Students can get a good research experience by carrying out the first three items; the
last two are rather advanced and will stretch your best students. In addition, some
students will want to take a detour to look at things like the fundamental theorem of
arithmeticthe fact that every integer can be factored in essentially one way into
a product of primes. This is discussed on The Fundamental Theorem of
Arithmetic.
So, there are many ways to do this. Here’s just one possibility.
Phase 1. What does probability look like in infinite sample spaces? Discuss the
difference between “probability 0” and “impossible” (see Results). Work through
some “dartboard” examples of geometric probability. This could take two or
three classes. Then work through the Warm Up Problems. That will take
another class. It’s really important at this stage to get students talking and
writing about their thinking so you can catch the subtle (and inevitable)
misconceptions that develop. For example, get them to explain their intuition
about the probability of getting “3” on a random pick of integers. Why is this
different from “getting 3 if the pick is restricted to the integers between 1 and
100?”
Expect unusual and clever ideas. One (very advanced) student argued that the
probability of picking the integer n should be . She had two reasons why this was a
good choice:
 The sum of the probabilities over all the integers is 1.
 If someone asks you to pick a number at random, you’re more likely to
pick a small number than a very large one, so probability should decrease
with size.
Her model led to some very strange consequences.
Phase 2. Work through the perfect square probability. Ask students to come up with a
formula for the number of perfect squares between 1 and n (see Results). After
students have ideas that are general in principle, you may have to introduce the
greatest integer function and talk a bit about limits. Estimate three classes for
this.
Phase 3. Build and run a simulation. This could take anywhere from a couple days to
a couple weeks, depending on how facile your students are with the relevant
technology. Try to build the simulation so that the “test” is modularthat is, the
same simulation should estimate the probability that an integer is prime or square,
just by changing the tester. This is an especially nice project for a computer science
class.
If you are using Mathematica, a crude start on a program that lists the primes in a
given range goes something like this:
g[n_] := If[PrimeQ[n] == True,1,0]*n
 This models a function g defined by
Then define f that maps g over the integers from 1 to m:
f[m_] := Map[ g,Range[m]]
 So, for example, f(100) outputs a list of 100 numbers that contains all the primes
between 1 and 100 (and some zeros).
Phase 4. Here’s where more direct teaching begins. Building on the Warm Up
Problems, you’d like students to reason that the probability that a number between 1
and 100 is prime is
This is because a composite number between 1 and 100 has a prime factor
between 1 and . So, a number between 1 and 100 is prime if and only of it
is not divisible by 2, 3, 5, or 7. And you’d like things to generalize from
here.
You can expect things like this:
A brilliant (but incorrect) idea. But it’s the germ of exactly the right idea, and
you’ll have to find a way to tease out the subtle mistake (the ). What’s
worse is that this brilliant mistake is more than likely not the one that will
happen with your students. Think about several classes (maybe a week)
to get this solid, and beware that, even then, it won’t be solid for some
students.
The point here is to get students comfortable with writing down a mathematical
expression that, if evaluated, would give the right probability. So, this phase is as
much about building confidence in the technique as it is about developing skill at the
technique. For example, the problem above (pick a prime less than 100) can be
“checked” in a couple ways: Have one group of students evaluate the product and
another count the primes between 1 and 100. Then try the same experiment (using a
CAS) with a sample space of 1500. When you get to the infinite product, you have a
real laboratory for experimentation.
You might ask students to compare
with
for
some values of n between 1 and 1000. This “product notation” is similar to
summation notation. For example,
Here’s a table of the first 168 primes (the primes up to 1000). It was generated in
Mathematica by the command
TableForm[Table[ {n,Prime[n]},{n,1,168}]]
n  the nth prime 

 1  2 

 2  3 

 3  5 

 4  7 

 5  11 

 6  13 

 7  17 

 8  19 

 9  23 

 10  29 

 11  31 

 12  37 

 13  41 

 14  43 

 15  47 

 16  53 

 17  59 

 18  61 

 19  67 

 20  71 

 21  73 

 22  79 

 23  83 

 24  89 

 25  97 

 26  101 

 27  103 

 28  107 

 29  109 

 30  113 

 31  127 

 32  131 

 33  137 

 34  139 

 35  149 

 36  151 

 37  157 

 38  163 

 39  167 

 40  173 

 41  179 

 42  181 

 
n  the nth prime 

 43  191 

 44  193 

 45  197 

 46  199 

 47  211 

 48  223 

 49  227 

 50  229 

 51  233 

 52  239 

 53  241 

 54  251 

 55  257 

 56  263 

 57  269 

 58  271 

 59  277 

 60  281 

 61  283 

 62  293 

 63  307 

 64  311 

 65  313 

 66  317 

 67  331 

 68  337 

 69  347 

 70  349 

 71  353 

 72  359 

 73  367 

 74  373 

 75  379 

 76  383 

 77  389 

 78  397 

 79  401 

 80  409 

 81  419 

 82  421 

 83  431 

 84  433 

 
n  the nth prime 

 85  439 

 86  443 

 87  449 

 88  457 

 89  461 

 90  463 

 91  467 

 92  479 

 93  487 

 94  491 

 95  499 

 96  503 

 97  509 

 98  521 

 99  523 

 100  541 

 101  547 

 102  557 

 103  563 

 104  569 

 105  571 

 106  577 

 107  587 

 108  593 

 109  599 

 110  601 

 111  607 

 112  613 

 113  617 

 114  619 

 115  631 

 116  641 

 117  643 

 118  647 

 119  653 

 120  659 

 121  661 

 122  673 

 123  677 

 124  683 

 125  691 

 126  701 

 
n  the nth prime 

 127  709 

 128  719 

 129  727 

 130  733 

 131  739 

 132  743 

 133  751 

 134  757 

 135  761 

 136  769 

 137  773 

 138  787 

 139  797 

 140  809 

 141  811 

 142  821 

 143  823 

 144  827 

 145  829 

 146  839 

 147  853 

 148  857 

 149  859 

 150  863 

 151  877 

 152  881 

 153  883 

 154  887 

 155  907 

 156  911 

 157  919 

 158  929 

 159  937 

 160  941 

 161  947 

 162  953 

 163  967 

 164  971 

 165  977 

 166  983 

 167  991 

 168  997 

 
Phase 5. You’re looking at a month or two for this, maybe longer. Study the Results
and you’ll see that there’s some very difficult material here. We’ve found it
best to move aroundintroduce something new, then go back and reinforce
something from the earlier phases. Paul Goldenberg (a member of the Making
Mathematics team) distinguishes between the “clock time” and “calendar
time” it takes to understand something. This takes calendar timeslow
but steady hammering at the ideas, every day adding a little new territory
while and making more permanent structures on what has already been
settled.
