Tag Archives: algebra

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.

The goods [#NCTMDenver]

Good turn out for my session Saturday morning (EIGHT O’CLOCK!).

Thanks to Ashli Black (@Mythagon) for the shot of title screen.

I’ll get some more details up here sometime soon. In the meantime, here’s the handout (.pdf). And here’s the slide deck (.zip, and which—to be honest—was just a photo album on the iPad; the simplicity of this was liberating).

Here are Alison Krasnow’s notes from the session.

road.to.calculusOne last thing…this is the absolute best form of session feedback, as far as I am concerned—getting to read someone else’s notes on the session speaks volumes about what participants experienced (in contrast sometimes to what I think we did).

The slides:

UPDATE: This talk has been adapted to a paper submitted to Mathematics Teaching in the Middle School. I’ll keep you posted on its progress.

Does anybody at TI check the math?

I get that the images below are not real classrooms. These are combinations of staged and stock photos. I get that. But seriously, a waterfall with a straight-line cross-section? And just what answer do we expect to the “shade 1/6 of the hexagon” task? Is the resolution on that screen good enough to detect the difference between 1/4 and 1/6? And how will the teacher tell that difference at a glance? Do YOU know which of those responses is correct?

Note the right triangle on the NSpire screen. And the "Real-world" connection: "Diagonal distance of a waterfall".

We're filling from the bottom up. If height of hexagon is 1 unit, what fraction SHOULD we fill to? And how exactly do these images help me assess whether students can find it?

Modern Algebra or Common Core?

Let’s play a little game. We’ll call it Modern Algebra or Common Core?

For each of the following learning targets, determine whether it comes from my undergraduate Modern Algebra textbook or from the Common Core State Standards for Algebra I.

  1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
  2. Discuss… exponential functions with continuous domains.
  3. The polynomial ring…has features that have no analogues in the ring of integers.
  4. Polynomials form a system analogous to the integers.
  5. By “abstracting” the common core of essential features…develop a general theory that includes as special cases the integers…and the other familiar systems.
  6. With each extension of number, the meanings of addition, subtraction, multiplication, and division are extended. In each new number system—integers, rational numbers, real numbers, and complex numbers—the four operations stay the same in two important ways: They have the commutative, associative, and distributive properties and their new meanings are consistent with their previous meanings.

Answers in the comments.

My 6 year old’s definition of algebra

Overheard in a conversation between Griffin, 6 years old, and Tabitha, 4:

Griffin: I hate algebra.

Tabitha: What is algebra?

Griffin: Algebra is a piece of paper with math problems that are very hard.

So much prep work, and I got it (partly) wrong…

I know I messed this up.

My pH videos came out clean and distraction free. I am satisfied with their quality. I spent two hours getting 20 minutes of footage and have edited it down to tidy, manageable packages.

But I know I didn’t get a really important one into the mix.

See if you can guess what’s missing…

I have the pH of water, of 100 ml of water with 1 ml of orange juice concentrate, of 100 ml of water with 10, then 20, then 100 ml of concentrate. And I have the pH of straight concentrate.

So what’s missing? Orange juice made correctly-in a 3:1 ratio of water to concentrate. I have 1:1 and 5:1 but not 3:1.

Now I understand that the pH difference between 5:1, 3:1 and 1:1 is virtually unnoticeable. But that’s not the point. The point is that when I do this lesson on Wednesday I know one of the student questions in the “What can we ask here?” part will be about the pH of properly made juice. And it’s a less compelling answer to have me say “It’s between these other two” than to see it play out on screen (this is the whole point of the effort, after all, right?)

Too late for this time around. Lesson learned.


Sometimes it IS that easy

Both here and elsewhere, I have been part of a lively and robust discussion of the role of the real-world in mathematics instruction. One concern that is often expressed of instruction that is based in real-world problems is that of time and efficiency.

But sometimes it is really easy to turn the tables and make the standard word problem the motivational setup. An example.

In my College Algebra class, I started class with the following image the other day.

from http://www.ally.com 3/14/2011

And I asked What is there to wonder about here? What questions can we ask? Here is what my students came up with.

  • What is APY?
  • What is the difference between Rate and APY? And what does this have to do with interest?
  • How much does the principal grow over time in response to the rate?
  • How does the amount of the deposit affect the growth over time?

These are precisely the questions I would hope for. But they asked them, not me.

In order to get to the answer to the question about the difference between APR and APY, we have to consider this one:

If you offer a 1.54% APR, how should you go about compounding interest monthly?

I have no idea how to get my students to the place where they can ask this question, so I asked it myself. Their ideas:

  1. Figure a year’s worth of interest in dollars, then divide that by 12 and add to the account each month, and
  2. Divide the interest rate by 12. Apply this new interest rate each month.

Brilliant! Next year, I’ll use (1) as part of a homework assignment-how do we run the computations when money gets added to account midway through the year? It gets very tricky very quickly.

But (2) is how we have agreed to run the computations in the financial world.

And it’s what accounts for the difference between APR and APY.

Now they were ready to hear about and derive interest formulas and to do some computations on their own.

A small bit of preparation put my students in a much better frame of mind for the material.

Sometimes it is that easy.

But not always.