## MEASURING LENGTH

To me, this is the logical starting point for measurement. Measurement itself is a useful concept -- we measure length, area, weight, but also angle and probability. So this is a good preliminary for several different topics.

This exercise seems to complement the traditional teaching of measurement. All my students knew how to measure length using feet and inches, and apparently centimeters too. So they knew the facts, and this exercise assumes they know the basic facts of measuring length. This is about concepts. I used this for third-graders and sixth graders. The sixth graders sped right through it, as you might expect. But they seemed at first to stumble over the idea of measuring something without a store-bought ruler. So they too apparently had to learn the basic concept -- that the unit of length is an arbitrary decision.

### The Basic Exercise

This exercise is a variation of having students measure things -- first they have to choose their own arbitray unit of measure.

For the third-graders, I first explained about cubits, which was the length from your elbow to the end of your finger. Then I mentioned foot. Then I explained about them selecting their own unit of measurement. Naturally, they all chose parts of their body.

The third-graders asked if they could measure what they wanted. I said yes. After measuring a few things, they were amenable to direction in what they measured. The final project for the first day was simply them making a report of what they measured and the results.

The sixth graders were different. First, I brought some soft wood that I could cut in the shape of a ruler. So two of the sixth graders were content with an arbitrary length. Their first assignment was to measure themselves, so the third student chose himself as the arbitrary measure of length and cut off a ruler that was about 1/10 of his height. So they measured with rulers, not body parts. For the sixth-graders, I also brought in five rods of different lengths, and they measured that.

### Problems: Smaller Units of Measure

One immediate problem that will arise is that there is almost always be "something left over" when they use their unit of measure. In other words, the desk isn't two cubits long, it's two cubits and a little extra. My third-graders students solved this problem, to their satisfaction, by estimating fractions.

In real life, the standard solution is to define a smaller unit of measure (e.g., inches), and measure in these. It doesn't completely solve the problem of something being left over and having to use fractions, but it makes the problem less serious. My sixth-graders spontaneously adopted this solution.

But my third graders needed prodding. So, as teacher, what problem could I give them that a smaller unit of measure is the answer? I gave them the problem of measuring two things that were almost but not quite the same length (two different shoes). The two things measured out the same, even though the students could compare them and see that the two were slightly different.

Then I explained the concept of a smaller unit of measure. I explained the English system, which was to just pick a smaller unit of measure (in the same arbitrary way as the larger measure). I also explained about the French system of dividing into tenths. They both happily chose the English method, found a body part of the appropriate size, and happily remeasured everything.

I gave my sixth-graders five rods to measure. I started with three rods, and cut two of the rods into two pieces. So, if the students' measures were accurate, they should be able to add their measures of the two smaller pieces (of a rod) to get the measure of the larger rod.

In theory, this should be elegant and exciting. In practice, they didn't seem to like it -- it seemed to be a test that they weren't going to pass. And of course the addition can't be perfect, because measurement is never perfect; in practice, the errors seemed larger than that.

But this addition is also good practice at adding the mixed measures of length (such as 2 feet 8 inches plus 3 feet 9 inches). It also suggests one good feature of a measurement system -- there should be an integer number of small units of measure in a larger unit of measure.

### Other Possibilities

I like to present the history behind the arbitrary selection of units, when I have the chance. I can't say anyone was especially excited by the information, but they didn't look bored either.

If you spent more time at it, this is also a good exercise for the problem, what should a unit of measure be like? I think the French solved that problem, but it is nice to be able to appreciate the French solution (the decimal system based on meters).

In theory you should be able to do the same exercise with weight and time. In practice, that would probably be difficult. Measuring the size of an angle should be advanced but possible.

One student used the distance from her knee to her heel as her standard. This made measurement a little awkward. I think she enjoyed the silliness, but I also think she didn't realize she could have transferred her standard to an easier format -- made a piece of string the right length. Anyway, a potential problem for students is to make some unmovable distance the standard, then ask them to measure things. Maybe you could have string and scissors available.