A lattice for multiplication

When I was in Year 5, our teacher showed us an interesting way of multiplying. The image at left shows the lattice or gelosi method for mutipying 35 x 76. You put the numbers to multiply along the top and the right sides, with sufficient rows and columns for each digit. Each box is divided diagonally. You multiply (in any order) and the result goes in the appropriate box, with the tens above the diagonal and the units below. Once all the multiplications are done, you add along the diagonals (the red arrows in my pic), starting from the bottom right corner, working to the left and carrying where necessary.

As a method for long multiplication, it works very nicely, and is probably a little simpler to learn than the more usual columnar approach. It is certainly worth showing to students - some will no doubt prefer the standard columnar approach, but others will like the lattice (at least until they get calculators).

But the other day I found another use for the lattice. I was introducing a Year 10 class to multiplying polynomials. I used a lattice (minus the drawn diagonals) as shown at left. We collected the like terms (which conveniently lie diagonally). My students found this approach very accessible. While I did not suggest it on the day, an obvious way to speed up this approach is to omit the variables and work with only the coefficients.

Then I happened to look at a blog I've been occasionally reading, and the lattice was mentioned there as well. But in this case, it was about someone arguing against using the lattice method in the classroom. Say what? Sadly, it is true - there are those who would have teachers confine themselves to teaching only one way (in this case, the standard columnar approach), and who disparage any other approach and denigrate anyone who would teach any other approach.

I can't even begin to fathom the mental bankruptcy necessary to say to someone else "there is only one way that students should do it". For me, part of the beauty of Mathematics is that there is always more than one way. Some ways will be faster or more elegant. Some ways will be slow wandering paths, but you may learn other things along the way.

The person who insists that students do things only one way is no educator.

Magic squares? Mind-reading? High school maths, and nothing more.

I've come across an interesting site (which was advertised on another site I found through StumbleUpon), featuring a supposedly mind-reading magic square.

This site gets the user to choose a two digit number at random, and then subtract from this number the values of its digits, e.g. if you chose 87, you would then subtract 8 and 7 from 87 to get 72. You then look at a grid showing numbers and symbols, and find the symbol corresponding to your result.

The site tells you it will guess your symbol correctly three times, then if you pay $2 via PayPal, you can get five more guesses and a presentation about the "Secret to Life". And you can pay $20 to have the secret of the magic square explained to you.

Of course, there is no magic, and no mind-reading, and a bit of elementary maths can quickly reveal what's going on. This is actually a good exercise for both Mathematics and Computer Studies classes - construct a spreadsheet or computer program that shows all the numbers from 10 to 99 and the result after subtracting the number's digits from itself. Once you've done that, the trick is obvious.

What about the "Secret to Life"? I didn't pay to find out, but I suspect it might be that a fool and his money are soon parted.

Get me the poll position, stat!

The stirring stuff for the media this weekend, especially the News Ltd. outlets, was the Galaxy poll that shows the Government slipping behind again. The key points for the talking heads and their opinion-column analogues in the papers were the poll results showing most Australians were skeptical of the Government's motives for the current intervention in the NT, and the fact that voters were largely unconcerned about the influence of the Unions on the ALP.

The media love to make a big deal of polls. This would appear to be consistent with Ehrlich’s Law: “People pay way too much attention to things that are easily quantified.” But does this poll really tell us much?

For one thing, the sample was only 1000 people out of the 10 million in this country who can vote. I have some doubts about how much we can assume from so small a sample (~0.01%). (I'd love to see the pollsters publish the details about their sampling methods and confidence levels.)

Apart from the polling itself, the way that the results are presented in the media is worth a closer look. Let's consider the data about how the bad behaviour of union officials is likely to affect support for the ALP. On various TV news programs, the data was reported in a very simplistic manner: 46% of respondents indicated that the union officials' behaviour would not influence their vote, 43% said it would, 11% unsure. On two of those TV programs, this was cast as a bad result for the Government. Really? The 46% whose votes would not change are hardly the voters we should be looking at - they represent the voters who are already polarised on their choice of party. It's the other two categories that are of interest. 43% said it would influence their vote - but in which way? If a large chunk of the 43% are voters who might vote for the ALP but who may not because of the union officials, this is good news for the government, not bad. But of course, no breakdown of the data was given.

A column in the Herald Sun gives a different set of figures (which don't seem to entirely line up with what was appearing on the news programs mentioned above), specifically:

...20 per cent of voters said they were now less inclined to vote Labor because of [union official Joe McDonald's] actions, 9 per cent said they were more inclined to vote for the party, and 67 per cent said they would not be influenced either way...

and the overall interpretation in that column was "voters also said they were largely unconcerned by the bullying actions of union officials".

Again, the columnist is looking at the largest number in the figures and drawing a conclusion from that, but the interesting numbers are the small ones - 20% of the respondents indicated that they would be less inclined to vote Labor. What proportion of those respondents would otherwise vote Labor? How significant is their change in resolve? Could this represent an increase in the proportion of swinging voters who have previously come down on the left side of the political spectrum but who may now be leaning to the right? Again, there is no breakdown and thus no way to answer these questions.

What this does illustrate is the generally poor way that polls and statistical measures are handled by the media. They are interpreted in the most simplistic terms (and some columnists are clearly happy to interpret them in line with their own political leanings). What are the viewing public to make of these statistics and the interpretations doled out by the media?

Mark Twain quipped about "lies, damn lies and statistics", a response to the spurious use of surveys and statistics by the politicians of his day. If Twain were alive today, he'd no doubt use the same line, the difference being that now newspaper columnists would figure more prominently in the list of perpetrators.

But given that news has become like takeaway food, with an emphasis on speedy delivery of something palatable without too much attention to quality, should we be surprised?

Going Back

Reversing the digits of a number is a time-honoured exercise for student programmers. It gets the student to think about integer division and modulo operations, and about iterative processes.

An exercise I've often give to my students is to write a recursive procedure for reversing an integer's digits, even though I know that it can be done more simply as a loop. This is a good way to have the students think about parameter passing and variable scope. But up till now, I've only had students doing this in Pascal.

So when I went to do it in Python, I found myself momentarily stopped by the fact that Python does not have an exact equivalent to Pascal's variable parameters. So how do I get a recursive procedure for integer reversal in Python?

I came up with two solutions.

Solution 1:

revers(n):
global q #tell Python to use the global variable, not a local one
if n > 0:
  q=q*10+n%10
  revers(n/10)
return q

q = 0
x=int(raw_input("Enter a number: "))
print revers(x)

Here, I simply made the function calculate the reversed integer in a global variable. Now, when I was learning Pascal programming, all my reference works said the same thing: don't change a global variable inside a function/procedure. Since old habits die hard, I found myself looking for a different solution to the one above, one that didn't require the global variable inside the function.

Solution 2:

revers(n,q=[0]):
if n > 0:
   q[0]=q[0]*10+n%10
   revers(n/10)
return q[0]

x=int(raw_input("Enter a number: "))
print revers(x)

This solution uses the fact that arguments to Python functions are object references, so passing a mutable object (a list being an obvious example) allows you to progressively alter the value in that object with each subsequent recursive call to the function.

Python also allows you to declare a default value for an argument and thereby omit it from the function call. The default value is evaluated only once, so setting using q=[0] as an argument only affects the first call to the function, and subsequent recursive calls do not reset the list.

As a solution, I prefer this (since it doesn't violate the rules about variable scope I learned so long ago), but I wonder how well my students will understand how the mutability of an object relates to the scope of a recursive function.

Getting Keith Backwards

I stumbled over an interesting number sequence the other day - Keith numbers (named for mathematician Mike Keith) are found as follows: starting with a number, separate the digits, then add them to form another number; continuing to sum the last n numbers in the sequence (where n is the number of digits in the original number), if the original number appears in the sequence, it is a Keith number.

For example, starting with 14, we get 1+4 = 5, 4+5=9, etc. to form the sequence 1, 4, 5, 9, 14 - since the original (14) appears in the sequence, 14 is a Keith number.

It turns out that Keith numbers are very rare (Wikipedia's article states that there are only 71 Keith numbers below 10¹⁹).

I wondered about similar number sequences, in particular numbers that produce the original number with its digits reversed. Wolfram's MathWorld site has an article on Keith numbers that also mentions these Reversed Keith numbers, but prefers the ugly term revrepfigits (and also calls Keith numbers repfigits, from 'repetitive fibonnaci-like digits').

Before I read the Wolfram MathWorld article, I had written a small Python program to find Reverse-Keith numbers (which I had decided should be called Feek numbers), and generated all the numbers up to 10 000 000.

This turns out to be a nice little programming exercise, requiring one to think about how to manage both integer reversal and splitting an integer into its separate digits prior to building the resulting number sequence. If someone wants an interesting exercise for their students, I think this is worth a look.