Middle School Math Puzzle
Since there are middle school math parents and magazine editors among the core readership of this blog, I thought this might be interesting.Keenan recently had a whole page of trapezoid area problems. Depending on how you approach it, you can get different answers, which I thought was a bit unfair. I'll explain what I'm talking about later, in case anyone wants to take a crack at it.
Labels: precision, puzzles, recreational mathematics
posted by frank b. at
6:27 PM
11 Comments:
what are we solving for? that little square?
For the area of the whole trapezoid, I get 24.75 square units. I am not sure how you could get a different answer...The drawing isn't really very accurate given the numbers, though. Maybe that's where the confusion comes in.
24.75 is the official answer:
= 3*(11.5+5)/2
but it's not the answer I got when I first tried it. Hint: what if you decide not to ignore the lengths of the non-parallel sides?
(Yes, it's not even close to scale, but neither were the problems in the book. The book did show straight lines, though.)
do tell, frank. do you mean using a sine of an angle or something? How close are these 'other answers'?
Well, someone did e-mail me the alternate answer, so I'll post that later today.
Solving it another way, I found the area of the rectangular part and then the areas of the two triangular parts and then added all three parts. I used the Pythagorean Theorem to find the length of the sides of the triangles. Done this way, I came up with approximately 24.975 square units, which is slightly larger than the "official" answer. Maybe the trapezoid as drawn is not actually physically possible...
well, if you divide it up into three parts, the rectangle has an area of 15, on the right you have a 3-4-5 triangle, and then the left triangle, well, you run into some problems. I'm going to say DNE.
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Andy's correct. The preferred solution involves disregarding the lengths of the sloped sides to get the official answer of 24.75. For this to be true, however, at least one of those sloped sides has to have been rounded off.
The non-preferred solution is to treat the shape as a rectangle and two triangles (a technique the textbook had already covered, which is why I thought the problem was unfair!). Of course this way is harder as it involves square roots. It also requires different assumption about which measurements must have been rounded off.
Since the measurements are only precise to the nearest unit, and one or more values have been rounded, the correct answer is actually 25 - an answer can't be more precise than the inputs to the answer.
As a teacher (but only a fifth grade teacher, so maybe I'm not qualified to comment on middle school math) I often suspect that textbooks are not very well edited or proofed by the publishers. (The same goes for cookbooks actually.)
let's say no numbers are rounded off. you have a 3,4,5 triangle, a rectangle of sides 3 and 5, and a triangle of sides 3, 4, and root 7.
if you look at the longest side, it is 4 + 5 + sqrt 7. root 7 is about 2.645. that means the longest side is coming in at 11.645 (at the least)
so it can't be 11.5!
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