Telling a Story With Graphs

Task

Each of the following graphs tells a story about some aspect of the weather: temperature (in degrees Fahrenheit), solar radiation (in watts per square meters), and cumulative rainfall (in inches)) measured by sensors in Santa Rosa, CA in February 2012. Note that the vertical gridlines represent the start of the day whose date is given.

1. Give a verbal description of the function represented in each graph. What does each function tell you about the weather in Santa Rosa?
2. Tell a more detailed story using information across several graphs. What are the connections between the graphs?

IM Commentary

In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points, that they can tell a story about the variables that are involved and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points correspond to features in another graph, for example on a rainy day the solar radiation is very low and the cumulative rainfall graph is increasing with a large slope.

Some of the quantities shown are very familiar to students, such as temperature, where others might be less familiar, such as solar radiation. We can think about this quantity as the maximum amount of power that a solar panel can absorb. Depending on the experience of the students, teachers might want to discuss the idea of cumulative rainfall, i.e. the total amount of rain that has fallen since the beginning of the season.

All the presented data come from the website of the California Department of Water Resources and can be found at http://cdec.water.ca.gov/

Technically, the graphs only show some of the values of the functions they are meant to represent. A bivariate data plot is a representation of a function in the same way that a table is a representation of a function; while it has some gaps in information, there is an underlying function that the bivarate data plot is assumed to sample. (In this case, the data points are joined by lines which means we are interpolating between our given values.) So the tasks implicitly expect students to answer the question about the temperature (or solar radiation, or precipitation) that is a function of time based on the information about it provided by the sampled data. Given the qualitative nature of the tasks, this does not present a problem.

Solution

1. All graphs show functions that have the same independent variable, namely the time $t$, measured in days. All graphs have different domains, but they do overlap. They all show domains for different time periods in February of 2012. The independent variables are different in each graph. All graphs show a different weather feature in Santa Rosa, CA.

   The first graph shows temperature, $T$, in degrees Fahrenheit, as a function of time (by date and time) over a one-week period starting at midnight on February 6, 2012. On five days the temperature rose into the high 50s to low 60s during the day and fell to the high 40s to low 50s during the night. The maximum temperature during the given time period was 69 deg. F and it occurred in the early afternoon of February 9. The minimum temperature was 37 degrees F and it occurred in the early morning of February 12. February 7 and 10 were special insofar that the temperature did not change much throughout the entire day. Especially on February 7 the temperature stayed in the low 50s all day long.

   The second graph shows solar radiation in watts per square meter, as a function of time for 10 days starting on February 6, 2012. We can think of solar radiation as the power that a square meter of solar panel produces. This function shows some definite regularity. Every day the function values are zero for a certain time interval. This corresponds to the hours when it is dark and a solar panel would not produce any power. The function increases in the morning, reaches a peak in the middle of the day and decreases in the evening. On most days the maximum value is between 550 and 650 watts per square meter. Again, February 7 and February 10 are the exception. During those two days the maximum solar radiation was just over 50 and just below 250 watts per square meter, respectively.

   The third graph shows the cumulative amount of rainfall in inches as a function of time; this is the total amount of rain that has fallen since the season started. With the information given, we don’t really know when the beginning of the season was. This function is increasing on the entire time interval shown (February 1 through February 17, 2012), which makes sense, since we are keeping track of the total amount of rainfall. We can see that the function is increasing slowly from February 1 until February 7 and then the graph becomes much steeper. The cumulative amount of rain increased much more on February 7 than on any other day.

2. After analyzing all the graphs, it becomes clear what the weather was like in early February of 2012 in Santa Rosa. Most days it was sunny with temperatures reaching the mid 60 during the day and the mid 40 during the night. On February 7 it rained, but not very hard. We see that the cumulative rain graph is steeper during that time, but it only increases by 0.2 inches, so the rain can’t have been very heavy. Also, since the solar radiation numbers were very low, this shows that there was not much sunshine during the day, which we would expect for a rainy day.

   On February 10 it was cloudy and cooler during the day but not especially rainy. A cooler air system moved into the area after February 10 since daytime temperatures reach highs in the low 60s to high 50s and the nighttime low temperatures even drop into the 30s. We can’t say anything about the temperatures after February 13, but the solar radiation and rain graphs suggest continuing sunny days.

   There must have been a little bit of rain after February 7 as the cumulative rainfall continues to increase slightly, but it wasn't very much and seems to have been spread out over a number of days which is consistent with the information about solar radiation and temperature on those days.