# Graham’s Number

TL;DR: If you have an extra 45-60 minute class and want to expose your 9th/10th/11th/12th graders to a mindblowingly huge number and show them a bit about modern mathematics, this might be an option!

In one of my precalculus classes, a few kids wanted to learn about infinity after I mentioned that there were different kinds of infinity. So, like a fool, I promised them that I would try to build a 30 minute or so lesson about infinity into our curriculum.

As I started to try to draft it — the initial idea was to get some pretty concrete thinkers to really understand Cantor’s diagonalization argument — I decided to build up to the idea of infinity by first talking about super crazy large numbers. And that’s where my plan got totally derailed. Stupid brain. At the end of two hours, I had a lesson on a crazy large number, and nothing on infinity. You know, when that “warm up” question takes the whole class? That’s like what happened here… But obvi I was stoked to actually try it out in the classroom.

In this post, I’m going to show you what the lesson was, and how I went through it, with some advice for you in case you want to try it. I could see this working for any level of kid in high school. Now to be clear, to do this right, you probably need more than 30 minutes. In total, I took 35 minutes one day, and 20 minutes the next day. Was it worth it? Since one of my goals as a math teacher is to try to build in gaspable moments and have kids expand their understand of what math is (outside of a traditional high school curriculum): yes. Yes, yes, yes. Kids were engaged, there were a few mouths slightly agape at times. Now is it one of my favorite things I’ve created and am I going to use it every year because I can’t imagine not doing it? Nah.

We started with a prompt I stole from @calcdave ages ago when doing limits in calculus.

Kids started writing lots of 9s. Some started using multiplication. Others exponentiation. Quite a few of them, strangely, used scientific notation. But I suppose that made sense because that’s when they’d seen large numbers, like Avagadros number! I told them they could use any mathematical operations they wanted. After a few minutes, I also kinda mentioned that they know a pretty powerful math operation from the start of the school year (when we did combinatorics). So a few kids threw in some factorial symbols. Then I had kids share strategies.

Then I returned to the idea of factorials and asked kids to remind me what $5!$ was. Then I wrote $5!!$. And we talked about what that meant ($120!$). And then $5!!!$ etc. FYI: this idea of repeating an operation is important as we move on, so I wouldn’t skip it! They’ll see it again in when they watch the video (see below). While doing this, I had kids enter $5!$ on their calculator. And then try to enter $120!$. Their calculators give an error.

Yup, that number is super big.

Then I introduced the goal for the lesson: to understand a super huge number. Not just any super huge number, but a particular one that is crazy big — but actually was used in a real mathematical proof. And to understand what was being proved.

Lights go off, and we watch the following video on Graham’s number. Actually, wait, before starting I mention that I don’t totally follow everything in the video, and it’s okay if they don’t also… The real goal is to understand the enormity of Graham’s number!

I do not show the beginning part of the video (the first 15) because that’s the point of the lesson that happens after the video. While watching this, kids start feeling like “okay, it’s pretty big” and by the end, they’re like “WHOOOOOOAH!”

Now time for the lesson… My aim? To have kids understand what problem Ronald Graham was trying to understand when he came up with his huge number. What’s awesome is that this is a problem my precalculus kids could really grok. But I think geometry kids onwards could get the ideas! (On the way, we learned a bit about graph theory, higher dimensional cubes, and even got to remember a bit about combinations! But that combinations part is optional!)

I handed out colored pencils (each student needed two different colors… ideally blue and red, but it doesn’t really matter). And I set them loose on this question below.

It’s pretty easy to get, so we share a few different answers publicly when kids have had time to try it out. The pressure point for this problem is actually reading that statement and figure out what they’re being asked to do. When working in groups, they almost always get it through talking with each other!

One caveat… While doing this, kids might be confused whether the following diagram “works” or if the blue triangle I noted counts as a real triangle or not:

It doesn’t count as a real triangle since the three vertices of the triangle aren’t three of the original four points given. During class I actually made it a point to find a kid who had this diagram and use that diagram to have a whole class conversation about what counts as a “red triangle” or “blue triangle.”. Making sure kids understand what they’re doing with this question will make the next question go more smoothy!

Now… what we are about to do is super fun. I have kids work on the extension question. They understand the task (because of the previous one). They go to work. I mention it is slightly more challenging.

As they work, kids will raise their hand and ask, with trepidation, if they “got it.” I first look to make sure they connected all the points with lines. (If they didn’t, I explain that every pair of points needs to be connected with a colored line.) Then I look carefully for a red or blue triangle. Sometimes I get visibly super excited as I look, saying “I think you may have gotten it! I think you may… oh… sad!” and then I dash their hopes by pointing out the red or blue triangle I found. (So here’s the kicker: it’s impossible to draw all the line segments without creating a red or blue triangle… so I know in advance that kids are not going to get it… but they don’t know this.) After I find one (or sometimes two!) red or blue triangles, I say “maybe you want to start over, or maybe you want to start modifying your diagram to get rid of the red/blue triangle!” Then they continue working and I go to other students.

(It’s actually nice when students try to modify their drawings, because they see that each time they try to fix one thing, another problem pops up. They being to *see* that something is amiss!)

This takes 7-8 minutes. And you really have to let it play out. You have to ham it up. You have to pretend that there is a solution, and kids are inching towards it. You have to run from kid to kid, when they think they have a solution. It felt in both classes like a mini-contest.

Then, after I see things start to lag, I stop ’em. And then I say: “this is how you can win money from your parents. Because doing this task is impossible [cue groans… let ’em subside…] So you can bet ’em a dollar and say that they can have up to 10 minutes.!That it takes great ingenuity to be successful! What they don’t know is… you’re going to get that dollar! Now we aren’t going to prove that they will always fail, but it has been proven. When you have six or more dots, and you’re coloring all lines between them with one of two colors, you are FORCED to get a red or blue triangle.” [1]

Now we go up a dimension and change things slightly. Again, this is a tough thing to read and understand so I have kids read the new problem aloud. And then say we are going to parse individual parts of it to help us understand it.

And then… class was over. I think at this point we had spent 35 minutes all together. So that night I asked kids to draw all the line segments in the cube, and then answer the following few questions:

These questions help kids understand what the new problem is saying. In essence, we’re looking to see if we can color the lines connecting the eight points of a cube so that we don’t get any “red Xs” or “blue Xs” for “any four points in a plane.” Just like we were avoiding forming “red triangles” and “blue triangles” before when drawing our lines, we’re now trying to avoid forming “red Xs” and “blue Xs”:

So the next day, we go over these questions, and I ask how this new question we’re working on is similar to and different from the old question we were working with. (We also talk about how we can use combinatorics to decide the number of line segments we’d be paining! Like for the cube, it was $_8C_2$ and for the six points it was $_6C_2$ etc. But this was just a neat connection.) And then I said that unlike the previous day where they were asked to do the drawings, I was going to not subject them to the complicated torture of painting all these 28 lines! (I made a quick geogebra applet to show all these lines!) Instead I was going to show them some examples:

It’s funny, but it took kids a long while to find the “red X” in the left hand image. Almost each class had students first point out four points that didn’t form a red X, but was close. But more important was the right hand figure. No matter how hard you look, you will not find a red X or blue X. Conclusion: we can paint these line segments to avoid creating a red X or blue X. Similar to before, when we had four points, we could paint the line segments to avoid having a red triangle or blue triangle!

So now we’re ready to understand the problem Graham was working on. So I introduce the idea of higher dimensional cubes — created by “dragging and connection.” I don’t take forever with this, but kids generally accept it, with a bit of heeing and hawing. More than not believing that it’s possible, kids seem more enthralled about the process of creating higher dimensional cubes by dragging!

And then… like that… we can tie it all together with a little reading:

And… that’s the end! At this point, kids have been exposed to an incomprehensibly large number. And kids have learned a bit more about the context in which this number arose. Now some kid might want to know why we care about higher dimensional cubes with connecting lines painted red/blue. Legit. I did give a bit of a brush off answer, talking about how we all have cell phones, and they are all connected, so if we drew it, we’d have a complex network. And analyzing complex networks is a whole branch of math (graph theory). But that’s pretty much all I had!

In case it’s helpful: the document/handout I used: 2017-04-04 Super Large Numbers (Long Block).

[1] I like framing this in terms of tricking their parents. We’ve been doing that a bunch this year. And although I understand some teachers’ hesitation about lying to their students about math, I think if you frame things well, don’t do it all the time, it can be fine. I don’t think any student felt like I was playing a joke on them or that they couldn’t trust me as their math teacher because of it.

# How I Did Trig Review in Adv. Precalculus this year

### The Problem: Wasting Class Time Reviewing Trigonometry

Last year the Advanced Algebra II kids did a boatload of trigonometry, and this year I had to make sure my kids had a strong grasp of the basics (it’s been ages since they’d seen it) before we delved into trigonometry this year in Advanced Precalculus. In previous years, I always did a “trig review unit” which I always felt like wasted time. I like to use classtime to give kids things where they have to rely on each other — but during the review unit, kids didn’t need each other much. Different kids needed to review different things. I found ways around it, but overall, it felt like wasted time.

### The Solution: Mix Review A Little Each Night During the Prior Unit (Which Posed Another Problem…)

So the other teacher and I decided that while we were working on sequences and series, we would also give kids some basic trig questions each night, maybe 10-15 minutes worth. Although I can’t see myself using this curriculum in my classroom the teach the material for the first time, I really love eMathInstruction’s packets. They are well-thought-out, and their problems highlight drawing connections among tables, graphs, and equations, and they often give forwards and backwards problems.

So each night I gave a selection of problems from one or two of these lessons — all topics they had worked on last year — and had kids do them. I chose the problems and lessons based on the specific things I needed kids to remember for what we were doing this year. And then the next day, I gave the answers and let kids resolve any difficulties.

But I wanted to be thoughtful about this. It was review, but I needed to make sure that kids really had these basics down before we jumped with both feet into the depths of trigonometry. And remember, all of this was happening during a unit on sequences and series. And I was afraid without some sort of feedback mechanism, I was going to finish this review and find that kids didn’t interalize any of it, or regain the fluency with trig basics that they had last year. So I worked with another teacher (who has been acting as my “teacher coach” this year) to circumvent this problem.

### The Solution To The Problem My Solution Generated: Short Daily Feedback Quizzes

This is how it worked…

Let’s say on Friday students were asked to complete review problems from Lessons #1 and #2 from the review packet. Then on Monday, I give each group the answers, have them check their own work and talk with their group to resolve any difficulties (and if that doesn’t work, ask me!), and then the rest of the lesson is continuing on with sequences and series.

Then on Tuesday, we’d start class with a short 3-5 minute check-in super basic quiz on the trig review that was due on Monday and we had already gone over. It might look like this:

The back of the quiz would look like this (kids flip to the back when they’re done):

Then we have the rest of the class on Tuesday, consisting of going over the new trig review answers for a few minutes, and then working on sequences and series.

Tuesday night, I’ll mark up the quizzes. They are worth a whopping total of 1 assessment point (most of my assessments are 30-40 points). But here’s the catch: the score is either a 0 or a 1. To get the point, you need to get all parts correct. I’m okay with that because this is an advanced class and these questions are super basic. This is feedback for the student: do they really know the basic material, or do they merely think they know the basic material? [1]

On Wednesday, we’d start class with a short basic quiz on the review trig material we went over on Tuesday, kids would get their quizzes from Monday, and we’d go over the review trig material due today (before continuing on with sequences and series).

A note about timing… Most of our classes are 50 minutes. So about 4 minutes were spent taking the brief quiz, about 5-8 minutes were spent going over the trig review work and resolving any difficulties, and the remaining time was spent on the current unit of sequences and series.

At the very end of all the trig review, I had a mini-assessment on all the trig review material.

### Framing The Quizzes

When presenting the daily quizzes to students, I expected a lot of groans. Thankfully I didn’t get any audible ones, which I attitribute to taking my time framing the plan for them. I wanted them to understand the thinking and impetus behind this approach to the review material. I wanted to be transparent.

First, I acknowledged that it was a long time ago (last year!) that they had worked on trig. So it would be unfair of me to expect them to know things like $\sin(225^o)$ or how to graph $y=-2\sin(0.5x)+3$ immediately. I wanted us to build up to it, slowly, so they had time to practice and get feedback. It was my job to make sure that before we resumed trig that they had refreshed themselves with the basics.

Second, I talked them through the idea behind the daily quizzes. I made sure that kids understood they would be short and only on material they had reviewed and had time to practice first. I highlighted that the quizzes served three purposes.

1. That they were low-stakes feedback for you on what you truly know and what you don’t know.
2. They will provide specific places for additional help if you find you don’t know something.
3. They were feedback for me on what y’all are good with and what you need work with — so I know to talk publicly about anything I’m noticing the whole class needs help with.

I did mention the grading, but I didn’t put much emphasis on that. The score wasn’t super important, except as feedback.

You might have noticed that the back of the quizzes gave specific places for students to get help with the concepts/ideas on the quiz. This was an idea my teacher coach and I generated together. The conversation we had was about feedback in general. We teachers can be good about giving feedback, but we never teach students explicitly how to use that feedback. What do they do with it? By providing specific resources/places for kids to go to get additional help (along with their teacher and classmates, of course), we thought this might highlight that we really do want these quizzes to be part of a feedback loop.

### The Feedback

I took data on the quizzes. You’ll note that before each 1 or 0 are a few columns. Those are the concepts being asked. An “x” means that the student got that part incorrect. That data helped me look for trends, and what was more challenging for students, so I knew if I had to explicitly talk about any concept/idea in class.

I was planning on also using this feedback later. I was going to look at the assessment kids took at the end of the review, and see if there was any relationship between kids’ performance on the assessment and these feedback quizzes. I didn’t get a chance to do this, and truth be told, the average was so high for the review assessment (89%) I suspect it would have been a waste of time.

I also wanted to know how students felt about this process. This was an experiment for me, but to know if it succeeded, I needed feedback from students. I wanted to know (a) if they found the feedback quizzes were helpful, (b) if the feedback quizzes changed their practice in any way, and (c) if they used the feedback from the quizzes in any way. So my teacher coach and I wrote this short and pointed set of questions for them:

The results were interesting.

When asked if their preparation changed or not, it was interesting. Most kids said their preparation did change a bit, but even kids who said that it didn’t would then go on to say something that indicated that their preparation did change (highlighted in red)!

This is what kids did with the feedback (see list above from survey to see what each bar corresponds with):

And finally, here’s what kids wrote in the “anything else” box:

Overall, a success! Not only because kids found them useful on the whole, and because their practice changed because of them (for the better), but also because they did quite well on the trig review assessment (as I noted above, earning an 89% average).

More than figuring out how to deal with the annoying “how to review trig from the previous year before starting on trig in the following year” problem, this whole enterprise was an interesting excursion into feedback for me. I was hoping to find a way to create a feedback loop that was doable (and this was! it only took 10 minutes each day to mark up the simple quizzes) and created a change in student practice (which it did, because knowing there was something small students were accountable for each day changed how most kids prepared just a little bit).

To me this post and this experiment isn’t really about trig, but about now having another tool (daily feedback quizzes) in my teacher toolbelt to pull out at appropriate times.

[1] I debated whether I wanted to put a grade on these at all, or just let them be feedback with no score attached. I went back and forth about this for a long while. But ultimately, I knew that attaching a score, no matter how minimal, to the quizzes would effect more change than if I didn’t. But after introducing it, I didn’t mention the grade/score once when talking about them. I would mention common mistakes I noted and talked about ways to get extra practice with something or another. I kept my focus on the notion of feedback, and doing something with that feedback.

# Getting familiar with the Unit Circle

In our standard precalculus class, we’ve spent 4 days “getting ready” for trigonometry. Which sounds crazy, until you see what awesome thing we’ve done. But I’ll blog about that later. Right now I want to share what I created to help kids start learning the unit circle.

Here were the hurdles:

1. We are introducing radians for the first time this year. So they’re super unfamiliar.
2. The unit circle feels overwhelming.
3. Although I am familiar with the special angles in degrees and radians, kids aren’t. So I know when I hear 210 degrees that’s “special” but kids don’t know that yet.

Here is what we have done:

1. Filled in a “blank unit circle” using knowledge of 30-60-90 degree triangles, 45-45-90 degree triangles, and reflections of 1st quadrant points to get the points in the other quadrants.

In this post, I’m going to do here is to share what I’m going to be doing to help kids learn the unit circle.

Phase I: Get confident with angles

I am going to talk about these like pizza. And to start, focus on radians.

I’m going to remind kids that $\pi$ radians is a half rotation about the circle. Then we can see that each pizza pie slice is $\frac{\pi}{2}$,$\frac{\pi}{4}$, $\frac{\pi}{3}$, and$\frac{\pi}{6}$ radians. [The “top” half of the pizza is divided into two, four, three, or six pieces! And the top half is $\pi$ radians!]

Then I’m going to work on the “easy-ish” angles by pointing at various places on the unit circle and have kids figure out the angle. I am going to have kids not only state the angle in radians, but also explain how they found it. For each angle, I will ask for a few different ways one could determine the angle measure. Then I’m going to repeat the same thing with the “easy-ish” angles, except I am going to do it in degrees.

And then… you guessed it… I’m going to do the same exercise but with the “harder-ish” angles. Start with radians. Then again with degrees. Always justifying/explaining their thinking.

Finally, I am going to let them practice for 5-8 minutes using this Geogebra applet I made. The goal here? To focus on getting kids familiar with the important special angles. Not only what the values are of these angles, but also to get them to start finding good ways to “see” where these angles are.

Phase II: Start Visualizing Side Lengths — utilizing short/long

Next comes getting kids to quickly figure out the coordinates of these special angles.

We’ve already been working on special right triangles, so I think this should be fine. And then…

Kids are asked to visualize the side lengths/coordinates based on the drawing. So, for example, for the first problem, kids will see that the angle is $\frac{4}{3}\pi$. They hopefully would have mastered that from the previous exercise. They also will see that if they would draw the reference triangle, the x-leg is shorter than the y-leg, so they know the x-coordinate must be $\frac{1}{2}$ (but negative), and the y-coordinate must be $\frac{\sqrt{3}}{2}$ (but negative).

After practicing with this for these four problems, kids are going to practice some more using this second geogebra applet I created.

Phase III: Putting It All Together

It’s now time to take the training wheels off. No longer do I give the picture to help visualize things. Now, I give the angle. This is more like what kids are going to be seeing. They need to know $\sin(315^o)$ and $\cos(3\pi/4)$. No one is going to be giving them nice pictures!

So this is what they’re tasked with:

I have a strong feeling that breaking down the unit circle in this way is going to make all the difference in the world. Fingers crossed!

If you want the file I created for my kids, here you go (.docx2017-02-xx Basic Trigonometry #2.docx:  , PDF: 2017-02-08-basic-trigonometry-2)!

# Sequences and Series

A few days ago I posted a card sort I did to start my unit on Sequences. I figured I’d share my entire packet for Sequences (2016-10-31-sequences [doc] 2016-10-31-sequences [pdf]) in case it’s any help for you. This was designed for our standard precalculus classes, and I have to say it worked pretty well.

1. It allows for kids playing with math at the start (card sort, some visual patterns, a 3-act)
2. It doesn’t “tell” kids anything. They discover everything. And sometimes asks for a couple different ways to do things.
3. Kids messing up notation is always an issue with sequences. So I introduced notation way after kids started playing around with sequences — and got a handle on them. I did still see some kids confuse $n$ and $a_n$ (the term number and the value of the term), but it wasn’t a huge number. I actually was pretty proud of that.

Note: Problem #17 on page 23 is actually ill-posed. So I didn’t have my kids work on it. I have a replacement warm-up question (included at the bottom of this post) which worked wonders!

Also in the packet, there is a blank page (on page 4) which simply says “A Pixel Puzzle.” For that, I used Dan Meyer’s 3-Act.

For me, the 3-Act was a mixed bag. Mainly because I thought it would be easier than it ended up being for my kids, so I didn’t plan much about how to help groups get started. At the beginning of the 3-Act, kids asked great questions, but not the one I was hoping for (when does the pixel hit the border). So I didn’t have a smooth way to deal with that. And I anticipated it would take less time than it actually did, so the 3-Act got split up between multiple classes. And when kids were working, I don’t think I was strong at facilitating them working. I should have added in a wee bit more structure to help kids out. Even if it’s something as simple as coming up with ways to get kids to think about making a table or a graph — and what would be useful/important to include in the table/graph. And I didn’t have a good way for kids to share their findings. So I’d need to think about the close to the 3-Act better. I’d give myself a “C.” Would I do it again? Yes. I saw the potential in it. It got kids thinking about sequences visually. It had kids thinking temporally. And had them relate tables/graphs/equations. All good things!

One last thing I want to include in this post… This is related to the ill-posed Problem #17 on page 23 I made note of before. There is one type of question which tends to flummox kids when it comes to geometric sequences. It reads something like “The third term in a geometric sequence is 6 and the seventh term is 96. Come up with a formula for the $n$th term in this sequence.”

The reason this is tricky is because there are two possible sequences! (The common ratio could be 2 or -2.)

Thus, I created this warm-up (.docx) for my kids to check if they truly understood what we were doing. The conversations were incredible. Some groups were done in 7 minutes… Others had a solid 15 minutes of discussion.

# 2, 4, 6, 8, what do we appreciate? A Card Sort!

This year I’m teaching both Advanced Precalculus and Standard Precalculus. (Totally confusing on a daily basis? Yup.) And I’m working with two other teachers to write the Standard Precalculus curriculum from scratch. Of course this is something that is daunting, but I love to do when I have the time and like-minded colleagues.

I was in charge of spearheading our sequences and series unit. In this post, I want to briefly share how we started the unit. Instead of diving right in, or doing something intense, I wanted to gently get some good conversations percolating. So I handed out this set of cards:

and gave them this set of instructions:

I debated having kids use Desmos for the card sort, but since I have kids work in groups (mostly groups of 3), and I wanted the entire group to be working together, and I wanted them to actually physically move and shuffle cards, I decided to use physical cards. I also had all kids stand up while doing the card sort. I had a feeling that would be magical, in terms of getting kids talking, moving, and engaging with each other (even thought they were all at the same table), and it was! So I highly recommend that.

These cards end up having three different types: arithmetic, geometric, and recursive.

Most kids got the arithmetic sequences quickly, but it was interesting to watch them struggle with the geometric and recursive. There were great conversations, and because I demanded the next number for each of the sequences, kids had to really think through what the pattern was (and in geometric sequences, how to find the common ratio). I had thought that kids would finish this really quickly, but I was totally wrong. It took about 20 minutes. So plan accordingly. (A few groups needed to do a little bit more at the end of class, so I had them take a photo of their card sort and use that photo to finish things up!)

One note: Card H which has the sequence 0,0,0,0,0,0 fits all three categories. So it’s great fun to watch kids try to place it.

I wanted to share this activity because I haven’t really done many card sorts before —  and I was so pleased that this particular one generated productive conversations. So I need to keep this teaching tool in my arsenal for generating conversations about something new. (Example: I just thought of giving a bunch of graphs of rational functions to kids on cards, before we start that unit, and say “find different ways to sort these!” There are so many features, so that could lead to so many different ways to sort the cards. I suspect that Desmos would be good for that particular card sort, since there would be many different ways to sort those pictures, and I’d want to project the different ways kids did it to the entire class. I bet through that sort, we could actually recognize vertical asymptotes, horizontal asymptotes, oblique asymptotes, and holes!)

Here is the .docx (2016-10-31-card-sort-for-sequences) and .pdf (2016-10-31-card-sort-for-sequences) for the cards.

I will try to write up some more about my sequences and series unit soon!

# Binomial Expansion

This is going to be a super short blogpost. But I’m excited about a visualization I came up with today — as I was working on a lesson — for showing why Pascal’s Triangle works the way it does with binomial expansions.

I’m sure that someone has come up with this visualization before. It feels so obvious to me now. That that didn’t make me any less excited about coming up with it! I immediately showed it to two other teachers because I was so enthralled by it. #GEEKOUT

I am thinking how powerful a gif this would be. Start out with 1. Have two arrows emanate from that 1 (one arrow saying times x and one arrow saying times y) and then it generates the next row: 1x    1y. And again, two arrows emanate out of both 1x and the 1y (arrows saying times x and times y). And generating 1x^2    1xy     1xy     1y^2. Then then a “bloop” noise as the like terms combine so we see 1x^2     2xy     1y^2.

And this continues for 5 or so rows, as this sinks in.

Then at the very end, some light wind chime twinkling music comes up and all the variables disappear (while the coefficients stay the same).

Of course good color choices have to be made.

Who’s up for the challenge?

Okay, I’m guessing something similar to this already exists. So feel free to just pass that along to me. Now feel free to go back to your regularly scheduled program.

# Visualizing Standard Deviation

A few days I got an email from someone (Jeremy Jones) who wanted me to look at their video on standard deviation. And then today, I was working with Mattie Baker at a coffeeshop. He was thinking about exactly the same thing — how to get standard deviation to make some sort of conceptual sense to his kids. He said they get that it’s a measure of spread, but he was wondering how to get them to see how it differs from the range of a data set (which also is a measure of spread).

Of course I was hitting a wall with my own work, so I started thinking about this. While watching Jeremy Jones’s video, I started thinking of what was happening graphically/visually with standard deviation.And I had an insight I never really had before.

So I made an applet to show others this insight! I link to the applet below, but first, the idea…

Let’s say we had the numbers 6, 7, 7, 7, 11. What is the standard deviation?

First I calculate the mean and plot/graph all five numbers. Then I create “squares” from the numbers to the mean:

The area of those squares is a visual representation of how far each point is from the mean.[1] So the total areas of all those five rainbow squares is a measure of how far the entire data set is from the mean.

Let’s add the area of all those squares together to create a massive square.

As I said, this total area is a measure of how far the entire data set is from the mean. How spread out the data is from the mean.

Now we are going to equalize this. We’re going to create five equal smaller squares which have an area that matches the big square.

We’re, in essence, “equalizing” the five rainbow colored squares so they are all equal. The side length of one of these small, blue, equal squares is the standard deviation of the data set. So instead of having five small rainbow colored squares with different measures from the mean, the five equal blue squares are like the average square distance from the mean. Instead of having five different numbers to represent how spread out the data is from the mean, this equalizing process lets us have a single average number. That’s the standard deviation.

I’m not totally clear on everything, but this visualization and typing this out has really help me grok standard deviation better than I had before.

I created a geogebra applet. You can either drag the red points up and down (for the five points in the data set), or manually enter the five numbers.

https://www.geogebra.org/m/EatncEg2

My recommendation is something like this:

1. {4, 4, 4, 4, 4}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
2. {8, 8, 8, 8, 8}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
3. Set the five numbers to {2, 4, 4, 4, 6} and look at what you see. What is the standard deviation?
4. Consider the number {5, 7, 7, 7, 9}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #3. Then set the five numbers to {5, 7, 7, 7, 9} and look what you see. What is the standard deviation? Were you right?
5. Consider the numbers {3, 7, 7, 7, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #4. Explain your thinking. Then set the five numbers to {3, 7, 7, 7, 11} and look at what you see. What is the standard deviation? Were you right?
6. Consider the numbers {3, 6, 7, 8, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #5. Explain your thinking. Then set the five numbers to {3, 6, 7, 8, 11} and look at what you see. What is the standard deviation? Were you right?
7. What do you think the standard deviation of {4, 8, 8, 8, 12} be? Why? Check your answer with the applet.
8. Can you come up with a different data set which matches the standard deviation in #6? Explain how you know it will work.
9. Set the five numbers to {4, 4, 4, 4, 4}. Initially there are no squares visible. The standard deviation is 0. Now drag one of the numbers (red dots in the applet) up. Describe what the squares look like when they appear? Eventually drag that number to 15. What do you notice about the standard deviation? Use your understanding of what happened to describe how a single outlier in a data set can affect the standard deviation

Okay, I literally just whipped the applet up in 35 minutes, and only spent the last 15 minutes coming up with these scaffolded questions. I’m sure it could be better. But I enjoyed thinking through this! It has helped me get a geometric/visual sense of standard deviation.

Now time to eat dinner!!!

Update: a few people have pointed out that the n in the denominator of the standard deviation formula should be n-1. However that would be for the standard deviation formula if you’re taking a sample of a population. This post is if you have an entire population and you’re figuring out the standard deviation for it.

[1] One might ask why square the distance to the mean, instead of taking the straight up distance to the mean (so the absolute value of each number minus the mean). The answer gets a bit involved I think, but the short answer to my understanding is: the square function is “nice” and easy to work with, while an absolute value function is “not nice” because of the cusp.