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.

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).