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

Comparing Fractions

Comparing Fractions

© Math As A Second Language All Rights Reserved

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Taking the Fearout of Math

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Prefacenext

Once we have internalized the meaning of a common fraction, there are times when we can tell, simply by looking, which of

two fractions is greater.

For example, students who have not internalized the definition often believe that 7/15 is greater than 2/3 because 7 is

greater than 2, and 15 is greater than 3.

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next However, if we have internalized what a

common fraction means we see at once that 7/15 is less than half while 2/3 is more

than half.

In terms of manipulatives that are easy for even the youngest students to understand,

imagine that there are 15 pencils on the table and they take 7 of them (that is, they

have taken 7 pencils).

Classroom Note

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next They will be able to see that there are more pencils left (8) than the number of

pieces they have taken (7). So they have taken less than half of the pencils

7 pencils 8 pencils

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next And if there are 3 pencils on the table

and they take 2 of them (that is, they have taken 2/3 of the pencils), they have taken more than they have left on the table and hence they have taken more than half of

the pencils.

2 pencils 1 pencil

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However, there are times when it is not as easy to determine by sight which of the

two fractions names the greater amount.

For example, while it is relatively easy to see that both 2/5 and 3/8 are less than 1/2, it

is not as easy to see which of the two fractions is the greater one. So we will soon develop a technique that works

all the time.

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Rational Numbersnext

The study of common fractions begins with what happens when we divide

two whole numbers.

In such a situation, the quotient might be a whole number, but it doesn’t have to be. So to extend the whole number system,we agree to define the quotient of two whole numbers (provided the divisor

is not 0) to be a rational number.

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Rational Numbersnext

Thus, every whole number is arational number (for example, 6 is also

6 ÷ 1) but not every rational numberis a whole number.

Although 5 ÷ 3 is not a whole number, it is a rational number which we may denote by

the common fraction 5/3.

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In the previous lesson, we introduced the concept of unit fractions.

From a non-mathematical point of view, notice that the

two statements “John is taller than Bill” and “Bill is shorter than John” do not

sound the same. Yet, both say the same thing

from a different point of emphasis.

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The same relationship exists between non zero whole numbers and a special set of common fractions called unit fractions.

We “complement” the set of numbers 1, 2, 3, 4, 5, 6 etc., with a new set of

numbers, 1/1, 1/2, 1/3, 1/4, 1/5, 1/6 etc. To see the connection between these two sets of

numbers visually, consider the “corn bread” (i.e., rectangle) that appears below.

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Looking at the rectangle, it is impossible to tell whether we started with the smaller rectangle and marked it off 3 times to obtain the larger rectangle, or whether we started with the larger rectangle and divided

it into 3 smaller pieces of equal size.

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next In terms of our corn bread model,

if we put in the units, we do not confuse1 piece with 1 corn bread.

However, once the units are omitted it is impossible to know without being

told what the 1 is modifying.

1 piece 1 Corn Bread

1 1

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For example, if the adjective 1 is modifying “piece”, it means that the

adjective 3 is modifying the (whole) “corn bread”

Corn Bread

1 piece 1 piece 1 piece

3 pieces

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However, if we started with the corn bread as our unit and used the

adjective 1 to modify it, then each piece of the corn bread is modified by the adjective

1/3 because each piece is 1 of what it takes 3 of to equal the whole corn bread.

Corn Bread1/31/3

1/3

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What we’re calling our “corn breads” can represent any amount.

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Thus, if our “corn bread” represented $24, 1/3 of the “corn bread” would represent $8

because $24 ÷ 3 = $8.

The fact that 3 × 8 = 24 allows us to say that 24 is 3 times 8 and equivalently

that 8 is 1/3 of 24.

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Thus, in much the same way that “John is taller than Bill”, and “Bill is shorter than

John” are two different ways of transmitting the same information, “28 ÷ 4 = 7” and

“1/4 of 28 = 7 ” are also two differentways of transmitting the same information.1

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note

1 We should be cautious here. 4 pens at $7 each is not the same “event” as 7 pens at $4 each even though the cost is the same in both cases.

In this context, 1/4 of 28 = 7 is an answer to 28 ÷ 4 = ___.

And 1/7 of 28 = 4 is an answer to 28 ÷ 7 = ___. Hence, 1/4 of 28 = 7 and

1/7 of 28 = 4 are both correct ways of saying that the product of 4 and 7 is 28.

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An ordinary ruler, marked off in inches, uses other unit fractions as well.

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For example, if the ruler is marked off in eighths of an inch. We can use 1/8 as a unit fraction, meaning that any measurement on the ruler can be given in terms of the

number of inches and also in terms of the number of 1/8 of an inch.

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1 inch1/82/8

3/84/8

5/86/8

7/88/8

8/8 = 1

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That is, just as we can count 1, 2, 3, 4, 5 etc., we can also count, 1 eighth, 2 eighths,

3 eighths, 4 eighths, 5 eights, etc., or in more mathematical terms, 1 × 1/8 , 2 × 1/8 , 3

× 1/8 , 4 × 1/8 , 5 × 1/8 , etc.

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In this context, as shown in the figure below, the shaded region which we call 5/8 is the

same size as 5 × 1/8. In this form, it is easy to see why 5 is the called the numerator (it

counts the number of pieces) and why the denominator is referred to as “eighths”.

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1/81/8

1/81/8

1/81/8

1/81/8

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This leads to a conceptual problem when we define 9/8. Namely, if there are

only 8 pieces, then you can’t take 9 of them. However, you can take 9 of what it would

take 8 of to make the whole.

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There is a tendency to define 5/8 by saying we divide the unit into 8 pieces and then

take 5 of the pieces.

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1 inch 9/810/8

1/82/8

3/84/8

5/86/8

7/88/8

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The ancient Greek mathematicians knew how to divide a line segment into any

number of pieces of equal length2.

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Viewing the original length as representing 1 unit, and assuming that the segment was divided into n pieces of equal length, each of the smaller pieces was named by the

unit fraction 1/n. note

2 The concept of dividing a line segment into any number of pieces of equal length gave rise to the concept of the number line. In this context, we may think of the

corn breads we’ve drawn as being “thick” number lines. Thus when we talk about dividing the corn bread into 5 pieces of equal size, it corresponds to the

ancient Greeks dividing a line segment into 5 pieces of equal length.

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The ancient Egyptians were enamored with unit fractions, and they were particularly

interested in expressing all common fractions as sums of unit fractions.

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For example, they were intrigued by such facts as 5/6 could be written as the sum of

the two unit fractions, 1/2 and 1/3 (1/2 + 1/3 = 5/6) and in that context they viewed unit

fractions as the “building blocks” for obtaining all common fractions.

Historical Geometrical Note

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This is an interesting way of illustrating that 1 + 1 can equal 5 if the numbers are

not modifying the same noun.

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More specifically…

1 half + 1 third = 5 sixths

Anecdotal Aside

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In our next presentation, we will discuss

equivalent fractions.


Comparing Fractions

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

Documents

fractions math

rights reserved preface

rational numbers

number of pieces

number system

study of common fractions

pencils1 pencil slide

concept of unit fractions