Important note: The problems are stored as an image and it is currently appearing fuzzy and too small. Guess that makes this fuzzy math! TRY CLICKING ON THE IMAGE TO MAGNIFY IT. I will try to make the image larger but no promises. Blogger was down for awhile and they're having problems with images right now.
PLS READ COMMENTS FOR EACH DAY'S PROBLEMS. ANSWERS APPEAR IN THE COMMENTS AND I USUALLY POST THOSE MUCH LATER IN THE EVENING IF I REMEMBER! ALSO, SOME OF THE BEST CONVERSATIONS ABOUT DIFFERENT WAYS OF SOLVING THESE AS WELL AS TEACHING STRATEGIES, CONTENT, INSTRUCTION AND ASSESSMENT TAKE PLACE THERE.
Wednesday, January 24, 2007
Problems for 1-25-07
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5 comments:
For 6, the diagram is deceptive, since it looks like a parallelogram, but in fact, given the congruences stated in the problem, it probably isn't. (If it is, it's a square and all 3 statements are true.) Assuming it does not have to be a parallelogram, I agree with darmok's reasoning.
darmok, here's an alternate:
sqr(i) is going to be a complex number, right? so a + bi
a+bi = sqr(i)
(a+bi)^2 = i
a^2 +2abi -b^2 = 0 + 1i
Let the real part = real part and
let the complex part = complex part
a^2 - b^2 = 0
2ab = 1
Now, we have a quadratic system to solve. (I'll leave that for you)
I prefer this, at least at first, to using a formula.
The few responses suggest it was very hard to read the problems, yes? It couldn't be the problems themselves!
Answers to 1-25-07:
1) (D) d + 9m
2) 2
3) 18
4) E
5) 500 (if 'a' is assumed to be real)
6) D (II, III)
Comments:
1) As Darmok pointed out, #1 is ambiguous. These kinds of questions always are but mine could have been worded more clearly. Students should be encouraged to make a table to see the relationships. Thus, at the end of week 1 her salary was still $d or d + (0)m. Assuming she gets paid at the end of each week, her salary at the END of week 2 would be d + (1)m, or we could say 'after 2 weeks.' Thus, after 10 weeks, her salary would be d + (9)m. The problem of course is what is meant by the word 'after'. I guess I could have written it 'at the end of'. But then again, when you're born are you celebrating your FIRST birthday! In some countries, like Korea, you are 1 on the day of your birth!
2) straightforward but some students struggle with anything that looks like a fraction!
3) easy but good skill practice and many will still be off by 1 when counting
4) darmok's approach is beautiful; imagine if our students had the number sense and the skills to go with it
5) if we regard this as an SAT question, then only real values are allowed; if we don't specify domain, anything goes and a case could be made for -500!
6) wonderful analysis, darmok! i love the way you articulate your thought processes...
whoa!
where did all those other comments come from!
i apologize for not acknowledging mathmom and jonathan
nice discussion about complex roots; visualizing them on a circle is beautiful though...
Darmok, Dave,
About problem 5, you're trying to do too much, and your students will probably try to do too much too. You don't need to solve for all possible values of a to solve the problem. In fact, the point about whether a must be real is a good one. Yes, finding all the 8th roots of 25 can be a good test of persistence, but you don't need it. Just do:
a^8 = 25,
a^4 = 5 or -5 [if a is allowed to be complex],
4a^12 = 4 (a^4)^3 = 500 or -500 [if ...].
I used to ask my 'intermediate algebra' students questions like:
How many real roots does
3x^2 + x + 1 = 0 have?
Most of the students would use the quadratic formula, write both roots, and tell me whether they were real or not. I would explain that all they needed to do was to check whether the discriminant is positive, zero, or negative. They'd still compute both roots. I would tell them that I would deduct points from their exam scores if they did that. They would still compute both roots.
I think the students were so used to writing everything they could in attempts to earn 'partial credit' that they lost sight of the problem. Perhaps I was wrong to deduct points they way I had done--it was 17 years ago. But I warned them first!
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