Tuesday, May 18, 2010

Challenging Geometry Assumptions: Review for SAT I/II

The video below presents a more challenging 3-dimensional geometry problem which would be at the upper end of SAT I or SAT II - Subject Tests (Math I/II). The key here is to challenge students' assumptions about a quadrilateral being a square because it has 4 congruent sides, a common error. This question will also review a considerable amount of geometry: Pythagorean Theorem, Volume of cube, spatial reasoning, 45-45-90 triangles, area of a rhombus, etc.

As always, the focus is on the art of questioning, suggested instructional strategies and pedagogy, although this problem may be interesting enough to capture the attention of some students who are preparing for upcoming standardized tests. For students who need help with spatial visualization, a model could be provided or have enough empty boxes available (they don't have to be cubes!).  

I strongly urge using learning partners or pairs for the discussion. 

Benefits include:
(1) Students feel less tentative when offering ideas to one other person or in a small group.
(2) Instead of posing conceptual questions to individuals, receiving little or no response except from the most confident or capable, you can pose a question to a learning pair: "Julie and Jason, what is needed to insure that ABCD is a square?" They should be given a few moments to think and confer before responding. The stronger student will usually explain it to the other. If neither can respond, they can say, "Pass!"
(3) The biggest advantage of student dialog is that often our explanations simply don't click with several students, but they do make sense to others. Those who "get it" can usually explain it in terms that their peers understand better, a benefit to both the "explainer" and the "explainee"!

By the way, the question posed near the end of the video is worth pursuing if time permits:

"Without calculating the areas, ithe area of the non-square rhombus less than or greater than the area of the square?"

The answer is less for many reasons, but we would hope they would recall the base x height formula for a rhombus. The height is maximized when the angle between the sides is 90°. Why? Interestingly, the areas are quite close: 19.6 vs. 20. I believe strongly that this is the type of higher-order question that not only reviews important concepts but promotes deeper thinking, or should I say, thinking more than one inch deep!

What are your thoughts? Would you give students the e√3 formula before a standardized test or ever?Are these videos helpful to you? If you respond both on this blog and on my YouTube Channel, MathNotationsVids, and also rate these videos, that gives me the guidance I need to improve them.

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific


Sol said...

Dave - this is a great video. I really like how many concepts it ties together.

Sol said...


I liked this problem so much that I stole it and blogged about it:


Anonymous said...

There's a relatively easy solution using vectors and the cross product. I describe the solution in the comments here:


Dave Marain said...

Sorry for the delay in replying Sol and Mr. Chase--
Sol, Sharing is what the blogosphere is all about. I get most of my ideas from others! It's a compliment for you to discuss my question.

Mr. Chase--
I completely agree that vector cross products are designed for areas of parallelograms but I made the decision to demonstrate a method accessible to less advanced students. I should have mentioned a vector approach however ( or did I, I can't remember...).