# Lesson 21

Predicting Populations

## 21.1: Notice and Wonder: Changing Populations (5 minutes)

### Warm-up

The goal of this warm-up is to introduce the data students will examine in the lesson. They will eventually try to model these city populations with an appropriate linear or exponential function, but the starting point for any model is to ask questions about and observe general trends in the data.

### Student Facing

Here are the populations of three cities during different years.

City 1950 1960 1970 1980 1990 2000
Paris 6,300,000 7,400,000 8,200,000 8,700,000 9,300,000 9,700,000
Austin 132,000 187,000 254,000 346,000 466,000 657,000
Chicago 3,600,000 3,550,000 3,400,000 3,000,000 2,800,000 2,900,000

What do you notice? What do you wonder?

### Activity Synthesis

Invite students to share their observations and questions, and then to see if they could, based on the collective responses, describe more generally any trends in each city's population. For examples:

• Paris has been growing steadily but not at a constant rate.
• Austin has been growing very rapidly.
• Chicago's population has decreased, at a variable rate, but it increased in the decade from 1990 to 2000.

## 21.2: Population Predictions 1

### Activity

This is the first of two modeling tasks prompting students to apply what they have learned about exponential growth and decay as well as linear functions. Students have just seen how linear functions change by equal differences over equal intervals and exponential functions change by equal factors over equal intervals, so they are likely to look for these features. They investigate the growth of three city populations over a period of 50 years. The data is real and thus not exactly linear or exponential. But the cities are carefully selected so that one population can be approximated well with a linear function, one can be approximated well with an exponential function, and one is neither linear nor exponential.

When writing equations to model the populations, a natural variable to use is decades, because the populations are only given each decade. Students can choose to use years, but then finding the growth rate for Austin is problematic. If students get stuck, consider suggesting the use of decades to measure time.

Once students create their models and make predictions, consider sharing more recent population data so they can check their predictions.

City 2010 2017
Paris 10,500,000 11,000,000
Austin 790,000 950,000
Chicago 2,700,000 2,730,000

Monitor for different strategies as students decide which model to use. Students may:

• calculate differences and quotients over the decades to see if a linear or exponential model is more appropriate
• plot the population values to see graphically if the data is linear or exponential

Once students decide which model to use, they may use different ways to write the equations for the model. Monitor for students who:

• use coordinates of two points to find differences or quotients
• find differences or quotients and then find their averages
• fit a line or a curve through the data points to approximate the growth (or use technology to fit a line or a curve) and then find the equation for that line or curve

No time estimate is specified for this activity as the length would depend on the expectations about how students' models are to be presented and discussed. For example, if they are to create visual displays for their models and either present them or do a gallery walk, the activity would require at least one class period.

### Launch

Arrange students in groups of 3 or 4. Display the populations of the three cities from the warm-up.

City 1950 1960 1970 1980 1990 2000
Paris 6,300,000 7,400,000 8,200,000 8,700,000 9,300,000 9,700,000
Austin 132,000 187,000 254,000 346,000 466,000 657,000
Chicago 3,600,000 3,550,000 3,400,000 3,000,000 2,800,000 2,900,000

Ask students: Can this information be used to predict today's population in the three cities? How? What about the population in 2030? Give students a minute of quiet think time and then time to share their thoughts with their group. Invite groups to share their responses. Highlight responses that suggest that we observe whether they grow linearly or exponentially (either by calculation or by graphing) and then creating mathematical models accordingly.

Provide access to graphing technology. If students are to present their models on visual displays, provide access to tools for creating visual displays.

### Student Facing

Here are population data for three cities at different times between 1950 and 2000. What does the data tell us, if anything, about the current population in the cities or what the population will be in 2050?

City 1950 1960 1970 1980 1990 2000
Paris 6,300,000 7,400,000 8,200,000 8,700,000 9,300,000 9,700,000
Austin 132,000 187,000 254,000 346,000 466,000 657,000
Chicago 3,600,000 3,550,000 3,400,000 3,000,000 2,800,000 2,900,000
1. How would you describe the population change in each city during this time period? Write one to two sentences for each city. Then discuss with your group.
2. What kind of model, linear or exponential, both, or neither do you think is appropriate for each city population?
3. For each population that you think can be modeled by a linear and or exponential function:
1. Write an equation for the function(s).
2. Graph the function(s).
4. Compare the graphs of your functions with the actual population data. How well do the models fit the data?
1. Use your models to predict the population in each city in 2010, the current year, and 2050.
2. Do you think that these predictions are (or will be) accurate? Explain your reasoning.

### Anticipated Misconceptions

Students may continue to find modeling tasks uncomfortable and challenging. Remind students that real-world data is often very "messy" and we should use the tools we have to approximate and estimate values as best as we can, but it will probably not line up exactly.

### Activity Synthesis

Focus the discussion on how students decided on the model to use for Paris and Chicago, and the specific models (the equations) they chose. Either linear or exponential can be justified, though students can also argue that neither of these models is appropriate because neither the successive differences (linear) nor the successive quotients (exponential) are close to being constant.

Ask students what features they would expect in a good linear or exponential model. Important responses include:

• The population predicted by the model is close to the actual data.
• The general trend predicted by the model respects the data (e.g.,the overall rate of growth for Paris is slowing down as time progresses).

Ask students how they went about fitting the line or curve for their linear or exponential models. Possible responses include:

• plotting the points and using a ruler to sketch a line that fits the data well, and then finding an equation for the line
• finding a line or exponential curve that goes through two of the data points, and then using it to find the slope or the factor of growth
• finding an average of differences over equal intervals, and then fitting that line through the first data point
• finding an average of factors of growth over equal intervals, and then fitting an exponential function through the first data point

If no students chose to plot the data or to test their models using graphing technology, consider demonstrating this to show that the parameters in linear and exponential models can be varied to check the fit of a model. This will help students decide which, if either, is better and also analyze their own models.

Conversing: MLR2 Collect and Display. During the discussion, listen for and collect the language students use to describe linear and exponential situations. Call students’ attention to language such as “successive differences,” “successive factors or quotients,” or key parts of the equation. Write students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students compare and contrast linear and exponential models using real-world data.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing

## 21.3: Population Predictions 2

### Activity

This task provides students with a more open-ended modeling opportunity within a population context. Students consider the world population by decade from 1950 through 2000, and then examine the annual world population for the last 70 years. Data are not provided in the problem, allowing students to investigate using online resources.

If students use the online resource suggested in the lesson narrative for world population estimates, restrict their attention to one column of the table. The big list of numbers may still be intimidating. Ask students what an appropriate level of accuracy for reporting the world populations is (to the nearest 10 million will probably give the clearest pattern, to the nearest 100 million would also be mathematically appropriate). Because there is a lot of information, it could be worth uploading the data in advance so that students can see a visual display and the data could be distributed more readily.

No time estimate is specified for this activity as the length would depend on the extent of students' research and analysis, as well as on the decisions about how their models are to be presented and discussed.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Display the following table from the task statement for all to see. Invite students to observe the table and share what they notice and what they wonder.

After students have worked on the first two problems, consider pausing for a discussion and showing a short video about the growth of world population. A video like this will help explain the incredible human population explosion of the last 200 years and can be found by searching 'human population growth' in an online search engine. This growth cannot continue indefinitely due to limited resources and we are likely nearing a turning point in the history of human population growth.

### Student Facing

 year world population in billions 1804 1927 1960 1974 1987 1999 2011 1 2 3 4 5 6 7
1. Would a linear function be appropriate for modeling the world population growth over the last 200 years? Explain. If you think it is appropriate, find a linear model.
2. Would an exponential function be appropriate for modeling the world population growth over the last 200 years? Explain. If you think it is appropriate, find an exponential model.
3. From 1950 to the present day, by about what percentage has the world population grown each year?
4. From 1950 to the present day, by about how many people has the world population grown each year?
5. If the growth trend continues, what will the world population be in 2050? How long do you think the growth will continue? Explain your reasoning.

### Student Facing

#### Are you ready for more?

Another common model for population growth which fixes some of the improbable predictions of the exponential model is called a logistic model. A sample function $$f$$ of this type is the function $$\displaystyle f(t)=\frac{10}{1+50\boldcdot 2^{\text-t}}.$$

Evaluate this function for integer values of $$t$$ between 0 and 15. Describe qualitatively how this function differs from an exponential one. What happens to the world population in the long run according to this model?

### Anticipated Misconceptions

Students may not notice that the years in the table are not consecutive. Help point out that the table shows only the years at which the next billion person milestone was reached.

### Activity Synthesis

The way the data is presented shows that the world population is not growing linearly. If it were, then it would take about the same number of years to grow by 1 billion. To bring this point home, ask questions such as:

• How many years did it take for the population to grow from 1 billion to 2 billon? (123 years) What about from 6 billion to 7 billion? (12 years)
• Why was that the case? Does that make sense? (With a smaller population, it is a larger percentage change to increase by 1 billion. With a larger population, however, growing by 1 billion is a smaller percentage change, so it does not take as long to happen.)

To test whether or not the data might be exponential is more subtle. One good strategy is to check some doubling times. It took:

• 123 years for the population to double from 1 billion to 2 billion
• 47 years for the population to double from 2 billion to 4 billion (and it looks like it will take maybe another 50 years for the population to double from 4 billion to 8 billion)
• 39 years to double from 3 to 6 billion

Based on this information, an exponential model for the entire period of time is probably not the best (the growth rate was faster in the 20th century than in the 19th), but a couple of different exponential models might work well (i.e. one for each century). If students dig deeper into the data, they will find that a linear model is remarkably good for the past few decades. If the videos listed in the Launch were not shown earlier (after the first two problems), consider showing them here.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support the whole-class discussion of whether the world population is growing linearly or exponentially. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because….” or "I disagree because….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. This will help students share their strategies and understanding in choosing a linear or exponential model for a set of data.
Design Principle(s): Support sense-making