Monday, May 23, 2016

MVPs and Fair Teams

You will not catch me claiming that problems need to be real world in order to be relevant. I would much rather think of classroom materials as either mind numbing or thought provoking. This continuum can be applied to hypothetical, practical, or genuine scenarios (a classification system neatly summarized in a chart in this article).

I see the greatest potential in scenarios that provide elegant entrance to mathematical reasoning. If it happens to contain a real world context, fantastic. Either way, it needs to be thought provoking. 

Take a look at the chart below:

If you don't follow the NBA at all, meet Steph Curry--he's kind of a big deal. He is the first unanimous MVP in the history of the league.

As I look at the table, a few questions immediately make it thought provoking. Some provide a more obvious link to a curricular outcome (in this case, systems of equations), but all foster what us Western Canadians call the Mathematical Processes (see p. 6) or what Americans might call the Standards for Mathematical Practice

(1) How many points is each position worth?
(2) What is the minimum number of first place votes you can get and still win the MVP?
(3) What is the best way to create two fair teams of five players?

If you take a minute with each of these problems, you will begin to appreciate their elegance. The first two appear to have a single solution, but many possible ways to approach its value. The last one is a matter of opinion, but questions of fairness prompt students to create arguments based on some type of numerate structure. While it is harder to see these arguments as explicit outcomes in a curriculum, they are undeniably mathematical. 

A closer look at each of the three questions provides insight into anticipated student milestones and possible lines of student reasoning.

(1) How many points is each position worth?

I think most students would typically start at the players who received the fewest votes to find some type of clarity in the sea of numbers, but the unanimous decision in the favour of Steph Curry makes the top of the table the easiest entry point. It is easily discernible that a first place vote is worth 10 points, but that doesn't move us any further into the field because he was the only one to get first place votes. It does, however, make the linear progression of 2-4-6-8-10 points a possibility. This is a reasonable conjecture, but quickly debunked by Kyle Lowry's point totals. 

Actually, If we assume that each denomination of vote carries an integral weight, Kyle Lowry's point total and distribution can result in only one possible value for a 5th and 4th place vote. We can then walk up the chart determining the value of a 3rd and 2nd place vote. 

Before this becomes too mechanical, take a look at the relationship between Kyle Lowry and Draymond Green. I had one student call this a "partial tripling". I may be tempted to eliminate the two intermediate lines in the table and ask students to determine the values of the votes with only these two lines. 

**Side Bar: What if the point values weren't whole numbers? Can you find a combination that works for Kyle Lowry? What about Kyle Lowry and James Harden? For how many players does your system work? **

(2) What is the minimum number of first place votes you can get and still win the MVP?

Students may use the answer from question one and determine the total number of possible points. They might make the assumption that a player would need over half these points to guarantee the win, but there are more than two people competing for the award. I imagine they would begin by securing every possible 2nd place vote, and seeing if it was enough. If it wasn't, they may also award 3rd, 4th, and 5th place votes. 

It is worth reminding students that a voter cannot vote for a player in more than one position, therefore the maximum number of votes any one player can receive (in any position) is 131. Also, each 1st place vote must be awarded to someone (and that someone cannot be you). Compounding the issue, you can only be voted for a total of 131 times. How can the points be distributed so you still win the MVP? How many players would even be eligible to receive a vote?

This question is a natural extension of the first; they make a fantastic pairing. 

(3) What is the best way to create two fair teams of five players?

This question has a distinctly different tone. Debates of fairness always seem to tie themselves in knots as students wade into the murky waters of data vs. experience. 

There have been four algorithms for choosing fair teams that I have heard students fight for: 

First is the straightforward one-for-one draft where the top two players become captains and take turns choosing the highest player available. This results in teams of 1-3-5-7-9 and 2-4-6-8-10. Opponents to this thinking are quick to point out that if you look at pairs of players all the way down the line-up, Team 1 has the advantage every time (1 v. 2; 3 v. 4; 5 v. 6 etc.).

Second, students aim to fix the problems in option one by imposing a snake draft. This means that Team 1 selects first, then Team 2 selects twice, then Team 1 selects twice...etc. This process continues until all players are taken. (Remember each team must have five players). This results in teams of 1-4-5-8-9 and 2-3-6-7-10. Some students point out that Team 1 still has the best player and Team 2 still must take the worst player. 

Third, students force Team 1 to take the worst player along with the best player. Each round, the team must choose the best and worst player available as a pair; this continues until all players are taken. Keep in mind, each team must contain 5 players. When it comes down to the 5th and 6th best players left, Team 1 takes the 5th and Team 2 takes the 6th. This creates a book end effect which results in teams of 1-3-5-8-10 and 2-4-6-7-9. In a slight alteration, some argue that a trade of 5th and 6th best players would make the teams more fair. 

Last, students argue that a combination of the snake draft and book end draft is best. Here, teams select the best and worst players available but in a snake fashion. Team 1 selects the best and worst, and then Team 2 selects the best and worst twice in a row etc. This results in teams of 1-4-6-7-10 and 2-3-5-8-9. 

Each of these arguments is supported by some type of numerate reasoning and communication based on their hypothetical ranks from one to ten. Some refer back to the raw data to justify certain approaches, but others argue that this years' results are not representative of the typical distribution of talent. Whatever the case, the context provides many natural (and curious) avenues into mathematical reasoning. When building, adapting, or choosing tasks for the classroom, the focus should be on the avenues for thought provoking activity and not the context in which you attempt to occasion it. 

A real-world context is the cherry on top. 


Sunday, May 8, 2016

(Min + Max) imize: A Classroom Game for Basic Facts

This is a game that was adapted from a colleague in my department. He can't quite remember where it came from, but knows there was some influence from his undergraduate days. Nonetheless, he reinvented it to play with his Grade 9s, and this post represents yet another reinvention.

The game has a simple mechanism (dice rolling), and endless extensions to elaborate on and play with. These are both keys to a great classroom game (for me anyway). 

(Min + Max) imize practices basic operations within the framework of larger, conceptual decision making. While I rarely bring up the probability of the dice rolling, it is obvious that students are making decisions based on the chances of certain rolls being obtained. The idea is to practice basic skills and order of operations in a way that allows students to be active, numerate decision makers. 

The game:
Each game begins with a structure. The structure is composed of a series of blanks (for the 10-sided dice rolls) and operations linking the blanks together in various ways. The students can clearly see how many rolls of the die will occur (one per blank) and copy the structure before the game begins. When all students are ready, the die is rolled. After each roll, the value must be placed in a blank. It is illegal to wait until all the rolls are complete to make choices; it is also illegal to switch a choice after it is made. (Although, when students try this, it allows me to see their thinking very clearly). 

The goal:
Before each round, decide what the target should be. I usually play three rounds. In the first round, we try to maximize the result. In the second round, we try and minimize the result. In the third round, we try and get a result as close to zero as possible. Each round, the structure of the operations is altered and the die is re-rolled. After all the numbers are rolled, I circulate and collect "high scores" from students. 

The boards:
Be creative with your game structures. I have had the most success with 4-5 blanks. After that, calculations can become a hindrance. As your classes get better, division and exponents are a great way to stretch their thinking. I have provided a few of the structures I use, but most are improvised in the moments of teaching in response to student's needs. 



Monday, April 25, 2016

Limbo: An Integers Game

Rationale: Create a game that embeds the skills of adding and subtracting integers into a conceptual decision making structure.

Objective: Insert a set of integers into a 4-by-4 grid so that the sums of the rows and columns is a minimum. 

Game Set-up:
All the students need is the game board and the list of sixteen numbers.
The board consists of sixteen boxes arranged in a four-by-four array. Space is left between the boxes to insert the addition and subtraction signs. You can give the students a blank board and have them all fill in the operations to match a board projected in the room, or you can write in the operations before you make photocopies. Every space between boxes needs to contain either an addition or subtraction sign. 

The arrangement is somewhat arbitrary, but it helps to have a fairly even distribution of both signs to increase the chances of students having a variety of skills to practice. The goal of the game is to arrange the numbers so that the sum of every row and column is a minimum. If the board contains a large majority of subtraction signs, then a high frequency of positive numbers would allow the sum to reach lower levels. You can play with a pre-determined list of numbers you give to the students. You could allow them to choose between two lists, or generate the lists randomly on the spot using playing cards. 

After all sixteen numbers are placed, students calculate the eight sums (four rows and four columns). The results of these eight are then summed to get their final result. 

Lines of Reasoning:
Students usually play the first round of the game with a pseudo-random strategy. Some only look at the rows while making decisions (this is the most common in my experience) and others only look at the columns. Most begin with the foundation that to get the smallest sum, you need to add negatives and subtract positives. 

It doesn't take long for students to realize that the boxes are more interconnected than originally evident. If we read the grid from left to right, the boxes on the right edge are more connected than the left edge. This means negative numbers fit naturally on the left edge. This seems like a great strategy, but there could be boxes on the board that are connected by two addition signs (one in a row and one in a column). It may be more profitable to use the negative numbers to negate both of those addition signs rather than placing it on the left edge. 

Boxes connected with two negative signs seem like they should hold positive numbers, but if there are no positives available, perhaps the smallest negative number must due. Is it then more profitable to remove a negative number from the left edge and place it there to avoid adding the positive number twice? These are some of the decisions to be highlighted. 

When final answers begin to percolate in, I like to ask, "What caused that arrangement to be lower?"
Contrasting arrangements on two boards is no simple task. If you want easy extensions, make the target the largest sum or the sum closest to zero. You can also only provide 15 numbers and allow them to insert a wild-card entry between -6 and 6. 

Game Board Download:
You can alter the addition/subtraction ratio as you see fit. You can change their locations. The list of numbers in this sample game has been randomly generated randomly and there is nothing special about it. This download is a starting point; play with it to create subsequent rounds of the game. 
Download a blank game board here.
Download the sample game here.

My students had some experience with integers, but remained tentative with their skills. This structure allowed them to think big-picture about the four possibilities of adding positives, adding negatives, subtracting positives, and subtracting negatives. I have a feeling that this structure may be overwhelming as an introduction.

Sunday, April 3, 2016

Fraction Task Testing

The testing of a task went horribly right.


Graham Fletcher (@gfletchy) tweeted an Open Middle (@OpenMiddle) prompt for comparing fractions.
The thread debated whether or not a representation on a number line would be best. 

Many people liked the number line better, but I decided to stick with the inequality signs because:

  1. Students see this type of two-bounded inequality notation with domain and range.
  2. The number line gave the impression of a single, fixed answer (because the fractions appear a definite, scaled distance away from each other).
I gave this question as a starter to a group of my grade nine students. They completed it in their portfolios

Brief Summary of Action:

It is very hard to follow all student strategy without some type of documentation, but as far as I could tell, most of the student strategy followed two scaffolds. 

  • Start with the (open) middle

These groups began by focusing on the five digits necessary to fill in the equality in the middle of the prompt. Many centered their action around equivalent fractions to one-half. This actually prompted a shift in the reasoning to what was impossible rather than what was possible. One student noticed that if you chose one-half, you could not use two-fourths because the two had already been used. Another pointed out that this was doubly bad because none of the numbers were two digits. This triggered their group to create a list of other impossibilities. One-fifth proved to be particularly useless. Two-tenths, three-fifteenths, and five-twenty-fifths were all ruled out. 

This action continued until the group settled on one of the possibilities. There was little debate about whether the choices made were correct, because almost every group acted as though multiple solutions were possible. (This is one of the strengths of the problem; it places students in an investigative stance). After the middle was chosen, the students created the two boundary fractions and compared sizes with common denominators or a variety of other estimation techniques. 

  • Set wide benchmarks
This subsection of the class was much smaller in size. They decided that the best way to ensure success was to create a really large fraction on the left and a really small fraction on the right. The problem would then be solved if the remaining five numbers could be written as an equivalent pair of fractions. These conversations were very fruitful as well. After hearing their strategy, I countered with the question, "How can you make a fraction as large as possible?" 

The answers generally organized themselves around one of two possibilities: one-ninth and eight-halves or the reciprocal. Then the groups listed out the remaining middle digits and attempted to create an equivalent relationship (all the time confident that it would fall within the set range). If they couldn't find one, they swapped out a single digit, and re-doubled their efforts. 

Probably the best conversation of the day came when I challenged the fact that one-ninth was the smallest possible fraction. (Even if there was a fraction with a double digit denominator). 

New Problems Posed:

I knew this task was going to provide conversation, but many of those conversations also posed new problems. Nothing out of left-field, but great problems because they 1) were simple and elegant alterations and 2) required a serious re-evaluation of the original strategy. 

Here are three:

Needless to say, what was planned as a ten minute opener to initiate classroom inter-action ballooned into a period long buzz of reasoning, argument, justification, and re-posing. That, for me, is a beta test gone horribly right


Wednesday, March 23, 2016

Dice Auction

Every student has a gut feeling when it comes to probability, and I feel like I have been too quick to theorize their gut instincts in the past. This year to introduce Grade 9 probability, I decided to exploit gut feelings to introduce the topic.

To do this, I needed a semi-familiar situation, some friendly competition, and a time pressure to make decisions.

To fit these criteria, I invented the Dice Auction.


You are invited to an auction, and given a budget of $10. Everyone at this auction has an identical budget. You all are bidding on possible events when two 6-sided dice are rolled. After all the spots are auctioned off to the highest bidders, the two dice will be rolled 20 times. Each time the event you purchased occurs, you collect a single prize. 

Bidding always begins at $1 and goes up in increments of $1. You cannot bid against yourself. The order of the events up for auction will be known beforehand. If you choose not to spend your money, the auctioneer will sell you a prize for $2 after all the bidding is completed.  

Your task: optimize the number of prizes you receive

Classroom Set-Up:

To begin class, I informed the students that they were attending the auction described above. I had a class of 22 students so I came to school armed with about 150 total "prizes" (an assortment of candies). I counted out groups of 10 pattern blocks which became our currency. Each student also received a single handout with this description of the events up for auction on the front and this set of assessment questions on the back. 

I gave them five minutes to decide which events they wanted to target. I also warned them that sharing their reasoning at this point might result in auction sabotage. I don't usually discourage the sharing of reasoning, but I wanted to delay it in this instance. I wanted to accentuate each student's individual gut feelings. 

After the five minute preparation time, the auction began. I opened each item at $1 and two pre-service teachers (@Mr_Harms_ & @HeidiLNeufeld) collected the funds after each event closed. It took about 30 seconds per auction event, and the presence of more hands made collection much easier. 

Once all the spots were taken, the dice were rolled 20 times and the prizes were distributed. Students kept track of how many times their purchased events were rolled (in the tally section of the handout) and later we combined these results so everyone had the data for their reflection questions. 

Task Action:

Bidding began rather timidly, and that enabled some to pick up spots for very cheap. I quickly realized that their perception of probability was very skewed. Certain events (like both number greater than or equal to 5) were selling for higher prices than others (like a single one is rolled). I didn't bother mentioning this in the heat of the action, but as the finds were being collected, I made comments like, "wow, that went fast" or "I thought more people would want that". These were more or less done randomly with no attention to the chances of them actually being rolled. I wanted students to think and re-assess on the fly. 

I could see student marking certain events from further down the list as they lost out on ones they wanted. I assumed they were trying to find events that had similar likelihoods of occurring. The patterns of bidding became recurrent, with the same students jumping in early, and others braving the high amounts. The $5 plateau seemed to slow down many bidders. 

I allowed students to buy prizes at the rate of $2 per prize because I wanted to offer a way for the very reserved students to take in the action, and still receive some candy. As it turns out, many used this strategy to guarantee at least one candy. I had two different students mention they were going to spend $8 and leave $2 as a guarantee. This struck me as interesting. 


I began by asking who felt they "won" the auction. Naturally, those with the largest stack of prizes asserted that they had. Others then claimed that they just got lucky. This was exactly what I was hoping for. The pressure of the auction format had forced students to think on the fly about the value of certain events. Naturally, missteps were made. We decided that it may be unfair to call the winner lucky until we determined how likely his events were to be rolled. 

One student had won the event "A single one is rolled" and it occurred 11 times. I began by asking students how many possibilities for rolling two dice exist and this led us to the inevitable conversation about a (2, 1) being unique from a (1, 2). I anticipated (and even prompted them toward) this. 

We ended up listing all 36 possibilities in an ordered pair structure (no one suggested we use a table). After we had that, students were quickly testing to see if they overpaid. I purposely chose 20 rolls because it necessitated students to compare fractions with denominators of 20 and 36. 


I gave the student four assessment questions; they were briefly introduced before the period ended. I essentially wanted them to reflect on the events, and their likelihood. I got an incredibly high percentage of assignments back the next day, and they included very insightful comments:

" 'Numbers multiply to a prime number' was underpaid for because I think many thought that you can't multiply to a prime. We forgot about 1"

"I knew I was guaranteed 6 because of 'missed three in a row'. I am happy with this, so I didn't overpay"

"I've used dice a lot, and they have bigger chance of landing a 6 or a 1"

"Sum to 8 occurred less often because I mean it's a number that's in the middle with many chances"

" 'missed three in a row' was overpaid for because you can only win 6 prizes. You should invest somewhere with more potential"

" 'No one else collects' was rolled less often than I thought. It surprised me that someone collected every round."

It was clear to me that students continued to think about their actions, and some even provided how they would approach the auction differently. It might be a nice extension to ask them to "set fair prices" for all 30 events.

I love the conversations surrounding this task. Students were upbeat, blaming each other for overpaying, and then justifying why they weren't. I used the happenings to introduce the idea of favourable events, total events, probability, odds, and even got into the fundamental counting principle. For my Grade 9s, that is almost the entirety of the curricular outcomes (although we play with them a lot longer because... it is fun). 

This task slowed the theorization of their curiosity. By allowing them to follow their gut instincts, the decisions became personal and the task became vibrant. Notation arose out of the necessity to communicate how they conceptualize chance. That is the starting point for any unit on probability.


Tuesday, February 23, 2016

My Favourite Surface Area Question

Surface area is intuitive. Intuition is a natural hook into curiosity. When you think something might (or should) be the case, it begs the question, why? It just seems as though textbooks haven't gotten wind of that.

Perusing the surface area chapter of the assigned textbook for my Grade 9 math class offers a steady diet of colourful geometric solids all mashed together (at convenient right angles) in various arrangements. Without fail, the questions ask the same thing:

Find the surface area of...

Best case, students are asked to "create" a mimicked amalgam of standard solids and then calculate the surface area of their creation. Almost no mathematical decisions are made in the process of the creation. The question may as well read:

Do something random, and then follow strict procedures to arrive at a meaningless calculation.

I would like to afford students the opportunity to make meaningful mathematical decisions. That doesn't mean the questions have to be exotic or complicated, but chances are they don't involve trying to convince a teenager that they need exact calculations in order to purchase paint for their next re-modeling.

Here is my favourite surface area question ever (and I took it from a textbook*)...
...and here is what I ask the students to do...

Design an expansion for this house that doubles its surface area. The expansion must share some portion of a wall with the original house. 

*Actually, textbooks are a great starting place for inspiration.
The remaining attributes of good problems emerge out of a combination of 1) an eye for meaningful mathematical action and 2) teacher curiosity. (I mean, we want our students to be curious. The least we can do is be curious ourselves).

I group them randomly into 3s and give them a large whiteboard to work on. The results begin predictable, but the avenues for re-calibration of their actions are incredible. They quickly discover that the problem is not so simple. 

Most groups start one of two ways:

The "back-to-front" design
The "side-by-side" design
I think both of these are logical steps, and contain beautiful mathematical reasoning. It also uncovers a key understanding to surface areas and their overlap. What area is actually lost when the surfaces touch?

Most groups try to compensate by building additions to the existing structures of various levels of difficulty; most typical is the "accordion" strategy. This is where students push and pull the expansion (like a prism) until the surface area matches their goal.

The "accordion" strategy 
Again, this is filled with wonderful mathematical thinking. Watching a side-by-side group accordion is particularly interesting. Do they pull out both sides? or just one? Do you include the roof? or just pull out a rectangular prism?

The problem would be cool if the thinking stopped there, but it never does. If you give the student space and an intuitive problem on which to act, you will get super cool alternatives that are not necessarily practical in terms of actual building design. But in math class, they represent brilliance.

Take this group's Escher-like solution of balancing the house together. Their justification was, "We are not bound by the laws of Physics". I asked if these two houses shared a wall, and they reluctantly re-organized their design. What they didn't do was slide the roofs together a little. This surprised me. 

Creative. Correct. Didn't follow ALL instructions.
I've had students overlap and then design balconies. I've had some punch enough windows to compensate. I've had groups append random hallways. This group got more than they bargained for with this elaborate and realistic solution.

Student design before dimensions are added.
Surrounding all of these results is the sphere of possible actions. Most of the actions are based around the strategy "overlap an entire wall, and then compensate for the loss". Very few act on the strategy "overlap a small portion of a wall and make a small alteration". Sometimes I ask them why they overlapped so much, and they usually respond that it is easier to work with larger numbers because surface area grows so fast. This is also a great noticing. 

The procedure of finding surface area is embedded in all of this is. In that sense, the original instructions of "Find the surface area of..." are still there. They are just steeped in the possibility of student action. It is the simplicity of the intuitive hook that makes this my favourite surface area questions ever. Maybe that's because I don't know which strategy I like best.


Thursday, February 18, 2016

I have been thinking about extending the Fraction Talk love ever since I wrote this initial post in June 2015. 

I have used them with my grade nine classes as the starter during units on rational numbers. I have taken the larger questions (such as "What possible fractions can be shaded using this diagram?") as the prompt for entire lessons of student activity. I have used them to create great conversations with grade 7 and 8 students at our school's annual math fair. 

I finally found the time (honestly, I found the tech guy... many thanks to @evandcole) to begin a collection of images and house them in a central location. 

That location is

The site is modelled after other #MTBoS spin-offs like, visual, and

The goal is to create a usable resource for teachers to begin building fraction numeracy with their students. There is also the opportunity for you (or your students) to create and contribute groups of images on the CONTACT/SUBMIT page. 

The content is organized into categories based on the type of shape. A brief rationale is provided on a HOW TO... page and the FRACTION TALKS HANDOUT page has a downloadable .pdf recorder sheet for teachers hoping to create a fraction talk routine with students. 

As teachers, your feedback is very important to me. I want to make this as usable as possible. You can tweet (@FractionTalks) or email (fractiontalks at gmail dot com) your feedback. 
(Twitter is, by far, the most effective way to contact)

Enjoy, invent, contribute!