Author: Robin A. Ward, California Polytechnic State University-San Luis Obispo
Audience: algebra I or algebra II students
Mathematical Topics: graphing using the Cartesian coordinate system, understanding and discovering the functional relationship between degrees Fahrenheit and Celsius, finding the line of best fit, computing slope and y-intercept, finding the slope-intercept form of a line, converting between degrees Fahrenheit and degrees Celsius
Rationale:
According to the NCTM Standards 2000,one of the Focus Areas for grades 6-8 is that students should:
The Standards 2000also emphasize the appropriate use of technology in the mathematics classroom
In this activity, students will be given the opportunity to discover the linear relationship between degrees Fahrenheit and degrees Celsius. Students will first use graphing paper to plot two pairs of points and then manually compute the slope and y-intercept of the line passing through these points. Using this information, students will develop the linear equation (expressed in slope-intercept form) that describes degrees Fahrenheit in terms of degrees Celsius. Using this equation, students can convert from degrees Celsius to degrees Fahrenheit. In doing so, students are given practice with solving a linear equation with one unknown.
If students have access to graphing calculators, students can use the STAT PLOT feature to plot these same points and then calculate the line of best fit to determine the symbolic, mathematical equation that represents the relationship between these two units of measure. Thus, students can compare the equation they computed to the calculator-computed equation to see how well their prediction matched the actual equation representing the relationship between degrees Fahrenheit and degrees Celsius.
Finally, students will manually verify the conversion formula (representing the relationship between degrees Fahrenheit and degrees Celsius) by finding the corresponding Fahrenheit temperature, given a value in degrees Celsius. If students have access to graphing calculators, they can use the TABLE feature to verify the formula.
Additionally, students will learn why temperature is so important in flight. All aircraft tested at the NASA Dryden Flight Research Center constantly measure air temperature in order to determine their altitude during flight. Click here to learn more about the science of flight testing.
Materials:
Background:
As any aircraft flies higher and higher, the temperature in the atmosphere decreases at a nearly constant rate, known as the lapse rate, up to an altitude of approximately 36,000 feet. This region of the atmosphere is called the troposphere. Above 36,000 feet, the temperature remains nearly constant, at -67 degrees Fahrenheit, up to an altitude of approximately 75,500 feet. This region forms the lower part of the stratosphere. Through the remainder of the stratosphere, the temperature increases, reaching approximately 26 degrees Fahrenheit at an altitude of approximately 164,000 feet. The temperature increases in this portion of the stratosphere as a result of heating by absorption of solar UV radiation in the ozone layer. The relationship between
temperature and altitude, as described above, is shown below.
Knowing the temperature is very important to a pilot flying an SR-71, the world's fastest and highest-flying plane. The SR-71 Blackbird can fly at Mach 3+, which is more than 2,200 miles per hour. Sound travels around 740 miles per hour. This speed is called "Mach 1." Mach 3+ means more than three times the speed of sound. The SR-71 can fly at altitudes over 85,000 feet. The high speeds and altitudes have a dramatic effect on the plane itself, actually causing it to expand during flight and heat up to temperatures of up to 800 degrees Fahrenheit. This is another reason why knowing temperature is important!
SR-71 Blackbird was developed in the 1950's as a reconnaissance plane for the U.S. Air Force. Because it flies at such high speeds and high altitudes, NASA Dryden Flight Research Center and the U.S. Air Force have used the SR-71 for research since the 1960's.
Recently, the SR-71 has been used for experiments investigating the earth's atmosphere, particularly in protecting and rebuilding the shrinking ozone layer. It has also been used to establish a new personal communications network in conjunction with Motorola. Data received from SR-71 flights will be used in the future to engineer supersonic/hypersonic aircraft for use as commercial carriers.
View several photos of the SR-71.
View a movie clip of the SR-71.
To measure temperature in the Metric System, we commonly use units called Celsius. In the U.S. Customary System of Measures, we measure temperature using degrees Fahrenheit. The formula that describes the relationship between these two units of measure is:
where F represents degrees Fahrenheit and C represents degrees Celsius.
Where does this formula come from? In this activity, students will derive the linear equation that represents the relationship between these two units of measure using graph paper and/or graphing calculators. Students can also learn more about the SR-71.
The Activity:
The above graph was generated by storing the points (0, 32) and (100, 212) using STAT ==> EDIT and then choosing the following WINDOW setting.
The slope of the line is 9/5 and is computed below:
where the x represents degrees Celsius and the y represents degrees Fahrenheit.
So, using F (for Fahrenheit) and C (for Celsius), we get:
Thus, at this point, students have now made the algebraic connection (F= (9/5)C + 32) to the graphical representation (their sketch on the graph paper) of the relationship between degrees Fahrenheit and degrees Celsius.
98.6 = (9/5)C + 32
To solve for C, we need to first subtract 32 from both sides of the equation (to isolate C).
66.6 = (9/5)C
To solve for C, we need to multiply both sides of this equation by 5/9.
37 = C
Thus, normal body temperature is 37 degrees Celsius. To verify this graphically, students should check if (37, 98.6) is located on the line they have graphed. In doing this, once again students can see the connection between the algebraic representation of the equation and the graphical representation of the equation.
800 = (9/5)C + 32
To solve for C, we need to first subtract 32 from both sides of the equation (to isolate C).
768 = (9/5)C
To solve for C, we need to multiply both sides of this equation by 5/9.
426.67 = C
Thus, 800 degrees Fahrenheit corresponds to 426.67 degrees Celsius.
Enrichment Activity:
Ask students to change the labeling on the axes such that degrees Fahrenheit is on the x-axis and degrees Celsius is on the y-axis. Ask students to predict if this will change the shape of the graph, and if so, how? The graph will stay linear since the relationship has not changed between the two variables.
Next, ask students to re-plot the same two points, but students must now switch the points to (32, 0) and (212, 100). Ask students why we need to do this. We must change the points because the x-coordinate in the (x, y) pair represents degrees Fahrenheit (which is now located on the x-axis) and the y-coordinate in the (x, y) pair represents degrees Celsius.
Ask students to now compute the y-intercept and the slope of the line passing through these two points, as done earlier in the activity. Finding the y-intercept will be a little more difficult this time, since it is difficult to visually judge the exact point at which the line intersects the y-axis. Let's start by finding the slope first.
Thus, the slope of the line passing through the points (32, 0) and (100, 212) is 5/9.
Now, to find the y-intercept, let's choose one of our known points, say (32, 0), and plug this point into the values for x and y to solve for b.
0 = (5/9) * (32) + b
0 = 17.78 + b
b = -17.78
Now that we know the slope is 5/9 and the y-intercept is -17.78, we can now find the linear equation that defines degrees Celsius in terms of degrees Fahrenheit.
y = (5/9)x - 17.78
or
C= (5/9)F - 17.78
Students can verify the accuracy of their graph by checking if the point (98.6, 37) (which represents normal body temperature in degrees Fahrenheit and Celsius, respectively) is located on the line passing through the two points previously plotted. If this point is located on their line, this will confirm that their graph is correct.
Students can verify their linear equation algebraically by plugging in the Fahrenheit value for normal body temperature (98.6 degrees) into the equation y = (5/9)x - 17.78 to see if it results in a value of 37 degrees (normal body temperature in degrees Celsius). This will confirm that they have obtained the correct linear equation algebraically.
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NASA Dryden Flight Research Center
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