Making Sense of “Same”

Monday, Jackie’s classes wrote updated conjectures about what they thought it meant for ratios to be equal. The language varied a bit from class to class, but they generally looked something like this:

Two ratios are equal if one is scaled up or down to get the other one,  or when simplified, are the same ratio.

Jackie then introduced the word proportional, subbing it in for the word “equal” in their statement.

That’s not too shabby. Its taken quite a while to get to this point, but we’re counting on (hoping?) this investment in conceptual understanding to pay off in the long run!

Tuesday, students dove into investigating “same” shapes by making rectangles with areas of 24 square units. There was lots of debate as they looked at how their shapes were the same and different. So far, they have:

If two shapes are the same kind of shape, have the same area and same dimensions, they are congruent.

Since they were only looking at rectangles, there was no mention of angles in their conjecture. We asked them to compare the bases to heights because we wanted them using their recently written statement about proportional ratios.  They were somewhat surprised to found out that even congruent rectangles, when drawn with different orientations on paper (and therefore with different bases and heights), do not have the same base to height ratios. Their home assignment (because we ran out of time– WOW it took them so long to make those rectangles!) was to choose one of their rectangles, and make another that is not congruent but does have the same base to height ratio. I saw a lot of puzzlement on their faces, but by reminding them that they just said congruent rectangles have the same area, clouds parted (well, for some of them) as they realized the restriction on area was lifted.

(I’ve got to start remembering to snap photos of student work, img_1804 if only to insert and break up all this text!  How’s this alternative?)

My biggest concern was not enough discussion participation by all students; we’re really pushing sharing reasoning, and the pace of a whole class discussion feels sluggish to me.  Probably to most of the students, too.  The ones with Hermione Syndrome seem like the only ones getting anything out of a whole-class discussion. I realize that many other students may be actively tuned into the discussion, but are not comfortable with raising their hand to voice their thoughts. I know this is true because when we pose a question to the whole class, *crickets*, yet when we then say, “Turn and talk to your neighbor about…..” suddenly we see kids talking and gesturing with their hands. So cool to listen in!

Jackie is realistically expecting that not every students will follow through with the rectangle assignment. Will there be too few, making discussion and moving forward awkward? Her plan is for them to use their proportional rectangles to wrestle with defining a different kind of “same” (similarity). She predicts/hopes their conjecture about similar figures will include something about proportional ratios, and is going to push students to use their sketches to “see” and justify proportionality in the linear measurements. Students who need it can expand into wondering about the growth in area.

Then in small groups, they’re going to look at three sets of triangles to refine their notions of similarity. During prep, Jackie and I had a lengthy discussion about angles. They matter, but we don’t want measuring them to distract from the main focus on proportional ratios. We think/hope these sets will address the congruent angles issue pretty well on an intuitive, visual level.

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