Monday, June 29, 2015

(0,0),(2,3) opp vert of a rectangle. Max area? And it's not 6!

Just posted this problem on Twitter. Might not get seen by too many educators/students who are on summer break here in US but when an idea comes to me I have to publish it or it will perish!
My passion is writing questions which promote divergent thinking and dialogue while developing conceptual understanding of the Big Ideas of Math. And of course encouraging all 8 Mathematical Practices simultaneously!
Since most texts have a dearth of these nonroutine questions I found myself creating my own when I was in the classroom. Now I share them with my online "family".
---So the answer is 6.5. But to me the EXPLANATION is always part of the"answer"!
---Would you give this problem or a version of it to 6th graders? Earlier? Only students in a geometry class? Only accelerated/honors students? My belief is it's appropriate for many "levels" but how we PRESENT it will change!
---Of course students need to sketch or graph it but is there benefit from both hand graphing and use of software like Geogebra? I believe the software can open vistas and promote inquiry not possible with just a manual sketch but a balance is still important. Learning HOW to use interactive geometry software is an aim here but it's not an END!
---Can you predict how many of your students would consider rectangles other than the obvious one whose sides are parallel to the axes? Should asking for the  "maximum" area suggest there is more than one possible rectangle, in fact infinitely many? Would you give them the "answer", 6.5, and have them justify it?
---How exactly would you want them to draw and consider other rectangles? This is not an obvious issue at all in my opinion.
---Would it be too much of a reach to expect a DEMONSTRATION of WHY the square is the rectangle of maximum area with a given diagonal? Would you relate this to the important idea that the triangle of max area with 2 given sides is a right triangle?
---Do you think discussion in class would lead students to a deeper understanding of the diagonal properties of a rectangle and the square as a special case? It isn't necessary for us to anticipate ALL the BIG IDEAS which emanate from problem-solving. What do you see as the main ones here?
---I depend on your comments here otherwise I'm writing in a vacuum. Your thoughts and constructive suggestions are not only welcomed but strongly encouraged!

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