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Lesson 11: What’s the Trend?

Lesson 11: What’s the Trend?

Objective:

Students will understand that the regression line is a model for a linear association (trend). They will learn to identify the direction of trends and interpret the slope and the intercept of a linear model in the context of the data.

Materials:

  1. What’s the Trend? handout (LMR_4.12_What’s the Trend)

  2. Predicting Values handout (LMR_4.13_Predicting Values)

Vocabulary:

trend positive association negative association no association linear model

Essential Concepts:

Essential Concepts:

Associations are important because they help us make better predictions; the stronger the trend, the better the prediction we can make. “Better” in this case means that our mean squared errors can be made smaller.

Lesson:

  1. Distribute What’s the Trend? handout (LMR_4.12). Students will analyze the two scatterplots on the handout. The Profits per Explosion plot shows the relationship between the number of explosions in Michael Bay’s movies and the profit earned by each movie. The Scores Over Time plot shows the relationship between M. Night Shyamalan movies made since The Sixth Sense was released in 1999 and their Internet Movie Database (IMBD) scores.

  2. In teams, students will discuss and record their responses to the following questions for each plot:

    1. What kind of plot is this? Answer: Scatterplot.

    2. What do the numbers on the x-axis represent? What do the numbers on the y-axis represent? Answer: The x-axis shows number of explosions and y-axis shows profit in millions of dollars.

    3. What is this plot telling us? Answers will vary. One example could be that if there are more explosions in a movie, then the movie will earn a greater profit.

    4. What kind of plot is this? Answer: Scatterplot.

    5. What do the numbers on the x-axis represent? What do the numbers on the y-axis represent? Answer: The x-axis shows the number of years since 1999 and the y-axis shows the movie’s IMDB score.

    6. What is this plot telling us? Answers will vary. One example could be that as M. Night Shyamalan has produced more movies, their IMDB ratings have gone down.

  3. Allow students time to discuss and record their answers to the questions.

  4. Display both plots, if possible (students may also refer to the plots in their own handout). Discuss the following questions with the whole class:

    • What is happening in each plot? What seems to be the trend? Answer: Guide students to understand that the Profits per Explosion plot shows an increasing trend, while the Scores Over Time plot shows a decreasing trend. An increasing trend is called a positive association and a decreasing trend is called a negative association.

    • What does it mean to have an increasing trend and a positive association? Answer: In Profits per Explosion, it means that as the number of explosions increase, the movie profits also increase.

    • What does it mean to have a decreasing trend and a negative association? Answer: In Scores Over Time, it means that as the years after 1999 pass, the movie IMBD ratings decrease.

  5. Quickwrite: What if we had a plot with no association? Ask students to sketch what they think a scatterplot that shows no association looks like. Answer: A correct sketch will show a scatterplot with data points that show no positive or negative association; no trend or pattern. There would be no association or a very weak one. The data would be scattered.

  6. Select a couple of sketches to share with the whole class. Discuss why the sketches show no association.

  7. Ask students to discuss their thoughts about why a line was drawn through the points of the two plots and why there are equations for each plot.

  8. Conduct a share out of their observations. Guide students to the understanding that both plots follow a linear trend. This line then represents a model for the relationship between the two variables. The equations shown in the plots above represent the lines through the points. They provide a description of the data and the relationship between the variables.

  9. Ask student teams to refer back to the What’s the Trend? handout (LMR_4.12). They should discuss the following questions and record their responses on the Predicting Values handout (LMR_4.13):

    1. What do you notice about where the points are and where the line is? Answer: Some points are near the line, others are further away, and one point is exactly on the line. Data points are observed values and points on the line are predicted values.

    2. Recall from Algebra that every line can be represented by an equation in the form y=mx+b. In this case, the equation of the regression line is y=3.2536x+154.3654. What do the x- and y-values represent in this equation? Answer: The x-values represent the number of explosions and the y-values represent the predicted profit.

    3. According to the equation, what is the slope of this line? What does the slope mean in relation to the number of explosions? Answer: The slope is 3.2536. It is the rate of change between the number of explosions and the profit. It means that for every explosion increase of 1 the profit increases by 3.2536 dollars.

    4. When the number of explosions (x-value) is zero, what is the profit (y-value)? How do you know? What does this mean? Answer: The profit is 154.3654 million dollars. Students may use the equation to show that they substituted zero for x, so the y-intercept is the profit. It means that if Michael Bay were to make a movie with NO explosions, this would be his projected profit.

    5. If you wanted to know the profit for the point that lies the closest to the line, what would the equation be? Write the equation and solve it. Answer: Profit=3.2536(211)+154.3654. Profit=840.875 or 840,875,000 million dollars.

    6. What was the actual profit for the point that lies closest to the line? Answer: The actual profit was 836,303,693 million dollars.

    7. What if Michael Bay made a movie that had 325 explosions? What would his predicted profit be? Show how you arrived at the solution. Answer: By substituting 325 in the value of x in the equation, predicted profit will be $1,211, 785, 400 or $ 1, 211.7854, or by finding the point on the line or both.

  10. If time permits, have students answer the following questions about the Scores Over Time scatterplot in LMR_4.12_What’s the Trend.

    1. What do you notice about where the points are and where the line is?

    2. What do the y- and x-values represent in this equation?

    3. According to the equation, what is the slope of this line? What does the slope mean?

    4. When the x-value is zero, what is the y-value? How do you know? What does this mean?

    5. What would the predicted value of the score be if M. Night Shyamalan released a movie in 2015? How do you know?

Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.

Homework

Students will finish answering the questions above about the Scores Over Time scatterplot in LMR_4.12_What’s the Trend referenced above.