Fraction of a Set

Have you seen your students make these mistakes?

Why do these mistakes occur? In fractions, students often “self-invent” rules (especially for fraction multiplication and simplification) which are wrong, but do not even realize they were wrong in the first place!

It is therefore important to inculcate a practice of understanding before rules when teaching fractions, and the topic of fraction of a set provides many opportunities for conceptual understanding and visual representations.

Progression

The topic of fraction of a set fits in between fraction of a whole and fraction multiplication. It is a subset of the general fraction multiplication, but is usually taught as a pre-cursor to fraction multiplication.

The general progression of the topics involved can be viewed as:

Fraction of a whole
Fraction of a set
Fraction multiplications
Simplification

With a spiral approach, fraction of a set is generally taught across grades 3 and 4.

3rd Grade – Pictures and Reasoning

In 3rd grade, we use pictures to help students understand the concept of fraction of a set. For example, these two questions from Math in Focus workbook 3B (p. 113-114) introduce concept in a friendly and conceptual manner.

Using the same fraction and task/pictures (painting trucks), students apply their understanding of 2/5 of 5 trucks, to 2/5 of a larger set of 20 trucks.

Notice we did not introduce any rules in these questions, students use their own reasoning to deduce the answers.

Once we’re familiar with fraction of a set using pictures, it is time to introduce bar model representations, e.g. Math In Focus workbook 3B (p. 116),

This slightly more abstract, but still pictorial, representation is very useful when we start extending cases to more complex cases in general fraction multiplications. Students will be dealing with bar models more in 4th grade.

4th Grade – Bar Models

In 4th grade, we use less pictures and more bar models. For example,

Once students have mastered the conceptual understanding through pictures and bar models, it is time to move on to teaching procedures. This is a good chance to explain the links between conceptual understanding and procedures, before students move up to 5th grade, where they will be doing more fraction manipulations. For example, we can invite students to explore the reasoning behind the procedure of “canceling” numerator and denominator, by showing why 2/3 x 24 = 2 x 24/3 = 2 x 8, by drawing pictures such as

Hence, two- thirds of 24 is the same as 2 groups of 8.

In general, when trying to connect conceptual understanding to procedures, we start with numbers that kids can visualize easily, e.g. ⅔ of 12, before others, e.g. ⅔ x 13. In the latter, to use visual representation to explain how it works out to be 8⅔ , we can try putting 13 shapes in 3 groups. For example:

Difference between fraction of a set and fraction of a whole

The fact that we teach these two topics one after the other sometimes lead to confusion. In fact, they may represent different problems or physical meaning, even though the answers are the same. For example, ⅔ of 12 vs 12 x ⅔ can be interpreted as follows:

I. 2/3 of 12

There are 12 pizzas. ⅔ of the total number of pizzas is 8.

II. 12 x ⅔

Each person takes ⅔ of a pizza, the total number of pizzas eaten by 12 people is 8.

It is important for teachers to understand the difference in the meaning. Mathematically, they may be equivalent, but they represent two different views of the same problem. In order for students to gain a strong conceptual understanding, it is important for them to understand which of the two interpretation they are dealing with.

Other Common Mistakes

Some other common misconceptions we see in students for this topic includes:

i) Mixing up procedures with equivalent fractions

For example,

2/3 x 5 = 2×5 / 3×5 = 10 / 15 = 2/3

This usually results from too much emphasis on procedures before strong conceptual understanding are established.

ii) Ignoring the Concept of the Same Whole

For example, in 2/3 x 3,

$Mistake fractions of a set - ignore whole$

These mistakes provide excellent opportunity to revisit the concept of the same whole in fraction addition. We can also remind students that multiplication is repeated addition to let them see the error.

Teach for Understanding

Teachers can help the students understand the meaning of each of the numbers in fraction of a set problems, e.g. in 3/4 of 8,

the number 8 represents “a set of 8”
the number 4 represents “total 4 equal groups”
the number 3 represents “the number of groups we’re considering”

Also use pictorial means to help students understand concepts whenever possible. Can we draw a picture to find 3/4 of 8?

$fractions of a set$

From the explanation, the denominator 4 represents the number of groups we need to divide 8 into, and the numerator 3 tells us we need three of the 4 groups.

Conclusion

Fraction of a set is usually presented as the precursor to the general topic of fraction multiplication, and is an important topic in fraction on its own. Usually taught right after fraction of a whole, students are sometimes confused about their distinction. This prevents them from gaining strong conceptual understanding and results in some common mistakes which if not addressed, will lead to further confusion when general fraction multiplication is introduced.

This article is part of a series of blog posts on Fractions: