Tag Archives: unit

Counting brownies

Griffin (eight years old) and Tabitha (five years old) were discussing the day’s activities. The feature activity had been making brownies with Mommy. This occurred while Griffin was out the house.

Griffin: How many brownies did you make?

Tabitha: One big one! Mommy cut it up.

I have emphasized elsewhere the importance of the unit; that one is a more flexible concept than we might think.


What is one? Adventures in TED-Ed

Back in March, the Twitterverse was suddenly alight with an announcement from TED about a forthcoming website, TED-Ed. They were seeking educators to write and narrate short lessons that would then be animated by TED-Ed staff.

I nominated myself. And I lobbied Karim Ani to nominate me, too. (Side note: Have you heard Karim is trying to raise some bucks for a supercool project?)

And then-surprisingly-I heard back.

Here’s the result.

The Process

The TED-Ed staff is super supportive, friendly and patient. I talked with them by phone in March. They loved the blog post I submitted as the basis for the lesson. We talked about how to adapt it for the website.

I wrote a script and sent it in. They had a few notes-mostly having to do with making it easier for the animator to work with. (Example-putting “One bag, one apple, one slice” in that order so the apple could come out of the bag, then get sliced).

They sent me a recording kit: One iPad, one microphone for the iPad, a mike stand, a collapsible soundproof recording booth and a clip for holding the iPad to the mike stand.

I did two takes and uploaded them through DropBox.

The Ted-Ed people used very kind words to tell me that I sounded stiff as a board.

I spent the bulk of one afternoon in my office re-recording to sound more conversational. Incidentally I had never noticed how much door slamming goes on in my hallway.

I got that down and shipped it all off. Four weeks or so later, it’s online. They edited my audio, adding pauses for things to happen onscreen, and splicing together a couple of different takes. But the result is totally faithful to my original vision for the lesson. They did a bang up job.

The Product

I have to say that I have truly no idea whether my lesson will get used. And if it does, I have no idea how, nor whether it will be effective. But I love that its ideas are now out in the larger world.

For me, this seemed like an opportunity to get my goofy sensibility a bit wider exposure. I hate that the teaching profession is under such pressure in the US right now to be about predefined standards. There is so little space for ideas in the present discourse around math teaching and learning. So I saw TED Ed as a venue for getting some ideas some airplay. TED’s motto is Ideas Worth Spreading. I’m on board with that.

My lesson is intended to give viewers something to argue about. It doesn’t align nicely to standards, and that’s by design.

The lesson does what I do best. It points to something that you probably don’t notice as you go about your life, and it asks a question that will elicit strong opinions. It builds a case that things may not be as they seem.

And then it offers a multiple choice quiz. Ugh. TED wanted five multiple choice questions, so I wrote them. I don’t think of this as a fact-based lesson, so the questions are clumsy. What are you gonna do?

It’s the Think and the Dig Deeper parts that matter to me. Those are intended to provide a place for the argument to go. Is it always obvious what the original unit is? Are decimals harder than whole numbers because of our real-world experiences? Did it have to happen that our number system is base-10? These are debatable questions with intellectual heft. I can get my own students thinking about them. I would love to believe that students I have never met will be inspired to argue over them, too.

But please don’t put them in the Common Core.


Several sharp-eyed viewers noted the price of those apples about 26 seconds in. Including Robert Gonzalez over at io9.

Call me anal-retentive, but I find it more than a little ironic that a TED-Ed video about the importance of units, whole number place value, decimal place value, and fractions features an animation of apples being sold for 0.79 cents each (see 00:25).

Yeah. That’s a bummer. I wrote back to Mr. Gonzalez.

I noticed the price of those apples when I watched it for the first time yesterday. Should have said something about it in my blog reflection…

I was of two minds about it. (1) That’s a major pet peeve of mine, and (2) It could provide fodder for discussion in a lesson ABOUT UNITS!

I literally saw this video for the first time on Monday. I’ll rattle some cages over at TED and see if we can’t get that fixed up.

I’ll keep you posted.

Composed units, or Why I now have the best office door in the math department

composed adj. formed by putting together

unit noun a single quantity regarded as a whole

If there’s something I enjoy thinking about almost as much as fractions, it’s place value. And I have become convinced that the same idea is at the heart of each one: the unit. I’m not talking about measurement units, nor about unit fractions (although these are of course related).

No, I’m talking about answering the question, What counts as 1?

My math content course for future elementary teachers is just beginning the advanced study of place value for the semester. As a preliminary activity to this, we considered the idea of composed unit. In particular, they had an assignment to photograph a composed unit in their everyday lives, and to post those photographs on Canvas.

I implored them to be creative. And I used the results to set up a few distinctions.

1: Composed units v. partitioned units

Composed units begin with a thing and we assemble several things to make a larger unit. Here is an example:

The original unit is an egg. The composed unit is 12 of these-a dozen eggs.

Here’s an interesting contribution:

This is a lovely illustration of a partitioned unit. But it’s not really a composed unit in the sense I’m going for. We didn’t make the loaf out of slices. Instead, we started with a loaf and cut it into smaller unit. That is the process of partitioning. It’s important, to be sure, but it’s an example of where fractions come from, not how whole-number place value develops.

2. Natural v. conventional units

Natural unit refers to a composed unit that has to be the size that it is. Conventional unit refers to a unit that we have agreed to, but which doesn’t have to be the size that it is.

A dozen eggs is a conventional unit. I have no idea why we put eggs in groups of 12. But it certainly doesn’t have to be this way. The unit below (a pair) is more natural. Shoes sort of have to be grouped in twos.

3. Standardized v. variable units

Whether natural or conventional there is an agreement on how to compose eggs into groups. Same for shoes. But how many slices in a loaf of bread? How many flowers in a bouquet?

Unclear, and it depends, right? A bouquet is a composed unit, but its size is variable. This is very different from eggs and shoes.

4. Composing composed units

I showed this one in class.

I thought the composed unit was a box of eight Pop Tarts.


The student thought of the foil pack being the composed unit. Everyone knows that there are 2 Pop Tarts in a pack. Then there are 4 packs in a box. So a box isn’t composed of 8 Pop Tarts; it is composed of 4 packs of 2 Pop Tarts.

Just like we put ten 10s together to make 1 hundred, we put 4 packs together to make 1 box.

This is brilliant and really useful for the purposes of working on place value understanding for future teachers. I never would have thought of it myself.


My students had some really lovely contributions. They had smart things to say about composed units after looking at each others’ contributions and considering some critical questions I posed in class. And they made me laugh:

I have always been bummed about my nondescript office door. And I have never aspired to an office door covered in comic strips. No, I want it to look nice AND to have content. Problem solved.

Finally, a quick tip o’ the cap to Dan Meyer, who has taught me to visualize with multimedia. I had gotten very good at thinking in diagrams through careful study and practice. Dan has expanded my visual world and for that I am grateful.

And finally, finally…Here are the questions I posed in class about the full set of 25 photographs:

  1. What is the original unit in each case?
  2. What is the composed unit?
  3. Is the size of the composed unit generally agreed upon or is it variable? If it’s generally agreed upon, what is the unit size? And is it a natural unit or a conventional one?
  4. Are all of the units here truly composed units? That is, are you convinced that in all cases, we start with a unit and put those units together to make a bigger one? Or are there some units in our collection that started with a unit and cut the unit into pieces to make smaller units?
  5. One goal of this course is to become better visual thinkers in mathematics. In that spirit, which of these images could be improved to better show relationships between units and composed units?