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 fractiontalks.com.

The site is modelled after other #MTBoS spin-offs like estimation180.com, visual patterns.org, and wodb.ca

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.