Lesson 4

This optional sequence of three lessons can be done any time after the unit 6 in the course. Students develop and use a mathematical model to predict temperature given the latitude of a location. The activities in this sequence of lessons provide students a chance to go more deeply and apply grade 8 mathematics to a real-world context. Students get a chance to engage in many aspects of mathematical modeling (MP4). The activities in this lesson build on each other, and so should be done in order. It is not necessary to do all three lessons to get some benefit, although more connections are made the further one gets.

In the first lesson, students investigate whether there is a relationship or a pattern of association between the north-south location (in North America) of a place and the temperature. This is a vague question, and the first step is to clarify the variables that we will consider for a mathematical model.

• The first activity gives students a chance to think about different factors that influence outside temperature. Some are geographical (latitude, desert or sea climate, elevation), others are time of year, cloud cover, time of day, etc. As a segue into the second activity, we ask if it is possible to vary just one factor so that we can predict how the temperature will respond. In particular, if we vary latitude, can we predict what happens to the temperature?
• In the second activity, students investigate whether the concept of a function is a good tool to model this situation: Is temperature a function of latitude? There are several issues with this question. The biggest one is that for the same latitude, we will get different temperatures at any given time. If we really want a functional relationship, then we would have to make many restrictions. For example, we could fix time and longitude. Then for each latitude as input we can report a unique temperature as output. This brings up the question of how meaningful this model would be. We will look at a variety of possibilities and discuss pros and cons.
• In the third activity, students discuss if a more meaningful model might be to look at an association between latitude and temperature, much like what they encountered in an earlier unit on bivariate data. Data like this is easy to find, and we don’t have to worry about repeating “input” values.

In the next two lessons, students construct a mathematical model, analyze the model, use it to make a prediction, and discuss limitations of the model.

Throughout these lessons, students make and discuss choices, assumptions, and approximations as part of their work. While some of the choices and decisions are made as a class or through the sequencing of the classroom activities, students get a chance to grapple with all of these steps in the modeling cycle.

If it is feasible, students could be making more of the decisions themselves. For example, groups could come up with different proposals of how to investigate the relationship between temperature and latitude, present, get feedback, finalize, and then continue with different data for their model (for example, different months, different locations, different continents, or overall average temperature). For the full modeling cycle, they could then revise their assumptions and come up with a revised model.

There are many extensions possible for these activities. Students could investigate if there is a similar association between latitude and temperature in other parts of the world. They could look at different measures of temperature like yearly average, yearly average high or low temperature, or average high or low temperature in a different month.

For the first activity, consider finding out the latitude and average high temperature in September for your city.