Sunday, March 1, 2009

Updates: RSS Feed, Preparing for SATs on Pi Day, A Ratio Problem,...

As I await the blizzard of '09 here in the Northeast, some updates...

Any problems getting my RSS feed via Google Reader or other aggregators? Blogger, which recently purchased Feedburner, required all users of Feedburner to transfer their feed. This may have caused a temporary disruption of the feed for MathNotations. I updated the feed address for redirecting my readers as of 2-28-09 and I inserted a new 'gadget' into the sidebar allowing for re-subscribing if needed. I noticed that the number of subscribers was cut in half when I transferred over so I'm hoping it will correct itself in a couple of days. Please email me at dmarain at gmail dot com to let me know if you're having any problems getting the feed. Either way, let me know. Apparently other Bloggers are having similar problems with their feed.

SAT Day is March 14th. How appropriate. May all of your students score at least 250π on the math section!

An SAT Ratio Problem

Here's a common SAT type of problem (above-average difficulty) which all students should know how to approach. I've chosen this because it demonstrates different mathematical approaches and test-taking strategies:

In Mr. Jonas' AP Stat class, the number of left-handed students is three times the number of right-handed students. If one-fourth of the lefties and one-third of the righties in the class play an instrument, what fractional part of the class plays an instrument?

(A) 7/12 (B) 5/16 (C) 11/36 (D) 5/18 (E) 13/48

Notes, Comments, Solutions, Strategies,...

(a) To make the wording more convoluted and difficult for students, I could have used the noxious "three times as many as" phrase. I chose to avoid this as it would weaken the reliability of this question IMO. Your thoughts?

(b) How would you categorize this problem? Prealgebra as it can be handled by ratio considerations? Algebra since it can be solved algebraically? Are questions like these typically included in middle school texts here in the US? Singapore texts in their primary materials?

(c) In your opinion, would most juniors in HS approach this algebraically or would they use the most common SAT strategy, 'Plug in'?? Is this question so obvious that it's rhetorical! More importantly, do most students really know how to use this method effectively? How many students know how to organize the information in the problem using a tree model? A Punnett Square model?

(d) How would teachers of Singapore materials explain this question? What model would their students be encouraged to use?

(e) What % of your students would have the level of ratio sense to do this:
(1/4)⋅(3/4) + (1/3)⋅(1/4) = 13/48.
As an aside, how many would attempt the arithmetic without a calculator!

(f) What is your opinion of the distractors? Could the intuitive student eliminate some answer choices quickly, narrowing the options to one or two and making an educated guess? Would this question make a better grid-in (student-constructed response)?

(g) If you felt that the proportion of left-handed students in this problem was highly abnormal, I can only reply that since I'm left-handed, I must be in my right mind!

I would like to share some other methods I've seen successful students use as well as delve further into the conceptual and skill foundations for this problem but I'll stop here for now. If you would like me to go further with this, let me know by commenting or emailing me directly.


Maria Miller said...

I have just finished rewriting my book Math Mammoth Fractions 2, which has lots of material for fraction multiplication... so I instantly thought of your solution at (3)... just multiply them fractions.

For a model, seeing as it talks about a third and fourth, one could first try draw 12 blocks or units. Then color the lefties and righties...

Then you can immediately color in 1/3 of the righties (which is 1/12). To color in 1/4 of the 9/12, you can color 2/12 and then 1/4 of one more twelfth. Total: 1/12 + 2/12 + 1/48 = 13/48.

Dave Marain said...

Nice, Maria. Good luck with your new volume. BTW, are you able to get my feed?

I'm wondering if Singapore students might try various multiples of 3 blocks for the righties. If they use 3 blocks, then 9 would be needed for the lefties but it's difficult to show 1/4 of 9, so they might experiment with other multiples of 3, say, 12 blocks for righties and 36 blocks for lefties. This works nicely since 12 is div by 3 and 36 is div by 4. Of course 12 is the lcm of 3 and 4 so it makes sense to use multiples of 12 for the 2 groups. Playing around with different units seems helpful in the development of ratio concepts.

To summarize (without a graphic model):
Righties: 12 and 1/3 of 12 = 4
Lefties: 36 and 1/4 of 36 = 9
Thus 13 out of 48 play instruments!

mathmom said...

For what it's worth, I'm still getting your feed just fine.

I added a comment on your Four Factors post, since I tried that with my MS group.

Dave Marain said...

Thanks, mathmom!

I always leave it to the discretion of the professional educator to reword and revise my problems/investigations to accommodate the needs of each group of students. Asking students to find the 2 categories was highly challenging and the dialog which results is really the environment we want to foster. Thank you for the discussion of the Four Factors. Let me know how your students felt about this challenge.

mathmom said...

Professional educator, haha, yeah, I'm an imposter, remember? ;-)

I plan to write up another post about how finding the categories went. In short, only my son found them. After also giving them the "does doubling the number double the number of factors" problem to investigate, he thanked me for giving him something really hard to work on. :) The others are less likely to thank me for challenging them, but they did seem to be enjoying trying to find patterns, and not just frustrated. I was pleased with how both went.

Dave Marain said...

My characterization of you stands!

I deeply appreciate all the feedback. I'm no longer in the classroom every day to experience student reaction to the challenges I love to write. Hearing that your son and others "enjoy" these challenges is very gratifying.

BTW, I never expected my students to applaud after working through a difficult problem, particularly when they were unable to crack it. Very hard for adolescents to appreciate at that moment the 'no pain, no gain' approach to learning!

mathmom said...

Your comment about the kids applauding reminds me of this story: When I was in college, I took a class in problem solving from Prof. Ross Honsberger. At the end of each class, he'd ask for a show of hands: "How many are glad they came today?" I always was, as were many others. ;-)

My guys felt that they were finding interesting patterns on the double-factors one, even if they weren't able to find "the" unifying pattern. It was really a great class.