4.1: Temperature Changes (5 minutes)
In this activity, students 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. This is a chance to make connections with some science concepts.
It is not important that students come up with an exhaustive list. They should just get the idea that there are many factors so that they are open to the idea that we’ll have to make some choices for our model and clearly define any variables we want to consider.
Arrange students in groups of 3–4. Tell students that they are starting an investigation on how to predict the weather, in particular the temperature. Check that students understand the example given in the activity statement, that as the time of day changes, the temperature often changes in a predictable way. They will brainstorm other factors that also influence the temperature.
Design Principle(s): Support sense-making; Cultivate conversation
What factors or variables can influence the outside temperature?
Make a list of different factors.
Write a sentence for each factor describing how changing it could change the temperature.
Example: One factor is time of day. Often, after sunrise, the temperature increases, reaches a peak in the early afternoon, and then decreases.
Invite students to share some of the factors they have come up with. Note that many of them are geographical. Point out that making a model that takes into consideration all or even many of these factors is very complex (weather forecasting is really difficult!). In mathematical modeling, we often start by fixing or disregarding (or randomizing) all but one of the factors. In the next activity, we want to pick just one—latitude—and investigate how just changing the latitude changes the temperature.
4.2: Is Temperature a Function of Latitude? (15 minutes)
One mathematical concept students learned about in grade 8 that relates to variables is the idea of a function. In this activity, students discuss if this is a helpful concept to investigate latitude and temperature. To be able to use functions, variables have to be very well defined, and in this case, many restrictions are necessary to have a functional relationship where each input has exactly one output.
Keep students in the same groups. Ask students if they can explain exactly what latitude is. It was briefly discussed in the previous activity as one of the factors that can influence the temperature. Students should understand that latitude is a way to measure how far north or south of the equator a place is located. The unit of measurement is degrees north or degrees south.
Draw a sphere (representing Earth) for all to see and indicate the circle representing the equator. Draw a quarter circle starting somewhere on the equator and ending at the North Pole. A location on the equator is at 0 degrees north (or south), and the North Pole is at 90 degrees north. Note that the angle made by the arc is a 90-degree angle. Since North America lies entirely north of the equator, all latitudes there have units of degrees north.
In this activity, students look at locations at 47.6 degrees north. Ask students where on the highlighted quarter circle this latitude is located. Then draw in the corresponding longitude circle (parallel to the equator) to show that there are many locations that have the same latitude.
Supports accessibility for: Language; Conceptual processing
Andre and Lin are wondering if temperature is a function of latitude.
Andre says, “I think it is, as long as we fix the time when we are measuring the temperature.”Lin says, “But what if you have two places with the same latitude? Look at this weather map for Washington State. Seattle and Spokane have the same latitude but different temperatures right now.”
What do Andre and Lin mean?
Andre and Lin are discussing whether it is possible to define latitude and temperature in a way that makes sense to talk about temperature as a function of latitude. They are considering different options. What are some advantages and disadvantages of each option?Here are the options:
Finding the temperature right now in cities with different latitudes.
Finding the daily high temperature in cities that have different latitudes.
Finding the average high temperature in a specific month, for example, September in cities that have different latitudes.
Finding the average yearly temperature in cities that have have different latitudes.
Discuss why it is important to clearly define the variables.
- “Temperature” is a very general idea, and we have to decide what is the most appropriate measure of temperature for our investigation. There are many different choices, but some are more appropriate than others. We want to look at an average since this will be more representative than the temperature at one point in time. It also evens out some of the other factors that could influence temperature, such as random weather events.
- Latitude is a tricky variable since each latitude value represents infinitely many locations that all have different temperatures. We can fix one line of longitude, but then how representative will our results be?
There are two main takeaways of this activity:
- It is important to clearly define the variables of the model.
- The function concept might be too restrictive for us to use for our model.
Statistical methods are a better tool for investigating the relationship between latitude and temperature. We’ll look at these in the next activity.
4.3: Is There an Association Between Latitude and Temperature? (10 minutes)
The idea of a function is very limiting when we wish to analyze the relationship between latitude and temperature. Looking at the situation from a statistics point of view is more helpful. In this case, the question becomes: Is there an association between latitude and temperature? This activity asks students to recall the setup necessary to answer this question. As an important step of mathematical modeling, they think about what data they need to collect, how they can collect it, and what methods will help them to analyze the data. This step is often done for students to save time, but it is non-trivial and even though in the next lesson the data will be provided, it is worthwhile for students to think about this step and to come up with a plan. If it is appropriate for a class, students can collect their own data rather than use the data provided.
Tell students that as they saw in the previous activity, the function concept might be too restrictive in this case to create a useful model. Statistical methods are a better tool for investigating the relationship between latitude and temperature.
Brainstorm examples from earlier in the year on finding associations between variables, for example, year and price of a car, or weight of a car and fuel efficiency.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Lin and Andre decided that modeling temperature as a function of latitude doesn’t really make sense. They realized that they can ask whether there is an association between latitude and temperature.
What information could they gather to determine whether temperature is related to latitude?
What should they do with that information to answer the question?
To highlight some of the methods and ideas students will need, ask:
- “What are some ways we viewed associations between two variables in the past?” (We made scatterplots and fit trendlines to the data.)
- “What are some things we need to keep in mind when we collect our data?” (We should use a variety of north-south and east-west locations, or “enough” data to be able to draw conclusions.) (If not mentioned, tell students that three cities is definitely not enough, and 100 cities in the same vicinity is not necessary.)
In the next lesson, we will pick a particular measure of temperature—average high temperature in September—and analyze data to see if there is an association.