Algebra II

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!

Dan Meyer says JUMP and I shout HOW HIGH?

On a recent blog post, Dan Meyer professed his love for me. He did it in his own way, through his sweet dulcet tones, declaring me a reality TV host and a Vegas lounge act [1]. LOVE!

He was lauding a worksheet… well, a single part of a worksheet… I had created. You see, I’m teaching Precalculus for the first time this year, and so I have the pleasure of having these thoughts on a daily basis:

What the heck are we teaching this for? IS THERE A REASON WE HAVE KIDS LEARN [fill in the blank]? WHAT’S THE BIG IDEA UNDERNEATH ALL OF THIS STUFF?

[Btdubs, I love teaching a new class because these are the best questions EVAR to keep me interested and to keep my brain whirring!]

And I went through those questions when teaching trig identities. And so I concluded the idea of identities is that two expressions that look different are truly equal… and they all derive from a simple set of ratios from a triangle in a unit circle. Equivalent expressions. When things are the same, when things are different…

So my thought was to make graphing central to trig identities. For the first couple days, every time kids were asked to show an identity was true, they were asked to first actually graph both sides of the equal sign to show they truly are equivalent. (And half the time, they weren’t!)

To introduce this, I made this worksheet (skip to Section 2… clearly I had to polish some stuff off beforehand):

Dan asked, I blogged.

[If you want, my .doc for the worksheet above is here… and the next worksheet with problems to work on is here in .doc form too.]

To be honest, I still have some thoughts about trig identities that I need to sort out. I am still not totally satisfied with my “big idea.” I still have the “so what” banging around in my brain when thinking about equivalent expressions. I have come to the conclusion that the notion of “proving trig identities are true” is not really a good way to talk about proof. There’s also the really interesting discussion which I only slightly touched upon in class: “Are 1 and \frac{x}{x} equivalent expressions?” I have something pulling me in that direction too, saying that must be part of the “big idea” but haven’t quite been able to incorporate.

If I were asked right now,  gun to my head to answer, I think I suppose I’d argue that “big idea” that a teacher can get out of trig identities are teaching trial and error, the development of mathematical intuition (and the articulation of that intuition), and the idea of failure and trying over (productive frustration). Because I think if these trig identities are approached like strange mathematical puzzles, they can teach some very concrete problem solving strategies. (To be clear, I did not approach them like strange mathematical puzzles this year.) Now the question is: how do you design a unit that gets at these mathematical outcomes? And how do you assess if a student has achieved those? (Or is truly being able to verify the identity the fundamental thing we want to assess?) [2]

[1] Except I got my teaching contact for next year, and I’ll be making more than the tops of those professions combined. YEAH TEACHING! #rollinginthedough

[2] Different ideas I remembered from a conversation on Twitter… Teachers have contests where they see how many different ways a student/group/class can verify an identity. And another idea was having students make charts where they have an initial expression, and they draw arrows with all the possible possibilities of where to go next, and so forth, until you have a spider web… What’s nice about that is that even if students don’t get to the answer, they have morphed the original expression into a number of equivalent and weird expressions, and maybe something can be done with that? I also wonder if having kids make their own challenges (for me, for each other) would be fun? Like they come up with a challenge, and I cull the best of the best, and I give that to the kids as a take home thing? Finally, I know someone out there mentioned doing trig identities all geometrically, with the unit circle, triangles, and labeling things… I mean, how elegant is the proof that \sin^2(\theta)+\cos^2(\theta)=1? So elegant! So coming up with equivalent expressions using the unit circle would be amazing for me. Anyone out there have this already done?

Families of Curves

When I put out my call for help with Project Based Learning, I got a wonderful email from @gelada (a.k.a. Edmund Harriss of the blog Maxwell’s Demon) with a few things he’s done in his classes. And he — I am crossing my fingers tight — is going to put those online at some point for everyone. To just give you a taste of how awesome he is, I will just say that he was in NYC a few years ago and agreed to talk to my classes about what it’s like to be a real mathematician (“like, does a mathematician just like sit in a like room all day and like solve problems?”), and have kids think about and build aperiodic tilings of the plane.

Anyway, he sent me something about families of curves, and that got my brain thinking about how I could incorporate this in my precalculus class. Students studied function transformations last year in Algebra II, and we reviewed them and applied them to trig functions. But I kinda want to have kids have some fun and make some mathematical art.

First off, I should say what a family of curves is.


That’s from Wikipedia. A simple family of curves might be y=kx which generates all the lines that go through the origin except for the vertical line.


I made this in Geogebra with one command:

Sequence[k*x, k, -10, 10, 0.5]

This tells geogebra to graph y=kx for all values of k from -10 to 10, increasing each time by 0.5.

Okay, pretty, but not stunning. Let’s mix things up a bit.

lines 2

Sequence[k*x+k^2, k, -10, 10, 0.1]

Much prettier! And it came about by a simple modification of the geogebra command. Now for lines with a steep slope, they are also shifted upwards by k^2. This picture is beautiful, and gives rise to the question: is that whitespace at the bottom a parabola?

Another one?



And finally, just one more…



Just kidding! I can’t stop! One more!


Sequence[k sec(x)+(1/k)*x,k,-10,10,0.25]

What I like about these pictures is…





And then, if you’re me, they raise some questions… Why do they look like they do? What is common to all the curves (if anything)? Does something special happen when k switches from negative to positive? What if I expanded the range of k values? What if I plotted the family of curves but with an infinite number of k values? Do the edges form a curve I can find? Can I make a prettier one? Can I change the coloring so that I have more than one color? What would happen if I added a second parameter into the mix? What if I didn’t vary k by a fixed amount, but I created a sequence of values for k instead? Why do some of them look three-dimensional? On a scale of 1 to awesomesauce, how amazingly fun is this?

You know what else is cool? You can just plot individual curves instead of the family of curves, and vary the parameter using a slider. Geogebra is awesome. Look at this .gif I created which shows the curves for the graph of the tangent function above… It really makes plain what’s going on… (click the image to see the .gif animate!)

sec animation

Okay, so I’m not exactly sure what I’m going to do with this… but here’s what I’ve been mulling over. My kids know how to use geogebra. They are fairly independent. And I don’t want to “ruin” this by putting too much structure on it. So here’s where I’m at.

We’re going to make a mathematical art gallery involving families of curves.

1. Each student submits three pieces to the gallery.

2. Each piece must be a family of curves with a parameter being varied — but causing at least two transformations (so y=kx^2 won’t count because it just involves a vertical stretch, but y=k(x-k)^2 would be allowed because there is a vertical stretch and horizontal shift).

3. At least one of the three pieces must involve the trig function(s) we’ve learned this year.

4. The art pieces must be beautiful… colors, number of curves in the family of curves, range for the parameter, etc., must be carefully chosen.

Additionally, accompanying each piece must be a little artists statement, which:

0. Has the title of the piece

1. States what is going on with each curve which allows the whole family of curves to look the way they do, making specific reference to function transformations.

2. Has some plots of some of individual curves in the family of curves to illustrate the writing they’ll be doing.

3. Has a list of things they notice about the graph and things they wonder about the graph.

At the end, I’ll photocopy the pieces onto cardstock and make a gallery in the room — but without the artist’s names displayed. I’ll give each student 5 stickers and they’ll put their stickers next to the pieces they like the most (that are not their own). I’ll invite the math department, the head of the upper school, and other faculty to do the same. The family of curves with the most stickers will win something — like a small prize, and for me to blow their artwork into a real poster that we display at the school somewhere. And hopefully the creme de la creme of these pieces can be submitted to the math-science journal that I’m starting this year.

Right now, I have a really good feeling about this. It’s low key. I can introduce it to them in half a class, and give them the rest of that class to continue working on it. I can give them a couple weeks of their own time to work on it (not using class time). And by trying to suss out the family of curves and why it looks the way it does, it forces them to think about function transformations (along with a bunch of reflections!) in a slightly deeper way. It’s not intense, and I’ll make it simple to grade and to do well on, but I think that’s the way to do it.

What’s also nice is when we get to conic sections, I can wow them by sharing that all conics are generated by r(\theta)=\frac{k}{1+k\cos(\theta)}. In other words, conic sections all can be generated by a single equation, and just varying the parameter k. Nice, huh?

PS. Since I am not going to do this for a few weeks, let me know if you have any additional ideas/thoughts to improve things!