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.

Wrong.

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.

Summary

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?

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