** Edit

Thanks everyone for some great suggestions. I should have been more clear though. I am actually looking for a college level proof that pertains to algebra or leads to algebra in some form. My professor has already shot down any proof that would be easy enough to teach to a high school algebra I or II class.

I am currently in my final year studying mathematics education. I am doing a 1 credit math seminar project. I will be writing a paper and presenting on this topic. I have chose integrating mathematical proofs into high school math. However, I need a detailed proof for this topic. Anyone have any suggestions? I am looking for a proof that I can expand on and talk about for nearly 30 minutes.

• I would emphasize the psychological and pedagogical issues in teaching proofs. I think they are more relevant to your objective than a specific proof. An interesting topic would be the pros and cons and research done on the two column geometry proof. It is often reactively reviled here, but I wonder if it is really such a bad idea. Maybe it helps kids learn and is a useful stage. (The 5 paragraph essay is often reviled as well, reactively, but is a very powerful teacher of structure, which is the most important concept for intermediate writers.) – guest Sep 3 '18 at 14:50
• Why is your prof requiring a college level proof when your thesis is oriented to high school and essentially so is much of formal math education. (This is not to say pedagogy is not important in college...just that college teaching tends to be by people who are researchers more than teachers.) – guest Sep 4 '18 at 19:51
• How about something from trig, functions, analytic geometry, or calculus? That is above algebra 2, but at least still within high school (the latter arguably so). – guest Sep 4 '18 at 19:52
• Do you have any initial thoughts? Question feels very open ended for someone who is a graduate student. Can't you share more ideas of your own? (Just as a student asking for help would be asked to show what he had done first.) Feels strange that you ask in such a de novo fashion. – guest Sep 4 '18 at 19:53
• I am not a graduate student. I am undergrad mathematics education major. I am considering changing my topic. I wanted to tie into the education aspect since I am planning on teaching high school math, but my professor will not cooperate. She wants a difficult proof which really does not align with my topic. I am open for suggestions! – KMC Sep 4 '18 at 21:13

Edit - Following the edit made to the OP, I must admit it is a bit unclear what you are looking for exactly. Here is what I understand. The proof must:

1. Pertain or lead to algebra
2. Algebra must be harder than algebra I and II
3. Be college level
4. Be hard
5. Not be 'related to any subject'

First, I would ask your professor for an example of something she'd find acceptable, because there are contradictions here. No college level proof is hard, no proof is unrelated to any subject.

She is basically asking you to find a proof hard enough that it is unteachable at algebra level, but at the same time, you are somehow expected to find a way to make it teachable. This sounds as absurd as asking you to find a way to teach colors to blind people.

All I can think of is the usual calculus/analysis proofs:

1. Rolle's theorem
2. Intermediate value theorem
3. Unicity of zero theorem
4. Lagrange theorem

They 'pertain to algebra' because they use the concept of functions. They would be difficult to teach to algebra students, who think of $f(x)$ as a formula, and not some abstract object. They are college level.

I suggest you read section 1.1 of Stephen Abbott's Understanding Analysis. In it, Abbott talks about G.H. Hardy and two of the proofs he used as a 'defense' of mathematics (more precisely, about how mathematics is an artistic endeavour).

The first is the proof that $\sqrt{2}$ is irrational.

The second is the proof that there are infinitely many prime numbers.

Both proofs are quite short, relatively easy to understand and although they are very old, they are still "as fresh and significant as when [they] where discovered".

So you could talk about one or both of these proofs in the first half of your presentation and then explain why they are still relevant and how easily they could be included in the high school curriculum in the second half.

• The proof for the prime numbers can easily be extended to show that there are infinitely many primes that are congruent to three modulo four; either to fill time or as an exercise for the assumed high school students. – Dirk Sep 3 '18 at 8:27
• That $\sqrt{2}$ is irrational has the added benefit of having a bloody (if apocryphal) story attached to it. – Xander Henderson Sep 3 '18 at 14:59
• I requested a different professor be my advisor. I was getting nowhere with the current one. I am hoping to do research on the tower of hanoi and find an interesting proof related to it. – KMC Sep 6 '18 at 17:58
• @KMC I wish you good luck! – orion2112 Sep 6 '18 at 22:43

You could take a look at basic set theoretic proofs, like for example showing that $$A \cap (B \cup C) = (A \cap B) \cup (A \cup C).$$

You can draw a diagram here and you can show and emphasize formal proofs. This property is not hard to prove, but it teaches the students how to do formal proofs, step by step.

Another such sort of proofs would be the classical proofs by induction, like for example $$\sum_{i=1}^n i = \frac{n(n+1)}{2}.$$ Again, the proof is not the hard part here, you can use it to show and discuss formal procedures and rules.

Overall, check out your lecture notes from the first few weeks at university. Most courses start with a formal introduction to proofs and give proofs and concepts that are both easy to grasp and important to learn for beginning students, so they would fit well with high school students who just start to learn about formality and proofs.

Compared to the other answers you don't have the nice effect of showing off a great result, but you are teaching your (future) students valuable skills and are starting with the basics (of course one can argue if a fancy example to motivate them first would be good).

• I think that for the seminar, a "great-result-proof" would do just fine. For the actual integration of proofs in the curriculum, I would see them as introduction material (in fact, I believe the first proof I ever saw was the irrationality of $\sqrt{2}$). Then, a teacher could (and probably should) move on to talk about procedures and different types of proofs (even though that might be "less spectacular"). Considering this, maybe this could lead OP to talk about how to select which proofs would be convenient for high school students. – orion2112 Sep 3 '18 at 8:55
• I was also going to suggest the sum of the first $n$ integers, but for a different reason: it has a rather nice "proof without words". – Xander Henderson Sep 3 '18 at 14:51

The binomial theorem is a great way to introduce proofs, since it provides a "shortcut" for high school algebra problems using Pascal's triangle (which students enjoy drawing). You can ask why it keeps working and lead them to proof by induction.

You can also pivot to a question about choosing from sets, and let students realize they're the same numbers. There's a ton of fun stuff you can follow up with in future sessions.

In high school students usually see the statement that if a continuous function changes sign in a closed interval then it has zero in the interval. This is usually given some sort of informal proof in the form of a drawing, but it can be given a constructive proof that is interesting in its own right. Namely, what is called the method of bisection for root finding yields a proof of the theorem. (Recall that the bisection method consists simply in choosing a half of the given interval in which the given function changes sign, and iterating; an explicit error bound is not hard to obtain.) This approach has the virtue that the proof is not hand-wavy (not some informal appeal to geometric intuition the student might not yet have) and yields an algorithm for actually calculating the root (given a concrete function) that the student can actually do by hand (or on a computer if one has such facilities available). Presenting the proof in this way also gives an introduction to numerical methods, and there is a lot that can be said there, e.g. $x^{2} - 2$ has a root at $\sqrt{2}$, but how does one obtain a decimal approximation of $\sqrt{2}$? (In practice, bisection is not the way to do it, but that's another story, that comes later.)

A virtue of this example is that students are made to spend a lot of time factoring polynomials and finding their roots. The context of the problem is not unfamiliar to them.

Might I suggest: Every polygon triangulation can be $3$-colored.

Wikipedia image.
See this earlier ME post for surrounding details and links. This can be motivated by and leads to the art-gallery theorem. Art-gallery-like theorems continue to be an active area of research today.

I recommend choosing an elementary proof that includes both elements of surprise and beauty - one that might spark students to further study mathematics. Along these lines I've had much success employing the striking geometric generation of the tree of primitive Pythagorean triples.

Here is a brief sketch. Given such a triple, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2$ we can divide by $z^2$ to obtain $\,(x/z)^2\!+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can generate all such triples by a very simple geometrical process. Start with the trivial point $(0,-1)$. Draw a line to the point $\,P = (1,1).\,$ It intersects the circle in the rational point $\,A = (4/5,3/5)\,$ yielding the triple $\,(3,4,5).\,$ Next reflect the point $\,A\,$ into the other quadrants by taking all possible signs of each component, i.e. $\,(\pm4/5,\pm3/5),\,$ yielding the inscribed rectangle below. As before, the line through $\,A_B = (-4/5,-3/5)\,$ and $P$ intersects the circle in $\,B = (12/13, 5/13),\,$ yielding the triple $\,(12,5,13).\,$ Similarly the points $\,A_C,\, A_D\,$ yield the triples $\,(20,21,29)\,$ and $\,(8,15,17).$ We can continue this process with the new points $\,B,C,D\,$ doing the same we did for $A,\,$ obtaining further triples. Iterating this process generates the primitive triples as a ternary tree

$\qquad\qquad$

Descent in the tree is given by the formula (whose reflective geometric genesis is given here )

$$\begin{eqnarray} (x,y,z)\,\mapsto &&(x,y,z)-2(x\!+\!y\!-\!z)\,(1,1,1)\\ = &&(-x-2y+2z,\,-2x-y+2z,\,-2x-2y+3z)\end{eqnarray}$$

e.g. $\ (12,5,13)\mapsto (12,5,13)-8(1,1,1) = (-3,4,5),\$ yielding $\,(4/5,3/5)\,$ when reflected into the first quadrant.

Ascent in the tree by inverting this map, combined with trivial sign-changing reflections:

$\quad\quad (-3,+4,5) \,\mapsto\, (-3,+4,5) - 2 \; (-3+4-5) \; (1,1,1) = ( 5,12,13)$

$\quad\quad (-3,-4,5) \,\mapsto\, (-3,-4,5) - 2 \; (-3-4-5) \; (1,1,1) = (21,20,29)$

$\quad\quad (+3,-4,5) \,\mapsto\, (+3,-4,5) - 2 \; (+3-4-5) \; (1,1,1) = (15,8,17)$

Continuing in this manner one may reflectively generate the entire tree of primitive Pythagorean triples, e.g. the topmost edge of the triples tree corresponds to the ascending $C$-inscribed zigzag line $(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13), (-5/12,12/13), (7/25,24/25), \ldots$

The linked elegant presentation using geometry of quadratic spaces is a bit beyond high-school level (though elementary and accessible to bright high-school students). Instead we can derive the formulas directly though simple algebraic calculations, e.g. below is a sketch John Conway gave in email (1998/3/9), slightly edited.

Take the typical rational point $(x,y) = (p/r, q/r )$ on the circle, and join it to the nearest corner of the square. The typical point on this join is $(x',y') = (1-c(1-x),1-c(1-y))$ which lies on the circle provided that

\begin{align} 1 \,=\quad\ &1 - c(2 - 2x) + c^2 (1-x)^2 \\ \ +\ &1 - c(2 - 2y) + c^2(1-y)^2 \end{align}

hence $\,0 = 1-c-c(3-2x-2y)+ c^2(3-2x-2y)\,$ after using $x^2+y^2 = 1$

giving $\ c\, =\, (1\ {\rm or})\ 1/(3-2x-2y) = r/(3r-2p-2q)$

Therefore $\,x' = 1-c(1-p/r) = (2r-p-2q)/(3r-2p-2q)$

and its denominator is strictly between $0$ and $r$ since $p+q$ is between $r$ and $\,r\sqrt 2 < 3r/2$

This shows that if you join to the nearest corner the denominator decreases, but also conversely that if you join to any other corner, it increases. By induction, we see that there is a path to any point, and then that this path is unique, because if an increasing move is followed by a decreasing one, they must cancel.

Remark The diagram is excerpted from the terse presentation on p. 172 of Conway and Guy's delightful The Book of Numbers - which is a good source to browse for other possible elementary topics worthy of presentation at this level. John Conway informed me that the entry was contributed by Richard Guy, and Richard informed me that it is based on a talk given by Roger Vogeler at 15:40 on Fri 89-04-07 at the Intermountain Section meeting of the MAA at Brigham Young University.

The example provides an elementary glimpse of some beautiful and deep connections between number theory and geometry (reflective lattices). See the linked MSE post for further discussion.