The purpose of this warm-up is to familiarize students with data in a context that they will be using in a later activity. While students may notice and wonder many things about these data, estimates of average rate of change and curiosity about models for the populations are the important discussion points.

This Warm-up prompts students to make sense of a problem before solving it, by familiarizing themselves with a context and the mathematics that might be involved (MP1). 

Ask students to share the things they noticed and wondered about. Record and display, for all to see, their responses without editing or commentary. If possible, record the relevant reasoning on or near the table. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If a general description of the trends in population and possible models for population do not come up during the conversation, ask students to discuss this idea.

For example:

• 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.

This is the first of two modeling tasks prompting students to apply what they have learned about exponential growth and decay as well as about linear functions. Students have just seen how linear functions change by equal differences over equal intervals and how 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 for 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.

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

Monitor for students who create their model by:

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

Plan to have students present in this order to support more accurate models based on multiple data values.

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.

Invite previously selected groups to share how they decided what models to use, including the values they selected in their models. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “What would you expect in a good linear or exponential model?” (The population predicted by the model is close to the actual data. The general trend predicted by the model follows the shape of the data in the graph (for example, the overall rate of growth for Paris is slowing down as time progresses).)
• “How do the slopes for the linear models compare to one another using these different methods?” (They are all very close, at around 0.7 for Paris.)
• “How do the growth factors compare for the exponential models?” (They are also very close, at around 1.38 for Austin.)
• Use technology to display the data on a graph along with some of the models that are shared. “Looking at the data on a graph and the different models that we came up with, which models are the best?”

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 population 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 so that the data could be distributed more readily.

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

The way the data is presented shows that the world population is not growing linearly. If it were, then it would consistently 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 billion?" (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 (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 suggested in the Launch were not shown earlier (after the first two problems), consider showing them here.