### 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.

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 year-to-year 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?
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 4-7 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 NOW-NEXTstatements as developed in the Checkpoint.

### MORE: Activities

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.

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 ten-year 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.

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?

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?