Algebra II

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.

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.

Quadratic Play

CAVEAT: There isn’t any deep math in this post. There aren’t any lessons or lesson ideas. I was just playing with quadratics today and below includes some of my play.

I’ve been struggling with coming up with a precalculus unit on polynomials that makes some sort of coherent sense. You see, what’s fascinating about precalculus polynomials is that to get at the fundamental theorem of blahblahblah (every nth degree polynomial has n roots, as long as you count nonreal roots as well as double/triple/etc. roots), one needs to start allowing inputs to be non-real numbers. To me, this means that we can always break up a polynomial into n factors — even if some of those factors are non-real. This took up many hours, and hopefully I’ll post about some of how I’m getting at this idea in an organic way… If I can figure that way out…

However more recently in my play, I had a nice realization.

In precalculus, I want students to realize that all quadratics are factorable — as long as you are allowed to factor them over complex numbers instead of integers. (What this means is that (x-2)(x+5) is allowed, (x-5.2)(x-1.2) is allowed, but so are (x-i)(x+i) and (x-5+2i)(x-5-2i) and (x-\sqrt{2}+\sqrt{7}i)(x-\sqrt{2}-\sqrt{7}i). (And for reasons students will discover, things like (x+i)(x+2) won’t work — at least not for our definition of polynomials which has real coefficients.)

So here’s the realization… As I started playing with this, I realized that if a student has any parabola written in vertex form, they can simply use a sum or difference of squares to put it in factored form in one step. I know this isn’t deep. Algebraically it’s trivial. But it’s something I never really recognized until I allowed myself to play.


I mean, it’s possibly (probable, even) that when I taught Algebra II ages ago, I saw this. But I definitely forgot this, because I got such a wonderful a ha moment when I saw this!

And seeing this, since students know that all quadratics can be written in vertex form, they can see how they can quickly go from vertex form to factored form.


Another observation I had… assuming student will have previously figured out why non-real roots to quadratics must come in pairs (if p+qi is a root, so is p-qi): We can use the box/area method to find the factoring for any not-nice quadratic.


And we can see at the bottom that regardless of which value of b you choose, you get the same factoring.

I wasn’t sure if this would also work if the roots of the quadratic were real… I suspected it would because I didn’t violate any laws of math when I did the work above. But I had to see it for myself:


As soon as I started doing the math, I saw what beautiful thing was going to happen. Our value for b was going to be imaginary! Which made a+bi a real value. So lovely. So so so lovely.


Finally, I wanted to see what the connection between the algebraic work when completing the square and the visual work with the area model. It turns out to be quite nice. The “square” part turns out to be associated with the real part of the roots, and the remaining part is the square associated with the imaginary part of the roots.

pic 8.png


Will any of this make it’s way into my unit on polynomials? I have no idea. I’m doubtful much of it will. But it still surprises me how I can be amused by something I think I understand well.

Very rarely, I get asked how I come up with ideas for my worksheets. It’s a tough thing to answer — a process I should probably pay attention to. But one thing I know is part of my process for some of them: just playing around. Even with objects that are the most familiar to you. I love asking myself questions. For example, today I wondered if there was a way to factor any quadratic without using completing the square explicitly or the quadratic formula. That came in the middle of me trying to figure out how I can get students who have an understanding of quadratics from Algebra II to get a deeper understanding of quadratics in Precalculus. Which meant I was thinking a lot about imaginary numbers.

That’s what got me playing today.


A New Insight on the Famous Painted Block Problem

There is a famous, well-known problem in the world of “rich math tasks” that involves taking an nnn cube and painting the outside of it. Then you break apart the large cube into unit cubes (see image below cribbed from here for n=2 and n=3):


Notice that some of the unit cubes have 3 painted faces, some have 2 painted faces, some have 1 painted face, and some have 0 painted faces.

The standard question is: For an nnn cube, how many of the unit cubes have 3 painted faces, 2 painted faces, 1 painted face, and 0 painted faces.

[In case you aren’t sure what I mean, for a 3 x 3 x 3 cube, there are 8 unit cubes with 3 painted faces, 12 unit cubes with 2 painted faces, 6 unit cubes with 1 painted face, and 1 unit cube with 0 painted faces.]

Earlier this year, I worked with a middle school student on this question. It was great fun, and so many insights were had. This problem comes highly recommended!

Today we had some in house professional development, and a colleague/teacher shared the problem with us, but he presented an insight I had never seen before that was lovely and mindblowing.

Spoiler alert: I’m about to give some of the fun away. So only jump below / keep reading if you’re okay with some some spoilers.


Radical Musings

This is a short post to archive some thinking I did on the subway home from work today. I had a Geometry class today and it was clear to me that their understanding of radicals was … not so good. And I don’t think it is their fault. I remember teaching Algebra II years ago and tried building up some conceptual understanding so puppies don’t have to die… and it was tough and I didn’t really succeed:


(Poster made by the infinitely awesome Bowman Dickson.)

I also remember having this exact same conversation with my co-teacher last year. We considered the following “thought exercise.”

How would you explain to a student in Algebra I why \sqrt{15}=\sqrt{5}\sqrt{3}?

I would like to add the corollary “thought exercise”:

How would you explain to a student in Algebra I why \sqrt{15}\neq\sqrt{10}+\sqrt{5}?

And so on the subway home, I thought about this, and had the same insight I had last year.

We define (at least at the Algebra I level) \sqrt{15} to mean “the number you multiply by itself that yields 15.”

I want to highlight the concept more than the notation, so let’s call that number \square.

So for us \square is “the number you multiply by itself that yields 15.”
Now let’s similarly call \heartsuit “the number you multiply by itself that yields 5.”
And let’s call \triangle “the number you multiply by itself that yields 3.”

We know from this \square \cdot \square=15. Why? Because that’s the definition of “square” for us.

But we also know \heartsuit \cdot \heartsuit=5 and \triangle \cdot \triangle=3 for the same reason.

Thus we know \heartsuit \cdot \heartsuit \cdot \triangle \cdot \triangle=\square \cdot \square.

Here’s the magic.

Let’s rearrange:

\heartsuit \cdot \triangle \cdot \heartsuit \cdot \triangle = \square \cdot \square .

Study this a minute. It takes a second (or it might for students) to see that \heartsuit \cdot \triangle = \square.

Now remember I used symbols because I wanted to focus on the meaning of these objects, not the notation.Let’s convert this back to our “fancy math notation.”

\sqrt{5} \sqrt{3}=\sqrt{15}

So that gets at our first “thought exercise.”

I wonder if trying the same with the second thought exercise might work? The tricky part is that we’re trying to show a negative statement. I know… I know… most of you probably say “hey, just show the kids \sqrt{1+4}\neq\sqrt{1}+\sqrt{4}.” But that doesn’t stick for my kids!

So let’s try it: for us \square is “the number you multiply by itself that yields 15.”
Now let’s similarly call \clubsuit “the number you multiply by itself that yields 10.”
And let’s call \spadesuit “the number you multiply by itself that yields 5.”

\square \cdot \square=15.
\clubsuit \cdot \clubsuit=10
\spadesuit \cdot \spadesuit=5

Then challenge students do something similar to show that \square = \clubsuit + \spadesuit. They hopefully will start failing in their endeavor!

I predict they will start with: \square \square = \clubsuit \clubsuit + \spadesuit \spadesuit. Yay. That’s true… So from that true statement, they are going to try to show that \square = \clubsuit + \spadesuit.

But they can’t really go anywhere from here. They’re stuck. I still predict some weaker students may say: “But clearly we can just say \square =\clubsuit + \spadesuit. It’s like you have “half” of each side of the equation!” But it is at this point you can ask students to do two things:

1) Ask ’em to show the algebraic steps that allow them to make that statement. There won’t be valid steps. And in this process, you can see what other horrible algebraic misconceptions your students have (if any).

2) Or say: okay, let’s see if you’re right. If \square =\clubsuit + \spadesuit, then I know \square \square=(\clubsuit+\spadesuit)(\clubsuit+\spadesuit). And as soon as you start distributing those binomials, they’ll see they don’t get \square \square = \clubsuit \clubsuit + \spadesuit \spadesuit (our original statement).

Okay I just needed to get some of my initial thoughts out. Maybe more to come as I continue thinking about this…



Rational Function Headbandz

TL;DR: An interactive activity having kids ask each other questions to guess the rational function graph they have on their foreheads.


I’m going to make a short post inspired by Twitter Math Camp 2013 (TMC13), rather than TMC14. Both @calcdave and I led morning sessions for precalculus teachers. Through that morning session, some nice end-products were created — an organization for the curricula, actual classroom activities — and you should feel free to check them out here. [1]

@calcdave and I brainstormed how we could get people in the morning session to know each other, but make sure we have math content in that activity. We came up with Rational Function Headbandz, which was inspired by this post on the agony and dx/dt.

The setup: There are a bunch of cards (they could be index cards). On the front of them is an graph of a rational function. On the back is the equation of the rational function. The cards are attached to ribbons or headbands, so that when attached to the forehead only other people can see the graph on the front of the card — not the person wearing it. Sort of like this image below. You can re-imagine how to create these cards/headbands so they work for you.


The Goal: Since this was an introductory activity, participants picked one of two goals for themselves… (a) to figure out as many features as they could of their rational function and to sketch a graph from those features, or (b) to figure out the equation of their rational function.

To Play: I put all the cards/headbands on the table, and covered up the graph with post-its so the participants couldn’t see the graphs. I wrote on the post it if the graphs were graphs I considered sort of challenging, pretty darn challenging, or wow-you’re-going-for-it challenging! Then they attached their headbands to their head, and had someone else remove the post-it note.

Before starting they were told the following things about their rational functions:

  • All the graphs are of rational functions.
  • Some might be plain old polynomials. (Rational functions with the a 1 in the denominator!)
  • If written in the most factored form, none of the terms has degree of more than two
  • If written in the most factored form, most of the coefficients are really nice

Each person carried around with them a notebook, and they were allowed to ask up to three questions about the graph to each person (and a get to know you question to each person!). The rub? All questions had to be answered with a single word or a single number.

A valid question: “How many holes does my graph have?”

A valid question: “Is my rational function a line?”

A valid question: “Does my rational function cross or kiss the x-axis at x=3?”

An invalid question: “What is the coordinate of the hole?” (Because the answer will have two numbers as an answer — an x-coordinate and a y-coordinate.) You could instead ask “What is the x-coordinate of one of the holes of my graph?” and then follow up with “For the hole with x-coordinate BLAH, what is the y-coordinate?”

After three questions, they move on to a different person. Then another. Et cetera. From these questions they were supposed to gather information about their graph, and possibly about their equation.

You stop the game whenever you want. Everyone looks at their graphs and equations, and ooohs!, dohs!, and aaahs! result.

And then if you have time, you can debrief it with students by talking about what they thought was important information to gather in order to sketch or come up with the equation for the graph (holes? x-intercepts? y-intercepts? vertical asymptotes? horizontal asymptotes? slant asymptotes? end behavior?). And then if you had time you could have individual students present their graph, their thought process, and their solution.

Our Graphs: We really varied the nature of the graphs because we were working with precalculus teachers and we didn’t know their ability level with the material. And also I know I emphasize in my class working backwards from the graph to the equation, but that isn’t a standard thing taught. So I would highly recommend creating graphs of your own based on the level of work that you’re doing in your class.

[.pdf, .docx]

Trouble Spots: One thing that was challenging for us when we played this was what someone does when they have figured out their own equation/graph. They came to us and we confirmed. But then what? We should have anticipated this because we had such varying levels of difficulty for graphs. I wonder if a good solution would be to then try to figure out the equation for the rational functions of others when they are being asked questions.

Another thing to keep in mind is that this will take a longer time than you think. We used this as a get-to-know-you activity, and so that extended everything even more. (In your class, your students probably won’t be using this as a get to know you activity.)

Alternatives: Just as I adapted this from a teacher using them for trig functions/graphs, these can easily be adapted for other topics. Some initial ideas:

Geometry vocabulary review: Students have a vocabulary word on their heads. They only can ask questions with one-word answers. (e.g. “Does it have to do with parallel lines?”)

Polynomial graphs (instead of rational function graphs), or even just parabolas [update: Mary did this!], or even just lines.

Students have derivative graphs on their heads, and they need to come up with a sketch of the original function (for this they should be allowed more than one-word answers).


[1] One thing I worked on in a group with four other people is how to get students to understand inverse trigonometric functions (a topic we collectively decided was challenging for students to wrap their heads around). I blogged about the result of our work here. I used it in class this past year, and although I didn’t use it completely as intended, it did really push home the meaning of what sine and cosine were graphically (the y- and x-coordinates on a unit circle corresponding to a given angle) and then what inverse sine and inverse cosine were graphically (the angles that are corresponding to a given y- or x-coordinate). Check it out!


A couple years ago, Kate Nowak asked us to ask our kids:

What is 1 Radian?” Try it. Dare ya. They’ll do a little better with: “What is 1 Degree?”

I really loved the question, and I did it last year with my precalculus kids, and then again this year. In fact, today I had a mini-assessment in precalculus which had the question:

What, conceptually, is 3 radians? Don’t convert to degrees — rather, I want you to explain radians on their own terms as if you don’t know about degrees. You may (and are encouraged to) draw pictures to help your explanation.

My kids did pretty well. They still were struggling with a bit of the writing aspect, but for the most part, they had the concept down. Why? It’s because my colleague and geogebra-amaze-face math teacher friend made this applet which I used in my class. Since this blog can’t embed geogebra fiels, I entreat you to go to the geogebratube page to check it out.

Although very simple, I dare anyone to leave the applet not understanding: “a radian is the angle subtended by the bit of a circumference of the circle that has 1 radius a circle that has a length of a single radius.” What makes it so powerful is that it shows radii being pulled out of the center of the circle, like a clown pulls colorful a neverending set of handkerchiefs out of his pocket.

If you want to see the applet work but are too lazy to go to the page, I have made a short video showing it work.

PS. Again, I did not make this applet. My awesome colleague did. And although there are other radian applets out there, there is something that is just perfect about this one.

Trig War

This is going to be a quick post.

Kate Nowak played “log war” with her classes. I stole it and LOVED it. Her post is here. It really gets them thinking in the best kind of way. Last year I wanted to do “inverse trig war” with my precalculus class because Jonathan C. had the idea. His post is here. I didn’t end up having time so I couldn’t play it with my kids, sadly.

This year, I am teaching precalculus, and I’m having kids figure out trig on the unit circle (in both radians and degrees). So what do I make? The obvious: “trig war.”

The way it works…

I have a bunch of cards with trig expressions (just sine, cosine, and tangent for now) and special values on the unit circle — in both radians and degrees.


You can see all the cards below, and can download the document here (doc).

They played it like a regular game of war:

I let kids use their unit circle for the first 7 minutes, and then they had to put it away for the next 10 minutes.


And that was it!