
Sample Lesson:
Course 1 Unit 2, Patterns of Change
What's
Next? (Launch)
The
United States government  like local, state and national governments
around the world  does lots of counting. Government agencies make hundreds
of census counts every year. They keep track of jobs and unemployment,
automobiles, accidents, animals, illnesses, forests, and farmlands. Every
ten years, the U.S. census bureau counts every American citizen.
Think
About This Situation 
The
population of our world changes rapidly  in ten years it will
grow by over one billion people.

a. 
What
are some of the major factors that might influence population
change in a country? 
b. 
Why
doesn't the U.S. Census Bureau make a complete count of the
population every five years or every year? 
c. 
Why
would it be important to know yeartoyear population changes,
and how you could estimate those changes without a full census
survey? 

INVESTIGATION
1 People Watching (Explore)
Brazil
is the largest country in South America, with a 1990 population of about
145 million people. Census statisticians can estimate the population of
Brazil from one year to the next using small surveys and the following
facts about patterns of change.
Population
Change in Brazil 
 Based on recent trends, births every year equal about 2.6% of
the total population.
 Deaths every year equal about 0.7% of the total population.
 The net change due to births and deaths is an increase of 1.9%
each year.

Working with your
group, use these facts to predict information about Brazil's population
in the future.
1. 
What
is the predicted change in Brazil's population by 1991 due to: 

a. 
births? 
b. 
deaths? 
c. 
all causes combined?

2. 
Use
your answers from Activity 1 to discuss the following: 
a. 
Explain
why the net population change due to births and deaths is an increase
of about 1.9% each year. 
b. 
What is the
predicted total Brazilian population in 1991? 
3. 
Using
your group's prediction for 1991 population as a basis, calculate
estimates for the change and total population in 1992; then for 1993,
1994, 1995, and 1996. Record your predictions in a table like that
below. 
Year 
Change
(in millions) 
New Total
(in millions) 
1992



1993



1994



1995



1996




The
1990 United States census report said that there were then 248 million
residents in the fifty states and territories. Before the ink was dry
on that report, the actual population had changed. So the Census Bureau
makes annual estimates of the change based on current trends.
Population
Change in the U.S. 
 Births every year will equal about 1.6% of the total population.
 Deaths every year will equal about 0.9% of the total population.
 Immigrants from other countries will add about 0.9 million people
each year.

Using the above statistics,
work with your group on Activities 47 to predict information about the
U.S. population in the future.
4. 
If
the 1990 population was 248 million, what is the predicted change
in population by 1991 due to: 
a. 
births
and deaths? 
b. 
immigration? 
c. 
all
causes combined? 
5. 
What
is the predicted total U.S. population in 1991? 
6. 
Using
your prediction for 1991 population as a basis, calculate estimates
for the change and total population in 1992; then for 1993, 1994,
1995, and 1996. 
7. 
Describe
the pattern of change in the U.S. population as time passes and compare
that pattern to the change in Brazil's population. 
Checkpoint
(Share and Summarize)
Checkpoint 
In
population studies of Brazil and the United States, you made
estimates for several years in the future, based on growth trends
from the past. 
a. 
What calculations
are needed to estimate population growth from one year to the
next in the two different countries? 
b. 
Using
the word NOWto stand for the population of the United
States in any year, write an expression that shows how to calculate
the population in the NEXTyear. 
Be
prepared to compare your expression relating NOWand NEXTwith
those of other groups. 

On
Your
Own

a.

Write
an expression using NOWand NEXTthat shows how
to use the population of Brazil in one year to calculate the
population in the next year. 
b.

If
the birth rate in the United States increased to 2.6%, like
Brazil, how would that change the population increase from one
year to the next for the U.S.? Find the predicted 1991 population.


Note: The
full Lesson 2 has a second investigation that emphasizes NOWNEXTstatements
as developed in the Checkpoint.
MORE:
Activities
Some
Sample Tasks
MODELING:
These tasks provide
opportunities for students to use the ideas they have learned in the investigations.
Each task asks them to model and solve problems in other situations.
2.

The
People's Republic of China is the country with the largest population
in the world, over 1.1 billion in 1990. Despite efforts to limit families
to one child, the population of China is still growing at a rate of
1.3% per year. 
a.

Predict the
population of China for each of the next ten years and record your
predictions in a data table. 
b.

When will the
population of China reach the 2 billion mark? 
c.

Using the word
NOWto stand for the population in any year, write an expression
that shows how to calculate the population in the NEXTyear.

d.

Suppose that
China allows 7 million people each year to leave for other countries.
How would this affect the growth of the population of China over the
next 10 years? (0.007 billion = 7 million) 
e.

Using NOWto
stand for the population in any year, write an expression that shows
how to calculate the population in the NEXTyear, assuming that
the 7 million Chinese leave annually. 
f.

Search for a
balance of growth rate and the number of people leaving China each
year that will lead to zero population growth in China. 
Return to outofclass
activities.
ORGANIZING:
These tasks will help students organize the mathematics
they have learned in the investigations and connect it with other mathematics.
1. 
The
studies of populations changing over time can be represented with
graphs if you form ordered pairs of (year, population)data.
Recall that the 1990 population of Brazil was 145 million people and
that the growth rate each year is about 1.9%. Use your calculator
to plot (year, population)data for each tenyear period from
1990 to 2050. 
a.

Make a sketch
of the plot and write a brief description summarizing the pattern
of the plotted data. 
Note: 
Sketches
for Parts (b) and (c) should be made on the same set of axes. 
b.

Sketch the pattern
of (year, population)data you would expect in Brazil if the
birth rates increased. 
c.

Sketch the pattern
of (year, population)data you would expect in Brazil if the
birth and death rates were equal. 
d.

Explain how
the pattern of points on each graph shows the NOWto NEXTchange
in the population. 
Return to outofclass
activities.
REFLECTING:
These tasks will help students think about what the mathematics
they have learned means to them. These tasks also will help students think
about what they do and do not understand.
3. 
In the population
model of this lesson, you made predictions based on given assumptions
about how the populations would change. Do you know anything about
differences between Brazil and the United States that would help explain
the differences in assumptions? 
Return to outofclass
activities.
EXTENDING:
Tasks in this section provide opportunities for students
to explore further or more deeply the mathematics they are learning.
1.

The
kinds of models of change are sometimes quite different from the ones
you have investigated so far. For example, many psychologists study
the way people learn and remember information. Suppose that when school
closes in June you know the meaning of 500 Spanish words, but you
don't study or speak Spanish during the summer vacation. 
a.

One model of
memory suggests that during each week of the summer you will forget
5% of the words you know at the beginning of that week. Make a table
showing (weeks, words in memory)for 10 weeks and describe the
pattern of data in the table. 
b.

A second model
suggests that you will forget 20 words each week. Make a table showing
(weeks, words in memory)data for 10 weeks following this model
and describe the pattern of data in that table. 
c.

Graph the data
from the two models and describe the patterns of data in those graphs.

d.

How would answers
to Parts (a)(c) be different if you knew only 300 words at the start
of summer? 
e.

Which model
do you think best represents memory loss? Explain your reasoning.

f.

Suppose 10 weeks
of summer are gone and you decide to do an intensive vocabulary review
for the remaining 2 weeks before school starts. If you are able to
regain 20% of your vocabulary each week, how many words will you know
when school begins? Which model of memory loss did you assume for
the first 10 weeks? 
Return to outofclass
activities.
Go to another sample lesson from Course 1,
Unit 7.
