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MathematicsNATIONALMATH + SCIENCEINITIATIVE

LEVELAlgebra 1 or Math 1 in a unit on statistics (bivariate data) Geometry or Math 2 in a unit on area

MODULE/CONNECTION TO AP*Bivariate Data*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

MODALITYNMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding.

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P – Physical V – VerbalA – AnalyticalN – NumericalG – Graphical

Collecting Linear and Quadratic Data with TubesABOUT THIS LESSON

This lesson provides students with a review of linear and quadratic functions as they collect data about the diameter and area of a circle

on a wall that can be seen through a cylindrical tube as a function of the tube’s distance from the wall. Students collect their data, graph scatter plots, use transformations of the appropriate parent function to create functions of good fit for the data, and make predictions based on those functions. In questions 1-6, the focus is on the linear relationship of the diameter to distance. In questions 7-11, students investigate the quadratic relationship between area and distance. In questions 12-16, students use similar triangles to determine the theoretical functions for the data, based on the physical measurements of the tubes used in the data collection.

OBJECTIVESStudents will

● collect and create scatter plots of real-world data.

● fit linear and quadratic models to the data using graphing calculators and transformations of parent functions.

● predict outcomes using the models created for the data.

● write functions based on similar triangle relationships.

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Mathematics—Collecting Linear and Quadratic Data with Tubes

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol (★) at the end of a specific standard indicates that the high school standard is connected to modeling.

Targeted StandardsS-ID.6a: Represent data on two quantitative

variables on a scatter plot, and describe how the variables are related. (a) Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.★ See questions 2-4, 6, 8-9, 11

S-ID.6c: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (c) Fit a linear function for a scatter plot that suggests a linear association.★ See questions 2-4, 6

Reinforced/Applied StandardsS-ID.7: Interpret the slope (rate of change) and

the intercept (constant term) of a linear model in the context of the data.★ See question 4

G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. See questions 12-13, 15

F-BF.1c: (+) Write a function that describes a relationship between two quantities. (c) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.★ See questions 10, 15-16

COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice.

MP.5: Use appropriate tools strategically. Students use a graphing calculator to

manually fit a function to a scatter plot based on transformations of the parent function.

Students must determine how to set up the data collection system to achieve accurate, consistent results. Students must determine how to keep the tube parallel to the ground and how to determine the diameter of the circular field of view.

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Mathematics—Collecting Linear and Quadratic Data with Tubes

FOUNDATIONAL SKILLSThe following skills lay the foundation for concepts included in this lesson:

● Plot points in the first quadrant● Write linear equations● Apply proportional properties of similar

triangles

ASSESSMENTSThe following formative assessment is embedded in this lesson:

● Students engage in independent practice.

The following additional assessments are located on our website:

● Bivariate Data – Algebra 1 Free Response Questions

● Bivariate Data – Algebra 1 Multiple Choice Questions

● Bivariate Data – Geometry Free Response Questions

● Bivariate Data – Geometry Multiple Choice Questions

MATERIALS AND RESOURCES● Student Activity pages● One tube (such as an empty toilet paper roll,

paper towel roll, or pvc tube) per group of 4● Centimeter rulers● Centimeter tape measures● 2 sheets of colored paper per group● Graphing calculators● NMSI video clip on creating scatter plots on

the TI-84

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Mathematics—Collecting Linear and Quadratic Data with Tubes

TEACHING SUGGESTIONS

Before beginning the lesson, divide the class into groups of four students. The teacher should model the process for collecting

the data and emphasize that the tube must be held parallel to the ground.

Use the following job descriptions and assign a job to each student in a group.

● Looker: Holds the tube parallel to the ground and looks at the target through the tube.

● Target Managers (2): Communicates with the “Looker” to identify the diameter of the visible target using colored paper.

● Enforcer: Makes sure that the “Looker” holds the tube correctly (parallel to the ground), measures the distance that the “Looker” is from the target, and records data.

If the students are using TI-84 calculators, have them store the distance from the wall in list L1, the diameter of the circle seen on the wall (the target) in list L2, and the area of the circle on the wall in list L3. Use the Y= menu to store the functions that are calculated throughout the lesson in Y1, Y2, Y3, and Y4. The functions that are not in use during a particular part of the lesson can be turned off by placing the cursor over the “=” sign and pressing “enter.” When the “=” sign is not darkened, the function is “turned off” and will not be visible on the graph screen. Repeat the same process to turn a function back on.

If the students are using the TI-Nspire calculators, have them store the distance from the wall in a Lists and Spreadsheet page in a list named distance_from_wall, the diameter of the circle seen on the wall (the target) in a list named diameter, and the area of the circle on the wall in a list named area. Students may use either a Graphs page or a Data and Spreadsheet page to graph the scatter plots and functions. Have students store the functions in f1, f2, f3, f4, and f5. The functions that are not in use during a particular part of the lesson can be hidden using the Hide/Show option under the Actions menu.

The class should discuss question 6 and question 11. The students should review the terms extrapolation (predicting outside the range of the data) and interpolation (predicting inside the range of the data). Discuss why interpolation usually produces a better prediction than extrapolation (One reason is that we do not know what happens beyond what is given in the data).

Part of the learning experience for students in this lesson is the process of manually fitting functions to data. Resist the temptation to have students use their calculators to calculate regression models of best fit for the data they collect. Instead, lead them through the process of starting with the appropriate parent function for each set of data and then adjusting the constants so that the function better fits the data. Valuable analytical thinking takes place as students decide whether the slope of the line (or the value of a in y = ax2) should be increased or decreased, etc.

You may wish to support this activity with TI-Nspire™ technology. See Displaying Data in a Scatter Plot and Transforming Linear and Quadratic Functions the NMSI TI-Nspire Skill Builders.

Suggested modifications for additional scaffolding include the following:

1 Provide selected entries in the data table.

2 Partially complete the scatter plot with the data provided in question 1.

12 Outline the similar triangles in Figure 2 with corresponding colors.

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Mathematics—Collecting Linear and Quadratic Data with Tubes

NMSI CONTENT PROGRESSION CHART In the spirit of NMSI’s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from sixth grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content.

The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence.

6th Grade Skills/Objectives

7th Grade Skills/Objectives

Algebra 1 Skills/Objectives

Geometry Skills/Objectives

Algebra 2 Skills/Objectives

Pre-Calculus Skills/Objectives

Plot points and analyze the pattern.

Create scatter plots and analyze the pattern.

Create scatter plots and analyze the shape using the correlation coefficient to describe the strength of the linear relationship between the variables. Distinguish between correlation and causation.

Create scatter plots and analyze the shape using the correlation coefficient to describe the strength of the linear relationship between the variables. Distinguish between correlation and causation.

Create scatter plots and analyze the shape using the correlation coefficient to describe the strength of the linear relationship between the variables. Distinguish between correlation and causation.

Create scatter plots and analyze the shape using the correlation coefficient to describe the strength of the linear relationship between the variables. Distinguish between correlation and causation.

Model points graphically using a line.

Model data graphically using a line.

Model data using linear and/or exponential regression equations.

Model data using linear and/or exponential regression equations.

Model data using linear, exponential, quadratic, power, and/or logistic regression equations.

Model data using linear, exponential, quadratic, power, and/or logistic regression equations.

Analyze the data based on the graph of the line.

Analyze the data based on the graph of the line.

Analyze the data based on the regression model.

Analyze the data based on the regression model.

Analyze the data based on the regression model.

Analyze the data based on the regression model.

Use interpolation and extrapolation where appropriate.

Use interpolation and extrapolation where appropriate.

Use interpolation and extrapolation where appropriate.

Use interpolation and extrapolation where appropriate.

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Mathematics—Collecting Linear and Quadratic Data with Tubes

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

Collecting Linear and Quadratic Data with Tubes

AnswersThe answers to this experiment will vary. The answers provided are samples from students’ work using a 16 cm PVC pipe with a diameter of 3 cm.

1. The dependent variable is the diameter of your view. The independent variable is the distance from you to the wall because your diameter view depends on how far you are from the wall.

Distance from the wall, in cm 50 100 150 200 250 300

Diameter, in cm 10.5 20 30.9 41 49 58.6

2.

3. b. y= 15x+1

c. Most of the points are close to the line, so I think the function is a pretty good fit.

4. The slope means that every time I stand 1 cm farther from the wall, the diameter of the circle that I can see on the wall will increase by 1/5 cm. The domain will be from zero to the maximum distance that I can be from the wall within the space where the data is being collected.

5. Ensure the data is as accurate as possible by using careful measurement, including the distance from the wall and the diameter. Also ensure that the tube is parallel to the ground, and double check measurements before recording the data.

6. a. If I stand 142 cm from the wall, I expect the diameter to be approximately 29.4 cm.

b. If I stand 7.22 m from the wall, I expect the diameter to be approximately 145.4 cm.

c. I am more confident in the answer to part (a) because 142 cm is within the measurements that we made.

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Mathematics—

Part 27.

Distance from the wall (cm)

Area(cm2)

50 86.590100 314.159150 749.906200 1320.254250 1885.741300 2697.026

8.

9. b. y= 132x2

c. The data is more curved than linear, and since the area involves squaring the radius, a quadratic function seems reasonable.

d. The points are very close to the function’s graph, so I think the function is a good fit.

10. a. r =15 x+12

and A= π15 x+12

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

2

b. The functions produce graphs that look almost the same when shown on this scale.

11. a. If I stand 142 cm from the wall, I would expect the area that I see to be approximately 678.867 square cm.

b. If I stand 7.22 m from the wall, I would expect the area that I see to be approximately 16,604.228 square cm.

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Mathematics—

12. The following triangles are similar, based on the drawing in Figure 2: ABC AED∆ ∆ ; AEF ABG∆ ∆ ; ADF ACG∆ ∆ .

13.

AF is the length of the tube, AG is the distance from the eye to the wall, ED is the diameter of the tube and BC is the diameter of the viewing circle. ED BCAF AG

=

14. In my experiment, the diameter of the tube is 3 cm and the length of the tube is 16 cm.

15. a. 316

d x=

Diameter of circle on wall = diameter of the tube distance from the eye to the walllength of the tube

×

c. The functions are similar, but the formula in part (a) is more accurate than the one in question 3b because it is based on the measurements of the tube.

16. a.

c. The functions are similar, but the formula in part (a) is more accurate than the one in question 10a because it is based on the actual measurements of the tube and the mathematics of similar triangles.

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Mathematics—

1Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics NATIONALMATH + SCIENCEINITIATIVE

Collecting Linear and Quadratic Data with Tubes

You are going to collect data to determine the effect that distance has on the diameter of the circle that you can see through a cylindrical tube. Use centimeters as the units. Collect six sets of data using different distances, and then complete the following questions.

● One member of the group, the “Looker,” will look through the tube. The “Looker” must hold the tube parallel to the ground and must hold the tube steady while the measurements are taken.

● The second and third members of the group, the “Target Managers,” have two tasks. The “Target Managers” communicate with the “Looker” to determine the location of the target circle, using two pieces of colored paper that the “Looker” sees on the wall, and measure the diameter of the target circle.

● The fourth member, the “Enforcer,” measures the distance that the “Looker” is from the wall, makes sure that the “Looker” is holding the tube correctly, and records the data.

1. On the data collection page at the end of the activity, list your data points in the first two rows of the table. Explain why the distance from the wall must be the independent variable.

2. On the data collection page at the end of the activity, draw a scatter plot of the data you collected. Be sure to label and scale the independent and dependent axes.

3. a. Enter the distance from the wall and the diameter of the circle on the wall in the lists on your calculator. Make a scatter plot of the data on your calculator.

b. Graph the linear parent function y = x on your calculator and then adjust this function until you determine a function that you think fits the data relatively well. Record your function on the data collection page at the end of the activity.

c. Explain why your function is a good fit and/or explain any problems you think there might be with the function.

4. In terms of the situation, explain the meaning of the slope. What is a reasonable domain for this function?

5. What sources of error do you think you might have had in collecting your data? How could you have reduced these errors?

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Mathematics—Collecting Linear and Quadratic Data with Tubes

6. Using your function from question 3b, answer the following questions.a. If you were 142 cm from the wall, what would you expect to be the diameter of the circle that you

could see?

b. If you were 7.22 meters from the wall, what would you expect to be the diameter of the circle that you could see?

c. Of the two answers you calculated in parts (a) and (b), which one are you more confident about and why?

7. On the data collection page, complete the third row of the chart by calculating the area of the circle that you can see through the tube at each distance included in the table.

8. On the data collection page, plot the scatter plot for the area of the viewing circle versus the distance from the wall. Be sure to label and scale the independent and dependent axes.

9. a. Enter the area of the circle on the wall in an additional list on your calculator. Make a scatter plot of your area and distance data on your calculator.

b. Graph the quadratic parent function 2y x= on your calculator and adjust this function until you determine a function you think fits the data relatively well. Record your function on the data collection page.

c. Discuss why this data should be modeled with a quadratic function instead of a linear function.

d. Explain why your function in question 9b is a good fit and/or explain any problems you think there might be with the function.

10. a. Use the function recorded in question 3b to determine an equation for the radius, then substitute the new equation into the formula for the area of a circle. Record the new area function on the data collection page and enter it into your calculator.

b. How does the new function in part (a) compare with the function you recorded in question 9b?

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Mathematics—Collecting Linear and Quadratic Data with Tubes

11. Use the function from question 10a to answer the following questions.a. If you were 142 cm from the wall, what would you expect to be the area of the circle that you could

see?

b. If you were 7.22 meters from the wall, what would you expect to be the area of the circle that you could see?

12. Name the similar triangles in Figure 2.

13. Identify AF, AG, ED, and BC in terms of the situation, and write a proportion using these lengths.

14. Measure the diameter of your tube and the length of your tube in centimeters and record these measurements on the triangles in Figure 2.

Figure 1

Figure 2

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Mathematics—Collecting Linear and Quadratic Data with Tubes

15. a. Using similar triangle ratios, write a function for the diameter, d, of the circle you see on the wall in terms of the distance, x, you are from the wall. Record the new function on the data collection page and enter it into your calculator.

b. Using your graphing calculator, graph the function from question 15a, the scatter plot from question 3a, and the function you recorded in question 3b.

c. How does the function in question 15a compare with the function you wrote in question 3b? Which function should be more accurate? Explain your answer.

16. a. Use the function you wrote in question 15a and the formula for the area of a circle to write a composite function for the area, A, of the circle you see in terms of the distance, x, you are from the wall. Record the new area function on the data collection page and enter it into your calculator.

b. Using your graphing calculator, graph the function from question 16a, the scatter plot from question 9a, and the function you wrote in question 10a.

c. How does the function in question 16a compare with the function you wrote in question 10a? Which function should be more accurate? Explain your answer.

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Mathematics—Collecting Linear and Quadratic Data with Tubes

Data Collection Page

1. and 7.

Distance from the wall

Viewing diameter

Area of viewing circle

2. Scatter plot for Diameter vs. Distance 8. Scatter plot for Area vs. Distance

3. b. Diameter: ___________________

9. b. Area: ___________________

10. a. Area: ___________________

15. a. Diameter: ___________________

16. a. Area: ___________________

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Mathematics—Collecting Linear and Quadratic Data with Tubes