## Cooled Data Exploration

### Helene Chidsey and Lou Ann Lovin

In this investigation we used a spreadsheet to analyze data from cooling water. We measured the water temperature using a candy thermometer with a scale from 130 to 390 degrees Farenheit. The temperature readings were taken over a thirty minute period, at one minute intervals. The initial temperature reading was measured in boiling water.

When measuring the water temperature, after the boiling water was poured into a cup, the temperature drastically dropped during the first minute. Due to the limitations imposed by the candy thermometer, we predicted that after 30 minutes, the temperature of the water would be off the scale. (In fact, we tried heating the cup to offset the drastic decrease, however, we did not heat the cup enough to keep the last five temperature readings on the scale.) So, we removed the pot from the heat source and recorded the temperature of the water directly in the pot. The room temperature was about 71 degrees Farenheit.

After plotting the data points using the spreadsheet, we constructed the following function that models the data: where x is the time the temperature was taken and y is the temperature. We used trial and error to construct this function, focusing on decreasing the measure of error between the model and the observed data. This measure of error was found by taking the square of the difference for each time, summing the squares, and dividing by the number of data points (i.e., the variance). The measure of error was decreased to 7.082. The following table shows the observed and predicted temperatures. The following graph shows the observed data (Series 1-blue) and the predicted data (Series 2-pink). Although the measure of error between the model and the observed data is 7.082, we get the following predicted temperatures after 45, 60 and 300 minutes respectively: 136.12, 155.84, and 3531.2. If we look at the function used to predict the data, we see that the time is squared. Thus at time = 300, we would expect the predicted temperature to be HUGE!!!!!!!! SO THIS IS A BAD MODEL to use for predicting later temperatures, yet it is quite good within our time range!!!!

In conclusion, there are many factors in this investigation that can have an effect on the data. For instance, the type of container used to hold the water, the temperature of the room, the type of thermometer used (e.g., the scale on the thermometer could be different from the one used in this investigation), and human error in reading the thermometer all can have some effect on the data collected. Thus, given different conditions as well as different data collectors, we could obtain different results.

For application to the classroom, we could have students perform this investigation at home and then compare data and make inferences from their findings. For example, they could look at the effects of different room temperatures on the cooling process. Also, they could look at the effects of the container on the cooling process. One important point to emphasize, is that they need to take the temperatures in a consistent manner with respect to other constraints (i.e., leaving the water in the pot or pouring the water into a heated cup, if the thermometer used has an appropriate scale, etc.). Ask the students what effects they are interested in investigating. With the use of the spreadsheet, there seem to be an endless array of interesting things to explore!