Easy math proofs or visual examples to make high school students enthusiastic about math

Question

I'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or demonstrations. It's meant for students who are in their last grade of high school and will be going to university next year.

So what are simple proofs or visual examples that made you love math? The more examples the better! Answers with pictures would be even better!

Thanks in advance!

P.S. Things that I did already teach my students the basics of are: complex numbers, probability theorem, prime numbers, vectors, functions of more variables, a little bit about group theory, set theory. These are all things that I tried to mix with the things they should actually know for their exams. It's meant to give them an idea of what math is really about, not just repeating formulas.

Answer 1

Here is one example that I find aesthetically pleasing, and which I have found effective in 8^th-grade classrooms.

Suppose you desire to cut out a triangle from the middle of a piece of paper, not by punching the scissors through and cutting the perimeter, but rather by folding the paper and then cutting straight through the folded paper.

The natural solution is to mountain-crease (red below) the angle bisectors, and valley-crease (green dashed) a "perpendicular" from the incenter $x$:

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        ^{(Figure from How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra.)}

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What I find so pleasing is that when you perform this physically, the angle bisectors meet at a point $x$ (the incenter), and one grasps Proposition 4, Book IV of Euclid viscerally. Naively, it could well be that the bisectors do not meet at a point. But careful creasing shows experimentally that they do.

I have found this tactile demonstration more convincing to (U.S.) 8th-graders than a two-column Euclidean proof.

^{(This repeats content from my answer to a related question.)}

Answer 2

My teacher on $\pi$ day during math club did the Buffon's needle experiment (except with little sticks) which we thought was extremely cool. And a plus is that the proof is relatively simple, requiring only basic knowledge of probability and calculus.

The probability that a stick will cross a line is $$P={{2l}\over{t\pi}}$$ where $t$ is the distance between the parallel lines and $l$ is the length of the stick, so if you want to approximate $\pi$ directly, let $t=2l$ then calculate ${{total sticks} \over {crossed}}\approx \pi$

Answer 3

I'll expand on my comment, now that I have some time.

For high school students, I really like discrete-math type ideas, particularly combinatorics. First, the vast majority of students are never exposed to these ideas (save binomial coefficients, and these, if introduced, are just strange symbols used to expand $(a+b)^n$, in my experience). And second, they often require relatively little prior knowledge, and offer the chance to think visually.

$\bullet$ First, that the sum of the first $n$ odd, positive integers is $n^2$. I'm currently teaching a remedial algebra class in college, and used the following as a bonus question on the first exam:
"This image shows that $1 + 3 + 5 + 7 + 9 = 5^2$. What do you think $1 + 3 + 5 + \ldots + 99$ will be? Don't worry about simplifying, I'd prefer you to leave the answer as $[something]^2$. You can test your conjecture to see if it makes sense for $1,\ 1 + 3,\ 1 + 3 + 5,$ and $1 + 3 + 5 + 7$, also conveniently shown in the picture."

I got a few right answers (one particularly impressive, as they used squares to count how many odd numbers there were from $1$ to $99$), and even more amazing, everyone stayed after class the following day to ask me how to solve it! You can rest assured this has never happened with the material in the curriculum.

$\bullet$ To expand on binomial coefficients, note that a subset of $\{1,2,3,4,5\}$, such as $\{1, 3, 4\}$, uniquely determines a sequence $$\underset{1}{L}\ R\ \underset{3}{L}\ \underset{4}{L}\ R$$ of left/right steps, and thus paths (of the sort found in this pdf) from the top of Pascal's Triangle, to a certain spot, are counted by the binomial coefficient $5 \choose 3$ in a very natural way.

This is also a very pleasing route to arrive at the famous identity $${n - 1 \choose k - 1} + {n - 1 \choose k} = {n \choose k},$$ as any path ending at the depicted point must pass through one of the two adjacent points above it; we simply add the numbers of those paths to see that ${4 \choose 2} + {4 \choose 3} = {5 \choose 3}$, as a calculation-free example. And of course, by exchanging L's with R's, we must have ${n \choose k} = {n \choose {n-k}}$.

Answer 4

Hilbert's hotel and other counterintuitive countability discussion are always fun.

A less abstract topic, yet still counterintuitive, could be the Monty Hall problem.

Answer 5

I really liked the proof that you can raise an irrational number to an irrational number and get a rational number.

Consider ${}\sqrt{2}^{\sqrt{2}}$. If this is rational, we are done (I think some answer on here shows that $\sqrt{2}$ is irrational). If it is not, $\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = 2$, so we are definitely done.

Answer 6

$$\begin{align}x = 0.999\ldots \\ 10x = 9.999\ldots\end{align}$$ so $$ \begin{align} 10x - x = 9x = 9 \\ \implies x = 1 \end{align}$$

Also in a right triangle with hypotenuse length $c$, legs $a,b$ and some angle $\theta$ opposite of $a$, then $$\begin{align}a^2+b^2 = c^2 \\ \implies \frac{a^2}{c^2}+ \frac{b^2}{c^2} = 1 \\ \implies \left(\frac{a}{c}\right)^2+ \left(\frac{b}{c}\right)^2 = 1 \\ \implies \sin^2(\theta)+ \cos^2(\theta) = 1\end{align}$$ A little more advanced for high school, you could show that there are as many natural numbers as there are integers.

$$\{1,2,\space \space 3,4,\space \space \space 5,6, \ldots \} \\ \{0,1,-1,2,-2, 3\ldots \}$$ The top list clearly contains every natural, and the bottom clearly contains every integer. Neither list is exhaustible so you can pair one natural number with every integer. If the students buy that, then maybe you could show them that the rational numbers are equinumerous to the naturals...

Answer 7

There are many interesting proofs of the Pythagorean theorem at Cut the Knot.

This is my favourite (as you might guess from my gravitar)

Let the length of the hypotenuses be $c$, and the length of the legs be $a$ and $b$, with $b$ being the longer (if either is longer). The area of the outer square is $c^2$, but it is also equal to the sum of the areas of the four coloured triangles and the area of the white square, and so the area is

$\quad4\times \frac{1}{2}ab + (b-a)^2$

$=\; 2ab + (b^2-2ab+a^2)$

$=\; a^2 + b^2$

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Another nice one (attribution information is at https://en.wikipedia.org/wiki/File:Pythagoras-2a.gif)

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A proof that I like to show Jr. High kids when I teach them programming is this. (It fits in with the programming since one of the things they do is to draw polygons by programming a simulated robot to trace out the polygon.)

Since the robot turns all the way around, the sum of $A$, $B$, and $C$ is 360. But we also know that $A = (180-a)$, $B = (180-b)$ and $C= (180-c)$. Thus

$\quad 360 = (180-a) + (180-b) + (180-c)$

$\therefore \; 2\times 180 = 3\times 180 - (a+b+c)$

$\therefore\; a+b+c = 180$

You can generalize the argument to show that for any polygon, the sum of the interior angles is $(n-2)180$.

Answer 8

$1+2+3+\dots +{(n-1)}$= ${n}\choose{2}$

Answer 9

I rather like the bean machine as a physical introduction to probability, normal distribution and the concept of randomness:

Now that I think of it, I rather like mechanical devices in general that demonstrate mathematical truths/ideas. Leibnitz's mechanical binary calculator is another that springs to mind.

Answer 10

There are several simple proofs I learned that made me love and appreciate math.

• the proof that $\sqrt{2}$ is irrational
• the proof that there exist irrational $a$ and $b$ such that $a^b$ is rational
• any simple inductive argument
• the proof that there are an infinite number of primes
• Cantor's diagonalization argument for showing the cardinality of the reals is greater than that of the integers (though this may be pushing it for people with no set theory)

Answer 11

I like the proof that the harmonic series diverge.

The french mathematician Nicole Oresme gave an accessible proof of this. The idea behind his proof, which is grouping terms by power of one halves, was used by Cauchy for his condensation test. There is a nice visualisation on Wikipedia of the process involved in the proof.

Oresme proof is explained in detail here and there (in video).

The following article by S. Kifowit and T. Stamps mentions several accessible proofs of the result (it is always enriching to see that there are several ways of proving a result). If you have already introduced the integral then you might consider Proof 9.

As I have first encountered this result (with a proof) at university, we used Cauchy's convergence test (Proof 4 in Kifowit and Stamps article), but I think the idea is similar (to a certain extent) to the one used by Oresme.

Edit : N.B. the above picture does not show the harmonic series. However I have made the following illustration to show the grouping process in the case of the harmonic series :

Answer 12

One mathematical fun fact that has left a strong impression on me when I was in middle school was the fact that you could state rules about random phenomena.

I remember I wasn't familiar with limits, but a teacher had decided to show us funny things about random walks on Excel (yep Excel... :)).

He basically made random walks on $\mathbb{R}^2$ with step size $1$, and showed us that the average squared distance after $n$ steps when we took a lot of walks was close to $n$.

I liked the fact that it felt like nothing could be said about the walks just by looking at them, but as soon as we computed the right quantity, order emerged.

Visual things are always nice. Plotting walks on the plane with colors, and showing how it looks so chaotic compared to the numbers/figures/functions we knew at the time did the trick I think.

Answer 13

If they have not learned it in high school, I would recommend showing them how to derive the quadratic formula. Since the proof only involves basic algebra, the students should be able to solve it on their own. Showing them that they already have enough math knowledge to prove a helpful mathematical formula may allow them to share in your enthusiasm.

If they have already learned that, I would recommend showing them iterative series as shown in dynamical systems and the Mandelbrot and Julia fractal sets. I always found this intriguing since the images were created with only mathematics. If you need images, there are quite a variety online.

Answer 14

When I was at high school I loved geometrical arguments, especially those which were both simple and profound. Later on, I discovered Combinatorial Geometry, which combines both Euclidean Geometry and Combinatorics. I think that a good idea would be to demonstrate the proof that
No matter how the plane is 3-colored, it contains a monochromatic unit distance segment.
This states that the chromatic number of the plane is greater than (or equal) to 4. In my opinion the most elegant proof is by using the Mosers Spidle. The proof goes like this:

Toss on the given 3-colored plane the Mosers Spindle. Every edge in the spindle has the length 1. Assume that the seven vertices of the spindle do not contain a monochromatic unit distance segment. Call the colors used to color the plane red, blue and green. Name the seven vertices A,B,C,D,E,F and G.
Let the point A be red, then B and C must be one blue and one green. Therefore D is red. Similarly E and F must be one blue and one green. Therefore, G is red. We found a monochromatic segment DG of length 1 in contradiction to our assumption.

There are several proofs of this result but in my opinion this will definitely bring enthusiasm to your students. I also love Ramsey like examples such as
Six people are waiting in the lobby of a hotel. Prove that there are either three of them who know each other, or three of them who do not know each other.
The statement is definitely far from being obvious. But, this is a beautiful application of the Pigeonhole Principle combined with a clever argument from graph theory, which marks the beginning of Ramsey Theory. It is often called The Party Problem (or Theorem on friends and strangers). The proof goes like this:

Take a $K_{6}$ (complete graph of 6 vertices) so that each person corresponds to a vertex. Color the edge joining A and B red if A and B know each other, and blue if they do not. Do this for all 15 edges of the graph. The claim will be proved if we can show that there will always be a triangle with monochromatic edges in our graph.
Take any vertex $V$ of our bicolored graph. We know that 5 edges "leave" the vertix $V$. Now, from the the Pigeonhole Principle $V$ must have at least three edges adjacent to it that have the same color. Assume without loss of generality that this color is red. Let $X$, $Y$ and $Z$ be the endpoints of three red edges adjacent to $V$. Now if any edge of the triangle $XYZ$ is red, then that edge, and the two edges joining that edge to $V$ are red. So, we have a triangle with three red edges. (and we are done) Otherwise, if the triangle $XYZ$ does not have a red edge, then it has three blue edges. (and again we are done)

I love those kind of examples because they use elementary tools, but the ideas behind them are profound. Many astonishing examples of this kind can be found at the first two chapters and chapter thirteen of Miklós Bóna's A Walk Through Combinatorics. I hope that I helped you.

Answer 15

I would go for the proof by contradiction of an infinite number of primes, which is fairly simple:

• Assume that there is a finite number of primes.
• Let $G$ be the set of all primes $P_1,P_2,...,P_n$.
• Compute $K = P_1 \times P_2 \times ... \times P_n + 1$.
• If $K$ is prime, then it is obviously not in $G$.
• Otherwise, none of its prime factors are in $G$.
• Conclusion: $G$ is not the set of all primes.

Answer 16

During my first semester of the career our calculus teacher taught us the concept of an infinite sum using some fractals. ATM I thought it was cool, since it introduced us to the wonders of "infinity" using visual motivations. The first example could be the visual proof that $\sum 2^{-i}$ converges.

After that she discussed some subtleties about infinite sums and proceeded to calculate the perimeter of the Koch snowflake (if you're feeling bold you could also calculate its area). This is extremely cool since you can see how the perimeter increases and opens an interesting debate about convergence.

The other example I remember, which is very cool too, is calculating the length of some infinite trees (not exclusively binary, but this would suffice to impress the students).

I'm sure there are other very cool examples involving fractals or limits which any student can understand. These problemas have the advantage of introducing the students to some basic tools to "manage infinity", which is fundamental for every maths student.

Answer 17

Gambling.

What got me interested in probability theory when I was in school was that I would be able to use probability to gamble. This usually makes probability much more fun.

(Of course, don't use it too excessively)

Answer 18

When I was in high school, I did not really have to do "proofs." In geometry, we used substitution many times to "prove" identities. I did not really understand these substitutions all the time in geometry because I only had one choice in most problems. When I learned the algebra side of substitution, the picture became much clearer. I grew to love math because of the substitutions one could do. All of these problems can be done with substitution or with a picture.

• The proof of the quadratic formula. Complete the square with a,b,c all letters (for review). Then use the substitution $x=y+m$, substitute, solve for $m$ in the middle term, substitute again, solve for $y$, then back substitute $y=x-m$ and finish.
• Using this equation $x^n -1 = (x-1)(x^{n-1}+ \ldots + 1)$ always intrigued me especially using the substitution "S".
• Speaking of sums $1+2+3+ ... + n = \frac{n(n+1)}{2}$. use "S" to prove that one again.
• Proof that every odd number is the difference of 2 squares. (just use $0^2=0, \ 1^2=1, \ 2^2=4, \ 3^2=9$ etc.)

Once I understood substitution, I could take definitions in math and substitute them into the problem I was working. If algebra seems too dry, then pictures can be used for the last 3 bullets.

Answer 19

In my $10^{th}$ mathematics book there was something given about the seven bridge problem. There was no proof given in the book so I tried it but I failed. I got a new book "Mathematical Circle(Russian experience)" in this book I found a chapter on graph theory. In that book I found the proof of the seven bridge problem, it was a very clean proof.

Another topic is invariant which nice and easy to understand.

In probability you may like to see Monty Hall problem.

Answer 20

Cutting a bagel into linked halves is impressive to any audience. Just make sure to make a full twist: cutting along a Moebius band would produce a bigger and thinner bagel rather than linked halves.

Answer 21

Infinite monkey theorem, Berry's paradox and then combine the two to show that while almost all files larger than 1 terrabyte cannot be generated by a self-extracting program to a size that will fit on a 4 GB memory stick, the proof that it can't be done in any particular case, cannot exist.

Answer 22

There's so many interesting subjects, that I can't even organize them systematically. Here are some that come in mind in arbitrary order:

• The geometric series convergence illustrated by a perspective view of a railroad track or blocks of buildings along a street.

• Classic geometry constructions by a straightegde and compass.

• Plane tilings.

• Ellipses constructed inside parallelograms by intersecting families of rays.

• Curves emerging from families of straight lines obtained by shifting and rotating a line.

• Various figures fitting problems (what conditions must a polygon fit to have an inscribed circle? or a circumscribed circle? Can we circumscribe a square on any bounded shape on plane?)

• Graph problems, including coloring a map (with a five colors easily proven with Euler characteristic of a convex polyhedron).

• A paradox of finite volume with infinite surface (Torricelli trumpet aka Gabriel's Horn).

• Pythagoras' theorem with a variety of proofs.

• Various examples of reasoning — and unsolvable problems, like a barber paradox.

• Explainig some optical results with a ray geometry (why you need a mirror half of your height to see your whole body – and why it's independent on the distance; why a ray falling between two perpendicular mirrors returns parallel to the original direction – and how it makes them give a reverted image).

• The Cantor theorem of a power set proof by constructing $\{a\in A: a \notin f(a)\}$.

• Pidgeon principle.

• Solids sections and unrolling (what shape is the surface of a tea in a glass? – an ellipse; why do squirrels chase each other around a tree along a helix? – because it is a straight line when unrolled; what shape is a toilet paper tube slant cut when unrolled? – a sinusoid; .........)

• Maze solving algorithms.

• Constant-width figures.

And many, many others.

Answer 23

Have you checked out Paul Lockhart's "A Mathematician's Lament" and "Measurement"? He describes his approach to teaching K12 math and - in the latter book - walks you through some elementary (but non-trivial) geometry.

Answer 24

My answer relates less to strictly mathematical proofs of specific concepts, but emphasizes drawing out the math behind stuff. There are 3 main topics which I would split my answer into: Math toys, Cool math tricks, and Math in everyday life

Math Toys

Spirographs: The mathematical basis of spirographs are simple and definitely within the grasp of high schoolers, if a bit tedious. https://en.wikipedia.org/wiki/Spirograph

Tower of Hanoi: A simple puzzle with loads of mathematics behind it. Tower of Hanoi http://www.activityresources.com/image.php?type=D&id=64 https://en.wikipedia.org/wiki/Tower_of_Hanoi

Rubik's Cube: Group theory can be applied to the algorithms approach to solving rubik's cubes. Probably a tad bit complicated for high-school math, but still, it's pretty cool :P https://en.wikipedia.org/wiki/Rubik's_Cube_group

Cool Math tricks

Approximating square roots of integer numbers to 2 decimal places: Say you have a non-perfect square integer x. Its nearest perfect square is y. By using the quadratic formula, it is easy to prove that

$\sqrt{x} \approx \sqrt{y} + \frac{x-y}{2\times\sqrt{y}}$

E.g. $\sqrt{101} \approx 10 + \frac{1}{2\times10} = 10.05$

With some practice, you can evaluate such square roots very quickly, and impress the less mathematically inclined people.

Telling the day of the week any arbitrary date is: Depending on the method you use, the trick's educational value can range from purely rout memorization, to using purely mathematical algorithms.

https://en.wikipedia.org/wiki/Determination_of_the_day_of_the_week

Math in Everyday Life

Geometries in Everyday Life

There are so many geometries embedded in everyday life that we don't give much thought to them. The reasons behind why certain geometries are used for various purposes are related more to structural engineering rather than mathematics, but there's great value in drawing out the importance of maths in real world applications. There are tons of examples you can give. My favourites are:

Antoni Gaudi's architectural designs

Hyperbolic Paraboloid Vaults

Hyperbolic Paraboloid Pringles

"According to Pringles marketing, the shape allows the snack to be securely stacked in a canister to prevent breakage during packaging and transport."

Golden Ratio

Surprised no one has mentioned this yet. You could talk about anything ranging from continued fractions, the fibonacci sequence, to its recurrence in nature/architecture/etc..

Answer 25

"Proofs without words: Exercises in Visual Thinking" is a book dedicated to visual proofs. The book has proofs about Geometry, Trigonometry, Calculus and also Sequences and Series. In case you run out of proofs for the class there is also a sequel of this book "Proofs without words II: More Exercises in Visual Thinking". Since, these are "exercises" no explanations are given for the proofs, and this is probably the fun part with this book.

Answer 26

Personally I love Euclidean geometry proofs. Try some simple IMO or similar contest geometry problems, for example 2002 IMO no.1 or 2012 IMO no. 1 or this year or last years number 4 are all super fun and I really love reading the solutions.

Answer 27

Not a direct answer to your question, since it's not about proofs, but it may help with the question behind your question.

The two books that turned me toward a career in mathematics were Steinhaus's Mathematical Snapshots (http://store.doverpublications.com/0486409147.html), which I read as a high school freshman, and Polya's Induction and Analogy in Mathematics (https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_, or from Amazon), which I read as a senior.

Answer 28

My two cents :

• Pigeonhole principle

• Real line

• Formal Languages (how to construct a word from a finite set, the
  alphabet)

• Simple harmonic motion physics and how the speed and acceleration of an object are derived as first and second derivatives of the
  displacement equation.

Answer 29

Proving Euler's identity has got to be one of the coolest things in math and probably perfect for high school seniors.

$$e^{i \pi}+ 1 = 0$$

Proof:

$$e^{i \theta} = i \sin{\theta} + \cos{\theta}$$

$$e^{i \pi} = i \sin{\pi} + \cos{\pi} = i (0) - 1 = -1$$

(You can prove the first line, "euler's formula" several ways depending on whether or not your students know power series or calculus)

Answer 30

I didn't see this problem until I was a freshman in college, reading Edwin Moise's Calculus I book. It was almost 40 years ago, so I am probably wrong about the details, but, here goes. He described how to prove $|x| + |y| \ge |x + y|$ using cases. Then he asked us to prove that $|u - v| \ge |u| - |v|$ and he hinted that it could be done without resorting to cases. I asked my professor about it, I asked my old high school teacher about it, I read that section of the book almost letter by letter. After a torturous week or so, I woke up one morning and knew the solution.

Let $x = u - v$ and let $y = v$. Then $|x| + |y| \ge |x + y|$ becomes $|u-v| + |v| \ge |(u-v) + v|$ . The conclusion follows.

Substitution seems to be such a trivial thing. But it is really an extremely powerful mathematical tool.

Answer 31

The proof of Thales' Theorem (that any triangle constructed using the diameter chord of a circle and a third point on the circle that doesn't coincide with the endpoints of the diameter is a right triangle) is a pretty nice one:

The triangle $\triangle OAB$ is an isoceles triangle because $OA$ and $OB$ are both radii of the circle and thus by definition must have equal length. That means that $\angle OAB = \angle OBA$, the measure of which we identify as $\alpha$. The same is true for $\triangle OBC$ and for the same reasons; the congruent angles are $\angle OBC = \angle OCB = \beta$.

Now, we know that the sum of all interior angles of a triangle is 180 degrees. The proof of that is also pretty intuitive in visual form:

We also can see by inspection that $\angle ACB = \alpha + \beta$. Therefore, the sum of the interior angles is $2\alpha + 2\beta = 180$, and when simplified by dividing everything by 2, $\alpha + \beta = 90$, and thus the third angle of a triangle of this construction is always 90 degrees, making the triangle a right triangle.

Answer 32

How about Galileo's paradox?

Consider all the square numbers. Since every square number has an integer square root and every integer can be squared to give a square number, then even though not all integers are square numbers and both are infinite, the number of square numbers must equal the number of integers!

Answer 33

The fact that we can give finite result even to divergent sums like for examples the sum of all natural numbers:

$$\sum_{n\in \Bbb N}=1+2+3+4+\dots =-\frac 1{12}$$

This fact really astonished me and continues to have a sort of fashion on me :)

Answer 34

These 3 triangles are similar, the two smallest pave the large one and their area is quadratic in the length of their hypotenuses...