Do Teachers Play with Mathematics?

Since my introduction to the twitterverse and blogosphere, I have been on the lookout for like-minded individuals who share my passion for the teaching and learning of mathematics. I have met numerous people who document their best strategies, and have already been very helpful to me. One such community of learners is the #mathchat gang that meets once a week (and re-opens discussion at a more European friendly time later in the week) to discuss a topic or theme in math education. Although it is often tough to express pedagogical beliefs in 140 characters or less, the conversation is incredibly fruitful. It was during one of the “mathchat”s that I was struck with a particularly convicting, and ironic, realization.


The topic of the conversation was:

“How do I promote deep, productive and creative mathematical play?”

and soon comments, ideas, opinions, and examples were pouring in from all corners of the globe. After about 20 minutes, the general consensus began to shift toward the definition of mathematical play, and more practical ideas on how to initiate it. You only have to hang out with teachers for a single Professional Development day to realize that they crave practicality. If a concept laces theory together with practice, it is quickly devoured. (Such is the partial aim of this blog). I noticed the conversation getting more and more impassioned as teachers began to take ownership; then a funny thought struck me:

Teachers seem very worried about initiating mathematical play for our students, but how often do math teachers engage themselves with “deep, productive, and creative” mathematics?

I am a self-professed math nerd. I spare no chance to initiate a mathematical conversation with my colleagues–often to the point of exhaustion. Despite my love for recreational mathematics, I almost always return to the knee-jerk reaction of relating what I am doing to two things:

1) The curriculum
2) The students

I am not saying that thinking about these two things should create guilt in the reader, but teachers seem to have the the inescapable urge to apply, apply, and apply. From their admittance to a College of Education, all focus changes. We become experts of the mathematics between the covers of our textbooks, and this is the exact phenomenon that many teachers are trying to disband in students. Math teachers need to take time to sit and marvel in the intricacies of their discipline, or risk becoming proficient only at the limited scope of math in today’s textbooks.

I work with a teacher who has seemingly freed himself from the necessity to only partake in mathematical activities that can become direct lessons. (Again, not necessarily a bad thing, as teachers may discover different perspectives and outcomes from authentic, mathematical play) I admire him greatly as a teacher and learner. You see, Mr. C. got into teaching a lot later than most of us. He experienced different vocations where he honed his skills in many mathematical, and practical, ways. When he discovered his love for education, he became a teacher, but never lost that spirit of investigation for investigation’s sake. It should be noted that Mr. C.’s pathway to education also gives him perspective that I have found extremely valuable in my young career.

After the mathchat concluded, I reveled in the irony of the situation for a while. The fact that math teachers could talk about mathematical play for close to an hour but never let the topic sway onto their own exploits in mathematical play was astonishing. I then looked back on a question put out by the folks at @NCTM and set out to play.

The topics of primes has intrigued me since my introduction to number theory a few years ago. They seem to be front and center on so many problems in mathematics. I started with this problem:

“Can the sum of 2 primes be prime?”

and set off to play. When my students set off to play, I ask them to make two lists: one of observations about the problem, and one of facts that may relate to the problem. I created these two lists about this problem:

Observations:
Primes are usually associated with multiplication.
Prime numbers are almost always odd.
We are going to work only with positive counting numbers.
The first couple trials work.

Facts:
2 is the only even prime.
Adding 2 odds always gives an even.
Adding an odd and an even gives an odd.

From here, I set out to “play with primes”. The first step that helped me was the fact that 2 was the only even prime, so it would become valuable with creating new ones in the future. Also, the first couple of trials including 2 created primes. (2+3=5 , 2+5=7) I then narrowed my focus by determining that all odd primes when added to one another would be even, and thus not prime. Initially, I used a standard number theory proof to show this, but mathematical play pushed me in another direction–a direction that would be crucial to the expansion of my perception of the problem.

I began to think of numbers as geometric bodies. We have “perfect squares” and “triangular numbers” so why not “trapezoidal” or “rhombic” numbers? Every odd number can be seen as an even with an extra point. This is the only point that has no ‘partner’ so to speak. I represented 7 this way on my page:

7 = :::.

So every odd prime, would have one left-over “dot”. When added, these two dots would line up and create a pair. That would symbolize an even (or rectangular) number. I then tried to fit my new representation of numbers (that has somehow been lost since I began primary school mathematics) onto the question of adding primes. We know that one of the 2 primes must be 2, and now we know that if the other prime can be arranged as a rectangle of dots with only 2 missing, than their sum would become a perfect rectangle. Essentially, any prime partner for 2 that needed exactly 2 dots to “complete the rectangle” would yield a non-prime sum. I encourage you to try the same representation system. The fact from it becomes:

Any number that can be arranged as a perfect rectangle is not prime, but divisible by both of its dimensions.

That led me to my conclusion. The sum of two primes could be prime as long as:

1) One of the primes was 2
2) The other prime was NOT of the form: an+b,
‘n’ is a natural number
‘a’ is a natural number > 2
‘b’ = a-2

i.e. 7 = 3(2) +1 w/ 1=3-2; so 7+2 is not prime.
but, 11 can’t be written as “an+b”, so 2+11 is prime.

This was my happy way to put my learnings back into my niche of number theory. Test this hypothesis out; I did. I continued to play around and refine my definition. This result only raised more enticing questions to play with; each of them became further and further away from the curriculum. I had to suppress my urge to find a student connection, so I could grow deeper in the learning of mathematics. Do Trapezoidal numbers exist? What about the sum of 3 primes? How many combinations of 2 primes are there? All of these have varying degrees of difficulty. The point is not to get caught up in practicality for students. Teachers need to understand that their learning of mathematics (often through exploration and play) is directly connected to their teaching of mathematics–even though all their thoughts may not fall neatly into a lesson plan.

NatBanting

The archive of the “mathematical play” #mathchat can be found at: http://mathchatarchive.wikispaces.com/2011.06.03+How%3F+-+Maths+play

Shouldn’t Probability be Vague?


I have always been drawn to probability because of its mysterious qualities. Maybe it is the result of the online poker fad that swept through my high school during the NHL lockout, but the calculation of odds still grasps my attention to this day. What fascinates me the most is how simple rules such as “AND” and “OR” can quickly create a mess of a situation. What begins in high school (or earlier) as a simple fraction that predicts the toss of a coin, soon balloons into factorials, combinations, Pascal’s Triangle, and Probability Density Functions. Despite the complexity of such calculations, they are still only theoretical; anything could still happen. This is a point that I stress to my students whenever we embark on a study of a game of chance.


Quickly back to the toss of a coin. I am assuming this is where most young gaffers get their start. You can imagine my surprise when I first learned that tossing a heads on the first throw does not guarantee a tails on the second, but we still say that the probability of tossing a tails is 1/2. The topic begins to muddy itself very quickly. Then, teachers present multiple tosses in a row. I was floored when I discovered that the sequence HHHHH was just as likely as HTHTT. How could this be? When teachers told me that there was a difference between tossing coins in sequence and throwing a handful of coins in the air, I was a willing skeptic. As if this wasn’t bad enough, during my time as a university student I learned that if I tossed enough coins, any imbalance between heads and tails will certainly happen! (See Stewart, How to Cut a Cake) Probability has its way of taking simple events and creating complex situations. It teaches me that my intuitive calculation of chance may be far off the mark.

I also find that it is this intuitive sense of chance in each student that makes lessons on probability so rich. Students explore the aspects of probability better than any other strand of mathematics. This is a very large statement to make, but probability’s convoluted nature lends itself to intrigue. Not to mention the mystique that pop culture places on risk assessment.

In my junior classes, we play “Homework Roulette”. This is a process (coined by my former teacher Mr. K. Peters) in which a random number is generated and the homework from that section is handed in. It is fairly easy for students to begin to calculate their odds of success, but they are constantly reminded that even if the odds are in your favour, you could be burned.

Probability moves away from this elementary calculation during the weeks of study dedicated to it. Call me crazy, but the challenge of calculating the odds of drawing a certain card from a deck wains quickly. Such problems have a very low floor (they are accessible to many) but a very limited ceiling (there is not much room for students to expand). The reason for this is the parameters with which they are stated. Students reach the end when they devise the nice fraction to represent the risk.

In order to initiate discovery in probability, the situations used must be intentionally vague.

Presenting problems in novel ways creates a battle between the students’ preconceived notion of chance, and the math that dictates otherwise. My favourite example of this comes from a problem I like to call, “Red Card, Blue Card, One Card, Two Card”. I will present the problem, how it is used in my class, and some student reaction.

There are 3 cards. Each is split in half down the middle like a domino. One has both sides painted red, one has both sides painted blue, and one has 1 red side and 1 blue side. If I choose a card at random and show you a side painted red, what is the probability the other side is blue?

The first urge that needs to be fought is revealing the answer. If students expect you to reveal the other side, the question is void. Every time a student asks to see if they are right, I make a very important distinction:

Student – “Mr. Banting, is it blue?”
Me – “You are answering the wrong question. I didn’t ask ‘is the other side blue?’ I asked, ‘What is the PROBABILITY that the other side is blue?’ “

When I begin this class, I always tell the students that I have a quick problem for them. I introduce it with props, and open up the floor for suggestions. I never let students off the hook when they provide a suggestion. Most will say 1/2 and I will ask them where the 1 and the 2 come from. This discussion usually leads to the development of the ideas of options, choices, and favourable choices. What we are doing is developing the ideas of events, sample spaces, and favourable events. I leave out the structure and definitions for now; they will only hamper the discussion. Other suggestions surface. 1/3 is popular. After thinking is explained, I make sure to agree with both.

My non-partisan stance coupled with the vagueness of the problem creates an atmosphere rich with query. Students begin rushing up to the board because they feel that if they only had a marker, the class would sympathize with their logic. Arguments of orientation begin to creep in.

“Let’s say that there are tops and bottoms”

I agree to every ounce of logic that is presented, but leave with a leading question. Are their really tops and bottoms? What if one card is rotated? What if they are on their sides? I pace around as students begin arguing in their desk groups. (I have never seen larger math rages than with this problem) Students grow red in the face as they wait with their hand up to resolve the problem and be the hero.

Soon some hard and fast facts are postured and proven.

“If we see a red side, then it can’t be the blue-blue card”

I agree, name the hypothesis after the student (i.e. the Smith Hypothesis) and then refer to their point periodically throughout the discussion. Students begin to make lists of possibilities and fractions that represent chances. The work is far more valuable because no one has shown them how to do the math. Their intuition leads the way.

I have chosen to end the problem a couple of ways. I have shown them a side of an unchosen card and asked if that changed the problem. Soon options are eliminated from the sample space, and new conclusions and theorems come forward. This is a profitable exercise. My biggest mistake was leaving the 2 unchosen cards unguarded. A student crawled on the floor out of my sight and attempted to rob them right under my nose. Another student actually kicked a desk at me to stop me from leaving before showing the card. Their desire to have it resolved drove them mad. I keep all cards close to me at all times now. I never show them the other side of the card…EVER. Doing this gives them an excuse to say, “I told you so”. It doesn’t allow the haunting to power new learning.

Of course, I could have taught sample space, events, and empirical probability with definitions and textbook questions. Each would have been tightly defined, and the answers would have appeared in the back of the book just as neat. The only problem is that would give the impression that probability is a tidy branch of mathematics. It is definitely not. Having the full-on math argument suits me just fine. The vague problem, creates an open learning atmosphere which emulates the study of probability itself–enigmatic. Until further notice, when it comes to probability in my classroom–vague is in vogue.

NatBanting

Must it Always be True?

This morning on twitter, there was a problem that I just had to solve before going out the door. It is safe to say that these types of problems are my vice. Number Theory has always held a special interest to me despite, according to G.H, Hardy, having “absolutely no practical use.” (A Mathematician’s Apology, 2001). This has all changed with the inception of encryption.


I wish just to present the problem and then muse on its educational significance both for my personal learning of mathematics, and for that of my students.



N is the 4-digit integer 6_9_. If these two digits are reversed, explain why the resulting number must be 2970 more?
(posted by @dmarain to @cuttheknotmath)

I immediately look toward the base system when digits are switching around. When a digit moves from one place to another, it takes on a new meaning. I expect this new meaning will give me the difference I desire.

First number can be represented as (with a, and b in the set of base 10 digits):

6X1000 + aX100 + 9X10 + bx1

When we switch the digits, the new number becomes:

9X1000 + ax100 + 6×10 + bx1

The second is always larger (which is an interesting discussion to have with students) so we subtract to keep difference positive. We se that because the a and b did not switch orientation in the place value system, their value remains constant. Therefore, they will cancel out upon subtraction and have no bearing on the final solution. This is why it is a constant regardless of the two digits.

9000 +60 – (6000 + 90)
9060 – (6090)
2970.

In this way, we show that for the general case (there are 100 cases in total–2 spots, 10 digits) this fact is always true.

The idea of proof to students is very elitist. In high school, a list of examples where it holds is often sufficient. Once the list gets long enough, the proof is concluded. In this case, it would be easy enough to show the students there are 100 cases; this may discourage a plug-and-chug method. Instead, number tricks like this help students realize two things:

1. The basic qualities of our base 10 number system
2. The many interesting patterns that numbers create

In my curriculum, both of these topics are mandated. Posing this problem to a class of grade 10s gives them opportunity to create hypotheses, test them out, and then dive deeper into the number system to look around. I would imagine that such an activity would pair nicely with one on scientific notation or binary numbers. If they are really keen, a proof like the one above may be deciphered. Then we drag algebra into the mix as well.

There are many areas of useful mathematics that are left out of textbooks. As teachers, our pursuit of learning can greatly effect our teaching.

NatBanting

Attaching a “Why” to the “How”

There has been plenty of recent twitter talk about the process of moving the focus of mathematics education away from the “how” and toward the “why”. Traditionally, students have been trained to approach a question–usually given to them by an outside source like a teacher, textbook, or test–with the express intent to show the grader “how” it is answered. Such responses often include the use of algorithms, formulae, or memorized facts we know to be true. (These facts are in no way axiomatic, but constant repetition reduces them to that state. Students have answered them so often, the process loses meaning. Take 2×2 for example.)


The focus on the how encourages a race to the finish. Thomas C. O’Brien calls this phenomenon “Parrot Math”. (Phi Delta Kappan, Feb, 1999) More specifically, it is the process that every student goes through when answering the question “how”–they attempt to repeat or imitate the process that has been shown to them. These carbon-copy answers are created without the knowledge of the mathematics that ensures their success, just as a parrot can possess a large vocabulary without understanding the intricacies of language.

In my class, I attempt to elicit the “why” as much as possible. The solution to “why” is much harder to come by. There are times in class when I challenge students on their mathematical statements–just to keep them honest. A student was working on sketching the graph of a rational function when she called me over. She explained to me that she could not get part c:

c) Find the equation of the Vertical Asymptote(s)

She was having trouble with a “how”. She could describe what a Vertical Asymptote was and how it effected the graph, by had simply forgotten the neat and tidy process to generate them. I explained to her that you set the denominator to zero and solve. Her face instantly renewed with vigor because her question had been solved. I then continued to challenge her with “why”s. I asked why the denominator can’t be equal to zero, but got a very standard, and hollow, response:

“Because you can’t divide by zero.”

Acting as confused as I possibly could, I asked her, “Why can’t you divide by zero?” Her joy instantly drained. It took a 5 minute conversation about piles and sticks before we decided that you could divide zero sticks into 10 piles, but couldn’t divide 10 sticks into zero piles. Division had become a “how”.

Moving on from the distinction, it is plain to see communication as a central element in “why”. This includes student-student and student-teacher communication. The Standards published in 2000 by the NCTM call communication an “essential part of mathematics and mathematical education.” It is these skills that build “meaning and permanence for ideas” in math. (NCTM, 2000) I think most teachers agree that both meaning and permanence are lacking. Just come back from a summer holiday with a group of grade 9s.

The simplest, and most useful, way I open the avenues of communication in my class is through yellow paper. That’s my secret. I grab a couple yellow, lined pads of paper from the office and hand each student a page. We complete the daily task on it, and they are to provide a reflection or explanation when they are through. The students know I read everything handed in on a yellow piece of paper. It took 3-4 practices before the students began to communicate their methods effectively in writing, but my persistence has paid off. I thought it would be interesting to include some student thought from my Mean, Median, and Mode task posted earlier on this blog. (It will be helpful to read that post if you have not already)

The writing created 4 distinct categories of learners, but I would have never made this distinction without the pursuit of “why” and the open communication of the yellow paper. The students were required to fix the set {1,2,3,3,3,4,5,5,6} so the mean, median, and mode are all 3. The mean was the issue.

Some students took out numbers from the high end until they met a target sum. These students knew that the average was dependent on the sum and the number of entries; they altered entries until the mean worked.

“There are 9 numbers in total. Not wanting to change this # that I know will divide by 3. 3 (the # I want to get) timesed by 9 will give me 27. I take the difference of 27 and 32 (32 because thats the original total of these 9 #’s) and I want to get 27 instead of 32 because 27 / 9=3. So I take away 5 (the difference) from 6. 1,2,3,3,3,4,5,5,1 = 27/9 = 3. Mean = 3.”

This explanation above (repeated verbatim) shows excellent mathematical communication. She has shown me the “why” in her process. She has also shown me that she understands how the total sum effects the mean. She uses mathematically rich words like sum, difference, and mean. She has also used less eloquent terms like “timesed” and “take away”. This shows me that she persevered through lapses in thought. She hit a wall in her explanation, and powered through it by sheer determination. This is how I know she can now answer “how” to find the mean, and “why” it is that way.

Other students tried lowering the entries one by one until they got the mean correct. The logic behind this dilution of sorts was they wanted to keep all entries in order so the median was not altered. Genius! I may have seen this process as primitive if they had not communicated their motives. Other students added entries so the large ones took less effect. Others talked of “balancing” the entries. I had never thought of mean as a balance before.

Creating meaning in mathematics is not about throwing the “how” out the window. Mathematicians have worked hundreds of years to establish hows. True meaning comes when students are given the opportunity to both attach a “why” to the “how” and communicate that connection to others.

NatBanting

Merit to Mathematics Labs

There is widespread turmoil among teachers and students when it comes to the practicality of mathematics. School mathematics, at the middle and high school levels, has moved out of the elementary niche of rudimentary skills, but has yet to make it into the realm of complexity necessary to apply it back into the world. Our happy compromise, as teachers, is to go with a two-pronged attack:


1. Tell the students that the practicality comes later
2. Create word problems about trains leaving stations or people tossing balls off cliffs

Every teacher of mathematics (from the wide-eyed rookie to the well-weathered veteran) has encountered “the question” numerous times. We hear it so often, that I would imagine many teachers have a well-rehearsed response to the query. I know I do. Even though we are prepared for this onslaught, the thought of having to employ our answer triggers chills down our spines and sends us retreating into the staff room for cover:

“When are we going to use this?”

There are sects of mathematics that present obvious answers to “the question”. Measurement has cooking, geometry has the trades, and probability has gambling; each to their own segment of society. I have already begun to envy the science teachers who bottle themselves up in their back room only to emerge with a concrete and engaging example of when people actually use their discipline. This takes the form of a specimen jar, an interesting chemical reaction, or a proposed perpetual motion machine. I still remember anxiously awaiting “Explosion Fridays” in my Chemistry 30 class in high school. It seems as though these subjects have endless applications, and they harness their practicality in a laboratory.

A lab is where the students get a hands-on, experimental crash-course in their discipline. It is my belief, born partially out of my aforementioned jealousy, that there are situations in the mathematics class where students need the opportunity to play around with mathematics in a laboratory setting.

Before we get too much further, I must give a word of caution. You may have noticed that I always use the term “mathematics lab” over “math lab”. Although the latter may be easier to say, speaking too quickly may give your students the impression that they will be creating methamphetamines. Awkward explanation for the administration; I digress.

There have been 3 deliberate labs in my math class this year. My hope is that by briefly detailing the setting and task, you too will begin to see the value in a practical, exploratory, and hypothesizing environment.

1. Monty Hall Problem Mathematics Lab

Students were presented the wildly popular problem verbatim from the game show. (I have found that the newer rendition found in the movie “21” works better as an introduction.) We spent a few minutes hypothesizing and gathering into our lab groups of 3-4 students. The majority of the class was spent re-enacting the situation and creating data. Not only did this build skills in the scientific method, it is, in my view, the only way to fully understand probability. Students must understand that the theoretical calculations we create are actually mimicked in their experiments. The second day, the class reported their data, and it was compiled. Most of the class was spent discussing what the data was telling us, which hypotheses were correct, and how we could alter the experiment to get differing results. (Finding the problem is a simple as googling “Monty Hall Problem”. I would encourage you not to research an answer until you have tried the very same lab.)

2. Route to the Commons Mathematics Lab

I gave the lab groups a miniature map of the school that is usually given to substitute teachers. I asked them to resolve the problem of hallway congestion by measuring out the fastest way to the commons and back to class. The problem was left intentionally vague, and the students were given a meter stick, a pencil, a calculator, and a centimeter ruler. Students began suggesting the route they took and locating it on the map. A scale factor was devised and tested on various lengths throughout the school. As students became sure of their results, I challenged them to create a map for their most efficient school day. They began charting their path from class to class. Students were encouraged to test this theoretical shortest distance during lunch in the hallway traffic; soon more crowded hallways needed to be weighted differently. Although I never got this far, a general “congestion constant” could have been developed for each corridor. In a main hall, one meter may be equal to 2 meters on an abandoned one.

3. Fermi Estimation Mathematics Lab

I have used this format a number of times this year to build basic numeracy skills. Students are asked to calculate large or small quantities in a general sense. It is entirely about numeracy-based estimation. How many hairs are on your head? How many hairs are on the floor of the school? Students enjoy creating outrageous questions and tackling them empirically. How many hairs does the average student get caught in their mouth in a given school day? The problems expand to include basic arithmetic, probability, surface area, etc. For ideas of Fermi estimations, see “Guesstimation” by Lawrence Weinstein and John A. Adam.

There are a lot of practical topics in mathematics. Creating disenchanted questions about choosing coloured tiles from a bag does not give math its full due. Setting up mathematics labs creates room for curiosity in math; students take direction of the content. Not to mention that giving students a practical platform to do mathematics will reduce the odds of having to field “the question”

NatBanting

Playing With Mean, Median & Mode

Teachers in Saskatchewan, Canada have had a lot to deal with lately in the classroom. The ongoing political battle has effected hours of direct instruction in a very real way. I quickly noticed my classes becoming disjointed with large amounts of time between each encounter with the mathematics. Needless to say, I entered today’s lesson in Math 9 with a little apprehension. A Friday morning after 2 days of job action and a long weekend didn’t sound like the most nurturing of environments. I decided that the time was ripe to attempt a lesson that has been in my mind for a couple of months; the following account is the story of the task, presentation, student reaction, and important learnings.


The students were introduced to the concepts of Mean, Median, and Mode earlier this week. It had been 3 days, so I quickly refreshed their memories with a standard problem. I gave them a list of data, and had them compute (in pairs) the three measures of central tendency. After some painful re-hashing and peer tutoring, the class was then alerted that we were going to take a major shift. With a quick swipe of the eraser, I eliminated the data set and left only 4 facts on the board:

n = 19
mean = 4.47
med = 4
mode = 2

I asked them if they could re-construct the data set using these facts. It should be noted that I unfairly took advantage of my students’ laziness. When asked if they should write down the set of numbers to solve the original problem, I told them not to bother. Do I feel guilty? Meh.

I expected the class to complain about this crazy task. Some began to rack their working memories for the last remaining traces of the numbers–but with little luck. After these efforts fizzled out (2 mins max), I posed the problem I actually intended them to solve:

Find the data set with the following attributes:
n = 1
mean = 3
med = 3
mode = 3

Soon the fear of wrong answers wore off and partners were conversing with other groups. After about 3 minutes, we decided that the only data set that fit all four was :

{3}

I asked very quickly for an explanation and verification of the facts, and then erased the “n=1” from the board. I turned toward the class (with a dramatic pause) and repeated the exact same question:

Find the data set with the following attributes:
mean = 3
med = 3
mode = 3

Group work began immediately. An electric hush fell over the room as they worked for 10 solid minutes discovering numerous data sets that fit the description. As I circulated, I overheard phrases like, “what about the median” and “won’t that change the mode”. At this point, I began to ask questions that required students to search their numeracy skills and metacognition. “How do you know you need to add a large number?” and “Why did you decide to start your set with 1?”. In due course, we generated a class list of sets, including one that could go on “forever and ever”.

{1,2,3,3,3,4,5,5,6} *
{1,2,3,3,3,6}
{3,3}
{3,3,3,3,3,3,3,3…}
{1,3,3,5}
{2,3,3,4}
{1,1,1,3,3,3,3,4,5,6}

The list itself reveled a lot about how each group thinks about numbers. Certain groups have been trained to think that lists start with one. This seems natural; we start school in grade 1; we begin counting with 1–why not begin listing with 1? Other students took the ingenious route of understanding what a list of 3’s does. When we had the list, I asked for observations about it. I often do this with my class to begin the processes of pattern recognition, problem solving, and problem posing. The list of observations was excellent!

“Number 1 is wrong”
“None of the patterns include a zero”
“All use only whole numbers”
“There are no negatives”
“They are all in ascending order”

Although I would love to detail the conversation along each of these points, I would rather use them to illustrate the growing nature of such a problem. These student inquiries can be used to explore the concept of Mean, Median, and Mode in a much deeper way. What began as mathematical play, quickly turned into serious mathematics. I would have loved to set up think-tanks to explore how negatives can be used in sets of numbers. Maybe a particular ambitious student would take on the trouble of finding an integer mean with fractions as data points. (I imagine this would lead quickly to paris of fractions that sum to 1, but I can never be sure where student thought leads). I chose to go into an interesting conversation about zero. I touched quickly on the history of zero, what adding zero to a data set would do to each of the three measures of central tendency, and constructing sets with a mean of 0. The last topic ran naturally into that of negative numbers.

I assigned the class to “fix” the first entry on our list, and describe, in words, their thought process. The topic of writing in mathematics is one for another day.

My point is, changing the focus of a very routine mathematical exercise changed the way the students saw the topic. It began to grow right before them. It is quite artificial for me to separate the extensions; it has already been shown that the topics intertwine. Posing questions in a playful atmosphere unlocks student drive. For an approach often coined, “Fuzzy Math”, it sure led me and my students into very serious (and curriculum supported) mathematics.

NatBanting

Fractions From Digits

This week marked my baptism by fire into the twitter world. It was not long until I was neck deep in tweets, favorites, re-tweets, and followers. The eternal nerd awoke inside me when I was confronted with my first NCTM “Problem of the Day”. A simple, yet dangerously deep, question was posed. Wanting to cement my reputation as a responsible twit, I sat down and began to tinker with the theory.


The question was as follows:

How many different fractions can you write using only the digits 1,2,3 & 4?
Be sure to include fractions greater than 1.

Immediately, I changed the word “fractions” into “rational numbers”. That way there would be no debate whether a number with a denominator of 1 is a fraction. As I began to experiment with the obvious nature of the problem, certain problem solving strategies emerged. I began to list observations and questions. Every digit, when placed over itself, will create a fraction equal to 1/1. I then became concerned with the reducibility of the rationals involved. What operations (+,-,x,/) are allowed on the digits? Can we repeat digits? It is not hard to see that the problem was becoming very large, very fast.

I decided that ordinary operations were not possible, because there was no way to know if a multiplication of two numbers in the set X={1,2,3,4} would yield a number whose digits were also in X. I then began to explore any operations that I could think of that may be acceptable on this very unique set. After addition, subtraction, multiplication, division, and exponentiation failed, I decided to take a step outside of the box. I decided to show that there are an infinite amount of rational numbers that can be created with the digits by “smushing” digits together.

I called this new operation “composition”. If you compose 1/2 and 3/4 you simply get 13/24. This way I was ensured that every subsequent “composed” fraction would have only digits in the original set X. From there I created a list of 11 rationals that had a single digit in the numerator and denominator. (1/1, 2/1, 3/1, 4/1, 1/2, 1/3, 1/4, 2/3, 3/2, 3/4, 4/3). I used the fact that these 11 were unique to begin the proof of this new infinite set of numbers.

From this point on the “@” symbol will represent composition. I now had the fact that:

a/b @ c/d = (10a + c)/(10b + d)

The first thing to be noticed that both a/b and c/d are in lowest terms and unique as rationals. So the newly composed fraction is also in lowest terms. The new fraction would be reducible if a common factor was available from all 4 constituents (a,b,c,and d). If such were the case, c and d would have a common factor, and wouldn’t have been in lowest terms to begin with. This contradicts the construction of the new fraction.

The second thing to notice is the uniqueness of the newly composed fraction. We know no a/b = c/d = e/f. Let’s assume that for some reason, we find two composed fractions where:

a/b @ c/d = a/b @ e/f

then:

(10a + c)/(10b + d) = (10a + e)/(10b + d)

so breaking apart the fractions we get:

10a/10b + c/d = 10a/10b + e/f

From here it is easier to see that this can only be true if c/d = e/f. We know this is not the case. So from the original 11 fractions, we can now create a set of two-digit fractions that are all unique and irreducible. This list continues to grow when we consider “composing” these two-digit numbers together to get 4-digit ones that follow the same rules. Only now the 2-digit numbers p, q, r, and s follow the form:

p/q @ r/s = (100p + r)/(100q + s)

In general, let ‘x’ be the number of digits of {p,q,r,s}, then:

p/q @ r/s = [(10^x)p + q]/[(10^x)r + s]

This pattern continues to create an infinite set of rational numbers from the digits 1,2,3, and 4. Of course, this list is not exhaustive–like most lists of infinite length. In fact, using this method, every fraction must have 2^n digits, where ‘n’ is a natural number.

Today’s problem has been yet another example of how an innocent problem, can lead in various directions. How many possible 4 digit combinations are possible in this set of numbers? 8-digit? Such are problems for another day.

NatBanting