## Tuesday, August 14, 2007

### Mean & Median of Arithmetic Sequences on SATs/Standardized Tests - The BIG IDEA!

Here are two standardized type questions that students sometimes struggle with. Those who do well on these kinds of questions know the key is to understand the basics of arithmetic sequences. The 2nd question is a bit more sophisticated. What changes did I make to complicate the picture?

1. The median of a set of 20 consecutive integers is 14.5. What is the mean of the first 10 of these?

2. The mean of 98 consecutive odd integers is 44. What is the greatest of these numbers?

Notes:
BIG IDEAS for ARITHMETIC SEQUENCES:
(a) MEAN = MEDIAN!
(b) MEAN = MEDIAN = (LAST + FIRST)/2
(c) N = (L-F)/D + 1
Can you guess what these variables represent?
Hint: This is a variation on a well-known formula for arithmetic sequences.

Anonymous said...

Nice problems.

I don't get why the mean is equal to the factorial of the median, though :-)

TC

Jackie said...

laughing at the factorial comment...

anyway, I wonder how many will catch the fact that in the second problem, the constraint is odd integers?

I like the hints/big ideas.

Dave Marain said...

Gee, I guess I should be less excited when stating math formulas!!!! By the way, is it a coincidence that the answer to the 1st question is 5.5, exactly five less than the original median? [I was going to end that question with an exclamation point, but now I'm self-conscious!?!?!?! - ok, I'm over it...)

jackie- the more students are exposed to these the less likely they will overlook key words. The hints are really powerful ideas that are sometimes overlooked when discussing arithmetic sequences. One doesn't need to memorize formulas (or store them on their graphing calculators) if one understands the 'big ideas'. Just my two cents of course.

BTW, why do my investigations (the previous 2 posts) usually get zero comments but when I ask questions like these, I get reactions? Is it because the other posts are viewed more as classroom activities and less as math problems to discuss?

Dave Marain said...

Oops--
I meant 9.5 not 5.5 (no exclamation point needed here).
I should know better than to post anything in the PM when my brain shuts down. I am definitely a morning person...

Jackie said...

re: BTW - I don't know. I was away for a few days. I honestly expected to read a great discussion on the inscribed/circumscribed problems. Rereading this post, I wonder if it could be that no questions were asked at the end?

As for the factorial, had a similar conversation with math team on arrangements!