude because there’s much less atmosphere above you on a higher altitude than there is at a lower altitude.

There are other factors that determine atmospheric pressure, like temperature, humidity, and pollution.

Pressure is measured in a unit of measurement called Pascals or kilopascals,

which are abbreviated kPa.


The following data was generated for atmospheric pressure at 15° C and 0% humidity. For the purpose of this activity, let’s assume that the earth’s atmosphere ends (and outer space begins) at about 100 km (100 000 meters)

Altitude (meters) Atmospheric Pressure (kPa) 0 100

2000 80 4000 62 6000 48 8000 35 10000 28

a) Use the space below to create a graph that shows the relationship between altitude and atmospheric pressure. Use Altitude as your horizontal axis and atmospheric pressure as your vertical axis. Make sure your data includes all of the data points above, as well as the fact that at 100 000 m, the kPa will be zero (0).


b) Compare this graph to the one you created to represent population growth. How are they similar? How are they different?

c) Use the data/graph to estimate the altitude at which the atmospheric pressure will

be 10 kPa. d) Use the data/graph to estimate the atmospheric pressure at an altitude of 50 000

m. e)

In the picture above, the bottle on the left was sealed at an altitude of 14 000 feet. The second picture was taken at altitude 9 000 feet. The third picture was taken at an altitude of 1000 feet. Explain why the bottle was affected in the way that it was affected.


5. Folding Paper

In the last few examples, you have used data that was provided for you. In this example, you are going to provide the data.

Take a regular sheet of blank printer paper. Measure the thickness of it (it’s very small – decide on the unit of measurement you want to use, or create your own unit of measurement. It’s completely up to you). Fold the paper in half, and then measure the thickness again (use the same unit of measurement as you did before). Repeat this process until you are unable to fold the paper any longer.

Number of Folds Thickness of The Paper

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

If you are unable to fold the paper 15 times, that’s OK: Try to establish a pattern in the measurements you took to complete the table through 15 folds.


Using the information from the table you created, estimate the answers to the following questions:

a) How many folds would it take for the paper to be taller than you are? Does that answer surprise you? Why or why not?

b) How many folds would it take for the paper to be taller than your house? c) It is a distance of 149, 597, 900 km from the earth to the sun. How many folds

would it take for your sheet of paper to reach that distance? Explain your answer.


6. Exponents in Fractions

Let’s consider the expression !!!

, as the value of n increases. Let’s find the value of the expression as n gets bigger from one, and let’s find the successive sums of those values.

We can use a chart to help us organize our data. Some of it is done for you,

continue filling in the chart below.

n 12! Added to previous total:

1 12

12

2 14

34

3 18

78

4 116

5

6

7

8

9

10

11

12

13

14

15

16


a) Graph your data from the chart on the graph provided below. Let the value of n

be your horizontal axis and the running sum be your vertical axis. The first two data points are done for you.

b) Compare the shape of this graph to the graphs you created for World Population and Atmospheric Pressure. How are the three graphs similar? How are they different?


c) When will the successive sum reach the value of one (1)? Explain your answer.

d) Use a dictionary/encyclopedia (online or a physical copy) and look up the word asymptote. Can you see an asymptote in the graph you created? Explain your answer.

e) Can you generalize what the sum of !!!

will be for infinite values of n? Explain your answer.


7. Exponents With Integers

Now let’s consider the expression (−2)!, as the value of n increases. Let’s find the value of the expression as n gets bigger from one, and let’s find the successive sums of those values.

We can use a chart to help us organize our data. Some of it is done for you,

continue filling in the chart below.

n (−2)! Added to previous total:

1 -‐2 -‐2

2 4 2

3 -‐8 -‐6

4 16

5

6

7

8

9

10

11

12

13

14

15

16


a) Can you describe a pattern in the sums? b) Is there a way to predict the sum for large values of n? In other words, can you generalize a formula that will take you straight to the sum?

c) Is there an asymptote for (−2)! like there is for !!!

? Make a graph of the sums if you think it might help you. If there is an asymptote, where is it? If there isn’t, can you explain why there isn’t?


8. Problem Solving: Nice Job If You Can Get It

To close, what should be an easy problem (if you understand exponential growth.)

Suppose you were to work every day for 30 days. You have two payment options:

1) You make $1000 on the first day, $2000 on the second day, $3000 on the third day, and so on through 30 days.

2) You make $0.01 on the first day, $0.02 on the second day, $0.04 on the third day,

$0.08 on the fourth day, and so on through 30 days.

Which payment option do you choose? Explain your reasoning.

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