Teaching math to pre-service teachers (3)

Previously, I posted about teaching a class of students who, for all intents and purposes, lived in different mathematical time zones. Every new mathematical topic presented a challenge, most especially when it was first introduced. Mention a new math concept right off the bat and I would lose the weaker students. Make it so simple as to be trivial and I’d lose the stronger ones. 

I needed a way to introduce new mathematical material that appealed to all, one that was non-threatening to the weaker students, yet just perplexing enough to capture the interest of the stronger students. I wanted an opening with a mathematical floor so low that it didn't even appear to involve any math at all, but which at the same time contained the core idea(s) underlying the new material.

One tactic that worked for me was to introduce things via a good puzzle, and I gave an example in the previous post. Another tactic, which I'll write about here, was to start things off with a math-based magic trick. This was not quite as successful as using a puzzle, but it did work. 

As was the case for initiating a topic with a puzzle, before doing the trick I never announced what mathematics was involved.

Here, for example, is a trick that, although simple in the extreme, permits an approach to a topic that usually causes eyes to glaze over.  

The 1001 trick

You place three playing cards face down on the table.

You invite a volunteer to write a three-digit number on the board. She writes 732.

You ask the volunteer to write the six-digit number that is formed by repeating the three-digit number alongside itself. You may have to explain that abc becomes abcabc. 

She writes 732732 on the board.

Then you ask the class to choose one of the three cards. You turn over the indicated card and it is a Jack. We are using the cards as counting numbers so that  Ace = 1, Jack = 11, Queen = 12, and King = 13. 

You ask them to divide the six-digit number by the Jack: 

732732/11 = 66612.

You ask them to indicate another card. You turn it face up and it’s a 7.  You ask them to divide the new number by 7.

66612/7 = 9516.

Now you tell them that when they divide that number by their original number, the result will match the card that is left. So they divide 9516 by 732:

9516/732 = 13.

You turn the remaining card face up and it’s a King!


Explanation 

It doesn’t matter what three-digit number is chosen, but it does matter what three cards are used—they must be a 7, a Jack and a King.  (After doing the trick, I always told the students this fact. I would usually repeat the trick, having them select different cards from the 7, Jack, and King.)

The trick works because 

732732 = 732 x 1001,  

and   

1001 = 7 x 11 x 13. 

So, dividing 732732 first by 11, then by 7, and then by 732 amounts to this:

732732 x (1/11) x (1/7) x (1/732) 

which is really this:

732 x 7 x 11 x 13 x (1/11) x (1/13) x (1/732).

Doing the trick a few times lets the students see that the order in which they selected the cards does not make a difference. In the past, most of my students had heard the words “commutative law” and “associative law”, but many hadn’t remembered these laws or seen them in action. The trick would not work if those laws weren't true. So I used this trick as a jumping off point to discuss the basic laws of arithmetic.

(I should mention that the fact that 1001 = 7 x 11 x 13 is the basis of an interesting test for divisibility by 7, 11, or 13, so the trick had some relevance later in the course.)

 * * *

Here's another simple trick that I have used to introduce the same topic (I can't recall where I first saw it):

Your favourite number

Everyone has a favourite digit. You can’t have 8 because that’s mine. The digits that are left form the magic number 12345679. 

Ask a volunteer to tell you his/her favourite digit. 

Jan tells you 6. You tell her that her magic multiplier is 54, ask her to multiply the magic number by her magic multiplier,  and this happens:

12345679 x 54 = 666666666.

Henry tells you 4. You tell Henry his magic multiplier is 36 and this happens:

12345679 x 36 = 444444444.

Explanation

You get the magic multiplier by multiplying the favourite digit by 9.

It so happens that 111111111/9 = 12345679, so multiplying the favourite digit, n, by the magic multiplier (9 x n) does the following:

12345679 x (9 x n) 

= 111111111 x (1/9) x 9 x n 

= 111111111 x n =  nnnnnnnnn.

And the trick would not work if the associative law of multiplication did not hold.  

* * *

The cup of coffee swindle

You shuffle a deck of cards and place the top card face down on the table. You predict that card’s value will be the same as a randomly chosen number.

With your back to the board so you cannot see, a volunteer writes any three-digit number on the board. It can be any number as long as the digits are different. 

The volunteer is asked to scramble the digits to form a different number. The smaller number is then subtracted from the larger and some arithmetic is performed, all of this being done out of your sight. 

Apparently unknown to you, the computation results in the number zero. (Your back is still turned and you do not see the answer.)  The students are sure that you have made a mistake. However when the volunteer turns the prediction card face up, it is revealed to be totally accurate—its face is completely blank.

Explanation

(You can get a blank card from any magic supply shop or you can manufacture one yourself). 

Of course, the shuffle is fake—it’s done so that you retain the blank card on the top of the deck. And any three-digit (or two-digit or four-digit) number may be used as long as not all the digits are the same.

Here are the instructions that you give for the computation after the smaller number has been subtracted from the larger one: 

“Add  the digits of the result. Don’t tell me your answer. If your answer has two or more digits, add the digits of your answer together. Keep doing this until you get a single digit. Tell me when you get a single digit, but don't tell me what that digit is.

“Now, subtract that single digit from 9. Don’t tell me the answer, because that’s the number that the card is predicting.”

Here’s how the computation would proceed if the randomly chosen number was 128:

Scramble the digits:  812 

Subtract the smaller number from the bigger one:  812 - 128  =  684

Add the digits of the answer:   6 + 8 + 4 = 18

Two digits, so add the digits again:   1 + 8 = 9

Single digit, so subtract the answer from 9:   9 - 9 = 0.


That was how I sometimes introduced a module about digital roots and divisibility tests. My students all knew the tests for divisibility by 2, 5, and 3, but strangely, they hardly ever knew the test for divisibility by 9. 

In case you don’t know what a digital root is, it’s just the sum of the digits of the number repeatedly done until you end up with a single digit. If the result is 9 (or zero), the number is divisible by 9. 

This trick uses the fact that, no matter what number you start with, the subtraction always results in a number whose digital root is 9 (or zero). And after the trick, the students were usually interested in seeing why that happens and why the test for divisibility by 9 works.


I usually began my presentation of this trick by saying 

“I have won many cups of coffee with this test because it works almost all of the time, even though people want to bet me it can’t be right. But it’s not fair to swindle people so I won’t take any bets this time.” 

But that's not the reason I call it The cup of coffee swindle. The first time I used this trick, I didn’t preface it with anything, and one student bet me a cup of coffee that I was wrong. When I turned up for the next lecture, there was a cup of Timmy’s sitting on the table, but the student wasn’t there. She was embarrassed and angry, and she skipped the next three lectures. I feel bad about it to this day.