Assignment 12: Spreadsheets in Math Explorations
The spreadsheet is a utility tool that can be adapted to many different explorations, presentations, and simulations in mathematics. There are spreadsheets available on almost any platform. An essential feature should be the ability to make graphs and charts from the matrix of data. Try using a spreadsheet, such as EXCEL or ClarisWorks, for some of the following investigations.
Explorations
Open the A11 spreadsheet to see these explorations. The worksheet tab number corresponds to the problem number below.
Explored are: 1,2,3,4,7,9
Saved for another day are the others
1. Construct a graph of any function y = f(x)
by generating a table of values with the x values in one column and the y values in another.
Exploration
-> see tab 1 for simple plot of y = x^2.
2. Graphs of parametric equations.
One way is to place an initial value of the parameter t in cell A1 and increment t in the A column. Put the formula for the x-coordinate, in B1 and the y-coordinate in C1. Fill down to get the appropriate range of t and then graph. Construct some graphs of parametric equations using problems from Assignment 10.
Exploration
-> I like the equations x = a (cos(t))^3, y = b (sin(t))^3, so lets mess wid dem, as Uncle Remus would say..
3. Use the spreadsheet to graph equations in polar form.
Try some of the examples from Assignment 11.
Exploration
-> I don't see how to plot polar coordinates, so they must be converted to Cartesian, as in the parametric equations above.
4. Generate a Fibonnaci sequence in the first column using f(0) = 1, f(1) = 1,
f(n) = f(n-1) + f(n-2)
a. Construct the ratio of each pair of adjacent terms in the Fibonnaci sequence. What happens as n increases? What about the ratio of every second term? etc.
b. Explore sequences where f(0) and f(1) are some arbitrary integers other than 1. If f(0)=1 and f(1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.
Exploration
-> the ratio of adjacent terms approaches the Golden ration 1.618 after only 13 terms or so.
5. Explore problems of growth,
e.g. savings, interest compounded.
6. Explore problems of maximization
such as the lidless box formed from a 5x8 sheet with a square removed from each corner.
7. Problem:
Place four numbers in the first row as follows
A B C D
For each successive row replace the entries by the absolute value of the difference of the entry just above and the entry just to the right in the previous row. In the fourth position use the absolute value of the difference of the fourth and the first (i.e. cycle)
|A-B| |B-C| |C-D| |D-A|
Will the process lead to a 0 in all 4 entries for some row?
What is the largest number of rows before a zero row is generated?
(If your answer is less than 10, you should try again)
8. Use the following exploration to generate a function to predict observed data.
a.Take a
cup of hot water and measure its initial temperature (time = 0) and then record
temperature readings each minute for 30 minutes. Make note of the room
temperature . . .
b. Enter the data on a spread sheet and construct a function that will model
the data.
c. Using the function predict the temperature after 45
minutes, 60 minutes, or 300 minutes.
d. Calculate a measure of the error between your model and the observed data by
taking the square of the difference for each time, sum the squares, and divide
by the number of data points. You can use this statistic to guide refinement of
your function to model the data.
The following spreadsheet graph is from one set of "cooled data." The raw data is plotted in Series 1 (the black squares and line, nearly hidden). The theoretical curve is in Series 2 (the purple).
9. Similar to Problem 8 but
use the String data, the length of a guitar string to each fret. . .
That is, the set of data is formed by measuring the length of a guitar string
from a fret to the bridge of the guitar. The open string can be thought of as
"fret 0", and the length of the open string is the initial
measurement. Then make measures for frets 1, 2, . . .,
22 and plot a graph of the points where the fret number is the x coordinate and
the length is the y coordinate. To see a completed graph
click here. Write
a function that produces this graph. Hints, if wanted.
Exploration
-> Note: I cheated
on this one and had my almost 10 year old daughter
For equation: Length
= 65.2
– len first fret .059 (L)]
– len 2nd fret .059 (65.2 - .059 (L))
– len 3nd fret .059 (65.2
– {len first + len 2} )
and so on.
Looks a lot like the interest rate problem of (1+ i)
^n.
Equation should be
Length = 65.2 (1 – 0558)^n
And a check of this shows it to be good.
10. Similar to Problem 8 but use the coiled spring
data.
This data set has been obtained using a sensor (e.g. a CBL probe) to measure
distance and the data was recorded directly into a computer. A
"slinky" was held above the probe that had been placed on the floor.
When the lower end of the slinky was released it bobbed up and down and the
probe measured the distance from the floor 295 times in about 30 seconds.
Excel
file for Spring Data
11. Similar to Problem 8 but use the tree data.
This data is from the lumber industry, giving the approximate number of board
feet of lumber per tree in a forest of a given age. What function will fit the
data? Predict the harvest for ages other than those given.
Excel
file for Tree Data
12. Similar to Problem 8 but use the Stamps data.
This data set is based on the first class letter postage for the US Mail from
1933 to 1996. Plot the data and develop a prediction function. When will the
cost of a first class postage stamp reach $1.00? when
will the cost be 64 cents? how soon should we expect the next 3 cent increase?
Excel
file for Stamps Data