Equations of Line of Best Fit

Objectives

In this lesson, students will plot data and find the line of best fit. Students will:

· perform an experiment and create a scatter plot.

· graph lines of best fit and write the equations of lines of best fit.

Essential Questions

How are relationships represented mathematically?

How can data be organized and represented to provide insight into the relationship between quantities?

How can expressions, equations, and inequalities be used to quantify, solve, model, and/or analyze mathematical situations?

How can mathematics support effective communication?

How can patterns be used to describe relationships in mathematical situations?

How can probability and data analysis be used to make predictions?

How can recognizing repetition or regularity assist in solving problems more efficiently?

How does the type of data influence the choice of display?

How is mathematics used to quantify, compare, represent, and model numbers?

How precise do measurements and calculations need to be?

In what ways are the mathematical attributes of objects or processes measured, calculated and/or interpreted?

· How can we determine if two variables correlate linearly?

· How can we use data to make predictions about the future?

Vocabulary

· Correlation: A measure of the relationship between two variables. [IS.1 - Struggling Learners]

· Continuous: The representation of data for which no individual values other than a range between intervals can be established. Continuous data is usually associated with physical measurements such as growth.

· Discrete: The representation of data for which one-to-one correspondence is established between individual points of data and the medium of representation. Discrete representations are often associated with countable objects such as populations.

· Line of Best Fit: The line that most closely fits the bivariate data.

· Patterns: Regularities in situations such as those in nature, events, shapes, designs, and sets of numbers.

· Scatter plot: A graph of plotted points that show the relationship between two sets of data.

Duration

90–120 minutes/1–2 class periods

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

· Matching Stroop (rhymes with loop) Cards (M-A1-6-2_Matching Stroop Cards.doc)

· Not Matching Stroop Cards (M-A1-6-2_Not Matching Stroop Cards.doc)

· two Stopwatches (or other methods of keeping time) per group

· Stroop Lab Activity Sheet (M-A1-6-2_Stroop Lab Activity Sheet.doc)

· Stroop Lab Instructions (M-A1-6-2_Stroop Lab Instructions.doc)

· Stroop Lab Extension (M-A1-6-2_Stroop Lab Extension.doc)

· Calculate the Line of Best Fit Worksheet (M-A1-6-2_Calculate LOBF Worksheet and KEY.doc)

· Lesson 2 Exit Ticket (M-A1-6-2_Lesson 2 Exit Ticket and KEY.doc)

· rulers

· graph paper

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

Formative Assessment

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- The Stroop lab activity gives students an opportunity to sense the difference in response times between matching and nonmatching opposites.
- Lesson 2 Exit Ticket evaluates students’ knowledge of how to use sequential data, associate it with corresponding values, draw a scatter plot, and make a meaningful interpretation.

Suggested Instructional Supports

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Active Engagement

W:	Students will know immediately from the attention getter that this lesson is about data and using that data to gather information. They know they will be partaking in an experiment and using their own data for this lesson.
H:	This lesson is introduced by telling students they are going to take a psychological test called the Stroop Lab. At first students might not understand why they’re doing the lab, but after a few minutes they will enjoy the challenge. They will wonder what they’re going to do with the data and that keeps them interested to see what is next.
E:	Students will be actively engaged in this lesson because they will be responsible for collecting their own data. Students feel more attached to lessons when it pertains to their own experiences. One good thing about this lab is that the experiment itself is not difficult, so any level of student can perform it. If higher level students need enrichment, there is an activity for them. If lower level students need modeling or scaffolding, the teacher can provide it by beginning with simpler examples that use smaller quantities, and then building larger quantities, and leading to more complex examples and greater quantities. Scaffolding may also consist of starting with positive integers and then leading into using increasing levels of abstraction, such as by using fractions and decimals.
R:	Students will be given ample time to reflect and revisit their thought processes. The teacher will observe and give feedback throughout the lesson and all activities.
E:	Students will express their understandings while they are working throughout the Stroop Lab and will be given time to self-evaluate during the work time on the Calculating the Line of Best Fit Worksheet.
T:	This lesson is tailored to all learners. There is an enrichment worksheet for students who are at or going beyond the standards and the activity is simple enough for those students needing opportunities for additional learning.
O:	This lesson begins with a fun lab, leads to a worksheet for a think-pair-share activity, and finishes with an independent exit ticket.

IS.1 - Struggling Learners

If there are any struggling students who need introduction to the vocabulary words, spend some time giving examples, showing pictures and diagrams to help students understand the language of mathematics. Perhaps constructing an activity of shape matching might be appropriate for students who may need this. Students can also identify the shapes in their environment. Struggling students may need to have concrete examples in their hands.

Instructional Procedures

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“One of the main uses of data analysis is to make predictions about real-world situations. The first step is to collect data. We are going to perform an experiment from cognitive psychology (the study of how the human brain works). The experiment is named after the man who first performed it, J.R. Stroop.”

Part 1

Hand out the Stroop Lab Instructions and the Stroop Lab Activity Sheet (M-A1-6-2_Stroop Lab Instructions.doc and M-A1-6-2_Stroop Lab Activity Sheet.doc). Also hand out the Matching Stroop cards and the Not Matching Stroop Cards (M-A1-6-2_Matching Stroop Cards.doc and M-A1-6-2_Not Matching Stroop Cards.doc). These should be cut up beforehand so they look like decks of cards. One student at a time is going to name the colors of ink in the matching list. This means the word and the color of the word match. The other two students will be the timers. Then do this process with the nonmatching list. Remember, the student names the color of the ink, not what the word says. (Example: If the printed word is Blue but the ink used is red, the student would say “red.”)

Students will fill out the activity sheet, answering the questions and making the scatter plot. They might need some guidance at drawing the line of best fit or finding the equation.

“A line of best fit is a line that goes through a scatter plot and fits most of the data. It tries to go through as many points as possible, with half the points above the line and half the points below the line. Once we have the line drawn in, how do we write an equation for it?” Hopefully students will volunteer ideas. “We find the slope of the line, using two points on our drawn line, as well as the y-intercept, using a point on our drawn line. The equation will be in slope-intercept form and is called the line of best fit. We use this equation to make predictions.”

“The line of best fit is the line that most closely models the bivariate (two-variable) data. Consider the following example.”

Annual Salary	$54,000	$48,000	$36,000	$39,000	$51,000	$56,000	$60,000	$59,000	$45,000
Home Price	$178,000	$165,000	$125,000	$136,000	$175,000	$180,000	$185,000	$182,000	$150,000

“Let’s draw a scatter plot, draw in our estimated line of best fit, and write the equation of the line of best fit.”

“Let’s first find our slope by locating two points on the line we drew, or two points very close to that line.”

“We can use the points (45000, 150000) and (60000, 185000).”

m=y2-y1x2-x1

m=185000-15000060000-45000

m=3500015000

m≈2.3

“We can now use one of those points to find the y-intercept.”

“Let’s use the point (45000, 150000).”

“The slope-intercept form of a line is written as: y=mx+b, where m is the slope and b is the y-intercept.”

“Substituting our slope, we have: y=2.3x+b.”

“Substituting the x- and y-values of our point gives:

150000=2.345000+ b

150000=103500+b

b=46500.”

“Thus, the equation of the line of best fit is: y=2.3x+46500.”

“We can also find the line of best fit using the graphing calculator.”

“You can enter the x-and y-values into the L1 and L2 lists; then go to Stat, Calculate, and Choose LinReg (a+bx). Doing so will give you the slope, a, and y-intercept, b.”

“With our data, the graphing calculator gives the following slope and y-intercept.”

a≈38859

b≈2.5

“Thus, the exact line of best fit equation is given by: y=2.5x+38859.”

“This equation is slightly different than the equation we found, simply because we chose points on the line to find the slope and y-intercept. The equation we found is to be considered a close estimate and is perfectly permissible for our current desired level of understanding.”

“What else can we infer from our scatter plot?”

“The correlation is positive.”
“The slope indicates that for every dollar increase in annual salary, the home price increases by $2.50.”
“The y-intercept indicates that at an annual salary of $0, the home price would be $38,859.”

Part 2

Hand out the Calculate the Line of Best Fit Worksheet (M-A1-6-2_Calculate LOBF Worksheet and KEY.doc). Have students work on it individually, then with a partner. After the partners are finished, have them team up with another pair and discuss their findings.

Part 3

Hand out the Lesson 2 Exit Ticket (M-A1-6-2_Lesson 2 Exit Ticket and KEY.doc) to evaluate whether students understand the concepts.

Extension:

If students have access to computers, they can use the Web site http://illuminations.nctm.org/ActivityDetail.aspx?ID=146 to input their data and have the computer find the line of best fit. Students can compare and contrast the line of best fit that they wrote and the one that the computer gave them.
If students get done early with the Stroop lab, give them the Stroop Lab Extension (M-A1-6-2_Stroop Lab Extension.doc).

Equations of Line of Best Fit