Getting Warmer

Objective: To find the temperature at which Celsius and Fahrenheit are equivalent. Extension: formulate the conversion equation for changing from Celsius to Fahrenheit.

Prior knowledge: Students should know how to write a linear equation.

Materials Needed:

Coffee maker or beaker of water with hot plate
Container of ice water
2 Medium to large Styrofoam cups

Group Size: 2

Setting up the Lab Pro and calculator:

Insert the Lab Pro and TI calculator in plastic holder. Connect with the short link cord into the bottom of the calculator and Lab Pro. Lab Pro should be connected to an electrical outlet.
Insert two temperature probes into Channels 1 and 2 on the side of the Lab Pro. Fasten the probes together with twist-ties.
In the TI calculator, go to “APPS” and find “DATAMATE”. The calculator should immediately recognize the probes. One of the probes should be measured in Celsius, the other in Fahrenheit. If not, choose Channel #1 for Celsius and Channel #2 for Fahrenheit. To modify the units, choose the first option, “setup”, and on the next screen use the arrow to choose the channel you need to modify, and press enter. Choose the first option, “temperature”, and then select the appropriate temperature unit. Remember to select the same probe the calculator recognized at the beginning of the application. (For example, if Channel 1 is “Stainless Temp (C)”, then choose “Stainless Temp (F)” for Channel 2.) Choose “OK” to return to the main menu.
Again, choose the “setup” option, arrow down to “MODE” and press enter. Select the second option “TIME GRAPH”, then the second option “CHANGE TIME SETTINGS” and enter “3” for the time between samples, press enter, and “20” for the number of samples, and press enter. Choose “OK” twice to return to the main menu.

Procedure:

Fill one Styrofoam cup approximately ¼ of the way full with ice water. Fill the other Styrofoam cup ¾ of the way full with hot water. Insert the connected probes into the ice water and stir until the reading on the Lab Pro stabilizes (stops decreasing). You will continue the stirring motion for the entire experiment.
Press the second option, “START”, on the Lab Pro to begin collecting the data. Pour some hot water into the ice water approximately every 10 seconds, making sure that you have enough hot water for 5 or 6 additions.
The Lab Pro will beep when it has stopped collecting the data. You can remove the temperature probes from the water.
Choose option #4, “MORE”, to see graph options for your colleted data. Choose option #6 to see both sets of data graphed with respect to time. Check that the graphs are increasing in a linear fashion.
If your graph includes “flat parts” you will need to omit them from consideration before you calculate a linear model for your data. Hit “enter” and choose option #8, “RETURN TO GRAPH SCREEN”, select option #2, “SELECT REGION”. This will take you back to one graph and ask you for a left bound. Move the cursor with the arrows to the right until you have passed the “flat part” at the beginning of the graph and hit enter. Then choose a right bound in the same manner and hit enter. Hit enter once again & then choose option #1, “MAIN SCREEN”.
To calculate the equation of the line of best fit for each set of data, choose option #4, “”ANALYZE”, then select option #2, “CURVE FIT”. This time choose “LINEAR (CH 1 VS. TIME)”. Record this equation on your paper as temperature in degrees Celsius vs. time. Press “enter” and you will see the graph of this line with its data.
Choose “enter” and repeat step 5 to calculate the equation of the line of best fit for “LINEAR (CH 2 VS. TIME)”. Record this equation on your paper as temperature in degrees Fahrenheit vs. time.
Choose “ENTER”, then “RETURN TO MAIN SCREEN”, then “QUIT”. Choose “ENTER” once again to completely exit the application.
Turn off any active STATPLOTS. Go to “y =” and clear any functions from your graphing menu. Graph both linear models using y1 and y2. To find the temperature that is the same in both Celsius and Fahrenheit, find the point of intersection of the 2 graphs. If you need help using this calculator feature, link to Lessons/Calculator Tools/Using the Calculate Menu—Part III.
(Hint: a window of x_min = -100, x_max = 100 with a zoom-fit should work well.) The temperature where the 2 scales are equal is a whole number.

Extension: Let x = temperature in Celsius, and y = temperature in Fahrenheit. Use the point at which these two scales are equal, along with the point representing the freezing point for water on each scale (0° for Celsius and 32° for Fahrenheit), and find a linear model that converts from Celsius to Fahrenheit. Compare this to the actual conversion equation:

F = (9/5)C + 32

Ask students to define the actual meaning of the slope and y-intercept of the conversion equation.