              Sample Lesson: Course 1 Unit 1, Patterns of Change ©2008  Change Over Time

Every 10 years, the U.S. Census Bureau counts every American citizen and permanent resident. The 2000 census reported the U.S. population to be 281 million, with growth at a rate of about 1% each year. The world population is over 6 billion and growing at a rate that will cause it to exceed 9 billion by the year 2050.

National, state, and local governments and international agencies provide many services to people across our country and around the world. To match resources to needs, it is important to have accurate population counts more often than once every 10 years. However, complete and accurate census counts are very expensive. Think About This Situation The population of the world and of individual countries, states, and cities changes over time. a. How would you describe the pattern of change in world population from 1650 to 2050? b. What do you think are some of the major factors that influence population change of a city, a region, or a country? c. How could governments estimate year-to-year population changes without making a complete census?

Your work on investigations of this lesson will develop your understanding and skill in using algebra to solve problems involving variables like populations that change as time passes. If you study trends in population data over time, you will often find patterns that suggest ways to predict change in the future. There are several ways that algebraic rules can be used to explain and extend such patterns of change over time. As you work on the problems of this investigation, look for an answer to this question:

What data and calculations are needed to predict human and animal populations into the future?

People Watching Brazil is the largest country in South America. Its population in the year 2005 was about 186 million.
Census statisticians in Brazil can estimate change in that country's population from one year to the next using small surveys and these facts:

 Population Change in Brazil Based on recent trends, births every year equal about 1.7% of the total population of the country. Deaths every year equal about 0.6% of the total population. Source: CIA—The World Factbook 2005 1. How much was the estimated change in Brazil's population from 2005 to 2006 due to: a. births? b. deaths? c. both causes combined? 2. Calculate estimates for the population of Brazil in 2006, 2007, 2008, 2009, and 2010. Record those estimates and the year-to-year changes in a table like the one below. a. Make a plot of the (year, total population) data. b. Describe the pattern of change over time in population estimates for Brazil. Explain how the pattern you describe is shown in the table and in the plot. 3. Which of these strategies for estimating change in Brazil's population from one year to the next uses the growth rate data correctly? Be prepared to justify your answer in each case. a. 0.017(current population) - 0.006(current population) = change in population b. 0.011(current population) = change in population c. 0.17(current population) - 0.06(current population) = change in population d. 1.7%(current population) - 0.6% = change in population 4. Which of the following strategies correctly use the given growth rate data to estimate the total population of Brazil one year from now? Be prepared to justify your answer in each case. a. (current population) + 0.011(current population) = next year's population b. (current population) + 0.017(current population) - 0.006(current population) = next year's population c. 1.011(current population) = next year's population d. 186 million + 1.7 million - 0.6 million = next year's population 5. Use the word NOW to stand for the population of Brazil in any year and the word NEXT to stand for the population in the next year to write a rule that shows how to calculate NEXT from NOW. Your rule should begin "NEXT = ..." and then give directions for using NOW to calculate the value of NEXT.

 The Whale Tale In 1986, the International Whaling Commission declared a ban on commercial whale hunting to protect the small remaining stocks of several whale types that had come close to extinction.      Scientists make census counts of whale populations to see if the numbers are increasing. While it's not easy to count whales accurately, research reports have suggested that one population, the bowhead whales of Alaska, was probably between 7,700 and 12,600 in 2001.      The difference between whale births and natural deaths leads to a natural increase of about 3% per year. However, Alaskan native people are allowed to hunt and kill about 50 bowhead whales each year for food, oil, and other whale products used in their daily lives. 6. Assume that the 2001 bowhead whale population in Alaska was the low estimate of 7,700. a. What one-year change in that population would be due to the difference between births and natural deaths? b. What one-year change in that population would be due to hunting? c. What is the estimate of the 2002 population that results from the combination of birth, death, and hunting rates? 7. Use the word NOW to stand for the Alaskan bowhead whale population in any given year and write a rule that shows how to estimate the population in the NEXT year. 8. Which of the following changes in conditions would have the greater effect on the whale population over the next few years? decrease in the natural growth rate from 3% to 2%, or increase in the Alaskan hunting quota from 50 to 100 per year

In studies of population increase and decrease, it is often important to predict change over many years, not simply from one year to the next. It is also interesting to see how changes in growth factors affect changes in populations. Calculators and computers can be very helpful in those kinds of studies.
For example, the following calculator procedure gives future estimates of the bowhead whale population with only a few keystrokes: 9. Examine the calculator procedure above. a. What seem to be the purposes of the various keystrokes and commands? b. How do the instructions implement a NOW-NEXT rule for predicting population change? 10. Modify the given calculator steps to find whale population predictions starting from the 2001 high figure of 12,600 and a natural increase of 3% per year. a. Find the predicted population for 2015 if the annual hunt takes 50 whales each year. b. Suppose that the hunt takes 200 whales each year instead of 50. What is the predicted population for 2015 in this case? c. Experiment to find a hunt number that will keep the whale population stable at 12,600.

 Summarize the Mathematics In the studies of human and whale populations, you made estimates for several years based on growth trends from the past. a. What trend data and calculations were required to make these estimates: i. The change in the population of Brazil from one year to the next? The new total population of that country? ii. The change in number of Alaskan bowhead whales from one year to the next? The new total whale population? b. What does a NOW-NEXT rule like NEXT = 1.03 · NOW - 100 tell about patterns of change in a variable over time? c. What calculator commands can be used to make population predictions for many years in the future? How do those commands implement NOW-NEXT rules? Be prepared to share your thinking with the class. The 2000 United States Census reported a national population of about 281 million, with a birth rate of 1.4%, a death rate of 0.9%, and net migration of about 1.1 million people per year. The net migration of 1.1 million people is a result of about 1.3 million immigrants entering and about 0.2 million emigrants leaving each year. a. Use the given data to estimate the U.S. population for years 2001, 2005, 2010, 2015, 2020. b. Use the words NOW and NEXT to write a rule that shows how to use the U.S. population in one year to estimate the population in the next year. c. Write calculator commands that automate calculations required by your rule in Part b to get the U.S. population estimates. d. Modify the rule in Part b and the calculator procedure in Part c to estimate U.S. population for 2015 in case: i. The net migration rate increased to 1.5 million per year. ii. The net migration rate changed to -1.0 million people per year. That is, if the number of emigrants (people leaving the country) exceeded the number of immigrants (people entering the country) by 1 million per year.

Note: In Investigation 2 of Lesson 2, students use spreadsheets to study patterns of change and connect the recursive formulas used to define spreadsheet cells to NOW-NEXT rules.

Applications
These tasks provide opportunities for students to use and strengthen their understanding of the ideas they have learned in the lesson.

 7. If money is invested in a savings account, a business, or real estate, its value usually increases each year by some percent. For example, investment in common stocks yields growth in value of about 10% per year in the long run. Suppose that when a child is born, the parents invest \$1,000 in a mutual fund account. a. If that fund actually grows in value at a rate of 10% per year, what will its value be after 1 year? After 2 years? After 5 years? After 18 years when the child is ready to go to college? b. Using NOW to stand for the investment value at the end of any year, write a rule showing how to calculate the value at the end of the NEXT year. c. How will your answers to Parts a and b change if: i. the initial investment is only \$500? ii. the initial investment is \$1,000, but the growth rate is only 5% per year? d. How will your answers to Parts a and b change if, in addition to the percent growth of the investment, the parents add \$500 per year to the account?

Connections
These tasks will help students to build links between mathematical topics they have studied in the lesson and to connect these topics with other mathematics that they know.

 14. Sketch graphs that match each of the following stories about quantities changing over time. On each graph, label the axes to indicate reasonable scale units for the independent variable and the dependent variable. For example, use "time in hours" and "temperature in degrees Fahrenheit" for Part a. a. On a typical summer day where you live, how does the temperature change from midnight to midnight? b. When a popular movie first appears in video rental stores, demand for rentals changes as time passes. c. The temperature of a cold drink in a glass placed on a kitchen counter changes as time passes. d. The number of people in the school gymnasium changes before, during, and after a basketball game.

Reflections
These tasks provide opportunities for students to re-examine their thinking about ideas in the lesson.

 17. How are patterns in the data tables and graphs arising in the studies of human and whale populations similar to or different from those that related: a. weight and stretched length of a bungee cord (page 5)? b. price per jump and number of customers for a bungee jump (page 6)? c. price per jump and daily income for operation of the bungee jump (page 6)? d. number of plays and fund-raiser cumulative profit in the Take a Chance die-tossing game (pages 8-9)? e. average speed and race time for a 500-mile NASCAR race (pages 11-12)? f. hours worked and earnings at Fresh Fare Market (page 12)?

Extensions
These tasks provide opportunities for students to explore further or more deeply the mathematics they are learning.

 24. The Fibonacci sequence has many interesting and important properties. One of the most significant is revealed by studying the ratios of successive terms in the sequence. Consider the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,  . a. Modify the spreadsheet you wrote to generate terms of the Fibonacci sequence to include a new column C. In cell C2, enter the formula "=B2/B1" and then repeat this formula (with cell references changing automatically down the column C). Record the sequence of terms generated in that column. b. What pattern of numerical values do you notice as you look farther and farther down column C?

Review (from Lessons 1 and 3)
These tasks provide opportunities for students to review previously learned mathematics (in this unit, middle school topics) and to refine their skills in using that mathematics.

 1. Evaluate each expression if x = 1, y = 3, a = -1, and b = 2. a. a2x + b3y b. a2(x + by) c. x3(y + 1)/(by + a3)

NOTE The next Review task is Just in Time review for Unit 2.

 2. The dot plot below indicates the number of students in the 40 first-hour classes at Lincoln High School. a. What was the smallest class size? b. What was the largest class size? c. What percent of the classes had 30 students in them? d. What percent of the students had fewer than 25 students in them? 3. Convert each of these percents into equivalent decimals. a. 75% b. 5.4% c. 0.8% d. 0.93%