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.

More junk food math

You’ve seen the billboards for these, right?

Basically, they’re mini McNuggets.

McNuggets have a lot of fat. Most of it (as with all fried foods with the possible exception of cheese curds) is in the outer coating.

So what happens when we shrink the McNugget?

We could model the chicken by volume and the outer coating by surface area. In this case, shrinking the McNugget by a factor 1/2 would yield a doubling of the ratio of coating to chicken. As a result we predict a substantial increase in fat as a percentage of weight.

But that’s a crude model-the coating isn’t infinitely thin.

A more realistic model might treat the coating as having a thickness, and thus a volume. When we shrink the McNugget to make a McBite, do we shrink the thickness of the coating proportionately? This seems unlikely. But even if we do, the ratio of coating to chicken increases.

More likely is that the coating is of constant thickness. In other words, McBites are probably smaller chicken bits covered with the same coating as McNuggets. Once again, we increase the ratio of coating to chicken, and we predict a substantial increase in fat as a percentage of weight.

Heading over to McDonald’s online Nutrition Information, we learn that McNuggets have about 12 grams of fat for a 65 gram serving. That’s about 18% fat by weight. This figure is consistent across serving sizes.

Mc Bites? 19 grams of fat for an 85 gram serving. That’s about 22% fat by weight. Also consistent across serving sizes.

How different is 22% fat from 18% fat, you ask? Very different. They’re both bad news, but 22% fat is (coincidentally) 22% more fat than 18% is. So take a serving of McNuggets, which has a lot of fat. And put almost a quarter more fat in. That’s McBites. Oh, and also increase the portion size from 97 grams (6 piece McNuggets) to 128 grams (Regular size McBites).

Middle school math will get you far, kids.

Now if we can get an estimate of the McNugget to McBite scale factor, we can set up a system of equations and figure out just how much of that fat is in the coating.

Griffy counts

Dan Meyer developed a lovely challenge for math teachers and curriculum writers last week:

Give yourself one photo or one minute of video to tell a mathematical story so perplexing that all of your students will want to know the ending, without you saying a word or lifting a finger.

He used it to start a Twitter discussion [#anyqs] that has yielded some really interesting discussion both on Twitter and on Dan’s blog.

I want to follow up on one of my entries in this fun and challenging game.

The following question have all come via Twitter.

1. How many times will Griffy make it around?
2. How many times would he make it around if he counting at the correct speed?
3. How close will his counting be to 2 minutes?
4. Which will occur first; 2 minutes or his counting to 120?
5. What number will he be on when the 2 minutes is up?
6. How many numbers/minute is Griffy counting?

Mission accomplished. Each of these is relevant to my intention; they overlap and interact in ways that would make exploration of any one of them relevant to any of the others.

These questions get at mathematical modeling. What assumptions should we build into the model? Should we assume that his counting continues at the same pace up to 120? Should we assume that his running continues at the same pace? If we answer no to either of these, do we think he’ll speed up or slow down? Etc.

Answers in video below. Note that he starts counting 10 seconds into the video.

Salt, continued

Callum writes:

Filling a container with salt, sorry but…is this kind of thing inspiring to students learning maths? Isn’t this all rather pointless – you filled a container with salt – well done!

Touché.

But can we agree that my salt problem is no worse than any of the following?

These were gleaned from a quick sample of middle-school, remedial college and mathematics for elementary teachers textbooks on my shelf. I grabbed four books off my shelf that I thought might have volume problems in them and found a problem for the gallery in each of them. No cherry-picking here. And I swear I didn’t leave out any compelling applications of volume of a cylinder.

But even if we agree it’s no worse, that’s not a very strong argument in favor of the problem.

So is there any aspect in which it might be better? I think there are several.

intuition

The initial question-will it fill or spill? admits student guesses. Students will have a hunch about the answer, and an intuitive sense of why it will or will not fill or spill.

This contrasts with these other problems. Students are asked to find a volume for the sole purpose of finding a volume. Not in order to answer anything some more meaningful question. And not even I have an intuitive sense of the volume of 678 flapjacks.

Plus, the question can come from the students. They can ask whether it will fill or spill; I don’t think I’ll have to. And then we’ll need to find some volumes in order to make a good prediction.

reality

These are real containers. Perhaps not very compelling containers (although I’m a big fan of vintage Tupperware). But real containers nonetheless. Unlike anything in the problems above, these are objects in their daily lives.

Perhaps this is a sign of my hopeless math geekdom, but I am pleasantly surprised every time I refill my salt container that it fits perfectly. No leftover salt; no space left in the container. A perfect fit. I imagine some of that enthusiasm will be contagious in the classroom. And perhaps inspire some students to look at the containers in their own homes a little bit differently and wonder which ones are “bigger” than others.

the answer

Did I mention that the salt fills the container perfectly? And that we can see it happen before our eyes?

I’m not looking to draw eyes away from the Super Bowl with this problem, nor to cause students to switch their major. But I hope they’ll be a bit more invested in the outcome than they are in the textbook problems above.

intuition again

Here’s an interesting task from the math for elementary teachers book.

from Beckmann, S. (2010). Mathematics for Elementary Teachers. Boston: Pearson.

My instinct is that, at middle school, where the salt task would be appropriate, this will still be part of some students’ intuition. It is much more abstract to run the calculations and see that they are very, very close than to run them and then see that closeness play out in the physical world.

I’m not hoping to draw students to mathematics with this problem; I’m hoping to get them engaged for a lesson on volume.

But we’ll see. I’ll be using the problem with my future elementary teachers in a few weeks. This is not a population that is already sold on math (although by this late in the semester, I’ve reeled them in pretty well). I’ll report back.

And I welcome further critiques.

How do we make volume compelling?

Math 2.0: A newbie tries his hand

Will it fill? Will it spill?

THe question[s]

Two questions: Will the salt fill the Tupperware? and How long will it take to empty the package?

Some useful information

answers

This is how much was rattling in the bottom of the container at the end.