Much of what appears in mathematics textbooks is what I like to call, downstream thinking. Downstream thinking usually involves two features that set the stage for learners. First, it provides a context (however doctored or engineered–often referred to as “pseudo-context”). Second, the problem provides a pre-packaged algebraic model that is assumed to have arisen from that context. I am imagining the reasons for this are three-fold: Instructional time. Developing these models is messy and takes time, especially with class sizes in the mid-to-high 30s. Accessibility. Developing these models isn’t always possible (i.e. can’t run trials and collect data on every …
Continue reading »Category: functions
Desmos Art Project
[Post Updated June, 2018] 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 …
Continue reading »WODB: Polynomial Functions
If you haven’t experienced the conversation stemming from Which One Doesn’t Belong? activities, you are missing out. As far as I can decipher (#MTBoS feel free to correct me), this all began with Christopher Danielson’s Shape Book centered around this structure. From there, a crew of tweeps (headed up by Mary Bourassa) established WODB.ca (YES! Canadian) to curate a collection of problems of this format. My unit on polynomial functions (either in Foundations of Mathematics 30 or Pre-calculus 30) requires students to decipher attributes of polynomial functions from their graph and vice versa. These include end behaviour, sign of …
Continue reading »Connecting Quadratic Representations
I always introduce linear functions with the idea of a growing pattern. Students are asked to describe growth in patterns of coloured squares, predict the values of future stages, and design their own patterns that grow linearly. Fawn’s VisualPatterns is a perfect tool for this.While stumbling around Visual Patterns with my Grade 9s, we happened upon a pattern that was quadratic. The students asked to give it a try, but we couldn’t quite find a rule that worked at every stage. While I knew this would happen, the students showed a large amount of staying power with the task. The …
Continue reading »Polynomial Personal Ads
Every year, my students study the general characteristics of polynomial functions. We investigate the various shapes of various functions and slowly shift parameters to watch changes in the graphs. Eventually, we deduce the roles of the constant term, leading coefficient, and degree. It should be noted that Desmos makes this process much easier than years previous. Just set up the generic polynomial, add sliders, set specific ones to play (depending on what you want to investigate), and have students discuss in groups. See sample here. (Sliding “a” to “0” invites an excellent conversation; same with “b” etc.)After we work with …
Continue reading »Linear Functions With a Bang
Many teachers tell me that it is their creativity that limits their ability to be adaptive in the classroom. Somehow the “reform” movement (or should I say re-movement) has pigeon-holed itself into a connotation where high-energy teachers give vague tasks to groups of interested students. Out of all this, curricular outcomes explode in no particular order. This can’t be further from the truth. In my view, the biggest steps toward changing student learning is changing teacher perception.When presented with a topic to cover, there are two dominant ends of the Math-Ed spectrum. First, you have the transmission approach which carefully …
Continue reading »Destroying Functions
I have spent the better part of 2 weeks going over various mathematical relationships in my Grade 10 class. They have been represented as tables of values, arrow diagrams, and sets of ordered pairs. Relationships, both qualitative and quantitative, have been defined, analyzed, and graphed. My focus on graphical literacy has been previously detailed on the blog. See the link below for details: http://musingmathematically.blogspot.com/2011/10/graphing-literacy.html Numerous relationships were handled. Students we required to create a family tree and then represent its branches as a table of values and set of ordered pairs. Throughout the various exercises, the words “input”, “output”, “domain”, …
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