A reader raised an interesting question in our previous post on Understanding the Concept of Same Whole in Fractions. Specifically, using a physical interpretation as an example: 3 out of 4 of Ellie’s pencils are red while 5 out of 7 of Tim’s pencil are red. Who has the “greater” fraction? After all, we can convert the fractions to the same denominator and compare 21/28 with 20/28.
Unfortunately, the examples we used in the said blog post to illustrate the use of the “same whole” in fraction comparisons are all pictorial illlustrations, e.g.
so there wasn’t a ready example that illustrate this concept of the “same whole” in physical quantities.
In this post, we’ll expand the illustrations to countable objects (fraction of a set) and physical interpretations of comparing fractions using the same whole.
Physical interpretation of fraction is incomplete without a definition of the whole
If one student was to say to another: “3/4 of my ruler” is longer than “5/7 of your ruler”, the immediate response would be “are the two rulers of equal length?” The notion of a fraction is incomplete without also specifying the object (the unit whole) or quantity of the entire set (fraction of a set).
It’s like say the property whose address is 4 Privet Drive in Little Whinging, Surrey is next to 3 Privet Drive in a different town. The address of the second property is incomplete.
The same goes with the question “Who has the greater fraction?” posted by the reader.
What about Fraction of a set?
Now, how about when we deal with the fraction of a set of countable objects?
Image a teacher has bags of pencils, each with 28 pencils. Each student in the class are given cards with a fraction. The students can then go to the teacher and exchange the card with the same fraction of a bag of pencils as indicated by his of her card. Then Ellie (who drew a card that says 3/4) and Tim (who drew a card that says 5/7) would be able to compare their cards and decide who is going to get more pencils.
However, if the teacher’s bags all contain different number of pencils, then there is no way of knowing how many pencils Ellie and Tim based on the fractions value indicated on their cards alone.
Can we compare two fractional quantities in this case?
So, back to the question, can we compare (i.e. decide which is greater between) these two cases – 3 out of 4 pencils with 5 out of 7?
The only meaningful comparison would be to compare the absolute quantities, e.g. 3 and 5 (if we’re comparing who has more red pencils) or 4 and 7 (if we’re comparing who has the most pencils), or 2 and 3 (if we’re comparing who has the most non-red pencils).
In this case, without a common whole unit, comparing the two fractions 3/4 and 5/7 does not offer any physical meaning.
So what does converting to the same denominator mean “physically”?
The act of converting the two fractions to the same denominator essentially establish the same whole, setting up a scenario where we can compare the “redness” of the two collection.
Physically, it would be like exchanging each of Ellie’s pencil (red or otherwise) with 7 smaller pencils of the same color. Likewise, Tim exchanges his each of pencils for 4 smaller ones of the same color. Then they will each have a total of 28 pencils, and a meaningful comparison can be made.
This seems hardly fair to Ellie, who was given a worse “conversion rate” than Tim. We agree.
What we’re comparing, technically, is the “red-ness” of their collection, i.e. by doing this, we can tell that Ellie’s collection is slightly “red-er” than Tim’s. We’re not comparing any meaningful quantity – who has more red pencils etc.
Conclusion
Before dealing with fractions purely in abstract form, teachers often try to offer physical interpretation for operations involving fractions, including addition/subtraction, multiplication/division and comparison. There, we run into two scenarios – 1) fraction of one or more divisible objects, e.g. pies, ruler etc, and 2) fraction of a set of countable objects, e.g. set of 28 pencils.
In both cases, it is very important to appreciate the fact that in order for a physical interpretation to take place, the description of the quantity has to include both the fractional value (fraction) and the unit (whole) to take a fraction of. Without a common unit (same whole), comparing just the fractions do not offer a meaningful physical interpretation.
Thank you for your reply. Very clear and it completely clarified my understanding. Great images and explanation.
You’re welcome! 🙂