## Thursday, September 23, 2010

### Optional Homework

Avery started an interesting discussion over at his blog, Without Geometry, Life is Pointless. He wants to give some challenging problems as part of the homework, and told the kids (6th grade) that they shouldn't spend more than 30 minutes on the homework - it's ok not to finish. One student complained about it, and "dislikes math for the first time". What to do...

Lots of good, interesting advice. Bowen Kerins said:
Another way to deal with speed demons is to give several problems in a row that are related and have the same answer. Speed demons may not even notice this happening, and the result is they're more likely to "look around" a little more before and after working a problem.
I liked that. And it made me think about an interesting thing that happened to me today.

I'm teaching beginning algebra at a community college. My first two mastery tests are on pre-algebraic topics. There is one fraction story problem. On version 3 of the test, it goes like this...
I have a lot of books at my house, especially after all the math books I bought during my sabbatical. Right now 3/7ths of my books are math-related; 3/10ths of those are kids’ books. What fraction of my books are kids’ math books?
I had marked my student's subtraction problem wrong, and was explaining to her why it would be multiplication. As I finished up, she pointed to her answer, which was the same as mine.

I said I hate marking people wrong when they have the right answer, but that this was a fluke. Subtraction doesn't solve this sort of problem. On the spot, I made up another problem, 1/2 of 1/3. Guess what.

Both answers are the same again.

Optional homework: When does this happen?  ;^)

(I give my students lots of optional homework. Most of it is: "Read this cool book and write a review.")

1. I liked his post, too.

My homework is now for a time period also. Much fairer among different students, and to all students to know what the expectations are. (Hour per workshop, but this is college.)

I like the idea of homework being for a purpose: if it's a skill, do it until you have the skill. Demonstrate the skill.

I do give choice home workshops, and have found that - on the whole - students make amazing choices.

2. ( let b be a rational number in (0, 1/2] and )

let a = b/(1-b). then

a - b = b/(1-b) - b

= [b - b(1-b)]/(1-b)

= b^2/(1-b)

= b*[b/(1-b)]

=ba = ab.

i got a last-second fill-in gig
at big state u starting yesterday
(hired the day before). so i'm
teaching again after 5 quarters
in exile. also madeline has set
up the net in her home again
(alas). so maybe i'll be back
to posting in some quasi-regular
way. doesn't seem likely.

http://vlorbik.wordpress.com

3. (Owen! It's so good to see you here! Will email.)

But, how'd you discover a? I couldn't think with your fractions being single letters, so I gave b the form n/d, and worked through your steps that way.

And I'm not seeing why b has to be less than 1/2.

4. i've assumed we want 1 > a > b > 0
so that a & b 'll be "common fractions"
(as i think i've heard said for suchlike
rational numbers a & b) and so the
difference a-b 'll be positive.

then you just set up ab = a-b
and fire up the algebra:

b = a- ab = a(1-b)

and out pops a = b(/1-b).
then you just shove it
into the old "theorem/proof"
obfuscation machine.

lose the parenthetical assumptions
for the general case of course
(or rather, replace with b\=1).

ot

5. much more interestingly (as i hope):
letting b = n/d in our context gives
a = [n/d]/[1-n/d]
a = n/(d-n).

(b, a) = (n/d, n/(d-n) )
is a more interesting formula
i think... particularly in light
of the *matching numerators*...
and students might be less
intimidated (subtractions
are easier than divisions).

one is reminded here of the theory
of "partial fractions" from (typically)
calc ii...

6. the character-string "(/"
does not appear
in elementary algebra
and should be replaced
with "/(":

"out pops a = b/(1-b)". 