Corrections and suggestions for
Instructor’s Manual for
Laboratories in Mathematical Experimentation:
A Bridge to Higher Mathematics
Chapter 3
Question 4:
A somewhat more tractable question is to choose a prime p and ask
what the probability is that integers a and b chosen randomly are not divisible by p
This chapter leads naturally to investigating the existence of multiplicative inverses modulo m. .
Chapter 6
It is useful to introduce this chapter by having students actually carry out a simulation of the randomized
response procedure, including choosing an “embarrassing” question to answer and actually doing the
needed coin flips. Tabulating the results in the format used in the programs prepares students to interpret
the output of the computer simulations, and doing actual tosses of coins or dice reinforces the ideas of
probability and randomness. Plotting the distribution of outcomes of repeated simulations is also useful.
Students may need to discuss the meaning of p versus the estimate p “hat”.
Introducing students to “tree diagrams” would help them with the expected value computations in this
chapter.
You might wish to divide this chapter’s investigations into two pieces. Since 6.4.1 and 6.4.3 on
randomization are more accessible, you might do them first. The sections 6.4.2 and 6.4.4 on estimation
could be viewed as a second , more challenging, project.
Chapter 9
This chapter has been successfully done on a TI-82. (The graphing proceeds slowly, so the viewer can see
the curve being traced out without requiring that the program include a moving cursor tracing the graph.)
Chapter 10
Section 10.5, Monte Carlo methods:
You may wish to have your students look at the distribution of errors after repeated simulations.
Another interesting question is: What happens to the error if you change the curve but keep the area
constant?
Chapter 11
Another way to frame the key question for investigation in this chapter is to ask how long it takes J(n) to
grow by 1; that is, what is the smallest k for which J(n+k) is at least J(n) + 1?
Chapter 12
This chapter leads naturally to the idea of the average value of a function.
Chapter 13
If not too many iterates are needed, a TI-82 can be used: just enter the desired function and hit the enter
key for each iteration.
Chapter 14
You may wish to introduce this chapter by telling your students about the applications to population
biology. (For example, the first chapter of P. Waltman’s book Competition Models in Biology,
published by SIAM in 1983, gives a good account of logistic growth models.)
You may find that your students are helped by collecting numerical data first and then graphical data, so
they are clear on what the graphical data represent.
Question 4:
The graphical representation is clearer if a smaller interval of a values is used in ITERGRAPH –
e.g., 3.4 <= a <= 3.6.
Section 14.4:
Students can be confused about what the program DIAGRAM shows them. A table of values might be a
good introduction to the graphical representation.
Chapter 15
You may wish to motivate this chapter by a brief discussion of an application, e.g., population dynamics.
Chapter 16
A nice preliminary by-hand exercise is to provide students with suitable grid paper and colored pencils so
they can carry out the analogue of the sieve of Eratosthenes. Depending on your students’ backgrounds,
you may also want to draw more pictures of vector addition etc.