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

Equivalent Fractions


© Math As A Second Language All Rights Reserved

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#6

Taking the Fearout of Math

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next In determining which amount of

money is greater, 17 pennies or 1 quarter, we do not use such “logic” as “Because

17 is more than 1, 17 pennies is more than 1 quarter”.1

Rather, we replace “1 quarter” by the equivalent amount “25 cents” and then conclude that the quarter is worth morebecause 25 cents is more than 17 cents.

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1 Of course, what is true is that 17 pennies are more coins than 1 quarter. Thus, when toddlers, who have not yet learned what the various denominations

mean, are offered the choice, they tend to choose the 17 pennies.

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next In other words, we did not compare the

adjectives until they modified the same noun; and it is this simple example that is the gateway to how we compare the “size”

of two fractions (at least when we view fractions as measuring rates).

The main point is that just as there are many different numerals (names) that

represent the same whole number, there are many different common fractions that

name the same rational number.

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To see this in terms of our “corn bread” analogy, imagine that the

“corn bread” is pre-sliced before we decide to take a fractional part of it.

For example, suppose we want to take 2/5 of a “corn bread” that is already

presliced into 15 equally sized pieces. In this situation taking 2/5 of the “corn

bread” means the same thing as taking 2/5 of the 15 slices.

corn bread1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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Since 2/5 of 15 is 6, it is clear that taking 2/5 of the “corn bread” is equivalent to

taking 6/15 of the “corn bread”. Visually, the figure below represents the whole

presliced “corn bread”.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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Taking 2/5 of the “corn bread” means that we have divided the “corn bread”

into 5 equally sized sections, and since 15 ÷ 5 = 3, each of the five new sections

contain 3 of the presliced pieces.

Hence, 2 of those 5 sections consist of 6 of the presliced pieces.

3 3 3 3 31 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 3 4 5 6

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In actuality, our “corn bread” can be viewed as a “thick” number line2.

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We could then have derived the previous figure by dividing the “corn bread” into 5 equally sized pieces and then taking 2 of

these 5 pieces, as shown below…

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2 Actually, the number line we draw is a “very thin” corn bread because when we draw the number line it has thickness (otherwise it would be invisible).

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We could then have divided each of the 5 equally sized pieces into 3 equally sliced

smaller pieces as shown below

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Notice that while the above figure still represents 2/5 of the “corn bread”, it also

represents 6 of the 15 smaller pieces.

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Thus…2/5 of the “corn bread” = 6/15 of the “corn bread”

and since the two numbers are modifying the same noun, we may conclude that 2/5 = 6/15.

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We read 2/5 = 6/15 as “ 2/5 is equivalent to 6/15”.

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They are different common fractions, but they name the same amount.

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More generally, we define two common fractions to be equivalent if they

represent the same fractional part3.

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3 From a more intuitive point of view, the command “Slice the corn bread into 5 pieces of equal size and then take 2 of them” is different than the command

“Slice the corn bread into 15 pieces of equal size and then take 6 of them” but they are called equivalent because they represent the same amount of

corn bread.

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In our minds a huge advantage that the corn bread has over the number line arises when we want to divide the corn

bread first into 5 pieces of equal size and then to convert the 5 pieces of equal size

into 15 pieces of equal size.

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More specifically, because the corn bread is 2-dimensional,starting with the

corn bread shown below…

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We may first divide it into 5 equally sized vertical strips to obtain…

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…and we can then take 2 of these vertical strips as shown below (the shaded portion

represents 2/5 of the corn bread).

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If we now divide the corn bread into 3 equally sized horizontal rectangles,

we obtain…

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In this way, we see quite easily that the shaded region represents 2 of the 5 vertical

rectangles and 6 of the 15 smaller rectangles (pieces).

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To reinforce the previous construction, let’s find a common fraction whose denominator is 20 that is equivalent

to the common fraction 3/4.

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If all we want is a “mechanical” way to find the answer, we only have to observe that since we have to multiply 4 by 5 to obtain

20, and if we multiply the denominator of a common fraction by 5, we must also

multiply the numerator by 5 in order to obtain an equivalent common fraction.

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We will obtain the result that…next

In terms of fractions as adjectives, another way to describe this result is by saying that

3/4 of 20 is 15.

34

=3 × 54 × 5

=1520

4

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4 Restated in terms of a more concrete rate problem, if you can buy pens at a rate of 3 for $4, then for $20 you can buy 15 pens.

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However, this is a “dangerous” way to teach students because they come away with the mistaken notion that as long as

we do the same thing to the numerator and the denominator we do not change the

value of a fraction.

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We can see that this is false by adding, 1 toboth the numerator and denominator of 1/2.

In that case we obtain…

1 + 12 + 1

or23

and clearly12

23

≠

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This is not to say that we shouldn’t have students see that if we multiply (or divide)

the numerator and the denominator by the same non-zero number we do not

change the value of a fraction, but ratherwe must have them internalize why it is

okay to assume that this is true.

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We can use the corn bread approach to find a common fraction whose

denominator is 20 that is equivalent to the common fraction 3/4.

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We may visualize 3/4 of a corn bread as the portion we obtain if we slice the corn bread into 4 equally-sized vertical pieces and then

take 3 of these pieces, as shown below…

34

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Multiplying 4 by 5 to obtain 20 becomes more visual by next slicing the

corn bread into 5 equally-sized horizontal rectangles as shown below…

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Thus, the same shaded region now consists of 15 (3×5) of the 20 (4×5) smaller pieces.

34

=3 × 54 × 5

=1520

34

1520

In other words…

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We may visualize the number line in terms of a “thin” corn bread. In that case, 3/4 of a corn bread is obtained by dividing the corn bread into 4 pieces of equal size and then taking 3 of these pieces, as shown below…

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We then divide each of the above 4 pieces into 5 equally-sized pieces to obtain the

figure shown above…

The shaded region now consists of 15 of the 20 smaller pieces.

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=1520

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As we can see from the diagram, what we did was to divide each of the larger pieces

into 5 equally-sized smaller pieces.

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This resulted (1) in producing 5 times as many pieces and (2) in our taking 5 times as many smaller pieces as we

did larger pieces.

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The same idea would apply no matter how many equally-sized smaller pieces we

divided each larger piece into.

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Thus, we may say that starting with any fraction we obtain an equivalent

fraction whenever we multiply both its numerator and denominator by the

same (non-zero) number.

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If we read the equality 3/4 = 15/20 from left-to-right, we see that we obtained 15/20 by multiplying both the numerator and

denominator of 3/4 by 5.

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And if we read the equality 3/4 = 15/20 from right-to-left we see that we obtained 3/4

by dividing both the numerator and denominator of 15/20 by 5.

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Two fractions are said to be equivalent if they modify the same amount.

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To obtain an equivalent fraction from a given fraction we multiply or divide both

its numerator and denominator by the same non-zero number.

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In comparing the size of two or more common fractions, we have to makesure that the adjectives (that is, the

numerators) are modifying the samenoun (that is, the denominator).

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This often requires that we have to replace one or more of the fractions

by an equivalent one.

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For example…

Which is the greater amount, 2/3 of a corn bread or 4/7 of the same corn bread?

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As adjectives 2 is less than 4. However, when the 2 and 4 modifydifferent nouns, we have to choose

equivalent fractions in which thenouns are the same.

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2/3 of the corn bread means that we would like to make sure that the number of pieces in the corn bread is a multiple of 3.

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4/7 of the corn bread means that we would like to make sure that the number of pieces in the corn bread is a multiple of 7.

Since we can only compare the adjectives if the nouns are the same, we want to make sure that the number of pieces in the corn bread is a multiple of both 3 and 7. One common multiple of 3 and 7 is 3×7 or 21.

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So assuming that the corn bread has 21 pieces, to find 2/3 of the corn bread, we

divide the number of pieces by 3 to determine that each of the 3 parts

consists of 7 pieces.

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21 pieces

Hence, 2/3 of the corn bread consists of2×7, or 14 pieces.

corn bread7 7 77 7 7

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On the other hand, since the corn bread has 21 pieces, to find 4/7 of thecorn bread, we divide the number of pieces by 7 to determine that eachof the 7 parts consists of 3 pieces.

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corn bread

Hence, 4/7 of the corn bread consists of4×3, or 12 pieces.

21 pieces

3 3 3 3 3 3 33 3 3 3 3 3 3

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Since the pieces all have the same size, the fact that 2/3 of the corn bread

consists of 14 pieces…

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…and 4/7 of the corn bread consists of 12 pieces…

7 7 7

3 3 3 3 3 3 3

…means that 2/3 is greater than 4/7.

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One might wonder what the practical value is of being able to compare

the sizes of fractional parts of a corn bread. The answer lies in the factthat the corn bread can be used to modify many practical quantities.

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For example, suppose you are looking for a partner to help you defray your

business expenses. Person A offers to reimburse you at a rate of $2 for every $3 you incur in business expenses (that is,

2/3 of your business expenses).

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And suppose Person B offers to reimburse you at a rate of $4 for every $7

you incur in business expenses (that is, 4/7 of your business expenses).

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What our solution shows is that Partner A is offering you the better financial deal because for every $21 you incur in business expenses, he will give you back $14. While, on the other hand, Partner B is giving you back only $12

for every $21 you incur in business expenses.

23

=1421

47

=1221Partner A Partner B

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Notice that in the previous statement, we are not saying that your business expenses

were $21. What we are saying is that Partner A is reimbursing

you at a rate of $14 for every $21 of your business expenses while Partner B is reimbursing you at the lower rate of $12 for

every $21 you incur in business expenses.

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In our next presentation, we will looks at rates in

greater depth.



1421

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