Lesson 12

Using and Interpreting a Mathematical Model

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12.1: Using a Mathematical Model (15 minutes)

CCSS Standards


Addressing

Routines and Materials


Instructional Routines

Optional activity

In the previous 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. 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.

Launch

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

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. 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 Facing

In the previous 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 at the following cities that were not included in the original data set:

    1. Detroit (Lat: 42.14)

    2. Albuquerque (Lat: 35.2)

    3. Nome (Lat: 64.5)

    4. Your own city (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: \(74^\circ\text{F}\)

    • Albuquerque: \(82^\circ\text{F}\)

    • Nome: \(49^\circ\text{F}\)

    • Your own city (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 four cities to the scatter plot, would you move your line?

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

Student Response

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Activity Synthesis

Invite some groups to share their results and compare predictions for the different cities. Ask if students think that the line is a good mathematical model to predict the temperature in a location if you know the latitude.

Representing, Speaking, Listening: MLR2 Collect and Display. Use this routine to collect verbal and written ideas about students’ mathematical models and the limitations of them that they see. During the synthesis discussion, chart key ideas that students share out about the slope, intercept, and differences in their predictions. Remind students that we call this a “mathematical model” (i.e., the line they drew). It is one way to look at data. Ask students, “What are the limitations of this model in making predictions?” Add their language to your chart. Reference this chart in the following lesson and make revisions as needed with student input.

Design Principle(s): Support sense-making; Maximize meta-awareness

12.2: Interpreting a Mathematical Model (15 minutes)

CCSS Standards


Addressing

Optional activity

Students interpret the slope and intercepts in the context of the situation. They also discuss limitations of the mathematical model. 

This activity can be extended by having students investigate if temperature on other continents or across continents fits the same pattern that we found for North America.

Launch

Keep students in same groups of 3–4.

Student Facing

Refer to your equation for the line that models the association between latitude and temperature of the cities.

  1. What does the slope mean in the context of this situation?
  2. Find the vertical and horizontal intercepts and interpret them in the context of the situation.
  3. Can you think of a city or a location that could not be represented using this same model? Explain your thinking.

Student Response

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Anticipated Misconceptions

Students may want to say “For every one unit increase in \(x\), \(y\) decreases by 1.07 units.” Ask them to use the specific units and quantities in the model, latitude in degrees north and temperature in degrees Fahrenheit.

Activity Synthesis

Invite students to share their responses. Discuss the limitations and uses of the model. Consider asking the following questions:

  • “What are some limitations of the model?”
  • “Do limitations mean that the model is not good?” (No, it just means that we have to be aware of when we can use it and when we can’t use it. Our model is pretty good for latitudes between 25 and 65 degrees north, and for locations in North America.)
  • “What questions do you have about predicting temperature?”
  • “How could you extend your investigation of predicting temperature or the weather?” 
Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention