On Aug 19 , at 8:43 AM, Stephen Davis wrote:
I do believe that algebra sums are going to particularly difficult
to work out using decimal fractions.
For instance a linear equation such as:
21y x 8 = 1
...is going to take some monumental working out using decimals.
21y x 8 = 1
1 / 8 =1/8
1/8 / 21 = 1/168
y = 1/168
I see no difficulty with such an algebra problem. I would solve it
just about like Stephan did, BUT ...
If it were part of a physics or other real world problem (rather than
a "pure", numerical algebraic problem), I would add one final step.
That step would be to
divide out the ratio "1/168" to get: 0.00595
(Note that I immediately rounded off appropriately*. The long
calculator calculation gives 0.005952381 ... etc.
I would never even bother to look at all those extra digits, but would
just note and write the rounded value. That might not be wise for a
pure algebra problem but it is certainly appropriate in the real world
of "real numbers" with finite precision.)
Then, if it is a physics problem, the answer would (almost surely) be
a measurable quantity and therefore would (almost surely) have units.
If the quantity is a length, the units might be metres, for example.
That would then allow me to simplify the number by changing from
metres to millimetres (or even micrometres), thus: 0.00595 m = 5.95 mm
(= 5950 µm).
In a purely algebraic problem, I would agree that one of the reasons
for keeping the answer in ratio form (1/168) is the fact that, without
units, there is no other way to express the answer in compact form.
But real problems with real numbers are a different story.
So I agree that expressing such numbers in ratio form might be better
than decimal form in some cases (algebraic). However, I continue to
think that one should not teach common fractions and all the related
arithmetic (adding them, dividing them, finding least common
denominators, etc., etc.) early in students' education. That should
wait until the students take an algebra course (usually high school,
not elementary or middle school). In the elementary grades, students
should be introduced ONLY to decimal fractions and decimal arithmetic
(along with decimal based metric units of measure).
Bill Hooper
1810 mm tall
Fernandina Beach, Florida, USA
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*The degree of rounding off would depend on the precision of the
numerical values in the original problem (the "21", the "8" and the
"1"). I have rounded the answer to three significant figures as is
common for ordinary measures or where the exact degree of precision is
unknown or unimportant.
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SImplification Begins With SI.
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