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E:\PAGESEW\RUFFTEXT\ROUGH034.TXT
A CRASH COURSE IN MATHEMATICS
Contents:
Notation Note
Simplify your expressions
Symbols
Units of measurement
Square Units
Computing Areas:
How much fabric will make a given length of tape?
How much is lost to seams?
NOTATION NOTE: mathematicians do not use the
times sign. When written by hand, it's likely to
be mistaken for the letter X. When written on a
typewriter, it's likely to *be* the letter X,
which is even more confusing.
Worst of all, when you have an expression such
as "3 times 4 + 5", you don't know whether that's
"multiply four by three, then add five to the
product" or "add four and five together, then
multiply the sum by three".
So what they do is just set the numbers side
by side, saying "3(5)" the same way you'd say
"three boxes" or "three dozen". There's no way
you can mistake "3(4) + 5" for "3(4+5)"
==============================================================
SIMPLIFY YOUR EXPRESSIONS
Little Johnny has five quarters in his pocket.
5($0.25) = $1.25
Little Johnny's pocket contains $1.25.
Mommy hands Johnny a ten-dollar bill and sends
him to the store. Johnny puts the bill into his
pocket and hops onto his little bike.
$1.25 + $10.00 = $11.25
Little Johnny's pocket contains $11.25.
At the store, Johnny takes the bill out of his
pocket and gives it to the clerk.
$11.25 - $10.00 = $1.25
Little Johnny's pocket contains $1.25.
The clerk hands Johnny a package and $1.67 in
change. Johnny puts the package into his bike's
pannier and puts the change into his pocket.
$1.25 + $1.67 = $2.92
Little Johnny's pocket contains $2.92.
When Johnny returns home, he gives his mother
the package and the change.
$2.92 - $1.67 = $1.25
Little Johnny's pocket contains $1.25.
If all you wanted to know was how much money was
in Johnny's pocket after all these transactions,
you didn't need to do any of those calculations.
He put money into his pocket, he took it out
again, and he ended up just where he started. In
other words, the expression
5($0.25) + $10.00 - $10.00 + $1.67 - $1.67
can be simplified to
5($0.25)
Now let's complicate the situation a little:
on his way out of the store, Johnny stops at a
vending machine and buys a piece of candy with one
of his quarters. Now the expression summarizing
his transactions reads
5($0.25) + $10.00 - $10.00 + $1.67 - $0.25 - $1.67
The "+$1.67" and the "-$1.67" are no longer
right next to each other, but they still cancel
each other. The expression simplifies to
5($0.25) - $0.25
If you are muttering impatiently "Look, he had five
quarters, he spent one, now he has four, that's a dollar --
what's with all the math?", then you have further simplified
this expression.
As you learned in grammar school, 5 times $0.25 is the
same as $0.25 times 5, so
5($0.25) - $0.25
= $0.25(5) - $0.25(1)
= $0.25(5 - 1)
= $0.25(4)
= $1.00
You can simplify a calculation even when only one
operation is involved. For example, suppose that sixteen
people go out for lunch and the bill is $128.00. I can't
divide by sixteen in my head, but I can divide by two, so I
think:
$128/16 = $64/8 = $32/4 = $16/2 = $8
==============================================================
SYMBOLS
Even in my over-simplified tale of the money in Little
Johnny's pocket, all those numbers are confusing. If things
get a little complicated, you can easily fail to see that
the same number occurs twice. This is exploited in all of
those magic tricks that run "Think of a number, any number,
add six . . . and the answer is YOUR ORIGINAL NUMBER!!!!!"
You can get around that by putting down a letter instead
of the number. (A star, a heart, or any simple picture
would do, but it's easier to pick an easy-to-remember letter
than to draw an easy-to-remember picture.)
So when someone says to you, "I have the scariest magic
trick -- write down the year you were born." You write,
instead, "B".
"Multiply by five"
5B
"Add twenty"
5B+20
"Multiply by two"
10B+40
"Divide by ten"
B+4
Subtract this number from 2008
(I'm telling this in 2004; in 2005 one
must say "2009", in 2006 "2010" etc.
2008 - (B+4) = 2004 - the year you were born
And the answer is YOUR CURRENT AGE!!!!!!!!!!!!!!
When this joke is told for real, the current
year is disguised a little better, and about
halfway through you are told, "If you've had your
birthday this year add this number, and if you
haven't, add (same number plus or minus one). You
want the victim to do several calcutions both
before and after, to keep him from connecting his
birth year with his birthday, and the birthday
with his current age.
Aside from solving puzzles, giving a number a
name without saying which number it is allows you
to make one calculation and apply it to all
possible values of that number. There are
thousands of pre-calculated "formulas" that you
can look up in various books.
This way to simplify calculations is
invaluable to the study of the relationships among
numbers -- it is ever so much easier, for example,
to say "for any two numbers, which we will call a
and b, ab=ba" than to list all the possible pairs
of numbers. Since the study of the relationships
among numbers is called algebra, we are apt to
refer to the system of using symbols other than
numerals to refer to unspecified or unknown
numbers as "algebraic notation".
==============================================================
UNITS OF MEASUREMENT
When you do real-life calculations, there are
nearly always units of measurements involved, and
it's very important to keep these units of
measurement straight. (I'm sure you've had the
experience of looking at your calculator in
consternation after absent-mindedly entering a
number of cents among a series of entries
denominated in dollars.)
In grammar school, we were taught to change
everything into the same units before starting to
calculate. This prevents the dollars-plus-cents
problem, but it's a great deal of work -- and
sometimes the units divide out anyway. It's much
easier to keep all the units that you started
with, treat them as though they were algebraic
quantities, and do any conversion needed at the
end.
Example: for reasons I'll go into later, you
might want to multiply four inches by ten yards,
then divide by fifty inches, and express the
result in yards.
Grammar-school method one: The easiest way to
get the quantities all into the same units is to
change "ten yards" into 360". Then 4 X 360 / 50 =
28.8, 28.8 is in inches, so you divide by 36: 0.8
yards.
(Please read "X" as the times sign, and "/" as
the division sign. Neither symbol is on my
keyboard.)
Grammar-school method two: You want the
answer in yards, so you change everything into
yards.
4/36 X 10 / 50/36 = 0.111111111 X 10 / 1.3888888888 = 0.8
Keep the units method: four inches times ten
yards is forty inch-yards -- a new unit of area
representing the area of a strip a yard long and
an inch wide. Then you divide by fifty inches,
and get 0.8 inch-yards/inch. The inches divide
out, the answer is 0.8 yards.
Carrying the units through also serves as a
check on mistakes -- and helps you to find the
mistakes. If, for example, your answer to the
second example comes out 1.543097, your only clue
is "This answer is ridiculous."
If you make the same mistake using the third
method, you get 200 inch-yard-inches, and it is
obvious that you multiplied by fifty inches
instead of dividing.
==============================================================
SQUARE UNITS
A square scrap that's one inch on each side
has an area of one square inch; a scrap half an
inch wide and two inches long also has an area of
one square inch, as you can see by cutting it in
half and placing the two halves side by side on
top of the first scrap.
A square piece of fabric that measures one
yard on each side has an area of one square yard.
You could divide this square into thirty-six rows
of thirty-six one-inch squares. A square yard is
equal to 1296 square inches.
Exercise: if you have half a yard of fifty-
inch fabric, it measures eighteen inches by fifty
inches. The area is eighteen inches times fifty
inches, which is 900 square inches.
Variant exercise: The same half yard of
fifty-inch fabric measures half a yard by 1.39
yards. The area is .69 square yards.
Or, working it in common fractions: It
measures 1/2 yard by 50/36 yards. Half of 50/36
is 25/36, so the piece is twenty-five thirty-
sixths of a square yard: it will give you the
same amount of bias tape as a piece a yard wide
and twenty-five inches long.
25/36 = 0.69444444444444444,
which agrees with the result above, so I haven't
dropped any decimals.
==============================================================
AREAS: HOW MUCH FABRIC D0 I NEED TO MAKE A YARD OF TAPE?
Down at the hardware store, a "yard" is the
yard appropriate to the item being sold. If you
buy a "yard" of rope, you get a linear yard, if
you buy a "yard" of concrete, you get a cubic
yard.
So my spouse and I were talking past each
other for quite a while when I talked about buying
yards of cloth -- something used by the area is
just naturally sold in square yards, so what do
you mean by saying that 72" muslin at $10/yard is
cheaper than 36" muslin at $6/yard?
If you know the area of a wall, and know how
many square feet a gallon of paint covers, you
know how many gallons of paint you need. If you
know the area of a floor, that is almost all the
information you need to decide how many foot-
square floor tiles to buy.
But if you know that you need three and a half
yards of fifty-inch fabric to cut out a given
pattern, that tells you nothing at all about how
much forty-inch fabric you need to make the same
pattern. Three and a half yards might be plenty,
you might need seven yards, you might have to
change the pattern to be able to use the narrower
fabric at all.
So we tend to forget all about area -- until
we need to know how much fabric will make a given
length of bias tape. Now, all of a sudden, length
and width don't matter -- all that matters is
*area*.
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The area of a rectangle is the length times
the width.
If both lengths are measured with the same
unit, the answer comes out in square units:
square inches, square yards, square miles, square
centimeters, square meters, etc.
Aside from seam-allowance loss, cutting up a
piece of fabric and sewing it back together does
not change its area.

------------------------------------------------------------
If you know how long and wide a piece of bias
tape you want, how big a piece of fabric should
you cut into bias strips?
The area of the bias strip is its length times
its width. You want to cut up a piece with at
least that much area.
I suppose the rather curious custom of cutting
up a square -- which I've seen suggested in
several places -- is based on the ease of the
calculations:
Example: I want ten yards of two-inch tape.
10 X 36 = 360 X 2 = 720
26.832815 -- a 27" square
should do the job.
Same example expanded: 10 yards times 36
inches/yard is 360 inches long. Multiply this
length by the width, 2 inches. The area of the
tape is 720 square inches.
The square root of 720 is 26.832815, so a
twenty-seven-inch square should be slightly larger
than the desired tape. (But remember that we are
ignoring seam allowances.)
====================================
If any rectangle will do, there is no clear
result. To get 720 square inches, you could use
two yards of ten-inch fabric, or ten inches of 72"
fabric, or one yard of 20" fabric, or twenty
inches of 36" fabric, or sixteen inches of 45"
fabric . . .
On the other hand, if you are buying fabric to
make the tape, accepting a rectangle of any shape
simplifies things marvelously. You just write on
your shopping list that you need 720 square
inches, put your calculator in your pocket, go to
the shop, select a fabric, note that it's marked
39 inches wide, whip out your pocket tape measure
to make sure it's not mis-marked, divide 720
square inches by 39 inches, get 18.46 inches, note
that this is slightly more than half a yard, and
buy 5/8 yard -- or three fourths, if you think it
might be cut crooked.
Then at home you wash it thoroughly,
straighten the ends, starch it, cut it on the true
bias from cut-edge to cut edge, sew the selvages
together, draw lines two inches apart parallel to
the bias edges -- there's another reason for
buying a bit extra: you might well end up with a
strip of 1.5" tape at one end of the parallelogram
-- sew the crossgrain edges together offset so
that the cutting lines join together into one
spiral line (being VERY careful that the cutting
lines cross at the seam line, NOT at the cut
edge), press the seam, and cut along the line that
spirals around the tube.
In the example given, the crossgrain seam will
spiral around the tube of fabric more than once,
which makes it hard to pin and hard to press. It
might be well to cut the piece in half along one
of the cutting lines, sew two of the crossgrain
edges together with the bias edges matched at the
seam line, then sew the fabric into a tube with a
cutting line spiralling around it.
If you have an odd number of strips, cut on a
line that gives you one more strip on one half
than the other, start cutting a strip off the
piece that has one too many, and stop halfway
across to leave it dangling. Then sew the other
piece to the edge that you have shortened
(matching cut edges and cutting lines at the
stitching line!). Now if you start cutting
another strip to make the other edge of the longer
piece match the shorter piece, sewing the
remaining crossgrain edges together will offset
the cutting lines automatically.
==============================================================
AREAS: HOW MUCH FABRIC D0 I LOSE TO SEAMS WHEN MAKING TAPE?
Calculating loss to seams when making
straight-grain or cross-grain tape is easy. The
area of a seam allowance is the length of the seam
times the width of the seam allowance. (It's a
little more complicated for curved seams, but
those are irrelevant to the present discussion.)
There are two seam allowances to each seam. There
are as many seams as pieces, minus one.
Suppose, altering the example given above, you
want to make ten yards of two-inch crossgrain tape
from 39" fabric.
If the seam allowances are a quarter inch
wide, you will lose one square inch to each seam.
(2" X 1/4"/allowance times two allowances/seam)
You will need 10yd/39" thirty-nine inch pieces
of tape. 360"/39"/piece = 9.23 pieces = 10 pieces
= 9 seams = 9 in^2. Since the tape is two inches
wide, that's 4.5" of tape lost to the seams.
Since you have .77 of 39" extra, all is well.
(Another way to calculate is that half an inch of
tape is lost to each of the nine seams.)
(A way to get out of calculating is to say
that 39" is 3" more than a yard, so ten 39" pieces
would be 30" more than the ten yards wanted, minus
seam allowances. )
When you make bias tape, the length of a seam
is the square root of two times the width of the
tape. Calculators have square root keys, and if
you are using your naked head, you can say that
the square root of two is one and a half -- making
it a smidgeon over gives you a little allowance
for error.
But how many pieces? You've got this little
piece in the corner that's zero on one side and
twice as long as the width of the tape on the
other, and then a piece that's cut from the first
piece on one side and twice that long on the
other, then . . .
Well, duh! There are only two seams! You sew
the selvages together, then you sew the crossgrain
edges together. So calculate the perimeter of the
piece of fabric you are using and multiply by the
width of the seam allowance. And then, if you are
fussy, deduct by the amount you save by not sewing
the two ends of the tape together. Then if you
are *really* fussy, deduct for counting a little
square at each corner twice.
==============================================================
EOF