From the same Common Core document:

I have no beef with the mathematics here. Close readers will recognize echoes of recent proportion discussions on this blog.

But really, what’s up with those triangles?

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The mathematics I encounter in classrooms

From the same Common Core document:

I have no beef with the mathematics here. Close readers will recognize echoes of recent proportion discussions on this blog.

But really, what’s up with those triangles?

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One can imagine continuing the discussion of beyond height and width to include the length of the diagonal, and the length of the line from the vertex of the rectangle to the midpoint of the diagonal. This would go well with the standards on scale drawings in Grade 7 and the standards on similarity in Grade 8. It may well be that an earlier draft of the document had such a discussion. This document is itself still just a draft, and we can take the triangles out if they are a distraction, or perhaps add a comment about the extension. Which do you think would be better?

Hmm. Yes. Triangles. I suppose my only question is, “Why??” Has this problem ever come up in real life?

Yes,

Bill, if you’re taking input at this grain size, I vote we lose the triangles. I’m happy to have them suggest a richer geometric exploration, but that’s not what the diagram is for. The diagram is trying to illustrate two ratios: one of corresponding parts of similar rectangles and one of two parts in the same rectangle. The glaring colors of the triangles make it hard to focus on these two relationships; they distract.I do not, for the record, share

Karim‘s concern about whether analysis of these rectangular-embedded triangles has ever come up in real life. I would be perfectly content to have kids explore some abstract geometric relationships here.But it’s a moot point in the present discussion, as the document in question is trying to illustrate a mathematical relationship for teachers and curriculum developers. It is not intended for classroom use.

Thanks, I’ll take input at any grain size. The more we can remove the burrs in the final product, the better. I agree there’s no need for every mathematical exploration to have a real life connection. But as it turns out this one (in its extended form) has quite a good one in scale drawings. There’s a nice problem (Shell Centre, I think) about a drawing of the beams for a pitched roof, with trusses somewhat like the drawing, where students have to calculate the amount of timber they have to buy from the drawing.

My immediate thought when looking at the picture was to wonder what the area of the red triangle was. That didn’t wind up appearing anywhere in the problem, but I had that thought distracting me (to use Christopher’s word) in the back of my head throughout the length of the illustration. I second the motion to omit the triangles if they are not going to be used.