## Tuesday, May 08, 2007

A colleague showed me a problem from a publisher-created test on the Algebra II topic of sequences and series. The problem was horrid.

It's one thing to ask for the next three terms of an arithmetic sequence or of a geometric sequence. But to give a random sequence, and then expect there to be only one approved solution, frosts me. Why? Because it instills in the students a belief that math is whatever the teacher says it is, and that the object is to read the teacher's (or the publisher's) mind in order to get the correct answer. In math, the exact opposite of that belief is reality--but that's not what problems like the following imply.

Here's the problem: Give the next three terms of the sequence 5, 7, 10, 15, 23, ...

The approved solution from the publisher is
5, 7, 10, 15, 23, 35, 52, 75
How did they get this? Like this:

The problem is, since we haven't identified what kind of sequence we're talking about, there's another easy-to-come-by solution that is just as mathematically justifiable:

Both are correct because, again, we haven't identified what type of sequence this is. The solution key gives only the first answer, even though there are reasonable answers aplenty.

Tony said...

It's funny, but before I scrolled down, your solution was the first one that jumped into my head. Maybe it's because I was just reading about Fibonacci and I have similar sequences on the brain.

Robert said...

Same here -- I was thinking Fibonacci the whole way. I always disliked these kinds of questions too and for the same reasons.

Anonymous said...

Wait a second! What's your rationale for the 7 to 10 jump of 3? I see your justifications for 10 to 15 and 15 to 23. The "approved" solution has a justification for 7 to 10, but yours does not, so I'm wary of your characterization "just as mathematically justifiable."

That said, it would have been a better problem if they'd included the 35 in the sequence.

Darren said...

My justification is the same as is used for the Fibonacci Series--the first two differences are both given before the rest of the series itself "takes off".

Anonymous said...

anonymous, you don't need a justification, it is a given. You use the given data to determine the series.

Darren, this seems like a typical problem from my interaction with the local math curriculum.

Chris

KYRIE said...

These kind of problems used to make me go crazy especially in those days in high school whn we had only learned Geometric sequence and arithmetic.
My country edu system was solely based on standardized exams and the govt had just announced that it had inculcated creative thinking into the exam system, so whn I approaced my teacher for help she told me to apply creative thinking without any explanation on how to solve the math problem!

Darren said...

At least they tested your creative thinking :-)