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 9⁷⁄₈ inches 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 9⁷⁄₈ .
Half of 9⁷⁄₈ is half of 9 plus half of ⁷⁄₈, which is 4¹⁄₂ inches plus 3¹⁄₂ eighths.
So measure 4¹⁄₂ inches and tack on an extra 3¹⁄₂ 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 ⁷⁄₁₆ inches beyond the 4¹⁄₂ inch mark (rather than 3¹⁄₂ eighth-inches).

There is another way which comes to mind: 9⁷⁄₈ is the same as 10 − ¹⁄₈,  so half of 9⁷⁄₈ is the same as half of 10 minus half of ¹⁄₈,  which is 5 − ¹⁄₁₆ ,  in which case you would probably find the midpoint by locating the 5 inch mark and backing up ¹⁄₁₆ 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 9⁷⁄₈ by 2 should be done as follows:

1. Convert the mixed number to an improper fraction: 9⁷⁄₈ = ⁷⁹⁄₈.
2. Convert the 2 to an improper fraction: 2 = ²⁄₁.
3. Do the division  ⁷⁹⁄₈ ÷ ²⁄₁. (by which they meant:  ⁷⁹⁄₈ ÷ ²⁄₁   = ⁷⁹⁄₈ × ¹⁄₂ = . . . ).
4. Convert back to a mixed number: ⁷⁹⁄₁₆ = 4¹⁵⁄₁₆.

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 ²⁄₃  divided by ³⁄₅  she would reason as follows:

  ²⁄₃   divided by ³ ⁄₅  is the same as ¹⁰⁄₁₅  divided by ⁹⁄₁₅ ,  which is the same as 10 divided by 9, or ¹⁰⁄₉ .

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 9⁷⁄₈ 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 4¹⁵⁄₁₆

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  9⁷⁄₈  inches and that centre line of the board is at the 4¹⁵⁄₁₆  inch mark. The point on the centre line can then be immediately marked on the board without actually doing any computations.

* * * * *

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. 

Apples and oranges and the number line

You can add apples to apples and oranges to oranges, but you cannot add apples to oranges.

One of my teachers used this old chestnut to explain that we had to convert to the same units before adding similar quantities. It doesn’t make sense to add 2 and 6 to find the combined volume of 2 quarts and 6 gallons — before adding, you have to convert everything to quarts, or everything to gallons (or, perhaps, everything to litres).

Apple + apples = ?

Of course there are circumstances where it makes perfect sense to add apples to apples, but there are also a lot of situations where it doesn't.

What do you get when you add my PIN and my wife’s PIN? Or my brother's telephone number and my sister's telephone number,  or the grocery product numbers for tomatoes and cucumbers? It seldom makes sense to add numbers that are used as identification labels.

But even when the numbers are more than just ID numbers, it may still not make much sense to add them. Would you perform the following additions?

• Hours to travel by bus + hours to travel by car from Edmonton to Regina.
• Average pay in Alberta + average pay in Ontario.
• Johnnie's  Algebra grade + Tina's Algebra grade. 
• Life expectancy for a person born in 1990 + life expectancy for a person born in 2015.
• Current Montreal time + current Vancouver time. 
• Cruising altitude for flight 107 + cruising altitude for flight 108. 
• Sunrise time + sunset time for today in Saskatoon. 
• Number of voters in the 2011 election + number of voters in 2015 election. 

Apples − apples = oranges ?

The odd thing is that in each of the above cases, although it makes little sense to add the numbers, it does make sense to subtract them. Here's another example.

I’m at the corner of 105 avenue and 34 street. If I walk along the street from 105 avenue to 112 avenue, how far will I have to walk?  

You would probably answer "seven blocks." Here you are subtracting an avenue number from an avenue number and getting a distance expressed in blocks.

Even when the result of a subtraction uses the same language to express the units of the answer, the units are sometimes measuring something different. Consider the following:

The temperature in Calgary at 10 AM today.
The temperature in Calgary at 4 PM today.
The difference between these.

Although the three measurements are expressed as degrees Celsius, the first two have a different meaning than the last.

Apples + oranges = apples ?

And there are case where you can add apples to oranges. (There’s the useful observation that adding 2 apples to 3 oranges makes sense because it gives 5 things, but I’m not thinking of that.)

Dinner Time 

Q: What time is it?
A: 5 O'clock
Q: How long will it take to cook the roast?
A: 2 hours. 

Did you just add hours to O'clocks?

Number Lines

I imagine that most people, when asked what 5 + 2 means, conjure up what I would call an aggregate model, like a collection of 5 tokens together with a collection of 2 tokens.

We tend to think of numbers as describing an aggregate or assemblage — the number of marbles in a box, or the total weight of your luggage.

In the Dinner Time example, above, the numbers are not being used in this way. Instead, one number describes a position (time of day) and the other describes a movement (a duration of hours). If you were asked to explain to a child why 5 O'clock + 2 hours is 7 O'clock, perhaps you might draw something like one of these:

These are variations of the familiar number line¹, where numbers are depicted in the two different ways that numbers are often used:

1. as a position on the line, with positive numbers to the right of zero, negative numbers to the left, and
2. as a directed distance or movement (a vector), drawn as an arrow pointing left or right. The length of the arrow represents the magnitude of the number and the direction indicates the sign of the number (right = positive, left = negative ).

Addition

On the number line, addition is thought of as

current (or old) position + movement = new position,

as, for example, in

current bank balance + deposit = new balance.

With a number line, addition of 5 + 2 and 3 + (−7) is represented as:

For addition, the movement arrow is drawn with its tail at the old position. Its head indicates the new position.

Subtraction

By far the most common application of subtraction is to describe the movement that separates one position from another:

new position − old position = movement,

as in

Best before date − current date = days left before milk goes sour.

With a number line, here's one way to represent 2 − (−4) and 8 − 5:

As with addition, the arrow is drawn with its tail at the old position and its head at the new position.


Children typically learn subtraction as taking-away, which is demonstrated with an aggregate model by removing a certain number of tokens from a larger group and counting what's left. In fact, so strong is this interpretation of subtraction that the language often persists throughout the rest of one's life.

The take-away model of subtraction, can be described as

old position − movement = new position.

It is also possible to use a number line to model this interpretation of subtraction. Here's what 8 − 5 would look like (8 = old position, 5 = movement):

In this case, subtracting means to travelling backwards along the movement arrow to reach a new position.

The arrow is drawn with its head at the old position while its tail indicates the new position.

This is almost tragically different from the previous illustration for 8 − 5. Apart from my own kids, I've never taught arithmetic to children, but I cannot help but think that the difference will confuse a student. Moreover, with the position-minus-movement interpretation, the overwhelming temptation is to reverse the direction of the movement arrow, thereby conflating 8 − 5 and 8 + (−5), which may lead to further confusion.

So what?

Most people work with numbers in an abstract way: 8 − 5  has no meaning unless the 8 and 5 refer to something "real," yet we have no hesitation in saying that 8 − 5 is 3. We do this without reference to any "real" setting whatsoever, and we do not revert to using a model to do the computation. We have completely abstracted the operation of subtraction from real life.

However, it seems that we need to give meanings to numbers in order to learn even the most basic arithmetic facts. But when we do assign meanings to numbers, we encounter the strangeness described in this post, namely:

• Two numbers can measure exactly the same thing but it may be nonsensical to add them. 
• Two numbers can describe quite different concepts, and yet it may make sense to add or subtract them.  
• Two numbers that don’t make sense when added can make perfect sense when subtracted.

I can't recall being taught about this strangeness when I was a student, and I don't know if it is explicitly dealt with by today's teachers, but it is conceivable that it could be a source of confusion.

To some extent, the number line model deals with this strangeness, but I haven't seen it used to this end.

*****

¹ Some number line models use only arrows. Positions are replaced by arrows whose tails are at zero.

Marilyn Burns and the number 11.

Recently I read an interesting post by Marilyn Burns about a pleasant discovery she made. It niggled at my mind (in a good way). It was not the discovery itself that struck me, but rather the reservations she had about how she reported it. This is what she said:

What I’ve done is given an explanation that falls into the category of "teaching by telling," which I avoid in the classroom when I want students to "uncover" knowledge that’s based in understanding relationships.

She had written a book about the number 11 for her grandson, and in it she had mentioned that 11 could be expressed both as 6 + 5 and as 6² – 5².

Marilyn's Problem 

Sometime after completing the book, she became curious about whether the same thing worked for other pairs of numbers that differ by one, and she found that indeed it did: for example, 

7 + 6 = 13  and   7² – 6² = 13.

He question was about how to generalize this. She explained how she did it both graphically and algebraically, but she was uneasy about the "teaching by telling" approach that she used. This lingered in my mind for some days because it reminded me that some of us (math educators) have not yet sorted out the relationship between "exploring the math" and "direct instruction," or, if you wish, between "learning mathematical reasons" and "applying the math trick."

Moreover, her two approaches (plus another that I have added) reflect the increasing distance between math via “discovery” and math via “direct instruction.” I think of the former as not requiring an extensive math background, and the latter as depending on already acquired knowledge. Perhaps I am being naive about this, but let me explain:

Her graphical approach, via (1) above, is related to what I think of as being "discovery based." You don’t need a whole bunch of content knowledge to understand it. The 11 blue squares visually illustrate the difference in areas between a 6 x 6 square and a 5 x 5 square, and the process can be extended to show the difference between 7² and  6² and so on.

Her approach, via (2) above, is a very natural transposition of the problem to the algebraic domain: compare the sum of the two numbers with the difference of the squares of the two numbers.

A third approach, via (3) above, is also a transposition to the algebraic domain. It is clean, and it leads very quickly to a solution (because, for consecutive numbers a - b = 1),  But, to me, it is a lot less natural than (2), for it depends upon having memorized a particular algebraic fact and having it always at the ready. For this particular problem, using (3) covers up the thinking about the math involved. It verges on being a mathematical trick.

Or does it? Do I really believe that it is a trick?

Using (3) certainly bolsters the argument that content knowledge helps you do mathematics. No mathematician would ever deny that content knowledge is important. An awful lot of mathematics takes knowledge from one area and applies it elsewhere, and one hopes students learn how to do this. 

I’m not sure how students can acquire content knowledge and learn to apply it without some teaching by telling. Yet, "telling" may lead to mimicry rather than understanding. (That may explain why teaching problem solving is difficult! And that’s maybe why we get a zing when we do solve a problem.)

Now, I think (or rather, I hope) fewer and fewer people are still invested in the total primacy of teaching content over everything else. An episode from my past suggests that content knowledge alone is insufficient.

Jim's Problem

In high school, my friend Jim would occasionally bug me with puzzles that exposed my poor abilities with mental arithmetic. He was fond of asking me things like 

Without using a pencil and paper, what’s 48 squared?  or  What’s 61 squared? 

(BTW, I existed as an entity long before calculators did, so that’s the pencil and paper reference.) 

But before I could even get started he would tell me the answer. 

48 squared is 2304.   61 squared is 3721.

How could he get it so fast? When pressed, he explained how he did it:

48 squared has got to be close to 50 times 46.  (I just added and subtracted 2.) The algebra goes like this:
50 × 46 = (48 + 2)(48 - 2) = 48²  -  4.
And 50 × 46 is easily seen to be 2300, so 48 squared is  2304.

Somehow, I didn’t see the trick until he showed it to me, even though by that time in my life factoring a² – b² had become an automatic reflex. If you're reading this, I’ll bet it’s an automatic reflex for you as well.

I find it interesting that both Marilyn’s problem and Jim’s problem can be resolved use exactly the same content knowledge. For Marilyn’s problem, that knowledge led me immediately to an answer, yet for Jim’s problem I did not make the connection.

Are you like me, or did you immediately see the connection with Jim’s problem? 

Rust remover

When you are trying to solve a problem, sometimes your worldview causes you to unintentionally import prejudices and assumptions that block you from the solution. 

To illustrate, here is an updated version of a puzzle that completely baffled me when I was a kid. In this day and age it won’t fool very many people, and in fact, many people would not see a problem at all.

A father and his son were in a car accident and were taken by ambulance to the hospital. The father was injured, but not seriously, and he was sent to the waiting room. However, the son needed surgery. After he was prepped for the operation, the surgeon came in, but said “I can't operate on him — he's my son!” How is this possible?

When I was young, I subconsciously pictured a doctor as being a man, and this prevented me from seeing the solution, namely that the surgeon was the child’s mother. (Of course, nowadays we recognize that there is a second solution: the surgeon was the son’s other father.)

While I was still teaching I often began one particular course with a small collection of puzzles like this. I used to call them Rust Removers because the students’ thinking always seemed a bit rusty after returning from their summer or Christmas break.

Here’s a dozen puzzles that were carefully posed to entice you into following your preconceptions or somehow cause you to make unwarranted assumptions. You may be able to solve most of them quickly, but very few of my students were able to solve all of them in one sitting. 

If you are absolutely desperate for an answer, a pdf file of the solutions is available here.

1. One night John was reading an exciting book when a power failure threw the room into complete darkness. Nevertheless, he continued reading without a pause. He was not using a laptop, tablet, or e-reader, so how could he do that?

2. I live in a modern high-rise apartment. A lady friend who regularly visits me always gets off the elevator five floors below mine and takes the stairs to my floor. Why?

3. Last night I turned off the light in my bedroom and managed to get into bed before the room was dark. My bed is 3 metres from the light switch. How did I do it?

4. A woman walked up to a counter and handed a book to the cashier. He looked at it and said “Ten dollars.” She paid the man and walked out without the book.  He saw her leave without it but did not call her back.  How come?

5. A man is found shot to death in the front seat of his car. A gun lies out of his reach in the back seat.  All the windows are closed and the doors are locked; there are no bullet holes anywhere in the car. How could this have happened?

6. An escaped prisoner was running along a forest road when he saw a cop car heading towards him. He sprinted into the woods, but before doing this he ran ten metres directly towards the approaching car. Why?

7. A woman had two sons who were born on the same hour of the same day of the same year. But they were not twins, and they were not adopted. How could this be so?

8. Jim Johnson lives in a four story apartment building. He is plant supervisor at an auto factory that is within walking distance of his apartment. Every morning at 0800 h, he walks down a flight of stairs, and when he arrives at his destination he settles back with a cup of Tim Hortons coffee that he purchased along the way and begins to read the newspaper. Halfway through the news his eyelids close and he falls asleep for several hours. Nevertheless, at the end of the month he looks forward to a nice pay raise. How does he get away with this?

9. Take 3 empty paper coffee cups, and put eleven coins in them so that each cup holds an odd number of coins. All the coins must be used. Once you have solved this, put 10 coins in the same cups so that again each cup holds an odd number of coins and all coins are used. (Remember, zero is an even number.) 

The following puzzle is now found throughout the web. I first saw it about 20 years ago. 

10. There is a light in a storage room on the second floor of a building. On the ground floor are three light switches, exactly one of which controls the storage room light, which is a standard incandescent 100 watt bulb. By turning some or all of the switches on or off, it is possible to determine which switch controls the light by making one trip to the storage room. How can this be done?

Outside the storage room, there is no way to determine whether the light is on or off, and disassembling the light switches will not reveal which one controls the light. You are not allowed to be helped by anyone.

11. Two grade six classes were going on a field trip to a museum. They were being transported by two buses each of which had 34 seats.  It so happened that there were 30 boys and 34 girls, and so they put all the boys on one bus and the girls on the other bus. The buses had to stop for a few minutes, and during that time 10 boys snuck onto the girls' bus. But the girls' bus driver noticed that there were too many on the bus, so he sent 10 children (boys and girls) back to the boys' bus. After this was done, were there more boys on the girls' bus than girls on the boys' bus? Or vice versa?

Another older puzzle that has also found its way onto the web:

12. Four people are being pursued by a menacing beast. It is nighttime, and they need to cross a bridge to reach safety. It is pitch black, and only two can cross at once. They need to carry a lamp to light their way. 

Mr. One takes a minimum of 1 minute to cross. Mr Two takes 2 minutes, Mr. Five takes 5 minutes, and Mr. Ten takes 10 minutes. 

If two cross together, the couple is only as fast as the slowest person. For example, if Mr. Ten and Mr. One cross the bridge together, it will take them 10 minutes. A fast person can't carry a slower person to save time. The person or couple crossing the bridge needs the lamp for the entire crossing, and the lamp must be carried back and forth across the bridge (no throwing, etc.). 

 If they don't all get completely across in strictly less than 19 minutes, who ever is on the bridge or left behind will be eaten by the beast. Is it possible for all of them to get across?

These puzzles are not mine. Some of them came from Martin Gardner’s book “aha! Insight!”  The book has a lot more than just these puzzles, and it can be read in selected short chunks. 

If you like short puzzles that challenge students to overcome fixations in their thinking, you should visit the WODB site. There you will find a collection of puzzles inspired by Christopher Danielson and curated by Mary Bourassa. Each puzzle presents four different things which are such that every three of them have at least one thing in common that is not shared by the fourth one. (WODB = Which One Doesn’t Belong.) 

Magic and Arithmetic Series

Now your high schoolers have learned the two formulas for the sum of an AP. (Do they still call it an Arithmetic Progression?)  So you ask them this:

*   *   *   *   *

Arithmetic sequences and series. I cannot think of a more mind-numbing introduction to them than the way it was done a century ago when I was in school. And a quick googling suggests that the situation may not have improved very much — what I see often begins with a caveat that “You won’t actually need this until you take Calculus.”  Hard on the heels of this are the definitions of the first and last terms, the common difference, and so on. Then comes the formula for the general term, and finally the iconic derivation of the formula for sum of an arithmetic progression.  

Like many math teachers, I also used to tell my students the story of the clever young Gauss. It probably firmed up their belief that you have to be born with a math brain in order to do math.  Raise a glass to Kate Nowak for what she did to introduce AP's. In fact it is her post that prodded me to write this.

*

If I were able to tardis back a few years, I would probably begin with this: 

Give me the sum of three consecutive numbers.

If the students were to tell me that the sum is 72, I would tell them immediately that the three numbers are 23, 24, and 25. 

And if then I might offer this:

Give me the sum of three consecutive even numbers.

If the sum is 84, I would tell them that the three numbers are 26, 28, and 30.

Perhaps even this:

Give me the sum of five consecutive numbers.

For example, if the sum is 45, I would tell them immediately that the five numbers are 7, 8, 9, 10, and 11.

*

The secret to this is that whenever you have an arithmetic series with an odd number, n, of terms, the sum is always n times the middle term.

It is easy to convince kids that this is the case for the simple cases given above. For example, three consecutive numbers with a middle term m must always be of the form

m - 1,     m ,      m + 1.

Adding, we get 3m. To perform the trick, divide the sum by 3, and subtract 1 to get the first number.

For three consecutive even numbers, the situation is pretty much the same. The three numbers would be 

m - 2,     m,     m + 2,

Adding, the sum is again 3m.

In case you are wondering about doing the trick when you are given the sum of four consecutive numbers: dividing that sum by 2 gives the sum of the middle pair of numbers from which you can easily deduce what the four numbers are.

For example, if the sum is 50, then the sum of the middle pair is 25, so the middle pair is 12 and 13, from which we get the four numbers 11, 12, 13, 14. 

The sum of an arithmetic series with an even number, n, of terms is always n/2 times the sum of the middle pair. Interestingly, the sum of the middle pair is also the sum of the first term and last term. 

You can pursue this far enough to derive the two formulas for the sum of an arithmetic progression, but I’m not sure that I would push it that far.

Instead I would switch the question around to finding the sum of a longer list of consecutive numbers, à la Kate Nowak.

*

The possibility of introducing arithmetic sequences and series in this manner grew out of a simple trick from William Simon’s book Mathematical Magic.  

A calendar trick

Ask a student to draw a rectangle around three consecutive dates on a calendar month, like so:

This is to be done so that you, the teacher, cannot see what dates have been encircled (for example, have the student stand behind your back while you are facing the rest of the class). Ask the volunteer to show the circled dates to the rest of the students (but not to you) and to tell you the sum of the dates. You can immediately announce the dates that have been circled. 

[A personal aside: I am notoriously poor at mental arithmetic — a brief description of my troubles is contained here. Using a calendar appeals to me because it forces the three numbers to be small, thus avoiding the floundering that would occur if some cheeky person asks me “What are the three numbers if the sum is 14691?”] 

After explaining how to do the three-in-a-row trick,  the calendar itself might prompt students to ask questions like this:

How would you do it if we gave you the sum of four consecutive dates?

How would you do it if we gave you the sum of three dates in a vertical line? 

What if we gave you the sum of five consecutive dates? 

The second and third questions the students could answer themselves.

When there are four consecutive dates, there is no middle date and the sum of the four dates is not divisible by 4. I imagine this might be a stumbling block for some students. But, fingers crossed, at least some of them will actually look at what happens when they do divide by 4, and thereby open up other avenues to explore. 

They might, for example, note that dividing by 4 gives the number that is the average of the middle pair of dates.  For the four dates circled above, when I divide the sum by 4, I get 5.5, which is the average of the two middle dates 5 and 6.  And not only that, 5.5 is also the average of the first and last dates.

That is: 

Which generalizes to 

But as I said earlier, I probably wouldn't push it this far.

*   *   *   *   *

What is the sum of the following arithmetic progression? Each square represents a term in the progression. 

What about this one? 

Or this one? 

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.

Teaching math to pre-service teachers (2)

In my previous post, I described a fundamental incompatibility between the students in Math 160: many students had consciously avoided math for most of their lives, while others were very confident in their abilities and even enjoyed the subject.

I ended the post with this paragraph:

When I began teaching Math 160, I had very specific ideas about how I should present the course. To me, the key to successful instruction was to concentrate on the clarity and explicitness of my lectures. It turns out that I was singing from the wrong songbook

I am reminded of that old but useful proverb: beauty is in the eye of the beholder. Or, in this case, ‘clear and explicit’ lives solely in the mind of the student.  For my Math 160 students, who differed so much from each other, there could be no interpretation of  ‘clear and explicit’ that would be suitable for all of them. Being ‘clear and explicit’ was not the way to engage them.

If you have tried to motivate students about mathematics, you know that the sticking point occurs at the very moment that you begin to talk about it. If you blow that moment, you will lose them for the duration.

So, after a period of time (too long a period), I shifted my attention towards what I could do to introduce new topics in an interesting way. 

For Math 160, there were certain constraints. I needed to make the introduction understandable without it depending upon prior knowledge. In particular, I wanted inexperienced students to be able to comprehend what was happening without a preliminary lecture or review. At the same time, I did not want the introduction to be obvious or trite to those who did have some previous knowledge. And one of my key tenets was that new terminology must be avoided so that students are not intimidated or distracted by strange jargon. And above all, it has to be interesting. 

Here’s how I usually introduced a certain challenging mathematical topic to my Math 160 students. It was my way of saying “Math 160, say hello to modular arithmetic.”

The Keystone Kidnapper

The police of Keystonia have a crack swat team. The team’s current mission is to rescue the prime minister's daughter who is being held hostage by an evil-doing kidnapper. But, through a sequence of mishaps, the swat team has allowed the kidnapper to capture them. He has herded them, along with the young lady, into a room containing seven caskets.

“Your puny intellect amuses me,” said the kidnapper. “You have one hour to figure out how to escape from this room. Sixty minutes after I leave, six of the seven caskets will disintegrate releasing one hundred angry venomous snakes. The other casket contains the key to the door. Find the key before the hour is up and you can escape. I'll give you a clue—the key is in casket number 54321. And I've done you a favour. The caskets are not locked.”

“But,” said the swat team leader, “The caskets are numbered from 1 to 7. None of them is numbered 54321.”

“I’ll give you another clue: start counting,”  and the kidnapper showed them how. 

Casket 1 was 1, casket 2 was 2, and so on until casket 7 which was 7. Then the kidnapper reversed directions: casket 6 was number 8, casket 5 was number 9, and so on until casket 1 which was number 13. Then the count reversed once more, and casket 2 was 14, casket 3 was 15. 

“You get the idea,” said the kidnapper, “Goodbye,” and he left them in the locked room.

“Well,” said the swat team leader, “Let's start counting.”

“Hold it,” said the young lady. “We have less than 60 minutes—that's 3600 seconds, and my calculator says that 54321 divided by 3600 is about 15. We would have to count 15 caskets per second and not make a mistake.” 

Luckily for the swat team, the prime minister's daughter figured it out. 

Which casket contained the key?

With its corny dialog and silly scenario, there is nothing real world about this puzzle, and there is no pretence that there is. But, as long as humans have walked the earth, people have willingly immersed themselves in tales set in unreal worlds. Using story puzzles to introduce new mathematics leans on this. It's a tactic that I used a lot, and for the most part my students let themselves be drawn in.

Spoiler Alert! If you want to solve this on your own, don't read below this line.

Here’s a short summary of how the lesson typically developed and how it fit the constraints that I mentioned earlier.

To set thing up, and to illustrate the counting process, I drew a diagram:

Usually the students asked me to write down a few more rows to clarify the counting process, and if they didn't, I wrote them down anyway, like this:

So far, all students were drawn in, and all had a clear grasp of the problem. And there was no alarming “mathematics” required. And the stronger students showed no sign of leaping to the solution. 

There was often some dead air time as they thought about the situation, and sometimes they had to be prompted with questions about what they noticed. Eventually, they started to see patterns. 

For example, some them noticed that in the leftmost and rightmost columns, successive numbers differed by 12. Or they noticed that in the second column the differences alternated between 10 and 2, and that in the third column the differences alternated between 8 and 4,  etc. 

The observations varied from class to class, and did not always occur in the same order. Sooner or later someone usually picked up on the ubiquity of the number 12, and said something like

“In all of the columns, the numbers increase by 12 for every two rows.”

With more prodding, they described what was happening by saying things like

“In the second column, all of the numbers are divisible by 12 or else their remainder is 2.”

Then, typically, someone will follow this up with:

“In the third column, when you divide by 12 the remainder is always 11 or 3.”

The key word remainder inevitably came up, but I avoided using the term until the students brought it into the conversation themselves. I sometimes had to find a way to push them towards thinking about remainders (by writing a few more counting rows and asking if they had any questions about any of the columns). I definitely never mentioned the words modulus or congruence or residue classes, even though they were grappling with these notions. They really didn't need them at this point, and there would be ample time to define them later.

As soon as someone mentioned remainders, they noticed the same sort of thing happening in other columns, which led eventually to making a chart of them:

Occasionally someone would express worries that this was leading nowhere. It is a difficult comment to handle. But, usually by this point in the course they will have learned that sometimes you need to go further down a road before you can decide where it leads. I would respond by saying something noncommittal like “Hmm.”

They noted patterns in the chart of denominators, for example, that although the remainder 10 appears repeatedly in the fourth column, it never appears in any other column. They usually also pointed out that the chart contained every remainder that you can possibly get when dividing by 12. (This sometimes initiated a conversation about whether zero should be considered to be a remainder.)

And finally a flash of understanding happened and they realized that they could identify the casket that contained the key by determining what column contained the remainder when 54321 was divided by 12. 

As I said, this is a summary, and it would be misleading to say that all classes followed the same script.  There were often lengthy pauses. There were unexpected observations that sometimes took us off-track. (As happened, for example, when someone noted that in many columns the sum of two consecutive remainders is always 14.) I tended to run with such diversions and let them play themselves out. Usually, someone interrupted and suggested something else that got us back on track.

In spite of the occasional detours, the classes always solved the puzzle well before the lecture period ended.  And I always pointed out that they found the casket well within the hour set by the kidnapper, which seemed to please them. 

* * *

The Keystone Kidnapper puzzle satisfied the constraints that I mentioned. The weaker students did not need either a preliminary lecture or new terminology to understand the puzzle. The quirkiness of the setting was captivating enough to conceal the solution from even the stronger students. And even if they knew something about clock arithmetic, they did not see the connection with the puzzle. 

The fact that the students at all levels worked through the puzzle together provided a common frame of reference. It helped overcome the non-uniformity of their backgrounds and provided a common base to build on.  Afterwards, when the new mathematical concepts of modulus, congruence, and residue classes were formally introduced, all the students could relate them to the same concrete setting. 

* * *

I don’t know the original name for The Keystone Kidnapper puzzle. I first saw it many years ago in a book by Martin Gardner I think, where fingers were being counted instead of caskets. (Several solutions for the finger counting version can be found on the internet. Almost all of the good puzzles have solutions somewhere on the web. If you use them, you might wish to alter the setting as well as the puzzle name so that students won’t be able to easily google them.)

As well as using puzzles to help introduce new topics in my courses, I occasionally used math based magic tricks. In the next post I will show how I used a couple of them, along with some pitfalls that have to be avoided.