The Pythagorean Theorem – an interactive lesson

This piece contains a mixture of reading and activities.  You can read the whole thing, or skim through for the activities.  Look for the word TASK to find the activities.


The Pythagorean Theorem.  It’s kind of a big deal.

Whether you know it or not, you probably rely on this theorem every day.  It defines a relationship between the three sides of a right triangle.  What’s a right triangle, you ask?  Here, let me introduce you:



Just like beauty, right triangles come in many different shapes and sizes.


TASK: Go play with the pile of triangles.  Separate the right triangles from the rest of the pile.

. . .

A right triangle will always have one side (called the hypotenuse) that’s longer than each of the other two sides (the legs).


TASK:  Can you be sure the hypotenuse is always the longest side? Try to draw a right triangle where the hypotenuse is not the longest side.  What do you notice?  (Don’t have any paper?  Do your drawing here.)

. . .

The Pythagorean Theorem gives us the following relationship between the three sides.

Pythagorean Theorem:  in a right triangle, the sum of the squares of the two shorter lengths (the lengths of the legs) is equal to the square of the longer length (the length of the hypotenuse).


The theorem enables you to solve problems like this one.


TASK: Compute the distance in the picture above.


. . .

(scroll down for a solution)

. . .



 SOLUTION:  Let’s call the ground distance x.  Since the 12-foot beam appears to be perpendicular to the ground (i.e. it makes an angle of 90 degrees with the ground), we can identify it as part of a right triangle.


Now we use the Pythagorean Theorem.  The two legs of the triangle are 12 and x.  The hypotenuse is 15.


Solve for x in the above equation:


Since x2 = 81, we have x = 9 or x = -9.

So… it looks like we have two solutions.  Except one of them doesn’t make sense in the context of the problem.  Is it possible to have a distance of -9?  Not in this situation.

So x = 9 feet.

The ladder touches the ground 9 feet away from the tower.



What does this have to do with your life?

Right triangles are everywhere.  All you have to do is look around.  They were probably used to construct the building you’re in right now.   They could be as close as the chair you’re sitting on.

I found one under my desk.  Do you see it?


I also found some right triangles outside my window.  I outlined one of them in pink.  Can you find the other ones?

Can you find the other right triangles in this picture?

TASK:  Identify at least three more right triangles in the picture above.

TASK:  Identify the closest right triangle in your physical space.  Pay attention to furniture, ceilings, staircases, clothing, electronics, roofs, etc.  Note that the nearest right triangle may be behind you.

. . .

Ok.  Were you longing for something more majestic?

See if you can find any right triangles in the Golden Gate Bridge.


Every time you see a right triangle in your world, either in a bridge, or the railing of a staircase, or a ramp, or the beams on the ceiling, it means someone somewhere was using the Pythagorean Theorem.

To practice applying this theorem to the triangles in your life, try these problems on Khan Academy:

Pythagorean Theorem – Practice

Pythagorean Theorem – Word Problems

Pythagorean Theorem – Challenge


Hamilton Lottery


What are your chances of winning this lottery?

Context: Hamilton is a sold-out musical on Broadway. When a show is sold out, the theater will often do a lottery. They save 20 front-row seats at $10 apiece. We went to the Hamilton lottery and there were so many people….so few seats…. I wanted to calculate our chances of winning. I told the people running the lottery that I wanted to make a video for a math problem, and they were game. They even let me stand up on their platform and take pictures of their little counter-clicker device that kept track of how many people were in the lottery.

Details: 20 seats available. You write your name on a slip of paper and you can choose either 1 seat or 2 seats if you win. Almost everyone chooses 2 seats. I have never seen anyone choose 1 seat. So keep that in mind. The final count was 554 – that means 554 people wrote their names on slips of paper.

Extension:  What if you didn’t know everyone was gonna choose 2 tickets? Suppose people choose 2 tickets p% of the time. And they choose 1 ticket the other [100-p]% of the time. Write a formula for your chances of winning in terms of p.

Any other math questions you can think of?

#MathProblemOfTheDay: Halloween Costume Puzzle


Remains of my last minute Halloween costume.

I dressed up as a puzzle. I taped numbers all over my pants. They were sorted into 2 clumps: one clump on each leg.  All the numbers on my left leg had something in common, and all the numbers on my right leg had something in common.

Hint: none of these numbers is divisible by 3.

Can you figure out why the numbers are sorted this way?

The mathematics of music: exponential growth

Tonight I came across this lesson on TedEd:

The Physics of Playing Guitar

TL;DW: Musical notes follow exponential growth!!!!!!!

What does that mean?

Well, in the Western scale–the scale used on a piano and guitar–the frequency of each tone is (approximately 1.06) times as high as the frequency of the tone before it. Since there are 12 tones in an octave, that means that by the time you’ve jumped up a whole octave, you have multiplied by a total of twelve times.

In other words, .

Which is equal to a perfect, clean 2.  So, the frequency doubles over the course of a full octave.

For instance a standard A has a frequency of 440 Hz, which means the guitar string or piano string vibrates 440 times every second.

The next A on the piano (one octave higher) has a frequency of 880 Hz.

Double the frequency.

How amazing that the tones that share musical harmony (two A’s for example) share some sort of mathematical harmony also.

Here is a listing of note frequencies where you can check if the mathematical rule works.

Move across any row and the numbers are doubling, as we jump up by octaves.

Now move one box up, starting in any location.  This represents moving up by one tone of the Western scale.  Sure enough, the number gets multiplied by , which is about 1.06.

Understanding the Western scale opens the door to different types of scales.  There is something universal about the sound of an octave.  But why take 12 notes to get from one octave to the next?  Why not 13?

One characteristic that I have learned about Indian classical music is that the notes are less discretized, meaning a singer may move “in between” an A and a B on the scale, thereby fitting many more tones into each octave.  You can hear it in this example: the singer hits all the notes of the Western scale, but also sings around and in between them, sliding between tones continuously.

#MathProblemOfTheDay: Sitting at a Circular Table

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Four people are gonna have lunch at a circular table. There’s an umbrella pole at the center of the table. How might you arrange the chairs so that everyone can see everyone?
PS: The arrangement pictured, where each chair is 90 degrees away from its neighbors, does not work, because people across from each other have look through the center to see each other.
PPS: one solution is just to put everyone right next to each other, leaving a whole half of the table vacated. Then no one has to look through the pole to see anyone else. But are there any solutions that distribute the 4 people around the table more evenly, minimizing the portion of the circle that is left vacant?

Original instagram post:

Ordering Games

Ordering games involve estimation and comparison, two essential skills for mathematics.

Here are some ordering games I gave to groups of students at Greenwich High School.  Teams raced to arrange the slips of paper from smallest to largest.  (No calculator)

Some groups were very enthusiastic about this game; others not so much.  In the future it might help to do one “practice” round with numbers like 1,2,3,4,5, so that each group could get a feel for what they are trying to do and what a successful ordering looks like.

Each ordering game below is geared to a different skill level.  The cards are not necessarily in order as pictured.

Focus on numbers and exponents:

Focus on shapes and area:

A little bit of trig:

Finding areas enclosed by curves.  I gave this to students in a calculus class, but it could also be great for eager students who haven’t learned calculus yet.  It can be completed without any calculus.

I feel I have only scratched the surface of what can be done with ordering games.  It would be cool to create decks of cards (several different decks) like this.  You’d need an ordering game of 13 items.  The concept of suits (clubs, diamonds, hearts, spades) would be preserved.  Just as in a normal deck, there would be 4 copies of every number.  However, math skills would be required to sort the cards.  For instance, instead of relationships like “a jack is higher than a 10” you’d have relationships like “a card with log(11) is higher than a card with sin(89 degrees).”  Students could then play classic card games with each other, using the mathematical card decks.  They’d have to swap out decks periodically to avoid getting too familiar with one deck and memorizing the orderings.

Eventually students could design decks for each other.  This would involve thinking of 13 numbers that are obscure enough to make people think, but “easy” enough that no laborious calculations would be needed to compare them to one another.

#MathProblemOfTheDay: Nike Running App


Help me figure out how fast I ran!

I was gonna go for a walk and then decided to turn it into a yog. My routine was: walk a half mile, run a half mile. Repeat 4 times.

Nike Running tells me my average pace was a 14:17 minute mile. No magic there – they just took my total time (57:11) and divided it by my total distance (4 miles).

Anyway, here’s what happened. While I was walking I was relaxed. I was able to look at the screen and see my pace – about 19 minutes per mile. While I was running I had no energy or focus to spare – and I didn’t look at the screen. I want to find out roughly how fast I was running.

Given that my walking pace hovered around 19 minutes per mile, what was my average running pace?

Stats, Probability & Games – A Presentation for Draper University

This is a recent math workshop I did for the students of Draper University in San Mateo, California.  It presents basic probability theory in a fun and accessible light.

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Draper University is a two-month long intensive program for aspiring entrepreneurs.  Throughout the course, students work on small teams to complete challenges designed to build creativity, problem-solving skills, and business sense.

When I came, students were preparing for a carnival, where each team was designing a booth.  Teams were competing against each other in a winner-take-all model: whichever team made the most profit would get to keep all profit from all teams.

I wanted students to understand basic probability and outcome analysis in order to optimize their pricing and payout strategies at the carnival.

I interspersed Challenge Problems throughout the presentation.  Teams who submitted solutions to these problems were eligible for extra points on the program-long team scoreboard.


We reviewed the difference between probability and odds — two words which are often used interchangeably, but have different mathematical meanings.


I have always wondered if the concept of odds was invented by casinos and lotteries to make their games more attractive.  Instead of comparing the winning event to all possible outcomes, they can compare it only to the losing outcomes, making it seem more likely.  Doesn’t “2 to 1 odds” sound better than “0.67 probability”?


We brought mathematical precision to two commonly used phrases:


And then played a huge rock-paper-scissor tournament.



Some people thought the probability of winning a rock-paper-scissors game would be 1/3, since there are 3 choices in every game.  However, there are only 2 choices that end the game.  This can be illustrated with a tree of all possible outcomes:


Note that there are 9 possible outcomes, but only 6 possible outcomes that end the game, three of which are losing outcomes, and three of which are winning outcomes.draperu_feb26.010-003

I recently read Delivering Happiness, by Zappos founder and CEO Tony Hsieh.  Hsieh writes about the story of Zappos, as well as the research he did on human happiness.  He notes that one of the main factors contributing to happiness is perceived control.  Not necessarily control, but perceived control.  Rock-paper-scissors is a perfect illustration of this.  Why else would we prefer it over a coin toss?  We believe that by choosing between rock, paper, and scissors, we somehow have control over the outcome.

Another factor that contributes to happiness, according to Hsieh’s research, is human connection.  Since rock-paper-scissors involves synchronization, some eye contact, and body language, students remarked that it creates more opportunities for laughter, creativity, and human connection.  For instance, many players like to act out the outcome: if I played paper and you played rock, I might wrap my hand around yours to assert my victory.

The bottom line is, there is more than mere numerical probability in these activities.  By maximizing the psychological “payouts” of your game, you can increase its appeal without lowering your profits.

One team ended up utilizing this at the carnival by making a dunk tank: zero monetary payout, but a huge experiential reward — watching someone get soaking wet due to your throw.


The winner of our tournament won five games in a row.  We found that the probability of winning five games in a row is (1/2)^5, or 1/32.  That’s about 3%.

Do you agree?

Can you make a general formula for the probability of winning n games in a row?

 (Hint: make a tree showing all the possible outcomes after 2 games, 3 games, 4 games, etc)


And now a new game to consider:


At first people said, “Charge 5 million dollars!”

Yes, if we charged 5 million dollars, we would make huge a profit every time someone played.  But no one would want to play.

So what is the least we could charge, while still making a profit?

The class was divided.  Some people said $3.01; others said $3.51.  Others were unsure.

This is was our doorway into discussing expected value.



In the example with a fair dice, the expected value is the same as the average, because all the outcomes are equally likely: all have a probability of 1/6.  If they had different probabilities we’d have to multiply the value of each outcome by its unique probability.  We’ll see an example of this later on.


Expected value has endless applications but we focused on its application to carnival games and casino-style games.



We used a numerical example to familiarize ourselves with process, and then generalized:


Here’s a simulation of how the expected profit per player is affected by the price to play, amount of payout, and probability of winning.  Click here to edit this spreadsheet and see how the results change. 

Now, what if you had multiple levels of payouts?

No one knew how to calculate the expected profit per player.  Why?  We don’t know any of the probabilities!

Suppose we did know the probabilities of hitting each zone.


Now we are able to calculate the expected profit per player (assuming these probabilities are legitimate) (are they?  where did they come from?).


To find the expected payout per player, we multiplied each payout by its likelihood, and added them together.  We expect to payout $3 one-third of the time.  We expect to payout $6 one-tenth of the time.  We expect to payout $9 one-twentieth of the time.  Thus,

expected payout per player = $3*(1/3) + $6*(1/10) + $9*(1/20).

This game ends up being a loss for the house.  The expected payout is 5 cents more than the price to play.  This means that on average, we are losing 5 cents for every person who plays.

How can we improve this?


We did another simulation.  To get more accurate probabilities, we built a real-life model of the game and each of us took a shot.  Surprisingly, no one was able to hit the green zone.  Hitting the blue zone was pretty easy but many people aimed for the green and yellow zones and instead got nothing at all.

Here are the results of our simulation.  We were emboldened to increase the green payout to $50 because no one was hitting it.


Here is a spreadsheet where you can simulate it yourself.  The red numbers are the ones you have access to change.  The black numbers are calculated automatically from the red numbers. Click here to access and modify the spreadsheet.

I spent the next 5 minutes letting teams work on the task below and answering questions.


That’s when we ran out of time.

Check out the bonus questions!




You’ll find that, without autocorrect, the probability of accidentally texting “I love you” is very, very low.  The probability of getting just the “I” would be around 1/50, because there are roughly 50 characters (26 letters + punctuation) on a typical texting keyboard.  By the time we get to “I” with a space afterwards, we’re already down to 1/2500.  That’s less than 0.04%!

However, with autocorrect, it’s a different story.  You could push something like “Umkocdeyi” and suddenly you’ve notified your boss that you love her.

#MathProblemOfTheDay: Skateparks and Concavity


Skateparks are a great place to observe concavity.

A curve is concave up if its slope is increasing. This could mean the curve has a positive slope and it’s getting steeper, or it has a negative slope and it’s getting flatter. In both cases, we say the slope is increasing because it’s getting more positive. (Think about how -10 –> -7 is an increase just the way 7 –> 10 is.)

A curve is concave down if the opposite is true: the slope is decreasing. Either it has a positive slope and it’s getting flatter…. or it has a negative slope and it’s getting steeper.

Visually, you can think of a positive slope as an uphill, and a negative slope as a downhill.

A point of inflection is where the concavity changes from down to up, or from up to down.

Where is the point of inflection on this skateboard ramp?

P.S.  What would be the concavity of a straight line?