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The problem:
The pyramid show above [not drawn to scale] has altitude h and a square base of side m. The four edges that meet at V, the vertex of the pyramid, each have length e. If e = m, what is the value of h in terms of m ?
(A) m ÷ square root of 2
(B) m x square root of 3 ÷ 2
(C) m
(D) 2m ÷ square root of 3
(E) m x square root of 2
answer: A
source The Official SAT Study Guide, 2nd edition
, p. 401
14 comments:
This is a Level 5 problem. Students have 1 minute & 30 seconds to work it if they're lucky.
Did you draw the picture wrong in the first place? Why did that picture help you?
I had to draw my own, and the one I drew was much more helpful to me. It showed a triangle made out of two opposing edges and the diagonal across the base. I only had to solve one triangle then to get the height.
oh!
I'm glad you asked.
I should have mentioned in the first post that this is one of those SAT problems in which the accompanying visual is identified as "not drawn to scale."
You learn from the problem that the figure is equilateral, but the pyramid in the drawing is isosceles.
For me, drawings RADICALLY trump words, partly because I'm staring at the drawing, not the words. But it's more than that.
I'll write a post about this at some point. For the time being, suffice it to say that it's fantastically difficult to 'detach' from the actual drawing you see on the page.
Betty Edwards' drawing classes are all predicated on developing an ability not to see what you naturally see (a whole) & these 'not drawn to scale' problems take you one step further; you have to force yourself to 'see' a figure that's not there at all.
In any event, the isosceles drawing kept me from grasping the implications of the fact that side e = side m, and I got sidetracked into trying to remember whether diagonals in a square might be equal to the side, etc., thus allowing me to use 1/2 of the diagonal as the leg of a right triangle.
When I saw the Cliff's Notes image it jogged my memory somehow -- it had to do with seeing the two right angles indicated in the Cliff's Notes drawing. But I no longer remember exactly why that caused me to go back to the problem and register the meaning of the fact that the figure was equilateral.
dang
The picture you posted isn't drawn to scale either. It also "shows" faces that you could interpret as isoceles. That's why I don't understand--what's different about this picture than the other?
re: being able to detach:
That's the point. Math class is not about working with what you see, but with what you're told. You're supposed to have enough facility to "draw your own picture" symbolically.
Now, I don't think pictures should be misleading, which is why you should draw your own. But competency in math isn't about drawing more accurate pictures, it's about manipulating the right math equation corresponding to what's known.
Here, you needed to figure out what triangles were built where one side was the height of the pyramid, and you knew the other angles involved.
How did you end up solving the problem?
Catherine,
You should *IMMEDIATELY* buy Polya's How to Solve it.
When you problem solve, think Dragnet and "Just the Facts, Ma'am".
The posted problem is a pre-algebra or Alg I problem, depending on when the rationalizing the denominator section of the radical operations unit was done. Don't let someone's labeling of 'it's hard' scare you or influence your opinion of its difficulty sight unseen.
As you read, label the facts on the figure. Keep in mind that one never uses the figure to determine angles or dimensions as they can be distorted during the printing process - only trust the facts given.
Once the figure has been labeled, the solver only needs to know the Pythagorean theorem, that all angles of the square are right angles, and radical operations. He should know what the diagonal of the square is in terms of the sides, but he has enough time to figure that out if he doesn't.
Dolciani et al's Pre-algebra: An Accelerated Course has a good unit on applying algebra to triangles and a good unit on beginning probability.
The picture you posted isn't drawn to scale either. It also "shows" faces that you could interpret as isoceles
I know.
What was different was that two right angles were specified - especially the right angle where h meets the base. The height as indicated in the CollegeBoard drawing looks like a slant height - although I knew it wasn't. But that added a second element of 'wrongness' to the image I was staring at.
I was trawling my memory for factoids about slant height (cones! triangular prisms! pyramids!) and diagonals of regular polygons....
fyi - for all of you who are teaching kids: I did not 'see' that the height is perpendicular to the base the first time I did a problem that involved the height in a pyramid.
If you had **asked** me whether the height in a pyramid was perpendicular to the base, I would have said 'yes.'
BUT when I did my first problem ever I was flummoxed.
The way in which the height is indicated in drawings makes it look like a slant height -- very slanted in some of the drawings.
I've owned a copy of Polya for years.
Haven't read it.
The posted problem is a pre-algebra or Alg I problem, depending on when the rationalizing the denominator section of the radical operations unit was done.
Wait!
Are you talking about the other problem?
Yes, that's a middle school problem (middle school competition problem).
Definitely algebra 1.
(Well, not algebra 1 when I took it back on the farm. I never saw radicals in a denominator until I worked my way through Saxon Math.)
That problem wasn't hard for me - it was fun.
That's a terrific book, I think.
lgm - thanks for the tip.
I worked my way through Dolciani's chapter on probability in her algebra 2 book -- had no idea there was a chapter in her pre-algebra book.
I'll take a look.
Polya writes in dialogue, so he's speaking out loud, and talking to his student. I think it would help you to turn your own verbalizations into the math.
Good advice!
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