Lesson: Exponential Growth
Unit: Population Growth
Materials: scrap paper(1 sheet each), pencils(1 each), large piece of
paper, markers
Age Level: 4th-6th
Space Requirements: classroom
Time: 45 minutes
Subjects: math and science
Objectives:
· Students will learn about exponential growth
· Students will see how quickly exponential growth can get out
of control
· Students will relate exponential growth to human population
density
· Students will experience the stress of overpopulation on individuals
· Students will journal on their feelings from the activity
· Students will realize that as space is limited, individuals
must invent more efficient and creative ways of using the space
Plan: 15 minutes: allowance
Directions:
· Ask the players if they would prefer to get $25 a week or a
penny the first day and doubling that amount every day. For example,
one penny the first day, two the second, four the third, eighth the
fourth and so one. Have them answer this question without deriving any
numbers mathematically.
· Once they have all answered have them figure out what the actual
amount for the second option would be. Make a chart of each day and
how much they would get each day, followed by their running total. (students
can add on the board and fill in the boxes).
· By doing the second option instead of just $100 for four weeks
they would receive $1,342,177.28. Explain to them that the second option
was exponential growth, which means the number continues to double,
and show them a curve demonstrating that.
5 minutes: Lily Lake
Directions:
· Tell the kids that there is a lovely lake with not too many
lilypads. (Draw the lake on the paper.) The people living on the lake
like to swim and fish in the lake. But the lilypad population is growing
exponentially. Put a tiny dot on a corner of the lake to represent the
current lilypad population.
· Tell them that because the lily pad population is growing exponentially,
that means it doubles everyday. Show day 1 through 7. (day 1-1 lily
pad, day 2-2 lily pads, day 3-4 lily pads, day 4-8, day 5-16, day 6-32,
day 7-64)
· Ask them, if the lake is totally covered on 31, on which day
is half the lake covered? If they are having trouble, ask them what
it takes to make a half a whole. Explain that multiplying by two is
doubling. So if the population is doubling everyday, the lake is half
covered on day 30. Show this with a dotted line on the lake. Keep cutting
the area in half and subtracting days until it becomes too small to
label.
· Ask them what they notice about the lilypad population in terms
of each day. They should notice that the population grew very slow at
first and it wasn’t until the last seven days that it grew extremely
fast. Emphasize the fact that it only took one day for the second half
of the lake to be covered. Talk about how quickly the population got
out of control. On day 25 the people might have said “what a lovely
bunch of lilypads we have.” And maybe one person is concerned about
how fast they are growing. But most people won’t recognize a problem.
But only four days later one fourth of the lake is already covered.
By then it’s practically too late to do anything.
15 minutes: hula hoop challenge
Directions:
· Start out with on large hula-hoop on the ground. Tell them
that the hula hoop is their habitat. In order to be in the habitat,
they must have both feet in the hoop. Tell them their population is
growing exponentially. Tell them they are going to have to be creative
to fit their population. This is a silent game. If they have an idea
of how they can fit, they can raise their hand and the teacher will
make them the temporary leader. Everyone must listen to each other and
work together. Put one person in the hula hoop. As the game progresses
tell the players that their population is increasing and they need to
share the space. Keep doubling the number of players in the hula-hoop.
The players will find that the more people per hoop the less space they
have. This shows the effects of population growth on individuals. (The
trick is they can sit down outside of the hoop, as long as they have
both feet in the hoop. Just keep telling them that the rule is both
feet must be in the hoop. They will figure it out). Afterwards ask the
players how they felt as the game progressed.
10 minutes: discussion
· Tell them that the human population is currently growing exponentially
and ask them what they think about that. Have a discussion about what
that means in terms of how fast our population is growing.
· Talk about the three things that humans can do to control their
population size and their impact on the environment. 1. Number of children
they have. 2. Technology. 3. Lifestyle. Discuss each of these sections
and things they can do to help reduce our impact. (i.e. buy low energy
technology, recycle, buy less stuff, reuse things).
· Draw three smaller, odd shaped circles on the board. Tell the
students that these are the wildlife reserves set aside for animals.
No one can develop these areas. Tell them that animals can only survive
on the inner part of these reserves, because they can survive right
next to humans. So draw another shape about an inch away from the previous
line. Ask students what would be a more effective way of designing these
reserves to best benefit the animals and plants. Make sure to accept
all of their ideas. If they don’t come up with it, tell them that
they could make one big reserve instead of three smaller ones, without
using up more land.
· Tell them that this is what environmentalists do. They solve
problems like the hula hoop challenge and the wildlife reserve problem.
These problems are often very controversial and hard. Make sure that
they see there is never an easy answer. All of us need to change how
we live and be good problem solvers.
Journal topic: how did you contribute to the progress of the group?
Were you helpful or
were you a distraction. What are good skills needed for teamwork?
Background Information: Exponential growth is a j-shaped curve that
can be described by the differential equation: dN/dt=rN, where dN/dt
represents population size (N) change with time (t), and r is the intrinsic
rate of increase. The mathematical equation that can be used to find
the amount given each day is x=2^(n-1). And the equation to find the
running total for any day n is x=(2^n)-1. There is a pattern between
the running total and the total each day (the running total is one cent
less then the total for the next day). Using this information to form
the above equations is called induction. Exponential growth is slow
at first, so we often don’t recognize the problem in its early
state. Growth increases rapidly and becomes out of control rather quickly.
