Why common multiples are needed for making and comparing equivalent fractions?

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Ami0B

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Hello all,

I am teaching fractions, but I am not sure how to explain to the children why multiplying the denominators to find a multiple is needed before comparing fractions.

Below is the problem and answer, along with my attempted reasoning, which I am not sure is sound.

Problem: Which of these fractions is greater? 3/4 or 2/6

Answer: We have to first multiply 4 and 6 together to get 24, and then multiply the numerators and denominators of 3/4 and 2/6 by 6 and 4 respectively to find out which fraction is greater. 3/4 will be greater than 2/6.

Reason: We need to find the multiple of 4 and 6 to make sure that the wholes which 3/4 and 2/6 are part of are indeed equal sizes. If we didn't know the size of the wholes being compared, 2/6 may be greater than 3/4.

2/6 of a 30cm long pizza/chocolate bar would give each person 10 slices/pieces, but 3/4 of a 10cm long pizza/chocolate bar would give each 7.5 slices/pieces.

Also, I am not sure how to explain why working out the multiple of 24 would ensure that a quarter of the 3/4 and a sixth of the 2/6 would be equal-sized, if 1/4 and 1/6 of 24 is 6 and 4 respectively.

Many thanks for the help.
 
My $0.02 worth:
I would compare 3/8 and 2/6 to stress that a larger numerator does not necessarily mean a larger fraction.
Next, I'd ask 3/8 and 2/6 of what? For example, if we use a 24 oz jar than 3/8 and 2/6 give as 9 and 8 oz respectively. I.e., the common denominator is something for which those fraction give us integer -- and easier to compare -- quantities.
What do you think?
 
Ami0B said:
I am not sure how to explain to the children why multiplying the denominators to find a multiple is needed before comparing fractions.
Click to expand...
Hi. We need a common denominator to ensure that we're comparing pieces of the same size. In other words, fourths are not the same size as sixths. After we obtain a common denominator, we compare only the numerators, to see which is larger.

Ami0B said:
Which of these fractions is greater? 3/4 or 2/6

Answer: We have to first multiply 4 and 6 together to get 24
Click to expand...
In that introductory exercise, the whole is 1.

24 is a common denominator, but we don't have to use it. We could also use 12. (12 is the smallest common denominator.)

To change fourths into twelfths, we multiply the fraction by 3/3.

To change sixths into twelfths, we multiply the fraction by 2/2.

[imath]\frac{3}{3} \times \frac{3}{4} = \frac{9}{12}[/imath]

[imath]\frac{2}{2} \times \frac{2}{6} = \frac{4}{12}[/imath]

Now we compare the numerators, and it's clear that 3/4 is larger than 2/6 (because 9 twelfths is more pieces than 4 twelfths).

Ami0B said:
If we didn't know the size of the wholes being compared...
Click to expand...
Perhaps, you're thinking of a different exercise. In your example, we're not comparing wholes.

:)
 
How to compare 3/4 and 2/6?

First I would suggest that you reduce the fraction as it is possible that the two numbers are 3/4 and 2/8. Now if this was the problem, since 2/8 is 1/4 then the answer should be obvious.

Ignoring reducing fractions you should explain that if one (positive) number is greater than another (positive) number then if you multiply both numbers by the same positive number then the initial larger number will still be larger.
I would explain that concept this way. If you deposit $7.25 each day for 11 days and I deposit $7.15 each day for 11 days than the total you deposited is greater then the total I deposited since $7.25>$7.15.

Back to your question (w/o reducing--which I think is wrong).
I would multiply both numbers by 24 (or 12 or any integer multiple of 12) and get 18>8 and conclude that 3/4 > 2/6


Alternatively you can prove the following. Compare [math] \dfrac{a}{b}\ and\ \dfrac{c}{d}[/math]
Now multiply both sides by bd and show what happens from there. The new result will be comparing two integers and that is simple to compare ....
 
How about this approach:
One pizza is divided into 4 slices, the other - 8. What's more: 2 slices of the first or 3 of the second?

2/4 or 3/8?

2 * (1/4) or 3 * (1/8)

Let's make the slices the same size: 1/4 slice is 2 slices of 1/8 size: 1/4 = 2/8

2 * (2/8) = 2 * 2 * (1/8) = 4 * (1/8) - that's more than 3 * (1/8)
 
Ami0B said:
Problem: Which of these fractions is greater? 3/4 or 2/6

Answer: We have to first multiply 4 and 6 together to get 24, and then multiply the numerators and denominators of 3/4 and 2/6 by 6 and 4 respectively to find out which fraction is greater. 3/4 will be greater than 2/6.

Reason: We need to find the multiple of 4 and 6 to make sure that the wholes which 3/4 and 2/6 are part of are indeed equal sizes. If we didn't know the size of the wholes being compared, 2/6 may be greater than 3/4.
Click to expand...
Possibly what you are thinking of is that (if we model fractions as number of pieces of a whole) we need to compare number of same-sized pieces, so it is the pieces, not the wholes, that have to be the same size.

So, if you choose to use 24 as your common denominator, you would picture cutting each of 3 quarters into 6 parts, so that 3/4 becomes 18/24; and cutting each of 2 sixths into 4 parts, so that 2/6 becomes 8/24. Now you are comparing 24ths to 24ths, and since 18 is more than 8, you know that 3/4 is greater.

As Otis said, we aren't comparing wholes; rather, in our comparison we are assuming that the wholes are the same -- maybe 3/4 and 2/6 of the same size pizza. And in comparing, we are looking for a common unit (24ths) so we can compare different numbers of the same quantity.

Ami0B said:
2/6 of a 30cm long pizza/chocolate bar would give each person 10 slices/pieces, but 3/4 of a 10cm long pizza/chocolate bar would give each 7.5 slices/pieces.
Click to expand...
This is a bad illustration, because you are comparing fractions of different wholes. The only way to do that would be to find the actual amounts: 2/6 of 30 cm is 10 cm (not 10 pieces!); and 3/4 of 10 cm is 7.5 cm (not pieces). So in this case, the former is larger, but you are answering an entirely different question.

Now, if you took 2/6 and 3/4 of the same 24-cm bar, you would get 8 cm and 18 cm respectively, and the latter is larger. This is what you are doing when you compare fractions.

It's worth observing that we have used two different models here: in the first I rewrote each fraction as an equivalent fraction (multiplying numerator and denominator of each fraction by the same number, modeled by cutting each piece into smaller pieces), and in the second I used a new unit (cm rather than bars) to change the form of the problem, multiplying each entire fraction by 24. (And I could have used 12 instead -- I'm avoiding that issue, and focusing on your basic concepts.)

There are many ways to think about fractions; a fraction really isn't any one of them, but just a division. It's important not to confuse the model with the real thing.
 
Thank you every body for replying to my thread, especially when its during your own time :) !
 
Ami0B said:
Hello all,

I am teaching fractions, but I am not sure how to explain to the children why multiplying the denominators to find a multiple is needed before comparing fractions.
Click to expand...


It is not needed. Look at this example:

Compare \(\displaystyle \ \ \dfrac{3}{5} \ \ to \ \ \dfrac{2}{3} \).

Multiply denominators by opposing numerators:

\(\displaystyle 3\cdot3 \ \ \ vs. \ \ \ 5\cdot2 \)

\(\displaystyle 9 < 10 \)

Therefore, \(\displaystyle \ \dfrac{3}{5} < \dfrac{2}{3}. \)
 
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