## Streamflow
Temperature

### Part
1: Streamflow Prediction

This module
is the first of three which are designed to lead to the prediction
of streamflow for a river in a particular region. Temperature
obviously affects the type of precipitation we get, and the amount
of precipitation affects the amount of water in the river. In
this module, we will construct a model for average temperatures
in the region.

**Comprehension**
Observations
of temperature trends, visualization of a model

- Write down
average monthly temperatures for the area in which you live.
You can estimate these or consult a website for accurate data.
- Draw a
coordinate system with months listed on the horizontal axis
and temperature on the vertical axis. Then plot your average
temperatures on this system.
- What kind
of function would best fit the points on your graph?
- Would the
function be different for different parts of the world?
- Why do
you think it might be important to know about the average temperatures?
- How does
temperature affect precipitation?
- Sketch
a graph of the function that might describe average monthly
temperatures in your area for one year.
- How would
mathematics be useful in the analysis of temperature changes
and the resulting effects?

**Acquisition**

**Mathematical
Topics**

Learn or review mathematical concepts and skills to study
temperature change. See the menu at the left.

** Information**

The table below gives the average temperature for the indicated
month in a certain location in the southwest.

Month |
April |
May |
June |
July |
Aug |
Sept |

Average
Temp (degrees F) |
25 |
36 |
53 |
61 |
61 |
47 |

We will use
regression to fit these data with a quadratic function. Functions
will be of the variable t, 0 __<__ t __<__ 12, where
t represents monthly time of year. For example, t = 5 corresponds
to the end of May, t = 4.5 corresponds to mid-May and t = 9.3
corresponds to the point in time which is .3 into the month of
October. Also, since these functions are cyclic (repeat year after
year) t = 0 and t = 12 correspond to exactly the same time, the
very beginning of January and the very end of December (midnight
on New Year's Eve).

In order to
approximate the average temperature for any month, we use the
mid-month value for t. For example, for the average temperature
for the month of May, we use t = 4.5. The table below shows the
values to use to determine averages for each month of the year.

Jan. |
Feb. |
Mar. |
Apr. |
May |
June |
July |
Aug |
Sep. |
Oct. |
Nov. |
Dec. |

0.5 |
1.5 |
2.5 |
3.5 |
4.5 |
5.5 |
6.5 |
7.5 |
8.5 |
9.5 |
10.5 |
11.5 |

The function
will be denoted A(t); A(t) will be the average temperature at
time t, 3 __<__ t __< __10.

**Application**
Apply mathematical
knowledge and Tool Chest Applets to data provided to analyze average
temperatures

**Questions**
- Round answers
to one decimal place.
- Make points
from the average temperature data provided; the first coordinate
will be the mid-point of the month and the second coordinate
will be the average temperature for that month. Plot these points
on the applet screen.
- Use regression
to fit the given data with a quadratic function A(t). Graph
the function showing an appropriate domain and range. Use this
function to answer the remaining questions.
- What will
be average temperature be on May 15? On August 15? On April
6? On September 21?
- When will
the daily average temperature be 50 degrees? When will it be
freezing?
- On what
day will the average temperature be the warmest?

**Reflection**

Examine the model and its implications
- Do you
think that this model would work for long periods of time, i.e.
decades or centuries? Why or why not?
- Could this
function be used to predict the temperature at a specific time
on a given day?
- What are
some properties of the quadratic function that make it appropriate
to use in this situation?
- What do
you see as advantages to the use of this function? Disadvantages?
- Do you
know other functions that might also be appropriate for use
in this study?