Twitter Problem 10-18-14
If (a,b),(-a,-b) are opposite vertices of a square, show that its area=2(a^2+b^2)
EXTENSION: What if (a,b),(-a,-b) are adjacent?
COREFLECTIONS
(1) What do you believe will challenge your geometry students here? The abstraction? "Show that"?
(2) Predict how many of your students would "complete the rectangle" by incorrectly drawing sides || to the axes?
(3) Even if not an assessment question, is it a good strategy to "plug in" values for a&b? This is worthy of more dialog IMO...
(4) How many of your students would question the lack of restrictions on a&b? Would most place (a,b) in 1st quadrant without thinking? So why doesn't it matter!
(5) Is it worth asking students to learn the formula "one-half diagonal squared" for the area of a square? I generally don't promote a lot of memorization but this one is useful!
(6) EXTENSION
Answer to extra question: 4(a^2+b^2).
Ask your students to explain visually why this area is TWICE the area of the original square!
(2) Predict how many of your students would "complete the rectangle" by incorrectly drawing sides || to the axes?
(3) Even if not an assessment question, is it a good strategy to "plug in" values for a&b? This is worthy of more dialog IMO...
(4) How many of your students would question the lack of restrictions on a&b? Would most place (a,b) in 1st quadrant without thinking? So why doesn't it matter!
(5) Is it worth asking students to learn the formula "one-half diagonal squared" for the area of a square? I generally don't promote a lot of memorization but this one is useful!
(6) EXTENSION
Answer to extra question: 4(a^2+b^2).
Ask your students to explain visually why this area is TWICE the area of the original square!
Posts
3 comments:
Dave,
In regard to memorizing the area of a square as d^2 / 2, I would more prefer generalizing to "the area of a rhombus is the 1/2 the product of the diagonals.
Agreed, Pat. In fact we could generalize further to any "kite". My suggestion was based more on the greater frequency of standardized test questions relating to squares. There is also a nice visualization for (d^2)/2 to which students should be exposed.
Nice to hear from you again...
Great Work Dave Marain! Thanks for this Blog that it helps me a lot to clear my concepts mostly we have very little bit knowledge about the formulas and now hope that i sought it out very clearly.
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