common fractions © math as a second language all rights reserved next #6 taking the fear out of...

Common FractionsCommon Fractions

© Math As A Second Language All Rights Reserved

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#6

Taking the Fearout of Math

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Inventing Common Fractionsnext

You have probably noticed how much more convenient it is to write

“5 inches” than to write “5 of what it takes 12 of to equal 1 foot”.

So our first abbreviation is to replace “5 of what it takes 12 of to equal 1 foot”

by “5 twelfths of a foot”.

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A “twelfth” means “1 of what it takes 12 of to equal

the given unit”.

So for example…

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--- 5 twelfths of a foot means 5 of what it takes 12 of to equal 1 foot.

Definition

--- 5 twelfths of a dozen means 5 of what it takes 12 of to equal 1 dozen.

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Since we use numbers as adjectives when we do arithmetic, we want to invent

a mathematical symbol to represent a twelfth.

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The symbol we invent to denote one twelfth is called a common fraction, and it

is written as 1/12.

The top number (1) is called the numerator and the bottom number (12) is called the

denominator.

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Notice that the word “numerator” suggests the word “enumerate” which

means “to count”. “To count” suggests “how many” and “how many” suggests

an adjective. Hence, in terms of our adjective/noun

theme, the numerator is the adjective.1

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1 There is a tendency for students to define the numerator as being the top. This masks the true meaning of the numerator. In fact if “numerator”

simply meant “top”, most likely we would have not replaced thesimpler word “top” by the more cumbersome word “numerator”.

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Notice that the word “denominator” suggests the word “denomination” which suggests the size of the quantity, and the

size is a noun. Hence, in terms of our adjective/noun theme, the denominator is

the noun.

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5

12

5 is the adjective.

12 is the noun.

5 is the numerator.

12 is denominator.

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The important point is that just as the word “inch” is a noun, so is the word

“twelfths”.

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In a similar way, we can define the following common fractions…

which we read as “a (or, one) half” and which means 1 of what it takes 2 of

to equal the whole.

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1

2

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which we read as “a (or, one) third” and it means 1 of what it takes 3 of to

equal the whole.

1

3

which we read as “a (or, one) fourth” and it means 1 of what it takes 4 of to

equal the whole.

1

4

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which we read as “a (or, one) fifth” and it means 1 of what it takes 5 of to equal

the whole.

1

5

which we read as “a (or, one) sixth” and it means 1 of what it takes 6 of to

equal the whole.

1

6

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The above common fractions are called unit fractions because they

behave the same way as other units.

For example, when we count “1, 2, 3, ...” the numbers are assumed to be

modifying a particular unit.

Therefore, 1, 2, 3 can refer to “1 half, 2 halves, 3 halves, ...” or “1 third, 2 thirds, 3 thirds, ...” or

“1 fourth, 2 fourths, 3 fourths, ... ”

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An Important Connection between Division and Common Fractions

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When we say to take 1 of what it takes 5 of to equal a given unit, it means the same

thing as dividing the given unit by 5.

In other words, taking a fifth of a number means the same thing as dividing the

number by 5.

For example, 1/5 of 30 means the same thing as 30 ÷ 5.

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More Notationnext

In the same way that we may think of 3 apples as 3 × 1 apple, we may think of 3 fifths as 3 × 1 fifth, and we write it as 3/5.

In this context, 3/5 of 30 means 3 × 1/5 of 30 or 3 × 6 or 18.

To take a fractional part of a number we divide the number by the denominator (to

find the size of each part) and then multiply by the numerator (the number of

parts we are taking).

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For example, to take 4/7 of 56 we would first divide 56 by 7 to obtain 8, and we

would then multiply 8 by 4 to obtain 32.

In terms of a picture, we may think of 56 as being represented by a rectangle

(which we personify by referring to it as a “corn bread”)

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corn bread

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We then divide the corn bread into 7 equally-sized pieces to obtain…

Thus, 56 is represented by our corn bread.

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56

8 8 8 8 8 8 8 88 8 8 8

And finally, we take 4 of the pieces.

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There is a tendency for some teachers (and some textbooks as well) to

define 4/7 by saying it means to divide the given unit into 7 parts of equal size and

then to take 4 of these equal parts.

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Roughly speaking, theysay it means to take “4 out of 7”.

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This is not a problem as long as the numerator is not greater than the

denominator. However, it can raise sort of a mystical question when the numerator is

greater than the denominator.

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For example, if we define 8/7 as meaning to divide the given unit into 7 parts of equal

size and then take 8 of these parts, it raises the serious question as to

whether we can take 8 parts from a group that has only 7 parts.

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However this problem is avoided if we use the adjective/noun way of defining a

common fraction. Namely, we define 4/7 by saying that we are taking 4 of what it takes

7 of to equal the given unit.

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In a similar way, 8/7 means that we take 8 of what it takes 7 of to equal the given unit. In that way, we see that it is equal to the

entire given unit (that is, 7 sevenths) plus 1 more part.

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While most of us might not have thought about it in that way, the ordinary ruler is a

very nice example of the “marriage” between arithmetic and geometry.

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The ruler is basically a straight line(geometry) with equally spaced points

(again, geometry) marked on it. The points are then given names such as

1, 2, 3, etc (arithmetic).

An Application of Geometry to Arithmetic

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In essence, the ruler is a model for the number line where geometric points

are given arithmetical names.2

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2 Notice that name “number line” itself indicates a combination of arithmetic (number) and geometry (line).

One of the constructions that’s described in Euclid’s elements is how to divide a line segment of any length into any number of

equally sized pieces.

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Most of us are aware of the simple case of dividing a piece of string or a sheet

of paper into two pieces of equal size.

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Namely, we essentially fold it in half by placing the ends together.

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Let’s look at Euclid’s way to divide a piece of string (of any given length)

into 5 equally sized pieces. 3

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Suppose you want to divide the line segment AB into 5 pieces of equal size.

A B

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3 We choose 5 simply for illustrative purposes. The same concept would work for obtaining any number of equally-sized pieces.

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Step 1: Through the point A draw a line of any length of your of your choosing.

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A B

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Step 2: Pick any size length and on the line you chose, mark that length off 5 consecutive times.

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Label the points you obtain in this way C, D, E, F, and G.

A B

C

D

E

F

G

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4 There is a subtle but important difference between writing AC and AC. Namely when we write AC we are referring to the set of points that constitute the line segment AC. However when we write AC we are

referring to the length of the line segment AC. Thus, to be precise, we do not write AC = CD because these two line segments do not consist of the same points. However what is true is that the length of these two

segments are the same; and to indicate this we write AC = CD.

By construction the line segment AG is divided into 5 pieces of equal length.

That is, AC = CD = DE = EF = FG.4

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Step 3: Draw the line segment GB.next

A B

C

D

E

F

G

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Step 4: Through each of the points C, D, E, and F draw lines that are parallel to

GB, and label the points at which these lines intersect AB by H, I, J, and K.

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KJIH

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A B

C

D

E

F

G

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next And the fact that the points on the line segment AG are equally spaced means that the line segment AB has also been divided into 5 pieces of equal length.

That is, AH = HI = IJ = JK = KB

In most text books, the “whole” is usually a circle (either a pie or a pizza). However, it is much easier to divide a line segment into pieces of equal length (5 pieces) than it is to

divide a circle into 5 pieces of equal size. Students might find it to be an enjoyable

activity to practice the above construction.

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Moreover, our corn bread

is a “thick” number line, and the same

Euclidian construction easily divides the corn bread

into any number of equal parts.

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KJIHA B

C

D

E

F

Gnext

Students seem to visualize a two dimensional cornbread more easily

than the one dimensional number line.

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By now you should be getting the idea that when treated in terms of the

adjective/noun theme, the arithmetic of fractions is a special application of the

arithmetic of whole numbers.

Final Note

All we have done is defined units that area fractional part of other units and

expressed these new units as common fractions.

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This is why it so important for students to internalize the arithmetic of whole numbers.

Final Note

If the students’ knowledge of arithmeticconsists of rote learning, it is very likely

that serious problems will arise when these students encounter the arithmetic of fractions.

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We will get a clearer insight to the arithmetic of fractions in our next

presentations.


Common Fractions

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