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Another response, this time to some questions about learning algebra:
I think part of why I love algebra is that I started much earlier than schools usually do (when I was 11) and because the class was completely optional.> But I didn't get the deep-down understanding that I should have had. Part
I think this is a big part of it. Playing with the concepts in your head makes a huge difference. Doug and I still do mental algebra in bed in the middle of the night sometimes. We'll get thinking and talking about something, usually trying to figure out how to manipulate some series of numbers, and then get all involved in working it out with each other. I'm trying to think of a specific example, but it hasn't happened in a while.Things like "If you want to add up the numbers between one and one hundred, what's the quickest and easiest way to do that?" There's a formula, I remember hearing about it in school, but I can't remember how it goes.
Let's see if I can figure it out, right before your eyes...
I remember that it has something to do with splitting in half, and adding the first half to the second half.
The first half are the numbers from 1- 50. The second half are 51-100 If you flip the second half around, and add them that way...
1 2 3 4 5 .....
100 99 98 97 96.....You can see that each pair adds up to 101, and there are 50 pairs, right? So to add all the numbers from 1-100, you just need to multiply 101 times 50
Now.. how can we express this generically, so we can do it for any sequential series of numbers?
If you call the number you want to add up to "n"
that can be expressed as (n+1) times (n/2)
in this case, at least... is it true for all cases?
Try a few, quickly...
numbers 1-10 does that equal (11)(5)? 55
1+2+3+4+5+6+7+8+9+10= 55 Why yes, it works.How about something without a zero on the end?
numbers 1-14 does that equal (15)(7) = 105
1+2+3+4+5+6+7+8+9+10+11+12+13+14 = 105Yes, indeedy- looks like we got ourselves a winner!
That's the kind of head games we play a lot. We talk a lot about grouping numbers in ways that make them easier to manipulate.