Grade Level: 10

 Subject: Coordinated Science

 Learner Outcomes:

• Students will use the Internet as a resource.
• Students will manually create a graph.
• Students will learn to use "Graphical Analysis" by creating a graph.
• Students will demonstrate understanding of the concepts: slope, independent variable and dependent variable.

Materials: Centimeter ruler, miscellaneous cylinders (coins to coffee cans), scissors, string, centimeter graph paper, pencil, and copy of the worksheet provided for the Internet site.
 
 

Technology Tools/courseware:

PC with Internet Connections, Browser of Choice, Vernier’s Graphical Analysis for Windows
 
 

Teacher notes:

• Students should be familiar with the use of a computer and how to explore the Internet.
• The worksheet for Lesson A should be printed out for each student or group.
• The worksheet for Lesson B should be printed out for each student or group.
• Teacher should collect and gather all materials for lesson B prior to lesson, remember there should be 5 different sized items, e.g. coins, juice cans, soup cans, petri dishes (anything that is a cylinder).

Students will complete the following lessons A and B. It is important to complete them in order.

 Lesson A:

 Teacher will divide the class into groups according to the number of computers available and complete the following discovery activity.

Students will go to the following site to manipulate the variables on line and to discover slope:

http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/lpsf.html

  Print the following student handout and give a copy to each student or group.

  _____CUT HERE__________________________________________________________________
 

Name

Date

Period

Group members
 
 

1. Click on the button "You Control".

2. Explore and notice that values for x1, y1, and m can be changed by using the mouse and the scroll bar

    directly below. Movement of the bar to the right raises the value and movement to the left decreases

    the value.

3. Change the x1 value to 56.7

   Change the y1 value to 69.3

   Change m to 1.2

  What is the value of the y intercept (where the x crosses the y axis)? _______

4. Change the x1 value to –20.7

 

 

Did the y intercept move? ______

If so, describe the movement.

5. Change the value of y to –22.5

 

 

Did the y intercept move?

If so, describe the movement.

6. Change the m value to –2.1

 

 

Did the y intercept move?

If so, describe the movement.

What happens to the slope of the line?

7. Now manipulate the values of x, and y. Record in the space provided the values of x and y used and the value of m for each of these changes.

 

 

a) x____ y_____ m____

Did the y intercept move? ______

If so, describe the movement.

b.) x____ y_____ m____

Did the y intercept move? ______

If so, describe the movement.

c.) x____ y_____ m____

Did the y intercept move? ______

If so, describe the movement.

d.) x____ y_____ m____

Did the y intercept move? ______

If so, describe the movement.

e.) x____ y_____ m____

Did the y intercept move? ______

If so, describe the movement.

8. Using the back of this paper write 4 paragraphs (minimum of 3 sentences for each paragraph) using the following leads:

As the value of y is changed what is the effect on the other 2 variables….

 As the value of m is changed what is the effect on the other 2 variables…

 The slope of a line is changed by…
 

____________CUT HERE________________________________________________________________

  Lesson B

Because graphs can be used to show whether data being measured are directly or inversely proportional to each other, graphs are one of the best ways to communicate the results of an experiment. If the data being measured is directly proportional to each other, as one increases the other will also increase or if one decreases the other will also decrease. However, if data being measured is inversely proportional to each other, for example, as one increases, the other will decrease. In this exercise graphs of directly proportional data will be studied.

If possible, the function to be graphed should be in the form of a straight line. The general form can represent the equation for a straight line (a linear equation):

y = mx + b

where y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept.

The independent variable (x) is the mathematical variable whose value is specified first and determines the value of the dependent variable.

• The dependent variable (y) is the mathematical variable whose value is determined by the independent variable. For example, if the volume of an object is known and its mass is to be measured experimentally, the volume would be the independent variable and the mass would be the dependent variable. The mass of the object "depends" on the magnitude of the volume.
• Slope (m) is the upward or downward slant or inclination of a curve. It is mathematically defined as the "change in y" divided by the "change in x" (m = D y/D x) or "rise over run".
• The y intercept (b) is the point where the line intersects the y axis of the graph--the point where x is zero.

If the slope of a line, the y intercept, and one variable is known, then the other variable can be determined. For example, when mass versus volume is graphed, the slope can be used to determine unknown values of volume for given masses or vice versa.

The equation for this graph (in the form y = mx + b) is M = (2.70 g/mL)V + 0 or more simply M = (2.70 g/mL) V since the y intercept (b) is equal to zero. The slope was determined from the graph as follows:

  Since D y = y₂ - y₁ and D x = x₂ -x₁

  D y = 135.0 g - 0.0 g = 135.0 g and D x = 50.0 mL - 0.0 mL = 50.0 mL

Thus
 
 

Materials

Vernier "Graphical Analysis" program Centimeter ruler

Scissors String

Miscellaneous cylinders (coins, juice cans, soup cans, coffee cans, petri dishes)
 
 

Procedure

Part A
 
 

LENGTH (cm) VS. LENGTH (in)
Length (in)	Length (cm)
0.0	0.0
5.0	12.7
10.0	25.4
15.0	38.1
20.0	50.8
25.0	63.5

1. Obtain centimeter graph paper from the instructor. Graph the data from the table above placing the length in inches on the x-axis and the length in centimeters on the y-axis.
2. Choose an appropriate scale for the graph. Label the axes, capitalizing the first letter of each word, place units in parentheses.
3. Title the graph in the form dependent variable (y-axis) versus independent variable (x-axis). Place the title in all capital letters. Place the title at top center of the graph.
4. Have the instructor to OK the graph before continuing to Part B.
5. Use this graph to answer the questions at the end of the experiment as homework.

1. Obtain five (5) various objects from those provided. Choose objects from small to large. Remember to make all measurements as precisely as possible in order to get good results.
2. Use a centimeter ruler to measure the diameter of the smallest object to the nearest 0.05 cm and record in the data table provided. Turn the ruler on its edge to get more precise measurements.
3. In order to measure the circumference of the smallest object, wrap a string around the perimeter of the object and measure the length of the string to the nearest 0.05 cm. Record the measurements in the data table.
4. Repeat the procedure for the remaining objects going from smaller to larger.
5. When all measurements are complete and recorded, put away all materials as directed by the instructor.
6. Proceed to a computer to graph the data using Graphical Analysis.

 

 
 
 
 
 

CIRCUMFERENCE VS. DIAMETER
Object	Diameter (cm)	Circumference (cm)
1.
2.
3.
4.
5.

7. In the Graphical Analysis program, choose File on the Tool Bar. Choose New and, if asked to save data in memory, choose No.
8. Choose File again on the Tool Bar and choose Printer Setup. Choose the name box and type in student name(s). Choose date box for current date. Choose Setup at the bottom of the Printer Setup window and choose Landscape. Choose OK and OK again.
9. Under the data table window, enter the data for diameter in the blue X column and the circumference data in the green Y column. Be sure to enter data from the smallest object to the largest object.
10. Double click on the blue X at the top of the data column and type the new name for the data column. Push Tab, type the correct units for the x-axis. Do not use parentheses for the units. Choose OK.
11. Repeat procedure 10 for the green Y column.
12. Under the graph window, double click on the title at the top of the graph and rename it using all capital letters in the form y-axis versus x-axis. Choose OK.
13. Click on anywhere on the graph. On the Tool Bar, choose Graph and click once on Connecting Lines to turn this option off.
14. On the Tool Bar, choose Analyze, choose Automatic Curve Fit, choose Linear, choose OK, and choose OK-Keep fit.
15. Record the slope and y intercept values for the graph in the table provided.

 

 
 
 
 
 

REGRESSION LINE DATA TABLE
Slope (m)	y- intercept (b)

16. On the Tool Bar, choose File, choose Print, and choose Selected Display. Type in the number of copies needed and choose OK.
17. When the graph is printed, choose File on the Tool Bar, choose New, and when asked to save data to memory, choose No.
18. Use the printed graph to answer the questions at the end of the lab for homework.

 Answer all questions as complete sentences on a separate sheet of paper! Show all work for problems!

  Part A
 

1. Calculate the slope for the graph plotted in Part A. Show all calculations neatly including the appropriate formula. Include units in the calculations and the final answer.
2. Determine the y-intercept value for the graph.
3. What use could be made of the slope value from this graph?
4. Use the slope value determined in question 1 to convert 15.5 inches to centimeters.
5. Use the slope value to convert 44.7 centimeters to inches.

 

1. From what has been learned from this activity, define slope and y-intercept in your own words.
2. Using the equation for a straight line, rewrite the equation substituting C for y, D for x, and the value from the slope of the graph for m.
3. What does the slope of this graph represent?
4. What unit(s) would be assigned to this slope value?
5. What is the common name that is given to this slope value?
6. Use the equation developed in question 2, calculate the diameter of an object with a circumference of 35.5 cm. Show all work including the original equation. Isolate the unknown variable before substituting numerical values into the equation.
7. Using the same equation, calculate the circumference of an object with a diameter of 7.75 cm.

Write a two-paragraph results/conclusions:
 

• The paragraph for results should include descriptions of the graphs that were made, the values of slope and y-intercept for Parts A and B, and the method used to determine slope and y-intercept.
• The paragraph for conclusions should include what was learned about slope, y-intercept, and proper methods of graphing.
• Discuss advantages/disadvantages for manual graphing versus computer graphing.

 
 
 
 

PRESSURE VS. VOLUME
Volume (mL)	Pressure (atm)
5.0	2.02
7.5	1.35
10.0	1.02
12.5	0.81
15.0	0.69
17.5	0.58
20.0	0.50

 
 

A graph with a curve as shown above suggests an inverse relationship. To confirm an inverse relationship, plot the reciprocal of one variable versus the other variable. In this case, pressure is plotted versus the reciprocal of volume, 1/volume. To plot this graph:
 

1. Enter the volume in the x-list and pressure in the y-list. Follow the directions given in Part B of the lab to label and give the correct units for the x and y-lists.
2. Click on Data on the Tool Bar, choose New Column and Calculated. Type in Inverse Volume under New Column Name and 1/mL under New Column Units. To type the formula "1/Volume" in New Column Formula., use the on-screen keypad to enter 1, /, and column ¯ , and choose Volume. Choose OK.
3. Double click on the x-axis label, and choose Inverse Volume and OK.
4. Double click on the title and type it in all capital letters. Click OK when finished.
5. Follow the steps for Printer Setup and Printing from Part B of the lab.
6. Write a paragraph explaining the mathematical relationship between pressure and volume and the mathematical relationship between pressure and inverse volume by comparing the pressure-volume and pressure-inverse volume graphs.

  Modifications:

• Seeing-impaired student's work should be printed on blue paper.
• Exceptional children may require more time and possibly need to work with an aide for assistance during lab activities (or paired with another student).
• Students may need to work in small groups with teacher directed instruction using a data projector and teacher going step by step with students as they enter data.

 Evaluation/Assessment:

• Teacher will grade the worksheets of the students and the graphs created. Teacher will look for the process of mathematical computation and unit placement.
• Students will place all work in a folder for their portfolio.
• Classroom participation will also be assessed.

• Science Ten

• English Language Arts

• Applied Math I-II

             AM2.3, AM2.5, AM2.13, AM2.15, AM2.16, AM2.18

• Algebra I-II

            A2.2, A2.21, A2.22

• Geometry and Applied Geometry

National Standards
 
 

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5