Some SAT-types for today (more reasonable)...
1. A rectangle is inscribed in a semicircle so that the base of the rectangle lies on the diameter. If the diameter is 4 and the base of the rectangle is twice its height, what is the area of the rectangle? (Sorry, no diagrams today!)
2. If 1 + 2 + 3 + ... + 2006 = x, then, in terms of x, 2 + 3 + 4 + ... + 2007 = ?
(A) x - 1 (B) x + 1 (C) 2x (D) x + 2006 (E) x + 2007
3. On a certain 20-question math contest, 12 points are awarded for each correct answer, 4 points are deducted for each incorrect answer and a blank receives zero points. What is the fewest number of right answers needed to score at least 100 points if no questions are skipped?
Note: Although many students will use guess-test here (which is fine), we're looking for an algebraic approach as well.
Wednesday, January 17, 2007
Problems for 1-18-07
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6 comments:
I had a horrible geometry year in middle school, and never learned to do these. Now I know, recognize a missing segment... That's number 1.
Number 2??? I hope that doesn't trip up too many kids.
The factoring problem from a day back? I told you I used it with sophomores (as a time killer). Today I shared it with some freshmen, at the end of a review class. I was surprised - they chucked it into the calculator, then argued about how to move the decimal. I think that problem is a keeper.
I liked all three of these :) The first one could again be used for 45-45-90 triangles (if I did it right heheh).
I agree with jonathan on #2, I would hope kids would notice how to add/subtract and not get hung up on the sequence
#3 took me a few minutes to think through, but as long as kids can identify their variables, they should be okay.
Again, great job! :)
Here are the 'official' answers (not that you need them!)
1) 4 2) D (x + 2006) 3) 12
Other than a careless error with the radius (equals 2 not 4), Darmok's analyses are wonderful. Doing these in your head is great mental gymnastics but I need pencil and paper since visualization is not my strong suit...
Joanathan, thanks for trying these in the class and reporting back. I'm not surprised some tried to enter the powers of 10 factoring problem into the calculator. Now while we might be taken aback, the fact is that it led to a nice discussion of how to interpret the result. This is an aspect of calculator use often overlooked. The technology raises new questions that we would never have considered!
BTW, #2, is a typical SAT question that, IMO, develops reasoning and is a good vehicle for demonstrating several strategies. I like the add/subtract method mentioned by mrs. temple and darmok, but the other method mentioned by darmok is very nice also. Since the ti-83,-84,-89 can do summations (using LIST operations), some resourceful student could actually find each sum. Then again someone might invent a method we couldn't anticipate. The dialogue is so powerful here!
Finally, I'm wondering if the 4 of you are the only ones trying these! (Yes, yes, I can see I have many visitors daily but I have no idea why they're visiting or if they exit rapidly when seeing these problems!). Am I going to eventually lose my readership with these or do you think a few problems and a few comments about my passionate belief in a standardized curriculum are enough? I really appreciate your comments. There is nothing more gratifying for an 'author' than that one good review, never mind 4!!
Dave
The wording of (3) is somewhat confusing. I can answer 9 questions correctly, choosing not to provide answers to the rest and still get 108 points. Does this mean I skipped the 11 questions? A wording I would have preferred would be 'What is the minim um number of correct responses needed if no question was left blank?'
I totally agree with totally_clueless! I got careless with my wording about 'skipped' vs. 'left blank'! The intent was that no question was left blank. Thanks!! Ah, the joy of writing unambigous math problems. That's why math people choose to communicate in cryptic symbolic code that is unambiguous to them! And I pride myself in using clear language!!
darmok--
Nice! Some students will always know those formulas for the sum of an arithmetic series! My goal is to show students how to solve these kinds of relationship questions without using the formula but the formula is always there if you need it...
In school we focus on procedures and formulas so much that when confronted with problems in which there is no obvious formula or it's forgotten, the student tends to give up too easily!
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