How did you learn your times tables?

When I was a kid, I liked playing ice hockey. I was actually not very good at it — no Connor McDavid here!  But I did acquire the basic skills. For example, I figured out how to lift the puck. (For you non-hockey players, that means shooting the puck in such a way that it flies off the ice into the air. It’s a essential skill if you want to be able to score goals.)

I practiced that skill a lot. Whatever I did, it worked. I could lift the puck consistently without thinking about it. I haven’t shot a puck for many decades, but whenever I imagine doing so, I swear can feel the memory in my triceps.

Of course, I really did not "figure out" how to lift the puck. I did not know the theory behind the lifting action. And to the extent that the skill was necessary, I didn’t need to understand the theory.

The lesson is this:

When learning something new that you will need for later use, master the mechanics first. You can learn why it works later.

* * * Warning: possible straw man ahead * * * 

It’s a useful lesson.  It helps me understand the approach to mathematics teaching advocated by the back-to-basics people: You can be successful by learning the how without understanding the why. Just learn the essential basic facts and algorithms. Don’t worry about why the puck flies into the air — just practice shooting enough so that you can lift it consistently and effortlessly.

Reasonable advice? Maybe. But, no matter how hard I practiced, I could not always "lift" the multiplication tables. As far as the basic multiplication facts are concerned, I do not have what some people call rote recall — I do not have the ability to rapidly and effortlessly retrieve all of the basic learned facts from memory.

A great chunk of my own elementary math education was founded on the contrary belief, that rote recall is, in fact, achievable by everyone — that all it takes is practice.  Accordingly, my classmates and I were regularly drilled and tested on the multiplication tables. I did not do well, and I argued with my teachers. Ultimately, I was punished for my inability to memorize the 12 x tables.

I don’t believe that my recall difficulties are exceptional. The more blogs I read, the more I suspect that there are many people who, no matter how much they practice, will never possess rote recall of the basic arithmetic facts. In that sense, those people can never know the basic facts.

So, it was with interest that I read that Nikki Morgan, the secretary of state for education in the UK, has decreed that:

"we are introducing a new check to ensure all pupils know their times tables by age 11"

An interesting post by @thatboycanteach asks what it means to "know" the times tables. Like me, he suffers from what might be described as rote recall deficiency. And like me, he survived (and even thrived) by using various work-arounds to compensate.

The UK times-table test will be computerized and time-restricted. It looks like it will be based on pure rote recall. For the flunkies, there will undoubtedly be some sort of penalty. It’s unlikely that they will be physically punished like I was, but even non-corporeal punishment can inflict great stress and harm and, in the end, may prevent them from learning mathematics.

What is of concern to me in Alberta is that, however sincere the back-to-basics people may be, they seem to be basing their reform efforts on the very thing that caused me difficulties, namely, the belief that all students can and must achieve rote recall, that this is the only way to know the basic facts.

That is the conclusion that I draw from reading their petitions and press releases. If I’m wrong, if I am raising a straw man, it is difficult to understand why they also want to banish the teaching of alternate approaches to the basic facts and algorithms that are needed so that people like me can compensate for our deficiencies.

Math fair workshop at Banff

The 14th annual math fair workshop at BIRS will take place over the weekend of May 6/7/8, 2016.
(BIRS = the Banff International Research Station.)

Right off, let me say that I have a pretty bad attitude about school science fairs.  You know — those competitions with poster sessions, baking soda volcanoes, and parent-created displays. The ones that end with an obligatory showcasing of a winner — a bright student who looks like he/she will go on to become the next Neil deGrasse Tyson, and who, for a short while, will be a poster-person for our education system.

OK, that's harsh, but it is still very much the norm to single out a winner.

How about having one that does not overly favour the highly talented? One that even a less confident student would enjoy and not end up feeling like a failure because he or she did not win a medal.

If you’re like me, you do not enjoy being tagged as a loser, and you would likely withdraw from a situation where that is liable to occur. Aviva Dunsiger touched upon this in her blog. Although her post is about phys-ed rather than mathematics, she paints a clear picture of the response to anticipated failure:

Yes, there were always strong athletes, but those that struggled (and I was one of them) wanted nothing to do with phys-ed. With my visual spatial difficulties, games like volleyball, basketball, and baseball were a tremendous struggle. I certainly never got picked for a team, and I couldn’t blame anyone. Why would I want to be physically active if I was only going to meet with failure?

[the emphasis is Aviva's]

Can we have a math fair where students can be mathematically active without the anticipation of failure?  One where students do not need a badge or ribbon to confirm that their efforts have paid off ?

Such math fairs do exist. They’re called SNAP math fairs because they are Student-centred, Non-competitive, All-inclusive, and Problem-based.

The fairs are built around math-based puzzles. The students first solve the puzzles^[1] and afterwards prepare the artwork and puzzle pieces that are required to display them.  

Visit such a fair and you will find students manning their puzzles. But, you will not see them exhibiting the solutions. Instead, they will invite you to try the puzzles yourself, and they will give you hints and help when you run into difficulty. The math fair is very interactive. It is much more than a poster-session.

* * *

Here are a couple of puzzles from past math fairs. The first one is for younger students to solve. 

Cats Pigs and Cows

A farmer has nine animal pens arranged in three rows of three. 

Each pen must contain a cat, a pig, or a cow. 

There is already a pig and a cat in two of the pens. 

The farmer wants you to fill the remaining pens so that no row or column contains two of the same animal.

The second puzzle is for older students.^[2]

The Sword of Knowledge

The dragon of ignorance has three heads and three tails. 

You can slay it with the sword of knowledge by chopping off all of its heads and all of its tails. 

With one stroke of the sword, you can chop off either one head, two heads, one tail, or two tails.

But the dragon is hard to slay !! 

• If you chop off one head, a new one grows in its place. 
• If you chop off one tail, two new tails replace it. 
• If you chop off two tails, one new head grows. 
• If you chop off two heads,  nothing grows.

Show how to slay the dragon of ignorance.

* * *

A SNAP math fair is remarkably adaptable to many different circumstances. If you are interested in learning about how you can incorporate a SNAP math fair into your own teaching environment, come to the BIRS workshop. You will meet teachers who have organized math fairs in their own schools. You will also meet a few mathematicians who have taught courses in which a math fair was key ingredient. 

As well, there will be math fair resources available, and the participants will be involved in puzzle-solving sessions.  

The BIRS workshop has room for about 20 participants, and it is oriented towards (but not limited to) K-9 teachers.

For more details about SNAP math fairs, visit the SNAP math fair site. And while you are there, take a look at the Gallery to see how students react.

For more information about the workshop, and who to contact, the link is here.

End notes

^[1] The solving part is a crucial element of the math fair.  Ideally, students solve the puzzle by themselves. They are surprisingly persistent.

^[2] I imagine the Sword of Knowledge puzzle would work with junior high or high school students. However, I once visited a SNAP math fair where two grade five students had solved it. Their teacher told me that they struggled with the problem but solved it after one of them grabbed a handful of pencils (tails) and erasers (heads).

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:

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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.

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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.

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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.

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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.

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

What about this one? 

Or this one?