MAE 301 Foundations of Secondary School Mathematics

MAE 301 Homework ~~ Fall 2000

1. Make up a problem involving the triangle inequality. (p. 47) Try to be different from the book!
2. Explain matrix multiplication (2x2) and show why in general AB does not equal BA. Is there a multiplicative identity matrix? Do matrices have multiplicative inverses?
3. Let [A,B]=AB-BA. Show that this operation is anti-commutative and non-associative.
4. What is the relation between absolute value and distance? Rephrase the inequalities |x-2| < pi and |x+3| > 1 in terms of distance.

For November 6:

1. Larson's Algebra II has problems on p. 143 about the number of solutions to pairs of linear equations in 2 unknowns. Analyze 42,43,45,46,58,51,52 BY ROW REDUCTION of the associated 2 x 3 matrix. Explain your work. (I.e. if you use a calculator, explain your input and output).
2. There is a set of problems on 3 equations in 3 unknowns on pp. 181-182. As above, analyze 14,16,20,22,28,33 BY ROW REDUCTION of the associated 3 x 4 matrix.
3. Make up a system of 3 equations in 3 unknowns where there is a 2-parameter set of solutions. It is hard not to have this fact obvious on cursory inspection of the system, but try.
4. Make up a system of 4 equations in 4 unknowns where there is a 2-parameter set of solutions. Disguise the nature of the system with row operations if necessary.
5. Explain why a system of 3 linear equations in 4 unknowns cannot have a unique solution. Argue by ROW REDUCTION.
6. Is it true that a system of 3 linear equations in 4 unknowns must always have at least one solution? Argue by ROW REDUCTION.

For November 13:

Prepare derivations of the following laws:

1. Law of cosines
2. Law of sines
3. Addition law for cosines
4. Addition law for sines.

For November 20:

1. Explain how a slide-rule performs multiplication.
2. Explain how a piano keyboard is like a logarithmic scale. If the A below middle C has frequency 440 Hertz (cycles per second), what will be the frequency of the sound made by a bird which sings an A four octaves higher? Which is the most convenient base for logarithms in describing the pitch of a musical sound?
3. Explain the Richter scale for expressing the strength of earthquakes.
4. The intensity of sound is measured in decibels. "A decibel is an arbitrary unit based of the faintest sound that a man can hear. The scale is logarithmic, so that an increase of 10db means a tenfold increase in sound intensity; ..." (from Time magazine).
   • An ordinary conversation is 60 decibels. A rock music cocnert (near the amplifiers) is 120 decibels. How many times louder is the concert?
   • The loudness of the water at the foot of Niagara Falls is a billion times louder that the least audible sound. What does that come to in decibels?
   • We know what a decibel is; what is a bel?
   • When do you think the quoted text was written?

For December 4

1. Use your calculator to generate 400 random numbers between 0 and 1. Make and hand in a histogram of the distribution of these numbers by tenths (between 0 and .1, between .1 and .2, etc.)
2. Use these numbers to simluate 20 repetitions of the experiment of tossing a coin 20 times (let [0,.5]=Heads, [.5,1]=Tails). Suppose the outcome of each experiment is p heads and q tails. Make and hand in a histogram of the distribution of p.
3. Find, copy and submit an intelligent applicaiton of probability theory to the Florida Vote Problem. Explain carefully why you think it is intelligent.
4. Write up an analysis of the Birthday Problem: how many people do you need in a room before the probability of two of them having the same birthday rises above .5?