Friday, 4 December 2015

How to divide by 2

Holy Moly! I thought I pretty well knew everything about dividing a number by two. I was about to hit the publish button, but I reckoned I should first do a quick scan of the web. Again, holy moly !!

This post was initiated by my watching a very skillful carpenter reface our fifteen year old kitchen cabinets. He was a "measure twice, cut once" sort of guy. In the course of his work, he did a lot of marking and checking of centre lines so that handles and panels could be precisely located.

Locating a centre line comes down to finding the midpoint of a measured length, in other words, dividing a number by 2. How he did this might surprise you, as did the advice I encountered on the web.

The measurements to be halved are usually mixed numbers. I happen to have a board that is 97inches wide, and I asked my wife (who is not a mathematician) how she would find the centre line. This is how she explained it using her usual yardstick (which has a resolution of one-eighth of an inch and which dictated her approach).
The midpoint would be half of 978 .
Half of 978 is half of 9 plus half of 78, which is 412 inches plus 312 eighths.
So measure 412 inches and tack on an extra 312 eighths to get the midpoint:

I imagine that a carpenter might do it in the same way, except that he or she would be using a tape measure with a higher resolution and would likely think of the midpoint as 716 inches beyond the 412 inch mark (rather than 312 eighth-inches).

There is another way which comes to mind: 978 is the same as 10 − 18,  so half of 978 is the same as half of 10 minus half of 18,  which is 5 − 116 ,  in which case you would probably find the midpoint by locating the 5 inch mark and backing up 116 inch. I would have used this way myself, and I suspect some carpenters would do it this way as well.

What did the web say?

On the web almost every "explanation" of how to divide a mixed number by a whole number reduced the problem to dividing two fractions using the invert-and-multiply trick. According to these posts, dividing 978 by 2 should be done as follows:
  1. Convert the mixed number to an improper fraction: 978 = 798.
  2. Convert the 2 to an improper fraction: 2 = 21.
  3. Do the division  798 ÷ 21. (by which they meant:  798 ÷ 21   = 798 × 12 = . . . ).
  4. Convert back to a mixed number: 7916 = 41516.
By the way,  my wife said she never really understood the invert-and-multiply thing. She said she would change the two fractions so that they had a common denominator and then divide the top numerator by the bottom one. For example, to find 2 divided by 3 she would reason as follows:
  2  divided by 5  is the same as 1015  divided by 915 ,  which is the same as 10 divided by 9, or 10.
What is somewhat astonishing is that I found two sites that described an algorithm designed precisely to solve our very specific problem, namely, how to divide a mixed number by 2. The algorithm differed according to whether the whole part of the mixed number was even or odd.

Here is how it applied to dividing 978 by 2:
  1. Divide the whole part of the mixed number into half (ignore the remainder): 9 ÷ 2 ➞ 4. 
  2. Add the numerator and denominator of the fraction: 7 + 8 = 15.
  3. Double the denominator of the fraction: 2 x 8 = 16.
  4. The answer is 41516

Most of the web stuff mentioned above never really explained why the particular algorithm worked. And I did not encounter any post on the web that used the distributive law to divide a mixed number by a whole number, that is, no-one suggested doing what my wife and I did:

Which method did the cabinet installer use?

Answer: None of the above.

Instead, he used a self-centering tape measure. This is a tape measure that has two number lines on it. The top one in black shows standard feet and inches, and the other one directly below it in red shows the half measurements. Here is the board being measured by a self-centering tape:

It shows that the width of the board is  978  inches and that centre line of the board is at the 41516  inch mark. The point on the centre line can then be immediately marked on the board without actually doing any computations.

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As I was rewriting this post, I encountered a couple of tweets by John Golden (@mathhombre) and Denise Gaskins (@letsplaymath) that directed me to their posts* about Richard Skemp's work which seemed to be relevant to what's going on here.  (Thanks.)

Skemp observed that people lean in one of two opposite directions when they describe what it means to "understand" mathematics. He called the one way an instrumental understanding, and the other, a relational understanding. (A detailed summary can be found in the posts mentioned below.) He contended that the way you tilt affects both how you learn math and how you think it should be taught. A person with an instrumental viewpoint would tend to think of math and teach it as a collection of rules to be memorized and applied. A person with a relational viewpoint would likely think of and teach math as exploring the connections between various parts of the subject.

If I grasp Skemp correctly, the stuff from the web that I mentioned above falls very much to the instrumental side while the approach that uses the distributive law is more relational.

The carpenter’s use of a self-centering tape would also appear to reflect an instrumental view of mathematics. But not necessarily—it could simply be a tradesman using a tool that simultaneously decreases the chances of making errors and increases the speed of doing the work.

* * * * *

* The posts by John Golden and Denise Gaskins are here and here. Also, the posts about Skemp by David Wees and Gary Davis are definitely worth a read.