Saturday, June 25, 2011

The Mathematics of Laundry Soap

The grocery store is a brain workout for the mathematically inclined. Not only do the varying metric and imperial conversions tease out the micro-savings of bulk, but neon yellow discount signs encourage percentages and good ole' multiplication tables. Often you find adults transfixed in a complex division trying to figure out which ham will be cheaper. Once that calculation is complete, they turn their attention to making sure the portion will be enough to feed their whole family. The sheer volume of available estimations overloads me; coupons just complicate the matter--significantly.

When you add the typical male intolerance to shopping, grocery stores become the closest thing to hell on earth for me. This condition has been worsened over the years by the free sample shortage that seems to be sweeping the nation. I guess we are in a recession. I found myself somewhere between the frozen meats and fresh produce when my wife posed a problem. Bless her heart; she knows that a few cognitive laps help pass the time. She gave me a coupon and asked me to go get the laundry soap. Although long journeys to the Dairy are my usual forte, I accepted.

She told me that the coupon was for Purex detergent. Upon further inspection of the aisle, the product came in three sizes; each size was measured in "loads". For simplification reasons, the detergent companies have coined a new unit of measurement--the load. No longer do humans have to toil in fluid ounces or milliliters. With this in mind, the task became simple:

Using the $3 coupon, which denomination of detergent was the best deal?

I stood there transfixed--very similarly to the ham-haze mentioned earlier--until I reached my conclusion. My trance was elongated by a re-evaluation of the numbers, because I could not believe my conclusion. Upon further rationalization, it makes sense, but it came as quite a shock amongst the toilet paper and facial tissues. Let's first define the problem a little more:
Bottle #1 - 40 Loads
$5.76 before coupon
Bottle #2 - 64 Loads
$8.57 before coupon
Bottle #3 - 96 Loads
$12.97 before coupon

Like every good grocery shopper, I kept the first rule in mind: bulk is ALWAYS cheaper. Initial calculations (in my head) didn't refute this conclusion. I correctly estimated the 40 load to be more expensive per load than the 96. If this is where the problem finished, I would have remained unastonished--but it didn't.

When the coupon was added, it seemed that the calculation switched. The 40 load mysteriously become far cheaper per load. I re-estimated 3 or 4 times to make sure I was correct before returning to my wife. I spent the next 5 aisles convincing myself. When I got home, I made the following chart with the help of my calculator:

Size | Original (Cost/Load) | After Coupon (Cost/Load)
96 Load | $.135 | $.104
64 Load | $.134 | $.087
40 Load | $.144 | $.069

There are two surprises:

1) It is actually cheaper to buy the 64 loads than the 96 loads.
2) The 40 load jug is the cheapest when the coupon is applied.

The first is just an entertaining observation. It would be interesting to see just how many shoppers bought the 96 load jug because they thought it must be cheaper. The difference is so miniscule (a tenth of a penny per load), but still creates another niche for numeracy in society. The second surprise creates a very interesting math discussion. Why does the coupon change the cost per load in favour of the 40 load jug? Posing that question to students would create a very interesting discussion.

Try and answer that for yourself. Did you see that coming? Where do the savings end? The elementary divisions open up discussions of average and even functions. My question was this:

What if I needed to do 300 loads of laundry? Is it still cheaper to buy 40 load jugs with the one coupon?

Why stop at 300? Seeing as the 40 load is only super cheap for a short time, then there must be a point where the 64 and 96 load jugs catch up in efficiency. How many loads will it take before it is more sensible to buy all 64 load jugs? all 96 load jugs? I set up functions for each relation, and graphed them on the back of my page. It was a great teaching point to see how the slope of the line changed when the initial "coupon jug" ran out. Each line got steeper because the cost per load increased.

So what's the big deal? The lesson here is that there is very interesting mathematics contained within mundane human exercises. Continue to fiddle with the problem; pose further questions!

What is the maximum coupon that keeps the 96 load jug the best deal?
What if we had more coupons?
Would it ever be a better deal to buy 96 load jugs over 64 load jugs?
What combination of jugs is the cheapest way to buy 10,000 loads?
How many coupons would you need to lower the cost per load to a specific amount?
How much should a jug that does 137 loads cost? (this is trickier than you think)
You have one coupon. Devise a general algorithm to get the cheapest 'n' loads.
What would prices act like if they were devised linearly, quadratically, logarithmically?

I can see this activity going on for 2-3 days in class. What a great way to cover averages, equations, proportions, unit conversion, percentage, sales tax, and consumerism. My answers have been intentionally omitted from this post in hopes that you too tackle some situations and pose further ones. They say that teaching is a 9:00-3:30 job. That may be true, but it is a 24-7 profession. Some of the most engaging and perplexing mathematics lessons are lurking somewhere amongst rotisserie chickens and contact lenses.


Tuesday, June 21, 2011

The Linear Relations of Hamburgers

Maybe you have seen the Burger King Stacker commercial where "Meat Scientists" work on an interesting problem. Needless to say, it piqued my curiosity the second I saw it; it was not long until I was trying to suck every ounce of mathematical value from the video. I am sure that I did not accomplish this goal, but I did manage to find some interesting problems and questions.

First off, the division of cow by pig seems very contrived. Their result ($) would seem to suggest that (pig)($) = cow. Are cows some sort of expensive swine? The representations of mathematicians as old, mustachioed men only reinforces the gender inequality in math today. (to BK's credit, one woman is included... barely) The burger abacus also reeks of mathematical stereotype. With those minor distractions now off my chest, we can get down to the heart of the task:

"How can we achieve maximum meat flavour, for minimum money?"

Sounds like calculus to me. Unfortunately, they give us no flavour-money function, and we don't really have a way (barring a wide-scale taste test) to plot one. There were, however, 2 more interesting pathways that a teacher could take this problem. One is a look at linear relations, and the other is a more open, problem solving take. Both could have value and fit curricular outcomes.

Let's start by examining the different linear relationships in the problem. I want to re-iterate that I have in no way exhausted all the possibilities. For example, I have only begun to scratch the surface of the possibilities of the "bacon" constant. The situation presents numerous variables. Some will be obvious to the students, others will be less obvious. Here is my preliminary list:

Number of Buns
Number of Patties
Cost of Burger
Strips of Bacon
Slices of Cheese

Combining these together, a teacher could begin to develop the relationships between them. For example, what is the relationship between the cost of the burger and the number of patties it has? This famously straight-forward example will begin the descent into the mathematics of the situation. Have them choose two other variables; Can they create that relationship? Table of values, graphs, and written word can all be employed. Group discussion and teacher modeling is also a great way to get the ball rolling.

It would be interesting to develop and graph the relationship between the number of buns and number of patties necessary to construct single, double, and triple burgers. Graphing these on the same plot brings up all kind of questions. Do we connect the dots? Why? What does the steepness of the line mean? etc.

I would be particularly interested in the Cheese Slice to Patty ratio. Go back and examine the clip again, and you will get my drift. It is not until close examination that these little nuances appear. The Burger Graphs could all be devised and presented in the class room. This will bring more meaning to their interpretation. Interpolation, extrapolation, and estimation all nicely follow this exercise.

Second, there could be many interesting problems that accompany this commercial. I have thought of one, and if you have another, a comment would be greatly appreciated. Consider the following situation:

You are the owner of Burger King, and an order comes in for 19 burger patties. The customer does not care how many burgers are created from the patties, but they want exactly 19 patties used. If you can make single, double, and triple burgers, how many of each type minimizes the number of buns? How many maximizes the number of buns?

This question could be a great starting point into an investigation of pattern. A teacher could choose to start here and then move into the examination of linear relationships. Can students explain their strategy for solving the problem?

Can they generalize their strategy for "n" ordered patties?
Will their final order ever include both a single and a double burger?
Will the minimizing solution always be unique?
Will it ever be unique?
In how many ways can 19 patties be sold?
In how many ways can 'n' patties be sold?

All these questions can lead into the listing of variables, and relationships between them.

Ironically, Burger King has created a fairly rich mathematical environment from which to study linear relations. I am sure they did not expect this when they founded the "Meat Mathematics Institute". As teachers, it is our job to create an inquisitive environment where the mathe-meat-ics can be encountered, teased out, and understood.


Sunday, June 19, 2011

NHL Dream Team

My thoughts have begun to turn to the new school year that will occur in August. This may be jumping the gun, but I like to enter prepared. This is partly due to the possibility of job action, and the surety of football, in the fall. I like to spend the first couple days of school working on basic numeracy skills with my grade 9s and 10s. I find a nice task is much more effective than a few worksheets. I do, however, keep a supply of worksheets on hand to offer to kids who just want the assignment. This idea came to me while I was reading an old edition of "The Hockey News" earlier this year. It has been taking up space on my desk, so I figured blogging about it would allow me to file it away for the beginning of next year.

I call these types of activities, "Numeracy Prompts". The term was inspired by Peter Liljedahl during one of his sessions at SUM 2010. The idea is to create a rich environment where students are required to make multiple mathematical decisions. The situation should be vague enough to provide freedom, but defined enough to allow mathematical modeling. A good numeracy prompt allows every student to begin by being accessible to elementary skills (a low floor). It also has a never-ending supply of mathematical depth (a high ceiling). These two attributes allow students to interpret the situation at a variety of skill levels. There are several reasons why I use numeracy prompts:

1) They are great formative tools at the beginning of the year.
2) They can be used over and over because answers and interpretations typically vary.
3) They are self-enriching. (the problem grows with the student's ability)

Some educators call these types of activities tasks, problems, or projects. I like the term "prompt" because is implies a vagueness that encourages play. A prompt may end in a product (much like a project) or give a list of requirements (much like a task) but leaves the process up to interpretation. The NHL Dream Team numeracy prompt can work skills like basic operations, percentage, unit conversion, and scientific notation. Again, the skills required to solve depend on the solution method chosen by the solver. The task is as follows:

NHL Commissioner Gary Betman has finally announced that the Winnipeg Jets are returning. In order to construct the team, he has enabled a fantasy draft. As the general manager, it is your job to put together a dream team. You can take any player from any team as long as you follow these rules:
1) No more than 65% of players can be from one conference
2) You can't take more than 2 players from a single team
3) You must chose 15 forwards, 8 defense, and 3 goalies
4) The total salary must be under 59.4 million dollars
5) No single player can be paid more than 10% of the salary cap
6) You cannot spend more than 50% of your money on one position (forward, defense, or goalie)

If you provide this question in the form of a memo to your class along with a list of every player and their salaries (available in any regular season edition of "The Hockey News" or online at www., they can begin right away. You may want to develop an alternative for those students who are not interested in hockey, or change the sport to fit your class' preference.

Mathematical discourse is best developed in groups, so pairs or groups of 3 would be effective. When groups begin to hammer down their final lists based on reputation of players, it may be interesting to extend the problem a little:

1) Get player stats for each of their picks. Whose team is statistically the "best"?
2) Fiddle with the rules to see if that would change any of their picks.
ie. Now a player can make 15% of the total salary cap
You can only have 6 players from any single country

The results are best reported in a chart format. I never miss an opportunity to model good statistical note-taking to my students. One of the options could look like this:

| Name | Position | Salary | % of Cap |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |

Let the students work around in the mathematics. If they need a refresher on percent or make a calculation error, let their group members work through it with them. Don't deprive them of a calculator! The mathematics lies in the decision to make a calculation, not in the actually calculation itself. Let students struggle with mathematical give-and-take, not with the onerous task of carrying 1s.

Numeracy prompts are excellent ways to tie engagement to basic skills. They alleviate the constant pressure for right answers. The basic skills become masked because they are no longer the end--they are now a means to an end. That is how basic skills need to be approached. Prompt, task, project--whatever you call it, it is just one more way to open up math, set it before your students, and allow them to be curious.


Saturday, June 18, 2011

Life's Not Fair

The school year is now over for me. That is a bittersweet statement, because I still have mountains of grading and report card comments to do, but there will be no more direct lessons in the 2010/2011 school year. I found myself nostalgic this morning, and began to recount the good times in the classroom. I recalled the probability mayhem that ensued with my Grade 11s. It was very amusing to see them come up with ways to describe "fair". I would always tell them that I would only do something if it was "fair". This, to them, meant a coin flip, draw from a hat, or a roll of the die. But whose hat? Who rolls the die? On what surface? Do these factors actually have an impact on "fair"?

Most questions with elementary probability include a fair clause. For example:

"Flip a fair coin 3 times, what is the probability that at least 2 flips will produce a tails?"

This fairness clause is supposed to wipe away all doubt that the theoretical probability matches the experimental. It also preempts those smug students who always find a corner case in the trial. Some kids would rather put their energy into being difficult rather than productive. I spent a considerable amount of time in class working on fair games, but never stopped to wonder if the pieces were fair themselves.

Take a die for example. We assume, as mathematicians, that each side has an equal chance of being rolled on a "fair" dice. Have you ever considered these 4 characteristics of a standard dice that may, in fact, make even a "fair" die act with bias?

1) If every marking on the die is concave (hollowed out), and each of these hollows are identical, then the die is actually weighted toward the 1 side. Isn't it logical that a heavier side would more likely land on the bottom than a lighter side? If that is the case, then 6 would be rolled far more often than 2 because the 6 is opposite a relatively heavy 1 side and the 2 is opposite a relatively light 5 side. This problem is easily remedied by large gaming establishments through scaling of the hollows. The gouges are all done proportionately so the same amount of material is removed from each face. Your typical board game dice however, will carry this "unfair" flaw.

2) The opposite sides of the dice sum to 7. This was surprising to me at first, but makes perfect sense. This means that if the dice (being 3 dimensions) has three axes, then an equal amount of material is removed (7 dots) from every set of opposite faces. This may seem to make the dice more fair, but think of this:

as the numbers grow further and further apart, the difference between their weights grows.

The difference between the 3 and 4 is only 1 dot--maybe this is negligible. The difference between 2 and 5 is 3 dots--three times worse than the first spread. The difference between 1 and 6 is 5 dots. This may be enough to ensure that the much lighter side shows up more often. It would take a large test to see if there is a relationship between the 3-4, 2-5, and 1-6 ratios. Again, this problem is easily solved by taking an identical amount of material from each side.

3) The center of gravity on each face affects the outcome. If we look at the 6 standard organizations of dots, we see interesting patterns:

Every edge of the face has the same pattern of hollows, except the 6. It is simple to see the symmetry in the 1, 4, and 5 faces. The dice look the same no matter how you rotate them. The 2 and 3 faces take some work, but it can be shown that each face is symmetrical using a combination of reflection and rotation. The 6 side however does not have this symmetry.

Two of the edges have 3 dots along them, and 2 have 2 dots. This results in an unbalance in the configuration. In a random toss, an unbalanced dice may turn toward a lighter edge. This could favour a 2 or 5 over a 3 or 4. It would be interesting to have students come up with a symmetrical design for 6 dots. What about 7? 8? I digress.

4) Each edge of the die does not have the same weight on either side. The die has 12 edges, and each one is formed by 2 sides. Along each edge there will be a certain number of hollows. For example, let's look at the 5-6 edge:

Here we see that the edge (which has been flattened) is not symmetrical. There is more weight on the 5-side because there are less hollows along the edge. (The 5-side has 2 and the 6-side has 3). If a dice was teetering between 5 and 6 on this edge, would this difference in weight make a difference? The 3-2 edge is balanced with 1 hollow a piece:

This exercise pleads to my curious spirit. It would be a great exercise to have a class find as many balance flaws in a die as possible. I am sure some of these would come up. After the problems are identified, the teacher could commission the groups to create a truly "fair" die. One that shows no favourites. It would also be interesting to use these 4 facts to guess which sides roll more often. A large-scale class test would reveal the results. This is only the introduction into a very interesting concept of fairness. Maybe it is fitting that, so close to Father's Day, I can almost hear my dad's voice, "Sometimes, life isn't fair." Maybe even the most objective tools we have in society prove to be unfair when put under intense scrutiny.


Thursday, June 16, 2011

Induction Squared

I came across an interesting problem recently that I gave to my students in need of enrichment. 

Given a square and the ability to divide that square into smaller squares, can you divide a square into 'n' smaller squares. The squares do not have to be the same size. For which values of 'n' is this possible? For which values of 'n' is this impossible?

Students initial reaction was to draw a square and experiment. I cannot think of a better way to begin this problem. It is organic, and contains some very speedy deductions. We begin with suspicion of a pattern, and move into a more regimented induction.
Here is our square. There is One of them. The rules state that we can divide any square into smaller squares. The first insight is that only "square" numbers can be created. So each square can be turned into 4,9,16,25, ... smaller ones. For example, dividing this square into 4 or 9 would look like this:
It does not take long to discover that 2 and 3 squares are impossible. Four is the closest we can get to 1. Five, Six, are also impossible; When you begin to stumble around in the problem, you find others that are possible. For example, 7 and 12 are possible:
The play reveals some interesting patterns. For example, we begin with 1 square always. Every time we divide it into the smaller squares we lose that original square and create a set of new ones. Therefore, we can only add specific numbers of squares. We can add 3 new squares by dividing an existing square into 4 (4 - the original 1). We can add 8 by splitting into 9. In general, we can add 'q' squares to any pattern if:

'q' = s -1 , where 's' is a perfect square

This accounts for why 3 was not possible. This also explains why 13 is not possible. The smallest number we can add is 3 (dividing into 4 new squares). We can use this new notation to describe 12 and 7 squares pictured above:

7 = 1 + 3 + 3 (cut original square into 4, then cut another square into 4)
12 = 1 + 8 + 3 (cut original square into 9, then cut another square into 4)

With this framework, we could try chopping forever. Instead, I had to make the realization that if we could represent any 3 consecutive numbers 'n', then we could represent every subsequent 'n' by simply cutting squares into 4 new squares. This sounds confusing; let me explain.
Because we can always add 3 squares to a formation by cutting one of its squares into 4, we can create 3 roots. Lets say we get a diagram with 14 squares, then 17 is achievable easily by splitting one more into 4--that adds 3 more and 14 + 3 is 17. Twenty is also simple, 23, 26, 29, etc. In fact, every third number would be possible. If we could find three "base cases" in a row we would be able to create any 'n' after them by adding successive divisions into 4. With a little playing, I found the first 3 consecutive successes are 15, 16, and 17:
15 = 1+3+3+8 16 = 1+15 17 = 1+8+8

From here, finding a formation is as simple as finding which "base case" it belongs to and cutting enough squares into sets of 4. This provides a proof that any solution above 15 is possible. Trial and error can be used to find those solutions that fit below 15.
Two important things can come from an exercise like this: 1) An appreciation and understanding that we owe mathematical exploration for the underpinning of the result. The process began with trial-and-error and pattern recognition. 2) A collection of further problems can be posed about similar situations:

What about dividing cubes into smaller cubes?
Could we restate this question with other shapes like triangles, rectangles, etc.?
Which numbers have unique representations?
Which 'n' has the most representations? Does it have to do with factors?

Mathematics is born from curiosity, suspicion, and stubbornness. All three of these elements are at play with this problem.


Monday, June 13, 2011

Maths' True Form

I teach mathematics at the high school level, and know all about the various theories surrounding school mathematics. I can still remember the intrigue when the term "Math Wars" was introduced to me through some undergraduate reading. I immediately took to the history of my art, and found a very convoluted and bloody past. The constant pendulum between retention math, new math, back to basics, and now the new-new math is dizzying. Whenever I converse with a colleague about a new way of thinking in math education, I am sure to remind them that we are in a war. It is this idea that has appealed to the more militant teachers (myself included).

I would love to stop this war. It discourages collegiality, and increases excuses. It also acts as a thin veil behind which many educators hide. One side accuses the other of being "useless"; the other retorts with chants of "fuzzy". One focuses on repeatable and measurable skills, while the other wants to develop thinking skills through mathematical investigations. If we were honest with ourselves:

Both sides need the other.

Students need tools for investigations. I am not saying they need to be transmitted to them, but there needs to be more than an extremist 'open discovery'. I would love to see an "open" discovery that accurately and completely covers a curriculum. Even the most effective ones are structured in "units" where teachers dictate what problems the students will investigate. The nature of the beast is that teachers have a job to do, and that job is dictated by our standards. We can still approach the standards with a spirit for understanding, but must also ensure we don't make basic skills the casualty of war.

After my exposure to the issues, it seems as though it was an immeasurable gap between the two sides. There was not much encouragement from either side, and negotiations were fruitless. Teachers sat in Professional Development days shooting down every possible suggestion. Eyes were rolled and the room filled with hurried whispers when a "reform" technique was proposed. The entire room was bursting with "here we go again". University standards require a rigorous understanding of many branches of mathematics, but the traditional teaching was not raising the scores at that level. The carousel continues to this day.

It was while looking through a number theory book that I found this paragraph. I was trying to find examples of mathematical induction, when I came across this introductory paragraph. It provides an interesting connection between the two armies. It is also very ironic that such a sympathetic view was found in a book written for study at the university level.

The following excerpt is from Elementary Number Theory: Second Edition by Underwood Dudley:

"Mathematics is notoriously a deductive art: starting with a collection of postulates, theorems are deduced by following the laws of logic. That is the way it is presented in print, but that is not the way that new mathematics is discovered. It is difficult to sit down and think, "I will now deduce," and deduce anything worthwhile. The goal must be in sight: you must suspect that a theorem is true, and then deduce it from what you know." (page 205)

I was very delighted to hear the ambiguous tone in the author's voice. Mathematics begins with suspicion. This is the nature with which mathematics must be presented in school. There is a common misconception that teachers who advocate for discovery want students to recreate every formula and theorem--no matter how difficult. We do not desire a re-invention of the wheel. To me, the new-new math is about developing the necessary atmosphere of suspicion in school mathematics. This is a feature that is crucial for investigations and also valued--as evidenced by Dudley--at the university level. Maybe the two camps are closer together than we think.


Sunday, June 12, 2011

Life Without Euclid

This post has nothing to do with geometry. I guess I can't say that exactly (because of the possible geometric representations), but I am not dealing directly with these. I am always intrigued when I think like I want my students to think. It is these moments that keep me going into the classroom hoping for new understandings. There have been times this year where students have made connections that I never have. These innocent realizations are mathematics manifested in its purest form. A similar experience happened to me this morning.

I had been mulling over a problem posed by the NCTM about Pythagorean Triples. I am very familiar with the theory around these special sets of numbers, but have fallen slightly rusty over the years since the number theory classroom. The problem was:

Can you have a Pythagorean Triple without at least one of the entries being even?

This would have been a rather dull problem for me to explore if it wasn't for that rust which has accrued over time. It allowed me to approach the problem with blessed ignorance. The first approach was to try and remember the formula that generates all triples; this failed miserably. I was sure that one of the stipulations was one of the terms must be even, but I could not be certain. I convinced myself not to google the formula, and set to thinking.

My mind immediately wandered to the idea of being relatively prime (or coprime). Two numbers, although they may not be prime, are coprime if they share no common factor. I knew that new triples could be "built" simply by multiplying each entry (a,b, and c) by some constant 'm' because:

if a^2 +b^2 = c^2
m(a^2 +b^2) = m(c^2)

I played around with common triples and new ones created by multiplying by certain constants, but I new that this was fruitless. Each triple I could think of had an even entry, therefore any multiple of that entry would also be divisible by 2. My time spent on this hypothesis was driven mainly by the intrigue that such a relationship must exist.

The existence of a formula paralyzed me. I new it was out there and that Euclid had done all the work for me. I could not bring myself to embrace the simplicity of the problem; I was attempting to find the general formula instead of showing that one entry must be even. Not only did the formula kill my desire to derive it, it closed down all other pathways to an answer.

Then, this morning, I had a spark of childlike thinking. It was so elementary that I smiled when it came to mind. (Elementary when compared to the derivation of Euclid's Formula). I tell my students that mathematicians are lazy; I also make it quite clear that this doesn't mean they do no work. It means that they always try to find the quickest and most succinct path from hypothesis to proof. Ideas should not be disposed as trivial because they are simple. Some of the greatest theorems of all time come from modest assumptions. With this spirit in mind, I sat down and answered the question.

My new angle was this:

Let's create a generic triple from 3 odds, and show that it cannot be true

I had reduced the question to a series of trials--an extremely short series of trials. I then set-up a situation where 3 odd numbers satisfied the Pythagorean relationship.

a^2 + b^2 = c^2
a = 2x+1, b = 2y+1, c=2z+1 | x,y,z are integers
(2x+1)^2 + (2y+1)^2 = (2z+1)^2
4x^2 + 4x +1 + 4y^2 + 4y + 1 = 4z^2 +4z + 1
collect terms and factor out a 2...
2[2x^2 + 2x + 2y^2 + 2y + 1] = 2[2z^2 + 2z] + 1

Now because all of x,y, and z are integers, the expression "[2x^2 + 2x + 2y^2 + 2y + 1]" is also an integer. So the Left-hand side (LHS) is an even number. (The entire thing is divisible by 2). The Right-hand side (RHS) is an odd number because it is of the form 2n + 1, with 'n' an integer.

If we clean it up a bit by letting 'm' and 'n' represent general integers:

2m = 2n + 1

This relationship is not solvable over the set of the integers. It essentially states that if a Pythagorean triple didn't contain an even entry, an Even number would have to equal an Odd number. This can't happen. The official reason is because the Odds and Evens form a bijection. Not important...

What other problems can be posed from this fact?

Can we have a Triple with 2 Evens and 1 Odd?
Can the Even be a,b, or c? Does it matter?
Can a triple be all Evens? Can I find one?
What organizations of Evens and Odds satisfy the equation:
a^2 + b^2 + c^2 = d^2 ??

I didn't need Euclid's formula to engage in meaningful mathematics. In fact, the knowledge of a formula acted as a roadblock. The feeling of enlightenment is one that I want my students to experience when they do mathematics. It is these "aha" moments that fuel my drive to create meaningful lessons. They invigorate me even when a students make a connection that I have already made, but I assure you this is not always the case. In this situation, life without Euclid provided me the opportunity to make numerous connections.


Thursday, June 9, 2011

Review: The Art of Problem Posing

I consider reading an essential part of my professional development. I enjoy a morning glance through a chapter or two, and like to wind down with a book and a cup of coffee by the fire during winter. Sometimes reading is the only way to relax my mind at the end of a day. (some professional literature is better at putting me to sleep than others.) To this point in my young career, no book has changed my perspective on the teaching and doing of mathematics more than The Art of Problem Posing: Third Edition by Stephen I. Brown and Marion I. Walter. The duo writes quite a bit for "Mathematics Teacher" (the high school journal for the National Council of Teachers of Mathematics) as well. The processes introduced in the book have been crucial to the penning of many posts on this blog. The book creates a framework from which creative mathematics flows.

The book is filled with examples that illustrate its points. As a teacher, it is encouraging to see a practical link to the problems which I encounter and present to my students on a daily basis. As the book continues, I found myself anxiously guessing which direction the author was going to take. Many of the problems are directly transferrable into a high school atmosphere, but it is the broad process of creating new problems that changed how I approach every problem solving situation.

Brown and Walter focus on a process called "Problem Posing". This process is designed to interweave the traditional problem solving process. The idea is that by asking questions and raising doubts about an original question, a student can learn about the intricacies of the original problem. In other words, playing with the question not only allows for various interesting pathways to be uncovered, it also reflects back upon the original attributes of the problem. The entire book, save a portion at the end dedicated to the teaching of mathematics education courses at the university level, is built around this theme. The process they use to introduce doubt into a problem is the "What If Not" method--or WIN.

A WIN begins with a problem. They use typical textbook problems, creative problem solving situations, and even objects like geoboards in the book. Any of these can be the subject for problem posing. The process begins by listing all the attributes about the problem or object. One particularly interesting example given in the text is that of the Fibonacci sequence:

1,1,2,3,5,8,13,21,34,55,89, ...

With some practice, this process becomes less regimented. The questions begin to flow out as you change the problem's attributes slightly. Brown and Walter's list of attributes for this sequence is as follows:

1) We start with two given numbers.
2) The two starting numbers are both 1.
3) The first two numbers are the same.
4) We do something to any successive numbers to get the next number.
5) That something we do is an operation.
6) The operation is addition.

Although some of these attributes seem redundant (2 and 3 for example), it is important to list as many as you can. From here, each attribute is doubted, replaced, or eliminated. This creates a whole plethora of unique and interesting problems. It is a great way to get instant extensions for students who need enrichment for the day's work. The new questions generated from this example could be:

1) What if we started the sequence with 3 numbers?
2) What if we changed the starting numbers?
3) What if the two starting numbers were different?
4) What if the operation that was performed was multiplication? exponentiation?
5) What if more than one operation was performed?

From here, the authors explore various pathways. Examples like these are repeated often throughout the book, and I find myself using the WIN framework in my own work with students in mathematics. I have made a habit of posing new problems using the WIN structure after I have solved a problem. This not only replenishes my store of new problems, it brings attention back to the original solution's attributes.

This book comes highly recommended to any teacher or learner who is interested in more than solving problems. Problem posing is an integral part in the process of doing mathematics. We cannot fully understand the nuances of our work until we change the parameters and redouble our efforts. Problem posing is an excellent way to allow students to encounter the never-ending web of mathematical connections.

The Art of Problem Posing: Third Edition
Stephen I. Brown, Marion I. Walter
New York, NY,


Tuesday, June 7, 2011

Balls and Bins

One of my pervious posts mentioned the problem of the balls and the bins. I got this problem from a source on twitter that I have since forgotten. Regardless of its origin, the question has been a fun one to pose to students and colleagues alike (I even asked my in-laws with some very interesting results). For those of you who haven't read "Practice What You Preach", The problem is as follows:

You have 8 balls, and 2 bins. 4 of the balls are Red, and 4 of the balls are White. Your job is to arrange the balls in the two bins however you like, but every ball must be put in one of the bins. (ie. no throwing balls away). I will then choose one of the 2 bins, and then draw a ball from that bin. If I draw a White ball, I win; if I draw a Red ball, you win. Which arrangement gives you the best chance at winning the game?

Ignoring my better judgement, I was immediately went down a fruitless path. What made things worse is that path involved pages of partial derivatives. There is no worse feeling than combining like terms and knowing that your method is going nowhere. In hindsight, my path toward a solution was an important one; the results I came up with surprised me, and many new questions were posed from them. I will explain all of this in due time.

I began with the observation that if all red were put in one bin, and all white were in the other, the chances are 50/50. This gave me base odds to judge my next trials. After I wrapped my head around my own question, I defined a value P(w)--the probability that the ball drawn was white. I also defined two more quantities:

let 'x' be the number of white balls in bin #1
let 'y' be the number of red balls in bin #1

Using these two variables, I set out to define a function F(x,y). If I could abstract the probability of drawing a white, I could maximize the function and find my best case scenario. For those of us who have been through calculus, maximizing a function comes naturally. It should not be a surprise that I chose this route. It may, however, surprise you to know that taking this route turned out to be less mathematically rich than listing the trials. When I got knee deep in partial derivatives, I lost all orientation of the mathematical meaning of my functions. Once I was presented with a function and knew where my destination should be, the middle--the actual process of mathematics--became a blur. Two pages of calculations and simplifications later, I started over. I re-focused with my function, and decided to try to list possible trials. This process unlocked the secrets to the problem. My function for the probability of drawing a white was:

Probability of drawing a white from bin #1: x/(x+y)
Probability of drawing a white from bin #2: (4-x)/(8-x-y)
Because each bin has the same chance of being chosen:

P(w) = .5[ x/(x+y) + (4-x)/(8-x-y) ]

The limitations of the computer type provides unnecessary complexity, but the function is relatively simple. It is based solely on the laws of multiplicative and additive probability. When I began listing the possible arrangements in both bins, I hit the major realization: having 1W and 3R in bin #1 was exactly the same as having 3W and 1R in bin #1 because the leftovers in Bin #2 would create the opposite arrangement. Each trial had a partner (or so I thought). It doesn't matter which bin has which organization of balls; the relationships and subsequent probabilities would be the same. Both those arrangements represent the same trial. I then made a master list to find how many trials I needed to plug into my formula.

Bin #1 Left Over
White Red White Red
1. 0 0 4 4
2. 0 1 4 3
3. 0 2 4 2
4. 0 3 4 1
5. 0 4 4 0
11. 2 0 2 4
12. 2 1 2 3
13. 2 2 2 2
14. 2 3 2 1
15. 2 4 2 0
23. 4 2 0 2
24. 4 3 0 1
25. 4 4 0 0

Twenty-five total options, but quickly scanning the two columns reveals what I thought. After case #13, each is a repetition with the "leftovers" mirroring the combination that was in Bin #1. Essentially, 12 of the cases were repeats. Notice that my initial guess that every trial would have a partner was false. Trial #13 turned out to be its own partner. Once this was revealed, it was as easy as plugging in the cases into the formula. The first column in the above table represents 'x' and the second, 'y':

Case (x,y) P(w)
(0,0) 1/4
(0,1) 2/7
(0,2) 1/3
(0,3) 2/5
(0,4) 1/2
(1,0) 5/7
(1,1) 1/2
(1,2) 7/15
(1,3) 1/2
(1,4) 3/5
(2,0) 2/3
(2,1) 8/15
(2,2) 1/2

From these calculations, we see that the probability of drawing a white is highest when Bin #1 contains 1 White and 0 Red and Bin #2 contains 3 White and 4 Red. This can be rationalized with the thought that half the time white is guaranteed a win (all of Bin #1 is White). If Bin #2 is chosen, there is still a 3/7 chance of drawing a White, almost 50%. I would encourage students to rationalize through each case. They will begin to look at the rule of multiplicity in a different way.

After I solved the problem, I began my favourite process: problem posing. What other questions could we ask about this situation:

1) What if there was a unequal amount of balls?
2) What if there were 3 bins? 4 bins? 'n' bins?
3) Which options create a "fair game"? (ie. 50/50)
4. What if you could not leave a bin empty?

There are numerous problems to be posed from this situation. The important part as teachers (and learners) is to not stop posing them. I enjoy posing and solving problems with my students. Divide and conquer! Although I am still sure that calculus could also solve this problem, it remains an open challenge for me. The best method is the one that is the richest in mathematical meaning.