Last week, we wrote about the importance of understanding fractions as equal parts. Of equal importance is the concept of the same whole in fractions. Fundamentally, fractions is only fully defined if the whole is specified. In the following example, without specifying what the whole is, it is not reasonable to ask what fraction is represented by the shaded area.
If each of the square is a “whole”, the shaded area represents the fraction 3/2. If the entire rectangle is the whole, the shaded area represents 3/4.
To emphasize the importance of specifying the whole, we can include a definition of the whole as the first step when working on exercises in naming fractions, e.g.
Now that we have this fundamental definition out of the way, let’s talk about why understanding the concept of having the “same whole” is so important in two aspects of learning about fractions.
1. Comparing Fractions
Fractions have to be compared based on the same-sized unit, i.e. the “same whole”. From the Common Core Standards:
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. – (CCSS.Math.3.NF.3d)
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. – (CCSS.Math.4.NF.2)
The following example is quoted from this excellent book on fractions by Dr Douglas Edge and Dr Yeap Ban Har – Teaching to Mastery Mathematics: Teaching of Fractions.
Which is more, 1/2 or 1/3?
The shaded area appears to be bigger in the right figure for 1/3 than the left figure for 1/2, but the fraction 1/3 is smaller than 1/2. To compare these two fractions pictorially, we need to draw them on the same base shape with the same area, e.g. using rectangles,
In the same sense, while 1/2 is more than 1/3, “1/2 of my salary can be less than 1/3 of yours”.
2. Adding and Subtracting Fractions
Adding and subtracting fractions can also be presented pictorially (CCSS.Math.5.NF.2). However, care must be taken to explain that the addition and subtraction is referring to the same whole.
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. – (CCSS.Math.5.NF.s)
A common mistake is explained in the following example.
1/4 + 1/4 is not 2/8, but it is easy to see how students can make this mistake if they count the total shaded squares and use it as the numerator, then count the number of equal parts and use it as the denominator. If we represent the two fractions as referring to the same unit, this error would be avoided.
While visual models are useful, care must be taken to present them correctly, otherwise it will only add confusion to our students. In drawing models for fractions, we need to emphasize the importance of the “same whole” when they are used for instruction on fraction comparison, addition and subtraction. This concept of referencing the same unit in fractions, along with that of “equal parts“, are both fundamental in helping students understand the meaning of fractions. These concepts will go a long way when students learn about fractions on the number line and equivalent fractions, which we will write about in the coming weeks.
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This article is part of a series of blog posts on Fractions:
- >> Read the next post on Fractions: Fractions on the Number Line
- << Read the previous post on Fractions: Understanding Fractions as Equal Parts
- Or start from the beginning: Understanding Fractions as Equal Parts
More Fraction Resources
For more fraction resources, refer to our main fractions page.