Experiencing Scale in Higher Dimensions

A colleague and I have often bemoaned our attempts to teach the concept of scale factor in higher dimensions. A topic that has such beautiful and elegant patterns and symmetries between the scale factors consistently seems to sail directly past the experience of our students. I have tried enacting several tasks with the students including some favourites from the #MTBoS (Mathalicious 1600 Pennsylvania and Giant Gummy Bear). Each time, the thinking during the task seems to dissipate when new problems are offered. It just seems like students have a hard time trusting the immense rate that surface area and volume …

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Solid Fusing Task

The progression followed by most teachers and resources during the study of surface area and volume is identical. Like a intravenous drip, concepts are released gradually to the patients so as to not overdose them with complexity. Begin with the calculation of 2-dimensional areas, and then proceed to the calculation of surface area of familiar prisms. (I say prisms, so a parallel can be drawn to the common structure for finding the volume of said prisms. That is, [area of base x height]). In this way, surface area is conceptualized as nothing more than a dissection of 3-dimensional solids into …

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Real-World: An Attack on “Relevance”

**deep breath**Last week, I caught myself saying something to a pre-service teacher as we planned a Grade 1 lesson for the making of 10s. I asked her,  “Why would the students need to know how to make up 10s?” When she was auspiciously silent, I filled the space with a statement said entirely tongue-in-cheek. It was only upon reflection, that I kicked myself for not being able to shut up and allow her to think. I said, “…because my job is to convince teenagers they need logarithms, and that is much more difficult.” Now, aside from the unwarranted attack on …

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100 Rolls Task

Most probability resources contain a familiar type of question: the two-dice probability distribution problem. Often times, it is accompanied with questions concerning the sums of the faces that appear on each dice. For example: Roll two fair, 6-sided dice. What possible sums can be made by adding the faces together? What is the probability that: a) the sum is 6 b) the sum is a multiple of 4 c) the sum is greater than 15? I think the obsession with this specific subdomain of probability questions stems from the elegant way in which a table of outcomes (pictured below) leads to a …

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TDC Math Fair 2016: A Summary

Background:On June 15th, my Grade 9 class and I hosted our second annual math fair. What started out as a small idea has grown into a capstone event of their semester. This year, we had 330 elementary school students visit our building to take part in the fair’s activities. Several people (following the hashtag #TDCMathFair2016) commented that they would like to do similar things with their student transitions. This post details the rationale behind the event, how we structured it, what stations we had, and feedback/advice from our exploits.Rationale:I pursued this opportunity with a two-pronged focus. First, I wanted to showcase a …

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Mathematics Is: Student Impressions

I have taught the second half of a Math 9 Enriched course for the last three years. The students generally finish two-thirds of the curricular outcomes during the first semester (with an different teacher). This alleviates the perpetual nemesis of time, and leaves me with no excuse to stretch the boundaries of what is possible in a classroom.  I spend most of the time developing a classroom ecology focused around conjecture, community, and curiosity. The result is a constant focus on problem shaping, solving, and re-posing.  At the end of the semester, I ask students to respond to a simple …

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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 …

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

**this post was elaborated on in the May 2016 issue of The Variable from the SMTS.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 …

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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 …

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Fraction Task Testing

The testing of a task went horribly right. Background: 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: Students see this type of two-bounded inequality notation with domain and range. 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 …

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