I'm basically teaching two levels of geometry this year even though we call them both "preAP". I have the 8th and 9th graders in one class, and for the most part they whiz through everything and delight in the challenging problems. I'm also teaching 10th graders that are in other sections, and they approach math in a different way. I have to keep reminding myself not to rush rush rush through topics and to keep remembering that they need more practice on everything to make it stick and to make it make sense.
I am doing one thing, though, to help them keep their algebra skills fresh: ALGEBRA :). For example, for the last few topics, I made sure to make up geometry problems that ended up having a factorable quadratic equation in it at some point. This led us to recall FOILing and ("claw"ing .... Thanks, Mrs. H) and factoring. Now I'm making sure to include quadratics that CAN'T be factored (hello quadratic formula review).
BUT. Here are some things that I didn't think would come up, but did/do, and REALLY I should make a mental (or better) note to myself of where things can go askew, so that I can make more problems throughout the year to have such examples, so that we can have a discussion about them and keep them fresh in our minds.
A student is solving: 2x + 4 = (1/2)x + 8, for example.
Hmmmm, I don't like that (1/2)x,
so I'll just remember that I can do the opposite to x to undo it,
and I multiply ONLY the 2x and the (1/2)x by 2 to end up with:
4x + 4 = x + 8. Eeeeek.
A student gets to a point in an equation where they have (13/2)x = 14. Well, heaven forbid we keep things in fractions and go the easier route of multiplying both sides by 2/13 to get x = 28/13. Boom. Done.
We convert to 6.5x = 14.
We stress that 14 is not easily divided by 6.5.
We chug through and do long division and create pain and suffering for ourselves.
We curse the teacher for such a hard problem and no calculator.
Teacher takes the expedient route frequently and makes up problems where the answers are integers. This saves time, she thinks, so that they're not struggling with messy things and they're concentrating on new material.
Students freak out the first instance an answer is not integer.
Gasp! I must have done something wrong. Things ALWAYS work out nice and pretty in "math world". Fractions are not REAL numbers, no matter WHAT my teacher says. Oh, and by the way, "your answer key says 3/2. Is it okay if I write it as 1.5? As 1 1/2?"
A problem comes up where you're asked to find the height of a person. The decimal answer is 4.666666666 feet. You think this either means 4' 6" or 4' 7" (if you round). You DON'T think that 2/3 of a foot is not either of these answers.
I have been having the same thoughts lately, and was thinking about writing a blog post...when I found time!ReplyDelete
The rounding KILLS ME!!!! Did you know that if you round 4.6666666667 to the nearest tenth you get 4.6?
I have been trying to emphasize keeping the integrity of an answer. We have been working with trig functions. LOVE it when they find the tan(59) = 1.663 and they round to 2 mid way through a problem! which then makes their final answer off by a foot or so and they say "What's the big deal 31 and 32 feet are basically the same thing?" My eyes usually bug out of my head!
Ooo! I was just thinking I should make a list of common errors students make in HS math. This is a gold mine for me!ReplyDelete
Hah! The funny thing is that JUST today, a homework answer was "3/2", and I was stopped short when a kid asked, "is 1.5 okay?"ReplyDelete
This is one of the first things I go over in my hands-on measurement unit -- the fact that decimals can only be used to represent base-10 systems! 4ft5in does NOT equal 4.5ft, and for the same reasons 2hours and 15 minutes does NOT equal 2.15hours! It's a good thing to discuss in 9th-grade before their misconceptions built further. (And they'll need to do a lot of conversions in high school science classes.)ReplyDelete
But yes, it drives me nuts!! I was happy to see that on the tests I gave at the end of the unit, most kids were able to do proper math to tell me that 13.2 hours was shorter than 13 hours and 15 minutes.