# Measurement, explored

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.

### The premise

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:

• Stick
• Woody
• Shroydelshnop
• Oompa Loomp
• BOG
• Ablue
• Pen
• 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:

1. Students nearly always partition in 4ths, 8ths and 16ths.
2. Students almost never partition into 10ths.
3. Students may group in threes or sixes, but they never ever partition this way.
4. 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:

1. Get to know the three units that have been handed to you.
2. Express relationships between your units and these new ones.
3. 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.)
4. 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:

1.  How do your three units compare to a standard measurement system?
2. How is using someone else’s units like (or unlike) converting between standard and metric systems?
3. How did your choices for partitioning, composing and naming support or impede your work?
4. 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:

1. How many square partitioned units in a square original unit?
2. How many square original units in a square composed unit?
3. How many square partitioned units in a square composed unit?
4. 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.

### summary

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.

### 15 Responses to Measurement, explored

1. Christopher, this is fabulous!

Benezet (have you heard about him – school superintendent in the 30′s), had the elementary schools skip math lessons until 6th grade. Instead they did estimation and measurement activities. They did better than kids who underwent the usual arithmetic curriculum of the time.

And don’t Russian elementary schools focus on measurement?

This seems like a great way to get kids started on really thinking about the meaning of math.

2. Laurie

This is perfect. Is there a Standard for Mathematical Practice that ISN’T covered here?! Just what I needed for next week’s lesson plans. Thank you!

3. I find VERY interesting that your students almost never partition into tenths. I assume that you are working with American students whose day-to-day measures and units are based on the Imperial System of Units and not the Metric System. I wonder what would MY students do (I am in Spain and here all we use is the metric decimal system…) Plus, mixed fractions are never or almost never used here in Spain as well…. so I’d bet Spanish students will partition into tenths more often that American. But I might be wrong, in which case it would be a nice hypothesis that fractions are more “natural” to children than the decimal system.

I’ll let you know!

4. This is brilliant – thank you! One quick, possibly “how long is a piece of string?” question: how long did the session take?

5. Lovely, thank you!

6. Christopher

Sue, this is the second time in a month or so that the Russian elementary program has come up here. Was it you both times? Either way, I clearly need to dig in; I am convinced that I will find it fascinating.

I am in love with Ignacio‘s hypothesis, and I dearly hope that someone will come through with either existing data or an action research project for us on the question of whether/how the metric conventions influence children’s or adult’s thinking when they create their own systems measurement. (And he is correct that my students are very experienced in the Imperial System, and much less so in the Metric System).

Piers, the activities I described here take approximately four 80-minute class sessions with first-and-second-year community college students studying to be elementary teachers.

Laurie and Jim, you are welcome. Let us know how you adapt things, and how things go with your students.

7. Nope. Just this time.

8. Laurie

I am in the middle of this with some high-powered (American) fifth graders and they are eating it up. One interesting observation for now: even though the students used metric examples when we discussed what “partitioned” and “composed” meant, none used tenths when creating their own, as expected. One clear reason(from my observations) was that it is very hard to accurately create fractional parts in tenths. One group did try to create fifths, but found it understandably frustrating. Fractional parts that can be created from a base of 1/2 (fourths, eighths, etc) are so much easier to produce with accuracy!
I will post more of what I have observed soon…

9. Late to respond, but it might have been me referencing the Russians via Twitter conversation on March 29…https://twitter.com/mathinyourfeet/status/317691974092210176

10. Christopher

Yes, Malke, it was you. Thanks for the link. I see I favorited that tweet, which means it is buried deep in my archives of things to return to. Thanks for the reminder.

11. Pingback: Some ideas… | ArkieTeacher

12. bcarrera

I’ll be using this activity in a small classroom on Friday. Several of my students are international students, so it might serve as a small (very small) sample size on the American v non-American students

13. Followup to the above comment;

i ran the project with 3 groups of 2-3 students each (I did mention the small classroom right?).

One group split their unit into 6ths because it was coincidentally the same length as 6 of the wood flats used to pave the classroom. They did however go 6ths all the way then. This group was 100% American students.

The German/American group split their main unit into 8ths, (fold it on itself to divide evenly and easily) but from there they did everything by tenths, They later admitted the should have used 10ths all the way.

The Peruvian/American group did 8ths all the way.

All in all, a wash I think.