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This lesson will examine the relationships, and the correct
method for labeling answers involving
length, area and volume.
Many times students are able to do the correct mathematical
operation(s), arrive at the correct numerical result, only
to mislabel their answer because they really don't
understand what it is they just found.
Through a variety of examples hopefully this confusion can
be eliminated, and students will have a better understanding
of what we mean when we talk about "units", "square units"
or "cubic units".
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What's a unit???
This seems like a
good place to start!
Whenever we "measure" something, whether it be our height, our
weight, the length of a piece of wood, the distance we ride our
bike, etc....we use a variety of different "units" of measure.
For height it's usually inches or centimeters, for weight it's
pounds and/or ounces, for the length of a piece of wood it'
probably inches or feet, and for a bike ride it's usually
measured in miles or kilometers.
In a math problem you will sometimes see a dimension, or some
other measurement, given in terms of "units." When this
happens all it means is that the author of the problem
has decided not to use a customary measurement
(for example: inch, feet , cm, m, lb., oz., mile) and simply
label it "units." This in no way changes how the problem
is done, it simply means that when you are finished doing the
calculations you are to label your answer as
"units", "square units" or "cubic units."
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Which
is Which???
So when do you use
the labels
"units", "square units" or "cubic units"?
This is easy!!!
It all boils down to this.....
"Length" is 1-dimensional.
That means that when you measure the length of your shoe,
for example, you only need one dimension because you are simply
trying to determine the how many "units" of measure there are
between the toe and heel of the shoe.
Your answer should be labeled in "units."
"Area" is 2-dimensional
That means that when you are calculating the area of your
bedroom floor, for example, you need 2 dimensions, the "length"
and the "width" to find the answer. The answer tells you how
many "unit squares" it takes to completely fill in your floor.
Your answer should be labeled in "square units."
"Volume" is 3-dimensional
That means that when you are trying to figure out the
volume of your fish tank, for example, you need 3 dimensions,
the "length", the "width", and the "height" of the tank. And the
answer you get tells you how many "unit cubes" it takes to
entirely fill up your fish tank.
Your answer should be labeled in "cubic units."
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Paul wanted to
calculate the perimeter of his backyard, and found it to be
23' along one side, 33' along the second side, 29' along the
third side and 39' along the fourth and final side.
After doing the arithmetic, he arrived at
P = 23 + 33 + 29 + 39
P = 124 sq.'
Is that the correct answer??
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NO!!
Do you see his
mistake??
23 + 33 + 29 + 39 IS 124!
But perimeter is a 1-dimensional concept. You are
determining the "length" of the boundary. Therefore his
answer should have been written:
124', NOT sq.'!
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Let's try
another...
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Sue needed to know
the volume of her son's sandbox so that she would know how
much new sand to purchase. The box was a square, 4' on
each side, and is 1 foot deep.
To calculate the volume of a rectangular prism she
remembered from 8th grade that V = (l)(w)(h), so she plugged
in the information:
V = (4)(4)(1)
V = 16 square '
Is that the correct answer?
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NO again!!
Sue made the mistake that
many people do, in that she assumed that because the sandbox is a
square, that the answer must be in "square units." However, as
we discussed earlier, volume is a 3-dimensional concept involving
length, width and height. And when calculating volume we are
determining how many "unit cubes" will fit in the sandbox. so,
again, the arithmetic is correct...4x4x1 is 16, but the answer
should have been labeled 16 cu'!
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Let's look at
one more....
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Bob needed to know
the area of one of his bedroom walls because he wanted to
put some really cool new wallpaper on it. The length of the
wall was 16.5' and the height was 7.5'. After doing the
math, he wrote down
on a piece of paper that the area was
123.75 ft2
Is his answer correct?
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Well,
first let's see if the arithmetic is correct.
A = lw
A = (16.5)(7.5)
A = 123.75!
So far so good...
Now, what does he mean by ft2?
If you say
that...you would say "feet squared"
Is 123.75 "feet squared" the same as 123.75 "square feet"?
Absolutely! The question is..should he label this answer
in square feet? And the answer is yes! He was
finding an "area" and that is a 2-dimensional concept.
Bob
got it right!
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