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I learned about their Illustrating the Standards for Mathematical Practice modules at the NCSM conference a few years ago. While I haven’t reviewed all of them, I’ve been using ideas from their Congruence & Similarity module ever since both with my students and in professional development with teachers.
One of my favorite tasks from the module is Hannah’s Rectangle Problem.
Students determine which rectangles are similar to rectangle a and explain how they know.
Tracing paper, scissors, and straightedges were available for students to use. (Although my straightedges really are rulers, I did ask my students to refrain from measuring.)
I set the class mode to individual and sent a Quick Poll to collect their results.
I think CS shared with the whole class first. (I’m writing the post two months after the lesson & can’t remember for sure.) CS had traced rectangle a & transformed the rectangle so that it shared a center with each rectangle.
I know that c, b, e, and g work because the space between the dilated rectangles has the same “width” all the way around. CS showed us what he meant, talking about the “frame” that forms.
AC added that she did that, too, but she eyeballed the length between corresponding vertices to check for similarity.
Another student noted that h works because the two rectangles are congruent, which of course made us think about why similarity does not imply congruence but why congruence does imply similarity.
What about one that doesn’t work?
CS shared what he did with rectangle d, and showed how the horizontal width of the “frame” isn’t the same as its vertical width.
Can we check whether one rectangle is a dilation of rectangle a using a different center?
At this point, we watched the video that accompanies the NCSM learning module, so that students could see Randy’s method. Had one of my students come up with Randy’s method, of course we wouldn’t have watched the video, but they didn’t, and so I was glad for a different source of learning other than me.
(Randy draws the diagonals of the rectangles from a shared vertex to show whether or not the rectangles are similar.)
I love that the rectangles not only gave my students the opportunity to remember how we define similarity from grade 8:
CCSS-M 8.G.A.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
But also gave them the opportunity to practice one of our high school geometry standards as well:
CCSS-M G-SRT.A.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.
Students then dilated a figure using our dynamic geometry software and played. I watched using Class Capture.
What happens when you move the center of dilation?
What happens when the scale factor is 3?
What changes when the scale factor is –3?
What is significant about a scale factor of –1?
And students transferred what they had learned to paper:
Dilate ΔXYZ about point C by a scale factor of 2.
Dilate points A and B about point O by a scale factor of 3.
And so the journey continues, with many thanks to NCSM for all of the work they do to provide worthwhile resources to the mathematics education community.