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Using Scatter Plots to Predict the Future

Subject:Math

Grade Level: 10-12

Description: In this unit, students learn to graph scatter plots. Scatter plots help us identify patterns, trends, and relationships for two quantitative variables. The stiudents explore the interaction between these two variables to reach meaningful conclusions about their predictability for future events.

How it Works: The relationship between variables is the driving force behind the mastery of statistics research. Students explore questions between two variables such as: is the shoe size of a man related to his height? Is horsepower in a vehicle related to its gas mileage? Does how long children remain at the lunch table help predict how much they eat?

Final Project/Product: A statistical research document predicting the quantity of cell phone users in the United States by the year 2015. The presentation should include scatter plots, and/or bar graphs, and pie charts or any other visual aid to help explain the results (a PowerPoint presentation is an acceptable alternative). Teachers are encouraged to create their own questions.

Overall Value: The student develop the ability to recognize relationships between two seemingly unrelated variables. The lessons provide the tools necessary to explore the conjecture and reach valid conclusions. They have the opportunity to use a graphing calculator, and use the Internet to research data and take a practice test.

English Language Learners: There are vocabulary/key words before each lesson. The lessons use visual aids such as transparencies, overhead graphing calculator, and LCD projector. In addition, questions should explore relationships that are meaningful to language learners. For example, does bilingualism help me to be a better student? Does educational level predict future earnings? Before collecting data, students should be encouraged to create their own survey questions and practice speaking skills by asking the questions to members of the school community.

Tips for the Teacher: Try to explore questions regarding variables that are meaningful to teenagers and/or ELL students. In addition, document all the projects by taking pictures, which can be used as visual diary of the class progress and can be displayed on a bulletin board or website.

 

 Standards Addressed
Students understand and become mathematically confident by communicating and reasoning mathematically, by applying math in real world settings, and by solving problems through the study of statistics and data analysis.
  Grade: 9-12 Subject: Math
As listeners and readers, students collect data, facts, and ideas; discover relationships, concepts, and generalizations; and use knowledge generated from oral, written, and electronically produced texts. As speakers and writers, they use oral and written language to acquire, interpret, apply, and transmit information.
  Grade: 9-12 Subject: Math
As listeners and readers, students analyze experiences, ideas, information, and issues presented by others using a variety of established criteria. As speakers and writers, they present, in oral and written language and from a variety of perspectives, their opinions and judgments on experiences, ideas, information, and issues.
  Grade: 9-12 Subject: English Language Arts
Students listen, speak, read, and write in English for critical analysis and evaluation. Students learning English as a second language use English to express their opinions and judgments on experiences, messages, ideas, information, and issues from a variety of perspectives. They develop and use skills and strategies appropriate to their level of English proficiency to reflect on and analyze experiences, messages, ideas, information, and issues presented by others using a variety of established criteria.
  Grade: 9-12 Subject: ESL

Day 1: What is a scatter plot graph?
Objectives
Students will define a scatter plot.
Students will draw a scatter plot for a set of data.
Students will explore the relationship between sets of data.
Materials
Graph paper and rulers
Transparencies, pens for writing on transparencies
Computers with Internet access and printer
Overhead projector, LCD projector
Keywords
Scatter Plot, Data, Increase, Decrease, Slope
Procedure 1
Ask students to work in groups of 4.
a. Ask students to get data from the website: Shoe size vs. Height (in inches). The website lists data from men and women. Select the men’s set of data. A printout of the data will be helpful. You might want to also print the women’s data for a later discussion/comparison.
b. Use the LCD projector to show the website on the screen to further explain procedure 1A.
Shoe Size vs. Height http://staff.imsa.edu/~brazzle/E2Kcurr/Forensic/Tracks/ShoeVsHeightIMSA.htm
Procedure 2
Use graph paper to plot coordinates. Horizontal line will represent shoe size and vertical line will represent height in inches.
a. Teachers should walk around to make sure that the axis are properly labeled and appropriately spaced out.
Graph Paper http://mathematicshelpcentral.com/graph_paper.htm
Procedure 3
Ask each group to complete their work on transparencies. Show these transparencies to the whole class to compare findings.
a. Begin a class discussion. It should include but not be limited to the following topics: Is there a relationship between shoe size and height? Can you have a man whose shoe size does not match his height (according to our findings)? Do you think that women follow the same patterns?
b. The teacher should try to elicit answers from the students based on logical inferences from the data.
c. In addition, an extension of the findings can be made to other relationships. The instructor should try to encourage the students to look for similar relationships such as waist size and height or head circumference and a person’s IQ.
d. If time allows, ask the students to go into the Internet and find data supporting or discouraging the previous statements. The main idea here is not to find the answer to every relationship. Instead, we want to begin encouraging the students to find patterns and relationships between variables that before today might have seen unrelated.
Extension
Collect the height (in inches) and shoe size of at least 10 people (select a group: men, women, or children) in your house to plot in your graph. How do these new coordinates compare to old ones? Any similarities? Any differences? (Maybe collect data from women only.)

Day 2: What is the correlation coefficient?
Objectives
Students will define the correlation coefficient.
Students will use the graphing calculator to help compute the correlation coefficient.
Students will use the correlation coefficient to measure how closely the data points cluster about a line.
Students will explain how the correlation coefficient for a data set describes how well the data points can be modeled by a line.
Materials
Graphing calculator and overhead graphing calculator for teacher
Overhead projector, LCD projector
Transparency of graphing calculator
Computer, Internet connection, printer
Keywords
Go over keys on the graphing calculator, correlation, and coefficient.
Procedure 1
Use the graphing calculator to enter last lesson’s data.
a. This is a great opportunity to familiarize the students with the calculator. My calculator of choice is the Texas Instrument TI-83 Plus. It is extremely friendly to new users.
b. A transparency that shows the keys can be very helpful. Show the keys that they will be more likely to use: Graph, Zoom, Second Function, and Window.
c. Follow procedures as indicated in the mathbits website below. Show steps to students by using the LCD projector. This website is extremely useful since it displays step by step the procedures to get the data into the calculator and display the graph.
d. Some students are likely to fall behind while following the instructions. You can ask them to continue working in their groups. I have noticed that students helping other students make the process a lot easier.
mathbits http://mathbits.com/MathBits/TISection/statistics1/scatterplot.htm
Procedure 2
Define correlation coefficient.
a. Draw from the experiences of the previous lesson. Help students notices that there are relationships such as shoe size and height that appear to be stronger than waist size and height. Why? Is it possible to find a person with a small waist size and short height or vice-versa? Do you think that there is a relationship between the eye color and how smart a person is?
b. All this questions should lead to a discussion on how some relationships are stronger than others. A correlation coefficient is an indicator of these relationships.
mathbits http://mathbits.com/MathBits/TISection/Statistics2/correlation.htm
Procedure 3
Predict the correlation coefficient from our graph (see procedure 1B). The closer you get to the absolute value of 1, the stronger the relationship.
a. This is a good point to stop and reflect on how we can find data from other sources and to predict the correlation coefficient.
b. You might even venture into the relationship between students’ ESL grades and their Math grades. Do you think that there is a connection?
c. If there is a connection, ask the students, what procedures they will follow to prove it.
Procedure 4
Calculate the correlation coefficient using the graphing calculator (see website above).
a. The correlation coefficient at this level is best left for the calculator to find. However, the result is a point of discussion. Students should be allowed to change the data temporarily to see the effects on the graph and the coefficient of correlation. What factors might contribute to the increase/decrease of the coefficient?
b. Particular attention should be taken at this point to explain in simple terms the process that the calculator takes to calculate this coefficient.
c. Does the direction of the line (up/down) affect the coefficient? Why? Is there a dynamic in the relationship between the two variables that causes some relations to show a negative or positive number? What do you think are those possible factors that affect the relationship?
d. Compare the two results. How does the graph from the previous day compare to the one in the calculator? Do you get additional information from the graph in the calculator? Ask them to name a few.
Procedure 5
The calculator is a tool that allows the students to further examine the relationship between two variables.
a. You can change the Window (in the calculator) to “Zoom In” into a particular area of the graph to further explain/discuss the details of each coordinate. Does the line formed by the graph follow a downward/upward path?
b. Ask the students to help explain the direction of the graph.
Extension
Find data relating to average family income according to postsecondary education. What is the correlation coefficient for the data? What conclusion can you make from this result?

Day 3: How do we find the line of best fit for a set of data?
Objectives
Students will state the properties of the best fit line or trend line.
Students will use the properties of the best fit line to find its equation.
Students will use the graphing calculator to find the equation of the best fit line.
Materials
Graphing calculators, overhead graphing calculator
Overhead projector, LCD projector
Transparency of graphing calculator
Computer(s) with Internet access, printer
Keywords
Horsepower, advertised, ratings, gas mileage
Procedure 1
Get data - Advertised horsepower ratings and expected gas mileage for several 2001 vehicles. (see website below)
a. Enter data in the graphing calculator.
b. Create a scatter plot.
c. Predict the correlation coefficient.
d. Use the calculator to find the actual correlation coefficient.
Pearson’s media, data and analysis http://media.pearsoncmg.com/aw/aw_deveaux_introstats_2/datasets/stat2dv_data.html
Procedure 2
Additional questions
a. Describe the direction, form, and strength of the plot.
b. Write a few sentences telling what the plot says about fuel economy.
Procedure 3
Find the equation of best fit line.
a. Use the algebraic method.
b. The line of best fit is one of many lines that can be used to model the relationship between horsepower and gas mileage.
c. Find the mean of both variables (x, y)
d. Select a coordinate from the data (x2, y2). Use the formula (y – y2) = m (x – x2) where m is the slope.
Procedure 4
Get the best fit line using a graphing calculator.
a. Compare the two results.
b. Why do we get two different lines? Is one equation better that the other? Explain. The possibility that both equations are exactly the same is remote. However, we should pay attention at their similarities. Both processes are attempting to approximate the best possible line (equation) to represent the relationship.
c. We should try to repeat the process with other coordinates to try to provide alternatives equations to the same relationship.
d. The students can try to input these new equations into the calculator to compare with the one provider by the calculator.
mathbits http://mathbits.com/MathBits/TISection/Statistics1/LineFit.htm
Extension
Visit Pearson’s website – Find the best fit line for exercise 19, chapter 7. Fast food is often considered unhealthy because much of it is high in both fat and calories. But are the two related? Here are the fat (in grams) and calories contents of several brands of burgers. Analyze the association between fat and calories. (Hint: Use a scatter plot, coefficient of correlation and the equation of the line of best fit.)

Day 4: How can we use the line of best fit to predict unknown values?
Objectives
Students will use the graphing calculator and charts of data to determine the equation for the line of best fit.
Students will use the equation found to predict the results for data points for which we do not have actual measurements.
Students will use the calculator generated correlation coefficient to determine how well the line fits the data.
Materials
Graphing calculators, overhead graphing calculator, transparency of graphing calculator
Overhead projector, LCD projector
Computer(s) with Internet access, printer
tape measure
Procedure 1
Arm span activity
a. In the previous lesson, we dealt with another body relationship (shoe size vs. height). Now we are dealing with the span activity (arm span vs. height).
b. Ask the students to predict the relationship between these two variables. Can they think of other type of body measurement that they will like to investigate? We will use this activity to prepare the students for a final class project.
c. Follow the instructions in the website below.
Regents Prep http://regentsprep.org/Regents/math/algtrig/ATS4/ARmSTretch.htm
Keywords
Charts, measurements, span, length, tip, across, aggregate, rounded, tenth
Procedure 2
Additional activity
a. A tourist has a pamphlet that contains the following list of taxi fares for various distances. How much should the tourist expect to pay for a cab ride of 18.4 miles? (An Internet mapping site, like Mapquest or Google Maps, can be used to determine the distance between locations.)
b. The miles the tourist wants to travel are not shown in the pamphlet. How can we use the line of best fit to predict this new value? Can lines be extended to include other values? Ask the students to write the steps necessary to complete the work either by using the calculator or just pen and paper.
c. At this point, ask students to work in groups of two. We want to hear possible different answers. This is a good opportunity to explore all possible sources of potential errors.
d. In addition, we can visit the Website of a Cab Company such Dial 7 to compare the actual prices with the above problem. Elicit from the students how can they convert the prices from Dial 7 into miles (prices are given between locations such between airports). Do you think that cab companies can use your method of predicting fares for distances not listed in their advertising? Are there any other factors beside distance that a cab company should also consider?
Cab company http://dial7.com/rates.html
Extension
When I was a little boy, my mother used to buy my pants by folding the waist of the new pants around my neck. If both circumferences were equal she would call that a perfect fit and she would buy the pants for me. Was my mother right? Ask members of your family to try this experiment (They don’t have to take their pants off. They can use a pair from their closet.) Record your findings in a table. Write an equation for a line of best fit? How strong do you think is the relationship between these two variables?

Day 5: In Class Project
Objectives
Students will be able to design their own experiment.
Students will be able to collect the data needed to prove or disprove their hypothesis.
Materials
Graphing Calculator, overhead calculator
Overhead projector, LCD projector
Computers, printer
Keywords
Typos, extrapolating, interpolating, label, hypothesis
Procedure 1
Hypothesis: Is there a relationship between the height of a person and his/hers arm span?
a. How do you suppose we can prove this hypothesis? Allow for the students to come up with a plan. Encourage their answers in the direction of collecting data to support the hypothesis.
b. How do you suppose we can collect data? Should we divide the class into groups and then place all data together or should each group work with their own data?
Procedure 2
Ask the students to go around the school (if it is allowed) and ask volunteers to provide their measurements.
a. When the data is collected encouraged the students to use all the available data together.
b.
Procedure 3
Place all data in a properly labelled table (on the board) so that everyone has access to it (maybe a transparency).
a. Discuss the possibility of transferring incorrect data (typos) into the table. How would incorrect data affect my hypothesis?
b. Allow the student to answer the following question by themselves: What is the equation of the best fit line? What is the correlation coefficient? What conclusions can you extrapolated from your results? (Calculators should be available.)
Assessment
Evaluating the final work.

Miguel Pineda

pineda3247@hotmail.com

Newcomers High School
28-01 41 Avenue
Long Island City, NY 11101

Miguel Pineda has been a mathematics teacher since 1991. He has worked in Queens, NY at Far Rockaway High School and at Newcomers High School in Long Island City. He graduated from Queens College in 1990 with a degree in computer science. He obtained his master’s degree in math education from City College, NY in 1994. His teaching experience ranges from 9th grade to college-level statistics.


Important documents for this lesson plan.

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