Transformation Geometry

When I was in elementary school, I was in the “gifted and talented” program. We were pulled out of classes once every 6 days or something like that for three years but to be honest I can only remember two things we did: logic puzzles and this mirroring thing where one person drew something and the other had to draw the exact mirror image.

One of the things I love about the IB MYP is its emphasis on investigation, so when it came time to do a unit on transformation geometry in MYP 4 (9th grade), I immediately thought back to that mirror project from elementary school. In the first day of our unit, we did translations — simple things like translating a point to more complex things like translating an equation (with its seemingly counterintuitive substitutions).

Then came our mirror investigation (a formative assessment, to practice for a real graded investigation). In each group of four there were two “objects” (the drawers) and two “images” (the mirrorers). In the first round, one pair used the x-axis as their mirror and the other the y-axis.

but how will that car go??

Object and image

 

Students then located corresponding points on their graphs and tried to come up with a rule about what happened to a point (x, y) when reflected over their specific axis. They conferred with the other two in their group to ensure that everyone knew what happened when reflecting over both axes.

ok but how beautiful is that student's handwriting? (also, i love my tape)

An example of the rule found by a group (notice that different groups express their rule differently)

 

Part two had them reflect over the lines yx and = –x, write the rules for these reflections, and share in their group. Part three was the practice assessment part where they had to work alone to then apply those rules, first to just a point, then to a line, and finally, I asked them to practice something unfamiliar, which was to reflect a point over the line y = 3, to check their reasoning skills. (Is it devious to on a test ask them to reflect over a line like y = 2x – 4? What other fun things can I ask them to do?)

I love this project and so do they, but I can’t help but wonder if the focus of this entire unit isn’t a bit too algebraic? The IB MYP is really unspecific on what exactly needs to be covered (just listing Transformation Geometry as one topic within the mathematical branch of Geometry that must be covered somewhere over the course of 5 years), but the Diploma Program (culminating two years, equivalent of 11th and 12th grade) has very little focus on geometry whatsoever.

So therefore I get that our textbook’s (and thus partially my) approach is to prepare them for the importance of translating graphs, for example, within the context of the vertex form of a quadratic equation and how the a value in a quadratic functions as a dilation or understanding how to manipulate trig function graphs. I see this even better now that I teach MYP5 (10th grade, the last year before the challenging DP) and I’ve worked to incorporate graphing calculators into our MYP4 class (we just got a class set this year!) so that even though the students haven’t yet tackled quadratic or exponential functions, they’re able to graph them and transform them already.

Still, am I missing something in this? Is transformation geometry really just algebra in disguise? (On that note, is everything we do just algebra in disguise? Sometimes I think so and I haven’t decided whether that’s a good or bad thing.)

Proof that the sun DOES sometimes shine a little in Holland.

A selection of finished products

 

 

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About katenerdypoo

Middle Years and Upper Grades math teacher in the Netherlands teaching both within the Dutch national curriculum and the IB MYP and DP.
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3 Responses to Transformation Geometry

  1. Susan says:

    You might want to think about how transformations are used to “move” 2-d objects on a screen for starters….here is a little lesson that uses them in coding. May be more than you want….
    http://processing.org/tutorials/transform2d/

  2. Hello
    I have been reading your blogs and found this. I have been thinking about transformation geometry lately, and it seems that everyone is stuck to rectangular coordinates. These are virtually useless for investigating rotations, until the dreaded topic of matrices is reached. I thought it would be a good idea to use polar coordinates for rotations around the origin and reflections in a line through the origin as well. The kids don’t need to know ANY of the heavy stuff, just use a polar grid. I just did one and you are the first to hear about it. Here it is:

    Let me know if the resolution is satisfactory. There are other ways of transmission.

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