What you should learn

To graph and interpret points on a scatter plot

To draw and write equations for best-fit lines and make predictions by using those equations

To solve problems by using models

NCTM Curriculm Standards 2, 5 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Scatter Plot

Positive Correlation

Negative Correlation

Best-Fit Line

Regression Line

Introduction: If you get a good score on you SAT does that mean you have a good chance of graduating from college? The table below shows the average SAT (Scholastic Assessment Test) scores for freshmen at selected universities and the graduation rate of those students.

To determine if there is a relationshp between SAT scores and graduation rates, we can display the data points in a graph called a scatter plot. In a scatter plot, the two sets of data are plotted as ordered pairs in the coordinate plane. In this example, the independent variable is the average SAT scorem and the dependent variable is the graduation rate. The scatter plot is shown below. The graph indicates that higher SAT scores do not necessarily result in higher graduation rates.

One way to solve real-world problems is to use a model. Scatter plots are an excellent way to model real-life data to observe patterns and trends.

Look for a relationship between x and y in the graphs below.

Exercise 1: The table below shows 13 of the fastest-growing cities in the United States and their latitude.

a. Draw one scatter plot to represent the correlation between the rankings and each city's latitude, and another to represent the correlation between the rankings and each city's longitude.

b. Is there a correlation between the cities' locations and their popularity?

 

 

You can use a scatter plot to make predictions. To help you do this, you can draw a line, called a best-fit line, that passes close to most of the data points. Use a ruler to draw a line that is close to most or all of the points. Then use the ordered pairs representing points on the line to make predictions.

A best-fit line shows if the correlation between two variables is strong or weak. The correlation is strong if the data points come close to, or lie on, the best-fit line. The correlation is weak if the data points do not come close to the line.

Exercise 2: Dogs age differently than humans do. You may have heard someone say that a dog ages 1 year for every 7 human years. Howver, that is not the case. The table at the right shows the relationship between dog years and human years.

a. Draw a scatter plot to model the data and determine what relationship, if any exists in the data.

b. Draw a best-fit line for the scatter plot.

c. Find an equation for the best-fit line (By find two points and using the information learned in Section 6-2)

d. Use the equation to determine how many human years are comparable to 13 dog years.

 

 

A regression line is the most accuarate best-fit line for a set of data, and can be determined with a graphing calculator or computer. A graphing calculator assigns each regression line an r value. This value (-1r 1) measures how closely the data are related. -1 indicates a strong negative correlation and 1 indicates a strong positive correlation.

Activity: Exploration: Graphing Calculators.

The table below shows teh years of rate increases and the fares charged to ride the subway in New York City.

a. Use the Edit option on the STAT menu to enter the year data in L1 and the fare data into L2.

b. Use the window [1950, 1995] with a scale factor of 5 and [0, 1.5] with a scale factor of 0.25. Make sure any equations are deleted from the Y= list. Then press 1. Make sure the following items are highlighted: On, the first type of graph (scatter plot), L1 as the Xlist, and L2 as the Ylist. Press , and describe the graph.

c. Then find an equation for hte regression line.

ENTER: 5 L1 , L2

The screen displays the equation y = ax + b and gives values for a, b, and r. Record these values on your paper.

d. Now graph the best-fit line

ENTER: Y= 5 7

YOUR TURN

a. How well do you feel the graph of the equation fits the data? Justify your answer.

b. Do any of the data points lie on the best-fit line? If so, name them.

c. Remove the ordered pair (1953, $0.15) from the set of data and repreat the process. What impact has removing this ordered pair had on the fit of the line to the data? (Hint: Compare the r values)

d. Use the equation and graph to predict the fares for the year 2000.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 11 - 29 odd, 31 - 35

Alternative Homework: Enriched: 12 - 24 even, 25 - 35

Extra Practice: Students book page 770 Lesson 6-3

Extra Practice Worksheet: Click Here.

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