Equivalent Fractions © Math As A Second Language All Rights Reserved next #6 Taking the Fear out of Math

next © Math As A Second Language All Rights Reserved 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 more because 25 cents is more than 17 cents. note 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. next

© Math As A Second Language All Rights Reserved 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.

© Math As A Second Language All Rights Reserved next 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 bread

next © Math As A Second Language All Rights Reserved next 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”

next © Math As A Second Language All Rights Reserved next 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

next © Math As A Second Language All Rights Reserved In actuality, our “corn bread” can be viewed as a “thick” number line 2. next Notes 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… next note 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).

next © Math As A Second Language All Rights Reserved We could then have divided each of the 5 equally sized pieces into 3 equally sliced smaller pieces as shown below next Notice that while the above figure still represents 2 / 5 of the “corn bread”, it also represents 6 of the 15 smaller pieces. 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.

© Math As A Second Language All Rights Reserved We read 2 / 5 = 6 / 15 as “ 2 / 5 is equivalent to 6 / 15 ”. next They are different common fractions, but they name the same amount. More generally, we define two common fractions to be equivalent if they represent the same fractional part 3. note 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. next

© Math As A Second Language All Rights Reserved 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. next More specifically, because the corn bread is 2-dimensional,starting with the corn bread shown below…

© Math As A Second Language All Rights Reserved We may first divide it into 5 equally sized vertical strips to obtain… next …and we can then take 2 of these vertical strips as shown below (the shaded portion represents 2 / 5 of the corn bread).

© Math As A Second Language All Rights Reserved If we now divide the corn bread into 3 equally sized horizontal rectangles, we obtain… next 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).

© Math As A Second Language All Rights Reserved To reinforce the previous construction, let’s find a common fraction whose denominator is 20 that is equivalent to the common fraction 3 / 4. next 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.

© Math As A Second Language All Rights Reserved 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 = 3 × 5 4 × 5 = note 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. next

© Math As A Second Language All Rights Reserved 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. next We can see that this is false by adding, 1 to both the numerator and denominator of 1 / 2. In that case we obtain… or 2 3 and clearly ≠ next

© Math As A Second Language All Rights Reserved 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 rather we must have them internalize why it is okay to assume that this is true.

next © Math As A Second Language All Rights Reserved 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. next 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… 3 4

© Math As A Second Language All Rights Reserved 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… next Thus, the same shaded region now consists of 15 (3×5) of the 20 (4×5) smaller pieces. 3 4 = 3 × 5 4 × 5 = In other words… next

© Math As A Second Language All Rights Reserved 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… next 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. 3 4 = next

© Math As A Second Language All Rights Reserved As we can see from the diagram, what we did was to divide each of the larger pieces into 5 equally-sized smaller pieces. next Notes 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.

next © Math As A Second Language All Rights Reserved The same idea would apply no matter how many equally-sized smaller pieces we divided each larger piece into. next Notes 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.

next © Math As A Second Language All Rights Reserved 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. next Notes 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.

next © Math As A Second Language All Rights Reserved Two fractions are said to be equivalent if they modify the same amount. next Summary To obtain an equivalent fraction from a given fraction we multiply or divide both its numerator and denominator by the same non-zero number.

next © Math As A Second Language All Rights Reserved In comparing the size of two or more common fractions, we have to make sure that the adjectives (that is, the numerators) are modifying the same noun (that is, the denominator). next This often requires that we have to replace one or more of the fractions by an equivalent one.

© Math As A Second Language All Rights Reserved For example… Which is the greater amount, 2 / 3 of a corn bread or 4 / 7 of the same corn bread? next As adjectives 2 is less than 4. However, when the 2 and 4 modify different nouns, we have to choose equivalent fractions in which the nouns are the same.

© Math As A Second Language All Rights Reserved 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. next 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.

© Math As A Second Language All Rights Reserved 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. next 21 pieces Hence, 2 / 3 of the corn bread consists of 2×7, or 14 pieces. corn bread

next © Math As A Second Language All Rights Reserved On the other hand, since the corn bread has 21 pieces, to find 4 / 7 of the corn bread, we divide the number of pieces by 7 to determine that each of the 7 parts consists of 3 pieces. next corn bread Hence, 4 / 7 of the corn bread consists of 4×3, or 12 pieces. 21 pieces

next © Math As A Second Language All Rights Reserved Since the pieces all have the same size, the fact that 2 / 3 of the corn bread consists of 14 pieces… next …and 4 / 7 of the corn bread consists of 12 pieces… …means that 2 / 3 is greater than 4 / 7.

© Math As A Second Language All Rights Reserved 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 fact that the corn bread can be used to modify many practical quantities. next 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).

© Math As A Second Language All Rights Reserved 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). next 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. 2 3 = = Partner A Partner B next

© Math As A Second Language All Rights Reserved 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.

next In our next presentation, we will looks at rates in greater depth. © Math As A Second Language All Rights Reserved Equivalent Fractions