Using and Interpreting a Mathematical Model

25 min

Teacher Prep
Setup
Keep students in same groups. Tell students the latitude and average high temperature in September in their city. Work time followed by a whole-class discussion.

Narrative

In an earlier activity, students drew a line that best fit the latitude-temperature data and found the equation of this line. The line is a mathematical model of the situation. In this lesson, they use their model to make predictions about temperatures in cities that were not included in the original data set (MP4). If the class used their own data, adjust the information in this activity as needed. In the next activity, they also interpret the slope of the line and the intercepts in the context of this situation. This leads to a discussion of the limitations of the mathematical model they developed.

This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.

Launch

Keep students in the same groups of 3–4. If available, tell the students the latitude and average high temperature in September in their city.

Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language students use to determine if the new data fits the model. Display words and phrases such as “close to the line,” “outlier,” “above the line,” and “lower than expected.”

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization, Attention

Student Task

In an earlier activity, you found the equation of a line to represent the association between latitude and temperature. This is a mathematical model.

  1. Use your model to predict the average high temperature in September for the following cities that were not included in the original data set:
    1. Detroit (Lat: 42.33 degrees north)
    2. Albuquerque (Lat: 35.09 degrees north)
    3. Nome (Lat: 64.50 degrees north)
    4. Your own location (if available)

  2. Draw points that represent the predicted temperatures for each city on the scatter plot.

  3. The actual average high temperature in September in these cities were:

    • Detroit: 74 degrees Fahrenheit

    • Albuquerque: 82 degrees Fahrenheit

    • Nome: 49 degrees Fahrenheit

    • Your own location (if available):

    How well does your model predict the temperature? Compare the predicted and actual temperatures.

  4. If you added the actual temperatures for these 4 cities to the scatter plot, would you move your line?

  5. Are there any outliers in the data? What might be the explanation?

Sample Response

  1. Answers may vary slightly depending on the equation of the line determined by eye.
    1. Detroit prediction: −0.8742.33+111-0.8742.33+111 , or 74.274.2 degrees Fahrenheit
    2. Albuquerque prediction: −0.8735.09+111-0.8735.09 + 111 , or 80.580.5 degrees Fahrenheit
    3. Nome prediction: −0.8764.50+111-0.8764.50+111 , or 54.954.9 degrees Fahrenheit
    4. Answers vary.
  2.  

    A scatter plot with data comparing latitude and temperature, with a line of best fit. Most of the data points are black, but three are blue.

  3. Two of the predictions are very close to the actual data. The actual temperature for the third city, Nome, is farther from the prediction.
  4. They would not cause a noticeable change to the line.
  5. There are a few points that could be considered outliers in the data. They tend to happen on the higher and lower latitudes. This could mean that the model is less reliable for extreme latitudes. The point that could be an outlier in the middle of the data represents San Francisco. This might be because of the geography in the area.
Activity Synthesis (Teacher Notes)
Direct students' attention to the reference created using Collect and Display. Ask students to share how well their model predicted the temperature for the cities and why their predictions were accurate (or not). In particular, invite students to comment on any outliers in their data and what might be causing them. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond.
Standards
Addressing
  • 8.F.B·Use functions to model relationships between quantities.
  • 8.F.B·Use functions to model relationships between quantities.
  • 8.SP.A·Investigate patterns of association in bivariate data.
  • 8.SP.A·Investigate patterns of association in bivariate data.

15 min