I have the oportunity to talk to a highschool class about mathematics, the topic I got to present are the integers modulo $n$, ie, $\Bbb Z/n\Bbb Z$ , however I don't want to be very heavy and formal, because I don't want to lose them. For the talk I also have limited time, since this is just talk, so my hands are a little tied.

At first I wanted to present the idea of equivalence class, so the fact that an infinity of numbers fit in a few boxes make sense, however I'm having a little bit of problem with that, the example I wanted to use was the hours of the day, or else, the hours of all time (sounds weird doesn't it?) which, for purposes of the discussion, we are assuming are countable infinite (and also, without them knowing we're also assuming the Axiom of choice).

Long story short, I find myself having a lot of issues, because whenever I encounter a very abstract mathematical concept (modular arithmetic, equivalence classes, infinites, etc.) I feel that I'm not actually going to help their intuition nor their interest (I aspire to get a: "I never saw it like that!") What recommendations and/or reference can you suggest about this topic?

  • 2
    $\begingroup$ Maybe Z-sub-p is not great notation if you mean integers mod p, because many people will read this as "p-adic integers". At the very least, don't use a blackboard bold "Z", maybe. $\endgroup$ Commented Mar 30, 2014 at 18:34
  • $\begingroup$ @paul yes you are right, I'll change it, thanks $\endgroup$
    – Ana Galois
    Commented Mar 30, 2014 at 18:38

6 Answers 6


I did this once with a high school group. My approach was to use the metaphor of a scientist doing a controlled experiment: You study a complicated system by introducing a single change, while trying to keep everything else constant, and watch how that change affects the system overall. In this case, the system is arithmetic, and the "one change" we are going to make is that we are going to replace the number line with a number circle: When we count, we are going to "loop back around" after a certain pre-chosen value.

Like Ittay (despite our names we are not related, by the way!) I pointed out that this approach is actually familiar, and used the "arithmetic on a clock" model to demonstrate what happens if after 12 you go back to 1. I also introduced the idea of a "7-clock", "5-clock", etc.

We practiced writing out some basic arithmetic on a 7-clock: for example, 4 * 5 = 6 on a 7-clock, and also 5 * 6 = 2, etc. I gave some puzzles, like: On what kind of clock is the equation 5 * 8 = 2 true? We discuss the fact that each number on a clock can be given many different names; for example, on a 12-clock we can think of 5 as 5, or 17, or 29, but also as -7, -19, and so on.

Then we started asking some more interesting questions, like "Which elements are invertible on a 12-clock?" and "Which elements are invertible on a 7-clock?" This starts to bring out the way in which prime numbers are different than composite numbers. Another interesting one is "How many square roots does 4 have on a 12-clock and on a 7-clock?" This latter one is an example of how changes in the ground rules for the system can have surprising ripple effects. In particular:

  • Students learn in high school that a quadratic equation has at most two real solutions, and on a 7-clock the equation $x^2=4$ has only two solutions (which can be recognized as ± 2, just like in "ordinary" algebra), but on a 12-clock the equation $x^2=4$ has four solutions. So that means that the Fundamental Theorem of Algebra -- a Theorem that high school students may learn about but do not learn the proof of -- must, at some deep level in its proof, rely on the difference between a 12-clock and a 7-clock.
  • Another difference that they can recognize is the fact that the "Zero Product Property" (which is the conceptual hinge on which turns the whole high school theory of solving quadratic equations) isn't true on a 12-clock.
  • On the other hand the equation $x^2 = -1$, which does not have any solutions in "ordinary" (i.e. real-valued) algebra, does have solutions on a 5-clock. So on a 5-clock $\sqrt{-1}$ is not imaginary!

Things that I avoided: I did not say a thing about equivalence classes or well-definedness of operations; nor did I try to provide a formal definition or proof for anything I said. If you are going to have the opportunity to come back and develop the ideas more systematically over a longer time frame those would be appropriate, but for a "done in one" lecture to a general audience I would stick with clear, specific, and surprising (if possible) examples.

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    $\begingroup$ It's a great lesson. I find it fun for it to dawn on us that 12 is behaving exactly like zero does for the integers. I think introducing identity elements with an example that isn't a do-nothing (even if it's an achieve-nothing) is more enlightening than just 0 or 1 all the time. $\endgroup$
    – AndrewC
    Commented May 1, 2014 at 16:20
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    $\begingroup$ @AndrewC I agree 100%. That is why I prefer to write the set as {1, 2, 3...., 12} rather than {0, 1, 2, ...., 11}... at least at first. $\endgroup$
    – mweiss
    Commented May 1, 2014 at 16:23

Modular arithmetic is beautiful and simple but tends to be frighteningly clotted with detail and notation at first. Whenever I present it I start off by telling the audience they already know modular arithmetic. I proceed to ask them: if it is now 1 o'clock (and then I write a '1' on the board), what time will it be in 2 hours (and I write a "+ 2" next to the '1'). Everybody answers '3', so I complete the equation "1+2=3". Then I say, if it is now 11 o'clock (and I write '11'), what time will it be in 2 hours? this gives rise to the equality "11+2=1".

Then I formalize this in the language of arithmetic modulo $12$, and then I give the general definition, compute some things, prove some things, show that $x=1 \mod n$ implies $x^n=1 \mod n$, and derive some divisibility criteria (i.e., a number is divisible by $3$ iff the sum of its digit is).

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    $\begingroup$ I'm not sure proofs would be a good idea in a talk like this one. Under the given time constraints, many high school students would become lost by the presentation of a drawn out proof. $\endgroup$ Commented Mar 30, 2014 at 20:28
  • $\begingroup$ @Draksis I tend to agree. What I usually do is just go over the proof that modulo is a congruence for addition and leave out subtraction and multiplcation. Then the proof the $1$-residues are closed under multiplication is trivial. $\endgroup$ Commented Mar 30, 2014 at 21:07

The motto I use for teaching is this:

Subsets are what you get when you throw elements away.

Quotients are what you get when you glue elements together.

In the case of $\mathbb{Z} / n\mathbb{Z}$, we glue together any pair of numbers if the number of steps to get from one to the other is a multiple of $n$.

I like this interpretation better because you don't have to be as abstract, considering whole sets as your elements. It is, for newcomers to math, very difficult to imagine "the set of all multiples of $n$" as a concrete, tangible number.

So instead, we just work with the representatives of the set as if they were the coset themselves. This is conceptually easier, now, because you never stop working with numbers.

The cost of this, of course, is you have to take a second to explain that, when working with elements that are glued together, everything has to treat them as equal. For instance, if you want to look at all the "even" numbers in $\mathbb{Z} / 3\mathbb{Z}$, you run into an unexpected surprise: 2 is glued to 5, and if 2 is even, that must mean that 5 is also an even number! Why? Because 2 = 5 once you glue them together.

So some ideas stop working when you glue. But other ideas keep working just fine. We know that 2 = 5 in $\mathbb{Z} / 3\mathbb{Z}$, so what happens if we add 1? Well, 2 + 1 = 3, and 5 + 1 = 6. But since 2 = 5, it better be the case that 3 = 6! And sure enough, the difference of 6 and 3 is a multiple of 3. So addition "respects" gluing.

Similarly, multiplication does too. An example of this would be that 2 * 2 = 4 and 2 * 5 = 10, and sure enough, 4 and 10 differ by a multiple of three.

This style of presenting quotients is much less common, but I have personally found it more appealing.


As others have suggested, it's good to begin with special cases that the students already know, at least implicitly, like $\mathbb Z/12$ (clocks) and $\mathbb Z/2$ (odd and even). Another implicitly known example is $\mathbb Z/10$. When you add two integers (by the usual procedure from elementary school), the last digit of the sum depends only on the last digits of the summands. So there is a notion of "last-digit addition". Similarly for multiplication.

I'd recommend discussing several such examples before bringing up the general notion of equivalence classes.

  • $\begingroup$ I would add that "last-digit addition" shows up in a number of texts under various names, and is a VERY natural way to introduce this, so I strongly suggest starting with this one. In principle you don't even have to talk about equivalence classes per se, instead saying "5 PM today, 5 PM tomorrow, 5 AM next Tuesday all count as the same." $\endgroup$
    – kcrisman
    Commented Oct 19, 2015 at 17:35
  • $\begingroup$ PS: You probably won't remember me from a visit looong ago, but a loong (only two o's) time ago I had your son in class, which I remember with pleasure. $\endgroup$
    – kcrisman
    Commented Oct 19, 2015 at 17:42

This is more of an "add-on" to existing answer than a full answer, but I think it needs more visibility than a comment.

I suggest beginning with something very concrete, like the idea of odd and even integers. You can observe that the sum of two even integers is even, the sum of an odd integer and an even integer is odd, etc. Then ask how one can be absolutely sure that these formal identities always hold. For the sum of two even integers, you can observe that $2$ can be factored out of each number, and hence out of the sum (by the distributive law), but what about an odd plus an even?

Then, to summarize the rules for adding even and odd integers, you can make a $2$ by $2$ Cayley table, using "O" for odd and "E" for even. Then you can say that since mathematicians like numbers, they replace "O" with $0$ and "E" with $1.$ Maybe mention $0$ and $1$ are the remainders upon division by $2$ to motivate somewhat what the rule $1 +1 = 0$ is saying. (Perhaps insert a joke at this point about how mathematicians can't even add $1$ and $1$ correctly.) The nice thing about having these two tables is that, later on, if you have to talk about what isomorphic means, you can come back to this example and say it's basically renaming the objects like we did with "even" and "odd".

To get past the even and odd example, you can then ask if there is any reasonable way someone can think of to classify integers into three different categories, such as "psudo-even", "psudo-odd", and "neither". Then talk about how this addition table works, then replace the objects with $0,$ $1,$ and $2.$

After all this, only then start talking about clocks and "mod 12" addition and the like.


I think there are lots of ways of doing this. One good and interactive activity could be to divide the class into four 'teams' (or more) and pick one member from each team and line them up at the front. The remaining folks in each team take it in turn to say a number larger than 12 (or some larger number) and the lined up folks have to count up to that number always going from left to right. The team scores points if they get the last number called out to land on their player. Eventually it should crystallise that it is only the remainder that counts and that numbers $4n + r$ and $4m+s$ are essentially the same in this context if $r=s.$

This could be extended by saying that now, instead of always starting from the leftmost player in the line-up, you carry on counting from the last player who spoke a number in each round. This should get them thinking about modular addition. I'm sure you could imagine loads of variations/extensions of this game. There are probably a few 'hacks' in this game where students can bypass thinking about modularity, but so long as there is a good discussion which explicitly brings these strategies to the fore afterwards I think it would be quite fruitful.

Some form of imagery such as

enter image description here

could be good to bring the point home visually because the 'groups of four' just go together to make more groups of four. Depends on their level but this could quite easily lead to some algebra. [I know ~ is more standard than $\widetilde{=}$ in mathematics to show equivalence of elements in classes but I think having the equals sign there expresses the idea of 'equal in a sense' - let's not argue about it just change it if you don't like it!]

This being done, it could be great to lead them to discover a proof that the sum of an odd number and an even number is always odd using the language of modular arithmetic (some could be amazed when they realise that odds and evens, which they have been using forever, are just a special case of a higher concept - 'even' in the above example would mean divisible by four, and there are three 'types' of 'odd': remainder 1, remainder 2 and remainder 3).

Just a few musings but I hope they might give you something. I'd be interested to know how you get on!


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