This idea started with someone else, but I do not remember his name. I believe he’s a shop teacher in a Twin Cities suburb. Inver Grove Heights, maybe? In any case, he was in a professional development session I was helping to run this year on the topic of fractions. We had a conversation over lunch in which he recounted a lesson he did that became the basis of the activity I am about to describe. If I can dig up the originator, I’ll revise to give credit.
In any case, while the kernel of this idea originated with someone else, I have given it the usual OMT treatment—expanding and complexifying in many ways.
Regular readers will know that I am always in search of ways to get my future elementary teachers to explore old ideas in new ways. Consider the cases of place value and the hierarchy of quadrilaterals. In that spirit, I give you the measurement exploration extravaganza. Do with it what you will.
Groups of three are each given a dowel (or, in this year’s case, a paper strip). The dowels vary in length. The lengths are chosen to provide a useful combination of compatability and incompatability. One may be 9 inches long, while another is 15 inches long. Choose numbers according to the skill level and age of your students (and yourself!)
But-and this is important-THESE LENGTHS ARE NEVER SPOKEN OF! You will never refer to these dowels using standardized lengths.
Each group names its unit. In recent semesters, we have had:
- Oompa Loomp
- Et cetera
The members of the group measure some stuff with their units. They make a tape measure to use for this purpose, and they decide how long a tape measure they would like to have.
For example How tall are you in Sticks? requires (in all likelihood) a tape measure that is several Sticks long. Well, it does not require such a thing, but such a thing facilitates this measurement.
At this point, students are measuring only with their own units. It usually occurs to them to subdivide the unit in some way, and they will frequently report out fractions of (say) a Stick.
Next, each group is responsible for creating a partitioned unit from their original. They choose how many of these smaller units make up the original, and they name the smaller unit.
And then they create a composed unit from their original. Again, the choice is theirs to determine the number of original units that make up a composed unit. And again they are tasked with naming the composed unit.
interlude for important observations
The fun has only just begun and already we stumble upon some beautiful insights. Among them are these:
- Students nearly always partition in 4ths, 8ths and 16ths.
- Students almost never partition into 10ths.
- Students may group in threes or sixes, but they never ever partition this way.
- Students rarely think to group the same way they partition. That is, if they made 8ths, they might very well group in sixes. The convenience that would be afforded by consistency does not tend to occur to them in advance.
back to the instructional sequence
Now that we have the units, we need to measure some stuff. I typically choose things in our classroom environment. It is important that we all measure the same things and that these things range from smaller than the original unit to larger than the composed unit.
We need to express our measurements in (1) partitioned units only, (2) original units only, and (3) composed units only.
This semester I had students look at this table and I asked What do you notice? and What do you wonder? (These questions are, of course, not original to me. But this was a productive place to ask them.)
Working across systems
Next, it’s time to switch things up. We put the table away. Each group passes their original unit, together with instructions for creating a partitioned unit and a composed unit (and the names of these) to another group.
Now each group is charged with these tasks:
- Get to know the three units that have been handed to you.
- Express relationships between your units and these new ones.
- For each thing you measured (table, licorice fish, etc.), make this prediction: If you were to measure that thing with these new units, would you end up with a greater or lesser value than when you measured in your own units? (In this step, do not compute; make a qualitative comparison instead.)
- Compute your height in these new units, and compute at least 6 of the measurements in the grid.
You have never seen such fraction computation work as proceeds from this sequence of tasks.
Now we list these computed measurements on the board, compare to the table we generated earlier and discuss reasons for discrepancies.
We write about these reflection questions:
- How do your three units compare to a standard measurement system?
- How is using someone else’s units like (or unlike) converting between standard and metric systems?
- How did your choices for partitioning, composing and naming support or impede your work?
- What do you need in order to be able to do these computations on your own?
On to area
Next, students build each of their units into square units.
We consider the essential questions:
- How many square partitioned units in a square original unit?
- How many square original units in a square composed unit?
- How many square partitioned units in a square composed unit?
- Most importantly: How do you know each of these?
Sample student observations at this point:
- Wow. The square partitioned unit looks a lot smaller relative to the square original unit than I expected.
- Oh no! Why did we decide to put so many original units together to make the composed unit?
Now we measure something.
This time around, I had them measure the area of a whiteboard in our classroom. Not the most exciting measurement to make, but straightforward and accessible. Working with these new square units is challenging enough; no need to get too fancy. It is important that the measurement be concrete and tangible, not abstract.
Students are encouraged to use known relationships in order to avoid tedious measurements, and to measure in order to avoid tedious computations.
Importantly (I think), most students want to use these square units to measure, rather than to measure with their tape measures and compute.
We use these experiences to discuss differences—both practical and conceptual—among measuring by (1) iterating and counting units, (2) using tools, and (3) computation.
We reflect on what these experiences can tell us about working within and across measurement systems.
We build on our fraction work and on the meanings of multiplication and division that were the focus of the preceding course.
I have not had students move to cubic units.