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Taking the Fear out of Math
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Fractions have long been a stumbling block for most students trying to internalize basic arithmetic. As surprising as it may seem, people a long time ago successfully avoided most of this issue by using what we are calling our adjective / noun theme.
1 note
1 Although they never used the term adjective / noun they did view numbers as quantities; that is, as a noun phrase in which the number was the adjective and the unit was the noun .
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For example , when they wanted to divide a 1 foot length into 12 smaller lengths of equal size, they named each of the smaller lengths 1 inch .
In other words, when they wanted to take a fractional part of a quantity they chose a different noun to denote the smaller quantity. In the language of fractions, the definition that an inch represented 1 of the
12 equal lengths into which a foot was divided simply became…
1 inch = 1 /
12 of a foot .
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As we have noted, when people wanted to divide a 1 foot length into 12 smaller lengths of equal size, they named each of the smaller lengths 1 inch .
In other words, when they wanted to take a fractional part of a quantity (e.g. a foot ) they chose a different noun (e.g.
an inch ) to denote the smaller quantity.
2 note
2 To the beginning student, writing 1 inch may be more user-friendly than 1/12 of a foot."
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If “ teen ” means “ plus ten ” , why wasn ’ t eleven named “ oneteen ” ?
In other words, why does the first teen appear after twelve rather than after ten?
Here are two thought questions that illustrate the previous idea…
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Prior to the invention of place value, 12 was preferred to 10 because it had more divisors than 10.
Because 12 has 6 divisors (1, 2, 3, 4, 6, and
12,), if we use 12 to a dozen, there are 6 whole number fractional parts, one for each divisor.
For example , 1/4 of a dozen is 3. But 10 has only 4 divisors( 1, 2, 5 and 10) so if we use 10 to a dozen there are only 4 whole number fractional parts. For that reason, 12 was chosen as the number of units in a dozen.
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If the prefix “ dec ” stands for “ ten ” why is the twelfth month named
December? The name of the tenth month is “ October ” and the prefix “ oct ” stands for “ eight ” . In other words, why wasn ’ t
October named December?
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At one time the calendar consisted of
10 months, each with 36 days (eventually the months July and August, named for
Julius and Augustus Caesar, were added to give stability to the seasons).
This made a year consist of 360 days.
360 has many divisors (1,2, 3, 4, 5, 6,
8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45,
60, 72, 90, 120, 180 and 360).
Thus, many fractional parts of a year were a whole number of days.
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Even though it turned out that a year had more than 360 days, the ancient Greeks were enough impressed by how many divisors 360 had that they divided a circle into 360 degrees; thus making many fractional parts of a circle a whole number of degrees.
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In most textbooks fractions are introduced as “ parts of the whole ” .
However, a much more important reason to introduce fractions is because they are a consequence of division.
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Although division is usually taught after addition, subtraction, and multiplication, in many ways it is the most important of the four basic operations of arithmetic.
In particular, there are many times when we are more interested in answering the question “ How fast?
” rather than the question “ How much?
”
Answering the question “ How fast?
” involves the notion of a rate .
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A rate is usually expressed as a phrase that consists of two nouns separated by the word “ per ” .
Examples of rates are…
“ miles per hour ” and “ dollars per pound ” .
3 note
3 There are colloquial equivalents for the word “ per ” .
For example , we might say “ $2 each ” rather than $2 per item ” .
Or we might say “ 4 pens for a dollar ” rather than “ 4 pens per dollar ” etc.
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For example , consider the “ Million
Dollar Lottery ” where first prize was
$1 per year for 1 million years versus the “ Million Dollar Lottery ” where first prize was $50,000 per year for 20 years.
Even though in both cases the amount of the prize is the same ($1 million), the rate at which it is distributed makes a huge difference.
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Or imagine that you driving on the highway and in your rear view mirror you see a police cruiser with flashing lights closing in behind you.
At that moment you are more inclined to look at your speedometer rather than at a road map. You aren ’ t as interested in where you are as you are in knowing how fast you are traveling. This is illustrated in a humorous way by the following dialog…
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Police Officer: Sir, do you realize that you were driving at a speed of 70 miles per hour?
Driver: Officer, that ’ s impossible. I ’ ve only been driving for 15 minutes!
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In this case, it is not enough to know that the driver was only traveling for
15 minutes. What is important is how much distance was traveled during those 15 minutes.
In more precise terms, what the officer really said was, “ Sir, at the rate you were traveling, you would have gone 70 miles in one hour ” .
To compute a rate we replace “ per ” by the division symbol ( ÷ ).
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Suppose a 4 pound package of beef costs
$12, and we want to find the cost per pound.
We have to find a rate, namely “ cost per pound ” . In this example, the cost is given in dollars. Hence, we must find
“ dollars per pound ” .
If we replace “ per ” by the division symbol, the solution is given by…
12 dollars per 4 pounds =
12 dollars ÷ 4 pounds =
(12 ÷ 4) dollars/pounds =
3 dollars per pound.
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Notes
Our solution was based upon the fact that we assumed that all of the meat was sold at the same rate (cost per pound).
However, if 2 of the pounds had cost
$2 per pound and the other 2 pounds had cost $4 per pound, the total cost of the package would still have been $12.
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Notes
As another example, suppose pears cost
$1.20 per pound. Most likely the pears are not all the same size. Hence, if you purchase some of these pears, the cost per pear will vary, depending on the weight of the individual pears. In other words, the cost per pound remains the same no matter how many pounds you buy. However the cost per pear will depend on the weight of the individual pear.
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So to make sure everything is clear let ’ s introduce some new terminology .
Some rates never change.
For example , there are always
12 inches per foot, 100 cents per dollar, etc.
When a rate never changes, we refer to it as a constant rate .
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On the other hand, in most real life situations, the rates are not constant.
For example , you do not usually drive at a constant speed.
4
So if you travel 200 miles in (per) 4 hours and you compute the quotient
200 miles ÷ 4 hours, you obtain 50 miles per hour as the quotient. In this case, we say that your average speed was 50 miles per hour .
note
4 The word “ speed ” is used to denote the rate of change of distance with respect to time ( for example , “ miles per hour ” which is usually abbreviated “ mph ” ).
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Average rates may prove to be deceptive.
For example , the fact that the average speed for the 4 hour journey was
50 miles per hour, only tells us that the total distance traveled during this time was 200 miles.
It does not tell us the speed at any particular instant during the journey.
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As an example that might be more relevant to students, suppose that your average score on 5 equally-weighted tests was 80 (points per test). All we can be sure of is that your total number of points for the 5 tests was 400.
It does not tell you, for example, whether all of your scores were 80 or if you obtained four 100 ’ s and one 0.
5 note
5 In the case in which the rate of change is constant, the average rate of change is the same as the constant rate of change .
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Rational Numbers, Rates and Division
To see why it was necessary to extend our number system beyond whole numbers, consider the following two illustrative examples.
We want to cut a piece of string that is
6 inches long into 3 pieces of equal length and compute the length of each piece.
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We probably see almost at once that the answer is given by 6 ÷ 3.
However, to keep the spirit of our present discussion, the problem is actually asking us to find the number of inches per piece (of string).
Remembering that we replace “ per ” by “÷” ,
6 inches per 3 pieces =
6 inches ÷ 3 pieces = we see that the answer to our problem is given by…
(6 ÷ 3) inches/pieces =
2 inches per piece.
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While the above problem was quite straight forward, it would have been just as logical to have this illustrative example…
We want to cut a piece of string that is
5 inches long into 3 pieces of equal length and compute the length of each piece.
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Except for the fact that we have replaced the length of 6 inches by a length of 5 inches, the solution is word for word the same as in the previous illustration, until we come to the last step.
We see that the answer is given by…
5 inches per 3 pieces =
5 inches ÷ 3 pieces =
(5 ÷ 3) inches per piece.
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The difficulty is that if the only known numbers are whole numbers, the example does not have an answer.
Namely, by definition 5 ÷ 3 is the number which when multiplied by 3 yields 5 as the product.
The fact that 3 × 1 inch = 3 inches, tells us that the length of each piece has to be greater than 1 inch.
And since 3 × 2 inches = 6 inches, we know that the length of each piece has to be less than 2 inches.
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However, there are no whole numbers that are greater than 1 but less than 2.
Yet, we know that it is possible to cut a 5 inch piece of string into 3 equally sized pieces.
6
Thus, we have arrived at a new obstacle that will require that we do something innovative that will allow us to reach the next plateau.
note
6 This will be demonstrated in greater detail in the next presentation
(in our note “ An Application of Geometry to Division ” ).
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In other words, unless we extend our number system beyond the whole numbers, we cannot perform the division problem 5 ÷ 3.
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5 ÷ 3 = ?
How we arrive at the next plateau is the topic that is studied in great detail in our next presentation.
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