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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 can grow (or shrink). In the past, I had used digital images of cubes growing after having their dimensions scaled by 2, 3, 4… etc.; students seemed to grasp the pattern yet under-appreciate the girth of 8, 27, 64… etc. times as many cubes.

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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 the now familiar 2-dimensional shapes.

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The Scale of Coffee Cups

A colleague is a religious McDonalds’ coffee drinker. One day she showed up with a medium coffee and a cream on the side. It was in two separate cups:

I asked her for her cups when she was done. (She is also a math teacher so understands that this is not a creepy request. It is no weirder than the time I bought 400 ping pong balls, or 1500 bendy straws). I then made her a request to buy a large and small coffee in the future and save me the cups.
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Stations in High School Math

One of the coolest experiences in my university training was the opportunity to invite a kindergarten class into our mathematics methods class for a mathematical field trip. Our class was divided into groups of three or four and were given the task of designing a mathematical activity that the students would try. The afternoon was a hit. Each group set up shop around the room and the kids freely moved from station to station as they mastered each activity.

Somewhere along the way, mathematics becomes formalized and stationary. I imagine it is around the time of fractions. I assume this for no better reason than teachers and students alike seem to blame most of their problems on fractions. That is until Grade 10, when polynomial factoring squeezes out fractions as the most hated mathematical procedure.

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Project Work Scaffold

There are two schools of thought when it comes to PBL:

1. Start with a large-scale project and fit the specific outcomes within it, or
2. Build toward a larger project with smaller tasks.
I love the idea of large projects, but also aim to make my work as accessible as possible for those who want to take it and improve on it. I just don’t see option one working within my traditional classroom of 35 students for one hour a day. The existence of an overarching curriculum only further decreases its accessibility.
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Soft Drink Project Part 5: The Show

This is the finale of a series of blog posts detailing a student posed project. To get the full picture, begin reading at part one:

Soft Drink Project Part 1: The Framework
As the project drew to a close, students began to place a valuation on their work. Very seldom did the topic of grades come up during the process, but even students know they are playing a game. They asked me how I would be grading, and I told them we would be using our self/peer/teacher model as always.
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Soft Drink Project Part 4: The Math

This is the fourth in a series of posts detailing a student-posed math project. To get the full picture, please read the previous posts beginning with:

Soft Drink Project Part 1: The Framework
This post is designed to dampen the fear of math teachers. I know, because I was very afraid that the project had missed the mark until students moved into this phase. For some reason, teachers feel like they have more ability to complete a list of outcomes if they dictate the exact way, pace, and form that the learning will take. My division states they want to create life-long learners; in this model, the only lifelong learners are teachers because they must continue to do all the learning for their students day after day.
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Soft Drink Project Part 3: The Design

This post is the third in a series of posts detailing the happenings of a math project. To better understand the whole story, please start reading at the beginning:

Soft Drink Project Part 1: The Framework
The next few classes after the brainstorming class were a blur. Students would come into class, grab their previous work, and get down to business. It was the best I could do to have supplies waiting for them. I learned quickly that students can become pretty demanding when it came to their learning.
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Soft Drink Project Part 2: The Brainstorm

This post will make a lot more sense if you read the framework for the project in “Soft Drink Project Part 1: The Framework“.

I left the classroom energized; I could not remember a time that I was more pleased with a lesson that I had taught. In fact, I wouldn’t even call it teaching. I was observing. The process of brainstorming began organically. I had my doubts that it would continue the following Monday. Typically, students can’t even remember where they sit after a weekend–let alone what task they ended on.

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Soft Drink Project Part 1: The Framework

This post is the first in a series describing a set of classes in my Grade 11 Workplace and Apprenticeship class. I have designed the course around the ideals of Project-Based Learning (PBL); students encounter a series of tasks, problems, and prompts that necessitate three crucial qualities: Collaboration, Critical Thinking, and  Communication. Each unit leaves ample room for student extensions and mathematical forays into more elaborate pursuits. This unit was no different. Students studied the topics of Surface Area and Volume through a series of tasks, problems, and prompts–one of which ballooned into the subject of this blog series.