2. What are the objectives of Unit 3? To learn how to critically analyze data using graphs. To learn how to critically analyze data using graphs. To learn how to use graphing techniques to arrive at mathematical equations that describes the relationship between two variables. To learn how to use graphing techniques to arrive at mathematical equations that describes the relationship between two variables. To learn to connect the mathematical slope of a line to a physical property To learn to connect the mathematical slope of a line to a physical property

3. Graphs compensate for shortcomings of data tables. Difficult to determine relationship between variables. Difficult to determine relationship between variables. Impractical to find equation Impractical to find equation

4. The most important purposes of a graph are Quick and efficient organization of large amounts of data. Quick and efficient organization of large amounts of data. Demonstration of relationship between independent and dependent variables. Demonstration of relationship between independent and dependent variables. Calculation of a mathematical equation that describes the relationship in the data. Calculation of a mathematical equation that describes the relationship in the data.

5. Scientists who used graphs to make important discoveries. Galileo’s falling body experiments showed all objects, no matter what they weigh fall at the same rate. Galileo’s falling body experiments showed all objects, no matter what they weigh fall at the same rate. Schwabe’s sunspot data showed that sunspots are at a maximum every 11 years. Schwabe’s sunspot data showed that sunspots are at a maximum every 11 years.

6. Dependent Axis and Independent Axis In any graph there are at least 2 axes In any graph there are at least 2 axes Vertical axis is called the y-axis Vertical axis is called the y-axis Plot the dependent variable on the y-axis. Plot the dependent variable on the y-axis. May refer to it as y. May refer to it as y.

7. Horizontal axis is called the x- axis. Plot the independent variable here except when time is a variable Plot the independent variable here except when time is a variable Always plot time on the x-axis Always plot time on the x-axis May refer to the variable as x May refer to the variable as x

8. (x,y) is the pair of data. Plotting this point allows you to see the relationship between the variables. (x,y) is the pair of data. Plotting this point allows you to see the relationship between the variables. Linear Data – data points form a straight line. Linear Data – data points form a straight line.

9. Best Fit Line Approximates the linear trend of the data. The line passes in between the data points. The line passes in between the data points. The line does not have to pass through all of the points. The line does not have to pass through all of the points. Points should be relatively close to one another. Points should be relatively close to one another. Random error causes about ½ the points to be above the line and ½ to be below the line. Use a straight edge to approximate. Random error causes about ½ the points to be above the line and ½ to be below the line. Use a straight edge to approximate. Can be computed by mathematical equation or computer program like Excel. Can be computed by mathematical equation or computer program like Excel.

10. Best Fit Curve Used when data points form a curve. Used when data points form a curve. Should pass between points but must remain smooth. Should pass between points but must remain smooth.

12. Types of Linear Relationships Direct Linear Relationships As the independent variable increases, the dependent variable increases. As the independent variable increases, the dependent variable increases. As the independent variable decreases, the dependent variable decreases. As the independent variable decreases, the dependent variable decreases. Example of both is ice cream cone - Melting vs. Temperature: Both variables move in the same direction. Example of both is ice cream cone - Melting vs. Temperature: Both variables move in the same direction. Have positive slopes and appear to go uphill. Have positive slopes and appear to go uphill. Equation is y=mx+b Equation is y=mx+b

13. Direct Linear Relationship

14. Direct Linear Proportion Direct Linear Proportion A direct linear relationship that passes through the origin (0,0). A direct linear relationship that passes through the origin (0,0). y-intercept is 0. y-intercept is 0. Equation is y = mx. Equation is y = mx.

15. Direct Linear Proportion

16. Indirect Linear Relationships Indirect Linear Relationships Indirect means as one variable increases, the other decreases. Indirect means as one variable increases, the other decreases. Have negative slopes and look like downhill lines. Have negative slopes and look like downhill lines. Indirect linear proportions are extremely unusual in the physical sciences since science deals with real situations. Indirect linear proportions are extremely unusual in the physical sciences since science deals with real situations.

17. Indirect Linear Relationship

18. Indirect Linear Proportion Data starts at (0,0) and decreases into the negative region of the graph. Data starts at (0,0) and decreases into the negative region of the graph.

19. Horizontal Line Horizontal Line 1. Indicates there is no measurable relationship between the variables. 1. Indicates there is no measurable relationship between the variables. 2. The independent variable is an irrelevant variable. 2. The independent variable is an irrelevant variable. Mass and Arc Size in the pendulum lab Mass and Arc Size in the pendulum lab

20. Indirect Linear Proportion

21. Let’s think a minute What would a vertical line tell you? What would a vertical line tell you? The values of the replicates are changing. The values of the replicates are changing. A variable has not been controlled. A variable has not been controlled. You’ve made a mistake in your procedure. You’ve made a mistake in your procedure.

22. Calculating the Equation of a Plotted Line In Algebra the general equation for a line is y =mx + b In Algebra the general equation for a line is y =mx + b x is the independent variable x is the independent variable y is the dependent variable y is the dependent variable m is the slope m is the slope b is the y-intercept b is the y-intercept We will determine a specific equation for a specific set of data where x and y will be replaced with more descriptive variables. We will determine a specific equation for a specific set of data where x and y will be replaced with more descriptive variables.

23. Calculating the Slope The slope of a line describes how quickly the dependent variable changes with respect to the independent variable. The slope of a line describes how quickly the dependent variable changes with respect to the independent variable. Large positive slopes - steep uphill Large positive slopes - steep uphill Small positive slopes – gradual uphill Small positive slopes – gradual uphill Large negative slopes – steep downhill Large negative slopes – steep downhill Small negative slopes – gradual downhill Small negative slopes – gradual downhill

24. To calculate slope Pick two points on the line (not data points) Pick two points on the line (not data points) Pick points where the line perfectly crosses a corner of the grid. Move from left to right. Pick points where the line perfectly crosses a corner of the grid. Move from left to right. Choose points as far apart as possible Choose points as far apart as possible Use Use

25. Significant Figures in Graphing The significant figures should be determined by the accuracy of the data. The significant figures should be determined by the accuracy of the data. Use the place value of the independent (x) and dependent variables (y) in the data table when selecting points off the graph Use the place value of the independent (x) and dependent variables (y) in the data table when selecting points off the graph

26. Calculating the y- intercept The y-intercept is the value of the dependent variable at the point where the best fit line crosses the y-axis. The y-intercept is the value of the dependent variable at the point where the best fit line crosses the y-axis. Estimate this value from the graph. Estimate this value from the graph. Express with units of the dependent variable. Express with units of the dependent variable. If the y-intercept is not shown on the graph, plug in values of a point and solve for b in the equation for the line. If the y-intercept is not shown on the graph, plug in values of a point and solve for b in the equation for the line.

27. Putting it all Together Change the variables from x and y to more appropriate variable. Change the variables from x and y to more appropriate variable. 2. Example pressure = P, temperature = T 2. Example pressure = P, temperature = T a. clarity a. clarity b. real situation b. real situation If there is not a standard variable, define a variable If there is not a standard variable, define a variable

28. Using the Equation Why use the equation? Why use the equation? So we do not have to redo the experiment each time we need a new data point. So we do not have to redo the experiment each time we need a new data point.

29. Melissa Hubert Example p 65

30. P 66 continued Select data points (2.0°C, 5.00atm) and (68.0°C,16.00atm) Select data points (2.0°C, 5.00atm) and (68.0°C,16.00atm) Calculating Slope: Calculating Slope: Slope = 0.167atm/°C Slope = 0.167atm/°C Equation of the line is: Equation of the line is: y = (0.167atm/°C )x + b y = (0.167atm/°C )x + b

31. Now we need b y = (0.167 atm/ºC) x + b y = (0.167 atm/ºC) x + b 5.00 atm = (0.167 atm/ºC) 2.0ºC + b 5.00 atm = (0.167 atm/ºC) 2.0ºC + b 5.00 atm = 0.34 atm + b 5.00 atm = 0.34 atm + b b = 5.00 atm – 0.34 atm = b = 5.00 atm – 0.34 atm = 4.66 atm = 4.66 atm 4.66 atm = 4.66 atm Plugging in the value of the y-intercept, the equation of the best-fit line becomes: Plugging in the value of the y-intercept, the equation of the best-fit line becomes: y = (0.167 atm/ºC) x + 4.66 atm y = (0.167 atm/ºC) x + 4.66 atm

32. Now change the variables P = (0.167atm/˚C)T + 4.66 atm P = (0.167atm/˚C)T + 4.66 atm

33. Finding Equations PWYR p69

34. PWYR p69 1. Which variable is being changed by the experimenter? 1. Which variable is being changed by the experimenter? Mass Mass Explain why you chose this variable? Explain why you chose this variable? It is the independent variable and is plotted on the x axis. It is the independent variable and is plotted on the x axis. 2. Find the equation of the best-fit line in the graph. 2. Find the equation of the best-fit line in the graph. (0g,0Kj) and (84g,250Kj) (0g,0Kj) and (84g,250Kj) E = energy E = (3.0Kj/g) M

35. P 69 continued 3. How many kilojoules (kJ) of energy would be released if 4,129 grams of ammonium dichromate decomposed? 3. How many kilojoules (kJ) of energy would be released if 4,129 grams of ammonium dichromate decomposed? E = (3.0Kj/g) M E = (3.0Kj/g) M E = (3.0 Kj/g) 4,129g = 12,387Kj = 12,000Kj E = (3.0 Kj/g) 4,129g = 12,387Kj = 12,000Kj 4. If researchers needed to limit the energy released in the chemical reaction to 975 kJ, what is the maximum mass of ammonium dichromate they can use? 4. If researchers needed to limit the energy released in the chemical reaction to 975 kJ, what is the maximum mass of ammonium dichromate they can use? 975Kj = (3.0 Kj/g) M 975Kj = (3.0 Kj/g) M M=975Kj/3.0 Kj/g = 325 g = 320 g M=975Kj/3.0 Kj/g = 325 g = 320 g

36. Derived and Empirical Equations Derived Equations can be found by graphing data or can be formulated by combining 2 or more already known equations. Derived Equations can be found by graphing data or can be formulated by combining 2 or more already known equations. C=∏d C=∏d d=2r where r = radius d=2r where r = radius Substitute 2r for d in the first equation Substitute 2r for d in the first equation C=2∏r C=2∏r

37. Empirical Equations Empirical Equations can be formulated by observing patterns in nature and creating an equation. Empirical Equations can be formulated by observing patterns in nature and creating an equation. Johanne Kepler observed the patterns of the planets orbiting the sun and concluded they had something in common. Johanne Kepler observed the patterns of the planets orbiting the sun and concluded they had something in common. If the periods of their revolutions were squared and divided by their average distances from the sun cubed, the result for all the planets was a constant. If the periods of their revolutions were squared and divided by their average distances from the sun cubed, the result for all the planets was a constant.

38. Extrapolation The extension of a best fit line beyond the plotted points. The extension of a best fit line beyond the plotted points. Allows you to make predictions. Allows you to make predictions. Use it cautiously and make sure the following conditions are met. Use it cautiously and make sure the following conditions are met. The plotted data is accurate. The plotted data is accurate. The data points were plotted correctly. The data points were plotted correctly. The relationship between the 2 variables continues. The relationship between the 2 variables continues. Small error can result in huge error over large distances. Small error can result in huge error over large distances.

39. Interpolation To go between data points. To go between data points. Can have a great deal of confidence that the interpolated data is as good as the experimental data. Can have a great deal of confidence that the interpolated data is as good as the experimental data.