Tag Archives: functions

New Desmos lesson(s)

You should seriously go check out Polygraph. Four versions of a delightful and challenging game:

  1. Lines
  2. Parabolas
  3. Rational functions
  4. Hexagons

The hexagons will be familiar to long-time readers of this blog.

Screen shot of hexagons

I have run the parabolas version in College Algebra, and the hexagons version in my Ed Tech course. It was a huge hit both times—lots of conversation happened both electronically and out loud in the classroom. It’s a ton of fun.

I am especially pleased with the rational functions version. It makes for challenging work—even among the mathematically astute Team Desmos in recent trial runs.

Read the Desmos blog post on the matter if you like.

A little gift from Desmos

Last summer, the super-smart, super-creative team at Desmos (in partnership with Dan Meyer, who may or may not be one of the Desmos elves) released a lovely lesson titled “Penny Circle“. It’s great stuff and you should play around with it if you haven’t already.

The structure of that activity, the graphic design, the idea that a teacher dashboard can give rich and interesting information about student thinking (not just red/yellow/green based on answers to multiple choice questions)—all of it lovely.

And—in my usual style—I had a few smaller critiques.

What sometimes happens when smart, creative people hear constructive critiques is they invite the authors of the critique to contribute.

Sometimes this is referred to as Put your money where your mouth is. So late last fall, I was invited to do this very thing.

I have been working with Team Desmos and Dan Meyer on Function Carnival. Today we release it to the world. Click through for some awesome graphing fun!


It was a ton of fun to make. I was delighted to have the opportunity to offer my sharp eye for pedagogy and task design, and to argue over the finer details of these with creative and talented folks.

Go play with it.

Then let us know what we got right and what we got wrong (comments, twitter, About/Contact page).

Because I just might get the chance to work on the next cool thing they’re gonna build.

A few quick words on a function context

Geoff Krall did us all the favor of preserving a brief Twitter conversation about a lovely applications of functions example found by Taylor Belcher.

Go have a look. Won’t take you long.

College algebra teachers! Please try this and report back!

My fellow teachers of College Algebra. I want to talk to you today about domain and range. For now, let’s leave aside all the analogies, vending machines, notation and ants crawling on graphs. Let’s get to the heart of the matter with an assessment/instructional task.

This is in the spirit of Eric Mazur, but it is low-technology. It will take 10 minutes. Then you’ll need to report your results back to me. We can talk about what those results mean.

Here is what you do.

Get yourself a College Algebra class that has studying (or even one that is still studying) domain and range.

Then get yourself some index cards in two colors. We used yellow and pink. You may use whatever you have on hand. Make clear that yellow means yes, and that pink means no.


Also make clear that they will raise their cards in unison and on the count of 3. This is to prevent conforming to the majority and will result in more honest representations of your students’ understanding.

Practice this routine on some non-mathematical questions. Some that will be universal nos, some universal yeses, some that are mixed.

Now have your students consider the function y=x^{2}, where x is interpreted as the input, and y as the output in the usual way. You are going to ask a series of questions about the range of this function.

Ask, Is 4 in the range of this function?

You should get near universal yellow. Ask someone to state the case and make a note of their argument on the board.

Ask, Is –2 in the range of this function?

My bet is that you’ll have a lot of pinks, but several yellows. Those yellows probably need clarification that we are asking about range, not domain. But don’t assume it. Ask the pinks to state their case. Ask the yellows to refute or question. Never say anything a [student] can say.

Likely, the yellows will argue that there is no number that, when squared, gives -2. They mean real number; I see no need to make a big deal out of this. But make a note of the argument. Perhaps writing something like this on the board:

( nothing )^{2}=-2

You will probably need to revisit this later on.

Ask, Is \frac{1}{4} in the range of this function?

This is probably all yellow and mostly unproblematic.

Now the fun begins.

Ask, Is π in the range of this function?

If you don’t get a good mix of yellow and pink here, I will eat my hat. And those pink people? They are going to tell you that there is nothing that—when squared—gives π.

Have them talk it out in pairs or threes. Then have them show cards again. And then have the pinks state their case. Nine times out of ten, it’s going to be that there is no number that can be squared to get π.

My fellow College Algebra teachers, I am not interested in your theoretical arguments about what a fabulous job you/your textbook/your online homework platform are doing at teaching domain and range. If you wish to claim that your students will not show pink for π here, the burden of proof on you is high.

Notice with your students the very important difference between:

(nothing )^{2}=\pi


(nothing I can think of )^{2}=\pi

Someone will point out that \sqrt{\pi} is a number, and that when you square it you get π. Highlight that contribution and estimate the value of this number.

Ask, Is 0 in the range of this function? 

Probably mostly yellow, but worth asking to make sure.

Ask, Is infinity in the range of this function?

Seriously. If you don’t get, like, 80% yellow here then I do not understand your school’s placement system

Reinforce that infinity is not a number. Connect it to the notation and wrap up with one more.

AskIs 12 in the range of this function?

You should get nearly all yellow here. Get back to your regularly scheduled classroom activities.

Then report and discuss your results with colleagues, on Twitter, on your blog and here.


There is lots of unexpected pushback in the comments on the value of teaching domain, range and functions in a College Algebra course. I had previously thought these to be de rigueur topics in such a course. I suspect sampling bias here.

Gregory Taylor mostly reproduced my results in his own course and had some lovely mathematical conversations along the way. You should go read his account.

Sameness in College Algebra

Two years ago, I began using unit as an organizing theme in the math content course for future elementary teachers. That led to many adventures, including a TED-Ed video and new ways of talking to my colleagues about fractions, decimals and place value.

That work continues, but it has become part of my instructional practice; one of my habits of mind.

This year, I am thinking about sameness, and about helping my students to notice and pay attention to sameness. The formal name is equivalence, but I am not so worried about the vocabulary and formal definitions here.

I am concerned with helping students understand something about how mathematics views and uses sameness.

It is awkward at first, as any new teaching moves are. But it got us some good stuff recently.

We are studying functions. Our grounding metaphor for functions is vending machines. We discussed the following collection of vending machines the other day. vending.machine.collage

  1.  This is my favorite vending machine of all time. The banana vending machine. It dispenses only bananas. It is like the constant function. More on this below.
  2. There are two ways to get the Pocari Sweat in a can. Two inputs, same output. That’s OK. It’s not one-to-one, but it’s a function.
  3. You put in a quarter, you turn the knob. Sometimes you get a die. Sometimes you get a top. Sometimes you get a ball. This is not a vending machine, really. Same input gets you different outputs. That’s a problem in the vending machine world, and in the world of functions.
  4. The battery vending machine is one-to-one. Each battery type has its own button to push.
  5. Put a dollar into this one, get a dollar out. Put in five dollars, get out five dollars. The output has the same value as the input. This is the identity function.

We discussed these in class one day. Then we opened the next class session by having students brainstorm with their partners specific functions with the traits exemplified by the vending machines. We divided up responsibilities for recording these functions on the classroom whiteboards.

Here is what our boards looked like after the large group (45 students) discussion. (Click to make legible.)

In order:

  1. Lots of good stuff here. x=2 is not a function because, as a vending machine, it would take your money and not put anything out. All input, no output. The idea that we can write y=5 as y=5+0x was important. More importantly, this led a student to ask* about y=5, “Can it be a variable if it’s always the same value?”
  2. Our example the previous day had been absolute value. They weren’t ready to venture much beyond this. As a class, they struggled to identify two x-values that would generate the same y-value. We need to work on that. But I have mentioned that this is College Algebra, right? Students have placed here, or worked their way here through developmental math. Either way, the idea of producing example points to demonstrate properties of a function has not been schooled into them yet. I’m on it.
  3. Again, +/– square root was the prior day’s example. I love +/– x as an extension of the technique. Love that. And square root of x is not right. We’ll come back to that. Having a permanent record of the difference will be helpful.
  4. Wow. Just wow. That was our example from the previous day. Not even a y=x+3 in the bunch! Work to do here.
  5. Now we’re having fun. I love the y=\frac{x}{1}. Same function, different notation. I finished off our work by asking whether y=\frac{x^{2}}{x} is the same as y=x.

Which (finally!) brings us back to sameness.

My students are highly accustomed to writing \frac{x^{2}}{x}=x. But they are not accustomed to thinking about what this means. Because when x=0, that equation is not true. The question then becomes, In what sense are these the same?

And that points us to the very heart of the discipline.

In mathematics, we decompose things according to their attributes, and we focus on one (or two, or…) of these attributes at a time, disregarding all of the others. Formally, when we write \frac{x^{2}}{x}=x, we mean “These two expressions are the same for all but a finite number of values of x.” We don’t say that, of course, but that is the essence of the equal sign here.

We returned to the sameness question with this video.

Are the two outputs the same? How? Are they different in any way? How? Again, mathematical sameness requires us to specify the precise ways in which two objects are alike.

We will return to machine number 3 above in class shortly. If you just want to get “a cheap plastic toy” out of the machine, then you get that every time. It’s a function. If you want to get “a top” out of the machine, then you get something different every time. Is it a function? Depends on what you mean by “same”.

Much more work to do. I’ll keep you posted.

*I recently argued that learning is having new questions to ask. This student was learning about what variable means, and had a question to ask that she maybe could not have articulated before this.

Christopher dipping toe in water of online education

UPDATE: The course described below has filled and a substantial waiting list has been closed. I will post updates to this blog as plans for future versions of the course become more clear.

This coming March, I am going to be offering an online course.

It will not be Massively Open. It will not be a MOOC.

Here’s the plan:

I want to gather a medium-sized group of middle-school math teachers to study the mathematics that lurks just beneath the surface of quality middle school math curriculum materials.

I’m talking about big ideas. I’m talking about what if? questions.

I’m talking about starting with the familiar and ending up with the strange. We will stretch our minds to consider new corners of the mathematical world.

If you want Common Core implementation help, go pay your $400 to NCTM. This won’t be that.

If you want machine-scored, lecture-based online instruction, go join a Coursera MOOC. This won’t be that, either.

If you want credit towards your Master’s Degree or other contract-based lane change, talk with your nearest graduate institution. Because this won’t be that.

This will be a community of 20 smart teachers learning with and from each other in a structured environment.

This will be a set of semi-synchronous activities and discussions designed to advance our own understandings of the foundational mathematical idea of function. Eventually, the course will expand to other topics, such as inversesymmetrychange and operation. But this first version will just be function.

What you’ll get:

An instructor (me) with over 15 years of professional development experience, 18 years of teaching experience, an open mind, a whole mess of math knowledge, and a deep curiosity about ideas and ways of thinking.

Also, you’ll get:

  1. Smarter than you are now,
  2. More connected to other smart math teachers, and
  3. A beautiful, suitable-for-framing certificate to assist in relicensure in your state/province/district/etc.

The details:

Course title: The Mathematics in School Curriculum.

Dates: I need 20 middle-school math teachers who are interested in spending about an hour a day thinking about functions for the two weeks March 17—30, 2013.

Platform: The course will take place on Canvas, an Instructional Management System (IMS) developed by Instructure. I have used this system in place of my institution’s adopted IMS (Desire2Learn, or D2L) and have been delighted with its design—especially the way it supports discussion and sharing of resources. Your Canvas account will be free of charge.

Content: We will pilot a functions unit—the first of what will eventually be five units embedded in a larger course. The goals of the pilot will be to broaden our knowledge of (a) curricular approaches to function relevant to the middle school, and (b) the ideas behind the formal mathematical function.

Cost: This pilot will be free. Eventually, I will charge a reasonable fee in compensation for my time and effort (each of which I imagine will be substantial).

Commitment: In signing up for the course, I will ask for your commitment to full participation. We will be looking to build community, and that won’t happen if we don’t commit to the effort together.

Full participation will mean agreeing to spend about an hour a day for the duration of the course. The “hour” is an average and is at your convenience. Course activities will include working through mathematics tasks, reading articles, seeking resources and participating in asynchronous online discussions.

But full participation won’t just be about seat time. It will be about committing to learning, and to supporting the learning of your classmates.

How to sign up: By filling out a Google form. I will cap the pilot course at 20 participants. First come, first served. I know that people’s lives are hectic and variable, so I’ll maintain a 10-person waiting list before ending registrations altogether.

Click here to register for the course.

What questions do you have? Send them my way.

I hope to catch up with you online in March. We’re going to have a ton of fun.

Diagrams, week 8 (late)


A College Algebra student wondered whether there could be a function such that its inverse is the same as its opposite. That is, can there be an f such that f^{-1}(x)=-f(x)?

I had to work graphically to think this through, which you see above.

That task is now an A Assignment.