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This question was just posted on another list I follow.
Hello all. I am totally blind, but my wife and children are sighted.
My son is nine years old and in the fourth grade, and he is having a little bit of difficulty with long division--Especially when dividing a double-digit number into another number (e.g. 5128 divided by 47).
Can any one give me some pointers on how I might explain and illustrate the concepts of how to perform these types of problems, with emphasis on how to explain how to estimate?
I hope that this question is clear enough and that someone may have some ideas that will help me.
Thank you for your assistance.
This request was posted on a list where most of the members are adult blind mathematicians who are unlikely to know what is currently going on with grade school math and to what extent the son's problem is likely to be related to the educational environment.
I don't think this Blogger interface is very accessible to persons who use screenreaders. However, if anyone has any advice or suggestions, I'm happy forward them to this father.
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8 comments:
If the problem is one of keeping track of steps in the algorithm itself I found it useful to have my son write zeros as place holders rather than leaving it blank and "bringing down the next digit." By using zeros as place holders the student subtracts and then moves directly on to dividing the remainder.
It will probably invoke some facial ticks in a few readers as they see how perilously close this is to fuzzy math methods but I write out a "times table" for whatever two or three digit divisor I'm working with and I do it without a calculator. I can't stand estimating, getting it wrong and having to back up erase and estimate again, so instead I write out the exact values of the multiples of the divisor.
When my son learned the long division alogorithm I showed him how to quickly write out a table as well and he uses exact values rather than estimation.
For example,
47 x 1 = 47
47 x 2 = 94; I doubled the 47 from above
47 x 3 = 141; I added the 47 + 94 from above
47 x 4 = 188; I doubled the answer from 47 x 2
47 x 5 = 235; halve 47 x 10
47 x 6 = 282; double 47 x 3
47 x 7 = 329; add 47 x 3 and 47 x 4
47 x 8 = 376; double 47 X 4
47 x 9 = 423; (47 x 4) + (47 x 5)from above.
When working with large dividends and divisors I waste more time estimating than writing out this reference table which takes me about a minute.
Allegedly Adrian's math webpages use LYNX which is a text reader although he says he rendered the long division problems in ASCII which makes them not useful in LYNX.
"...with emphasis on how to explain how to estimate..."
I like Myrtle's technique, but I'm afraid I don't have that much patience. But, then again, I deal with the problem of guessing the wrong number.
For multiplying numbers on paper, I always use the standard algorithm (right to left). For estimating or doing calculations in my head, I always go left to right.
For example,
5 X 47 = 5 X 40 + 5 X 7
= 200 + 35
I do this left to right because I can quit whenever I have enough accuracy. (My wife uses the standard algorithm in her head. She is able to remember the digits generated and then flip them around.)
Estimating like this isn't too bad for long division because you're only dealing with numbers from 0 - 9. It's kind of like two multiplication table facts with the first one shifted left by one digit. However, this gets more difficult with more digits, but I still do it two digits at a time, like this:
7 X 2693 =
7 X 2 = 14 + three zeros = 14000
7 X 6 = 42 + two zeros = 4200
Add them to get 18200
next,
7 X 9 = 63 + one zero = 630
Add to 18200 to get 18830
7 X 3 = 21 + no zeros = 21
Add to 18830 to get 18851
I do have a tendency to forget the previous sum, however.
Beyond long division, if I have something like:
17 X 23
I will take the 23 and change it into (20 + 3) X 17
20 * 17 = 2 X 17 + add a zero
to get 340
Then I will do 3 * 17 just like I do for the long division example to get 51.
As long as I haven't already forgotten the 340, I can add them together to get 391. Again, I do this left-to-right so that I can quit whenever I have enough accuracy.
Like Myrtle, I taught my children to write the zero place holders during long division. It helped make sure everything was lined up correctly, but it also made it very clear what was happening in each step. The steps of the algorithm made more sense because they were directly related to what my children knew about place value, multiplication and subtraction.
You guys are great, thanks!
This all sounds like exactly the kind of practical advice that's needed. I've forwarded your replies on to the father and I'll let you know if I hear back from him.
Blogger doesn't accommodate screen readers???
The trick they used in K-5, which is GREAT, is to turn the paper sideways so the horizontal lines are vertical. Keeps the columns straight.
I also like graphic paper. We have tons of the Mead Quad spiral notebooks in the house. I do all my math exercises in them.
Graphing paper is a good tool. That's just about all we use for math around here too.
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