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!


Friday, February 12, 2016

Candies, Pennies, and Inequalities

I want students to solve systems out of necessity.

I want them to feel the interconnectedness of the two (or three) equations. In the past, I've asked small groups to build a functional 4x4 magic square. Soon they realize that changing a single number has multiple effects; this is the nature of the system. Unfortunately, abstracting the connections results in more than two variables. This year, I wanted to create the same feeling with only two variables. (The familiar x & y).

Enter: Alex Overwijk.

We blitzed through a task of his for systems of equations when I participated in a workshop of his last year. His blog is fantastic, because he recounts classroom events. It is filled with straightforward stimuli for the practicing teacher. His ideas have occasioned many tasks of my own.

I took his post, and extended it into inequalities.

Day 1: Equalities

I purposely chose a system that
  1. Didn't use too many pennies
  2. Did not have a solution over the set of natural numbers
  3. Had 3-4 options of "close calls"
I settled on the following situation (directly from Alex's post):

Two children go into the candy store. Bob buys 3 JuJubes and 4 Smarties for 26 cents. Sally buys 2 JuJubes and 7 Smarties for 24 cents. If every JuJube costs the same and every Smartie costs the same, what is the price of both candies?

They grabbed a handful of pennies and coloured tiles and got to work setting up the situation in much the same way Alex's students did. 

After 3-5 minutes, students started to get close to solutions, but they continued to evade them (by design). My conversations with the groups went something like this:

Me: What's up?
Them: It doesn't work?
Me: What doesn't work?
Them: The prices. You can't do it. There is some left over.
Me: Then increase the price.
Them: [adding pennies] but then this one doesn't work.
Me: Then I guess take some away to make them match.
Them: But then we have too many...

...and it would continue on. They couldn't make a move (change a price) in isolation. This is the feeling I wanted from the system--interconnectedness.

[One student asked if they could break the pennies up, and I said that thinking was interesting, but not possible. I planned on bringing that line of thought up after class discussion]

As the discussion of strategy began, the class quickly realized that no solutions existed. Some began with high prices and then punched numbers in calculators in linear combinations until they matched. Others worked in tandem with one student calculating JuJube price and the other Smartie price. Some began with both candies being one cent, and then walked up until they reached the boundary. All contained great mathematical action, but none yielded a satisfactory answer.

I then began to alter the notation.

First, I defined JuJube to be "J" and Smartie to be "S". Then re-wrote the equations.

3J + 4S = 26
2J + 7S = 24

Second, I replaced "J" with "x" and "S" with "y".

3x + 4y = 36
2x + 7y = 24

Third, I numbered the equations.

(1)   3x + 4y = 36
(2)   2x + 7y = 24

At this point, several "ooooooohhhh" sounds came from the class. They had just finished solving systems of linear equations in grade 10. We then refreshed the vocabulary of this process and I gave them the next scenario (one with a nice solution). We ended the day talking, once again, about the interconnectedness of systems.

Day 2: Inequalities

I began the day by randomly grouping the students and placing a familiar problem on the board.

2 JuJubes and 2 Smarties cost 18 cents.
4 JuJubes and 3 Smarties cost 33 cents.
How much to JuJubes and Smarties cost?

They got to work arranging their pennies and quickly arrived at a solution of (6 , 3). I then posed a new scenario.

Two young gentlemen want to impress their girlfriends by giving them a dynamite Valentine's gift. The first can spend up to 18 cents, and wants to buy his sweetheart 2 JuJubes and 2 Smarties. The second can spend up to 33 cents, and wants to get his sweetheart 4 JuJubes and 3 Smarties. What could the prices be so that both gentlemen can get their desired gifts while remaining under budget?

The take up was slow until one student suggested that they steal the candy. This was a great starting point because I told them they wouldn't have to steal them if all the candy was free.

Him: "You can do that?"
Me: "Would that mean both gifts were bought under budget?"
Him: "Yes"
Me: "Then it seems to fit. Zero cents for a JuJube and zero cents for a Smartie is a possibility"

Then I asked groups to find all possibilities. I put them into a table in Desmos to show the pattern.

The result was a stippling pattern on the natural number solutions. This created a natural way to talk about the different domains and ranges that the problems require. I have never had such an organic entry into stippling; I always just said those were the spots that were "allowed" because of restrictions.

Day 3: Abstraction

On the third day, we went through examples of notation and mechanics of solving the problems. It was interesting to hear how many times a student would refer to candies or pennies when explaining things to other students.

My goal here is to mimic Alex's style and provide you with a start--an actual account of how it ran in my classroom. You know your students best, and have a intuitive feel for when they need more scaffolding or higher ceilings. Take this, adapt it, and extend it further.


Monday, January 25, 2016

Desmos Art Project

This semester I gave my Grade 12s a term project to practice function transformations. I began by sourcing the #MTBoS to see who had ventured down this road before. Luckily, several had and they had great advice regarding how to structure the task.

I use Desmos regularly in class, so it was not a huge stretch for them to pick up the tool. I did show them how to restrict domain and range (although most of them stuck exclusively to domain).

I gave them the project as we began to talk about function transformations, and they had 3.5 months to complete it. They complained, but the results were fantastic. (...bunch of drama queens).

Couple of important points, and then I'll let you peruse/steal the handouts and view the samples of student work (of which I am extremely proud).

Pointer #1: It was important that students copied a piece of art (this was typically a cartoon of sorts). Making them copy a pre-existing piece meant they must think about how the parameters shift to match. No lines are arbitrarily chosen.

Pointer #2: Illustrate how a variety of functions could model the same segment of line. When I do it again, I may even have weekly challenges as they are introduced to more and more function variety. Something small. I may project a simple image and ask, "What functions would you use to draw this?"

With all that out of the way, here are the materials I used:
  • Here is the handout I gave them. (It stresses the pre-drawing as well as the replication of a piece of art)
  • Here is a tutorial sheet that someone (whom I forget, but please remind me) gave me to show a simple example from quadratics.
  • Here is a .pdf of 9 samples of student work.
I was skeptical throughout the process because they resisted giving me updates on their progress. On the whole, they were fantastically done. I can also say that they did very well with function transformations on exams.

Now that I (and you) have samples of work, it will go all the smoother the next time.