# Fraction Multiplication

Compared to addition and subtraction, the procedures for fraction multiplication is fairly straightforward – multiply numerator with numerator and denominator with denominator. This may be why sometimes teachers do not allocate enough time to the topic in the classroom, especially if procedural fluency is the primary focus.

In a way, fraction multiplication and division is the opposite of addition and subtraction – in fraction addition (and subtraction), the concepts are easy to grasp but the procedures may be confusing for some students, whereas in fraction multiplication (and division), the opposite is true, i.e. it is simple to learn the procedures, but it is harder to let students get the correct conceptual understanding!

Without a firm conceptual understanding, it would be more difficult to interpret and solve fraction word problems, such as the “Mimi’s market” example in 5th grade:

Mimi’s market sold 24 heads of lettuce one morning. That afternoon 2/7 of the remaining heads of lettuce were sold. The number of heads left was now 1/2 of the number the market had at the beginning of the day. How many heads of lettuce were there at the beginning of the day?

(See our previous blog post for a video on how to solve this question using bar models)

## Progression

We break down the topic of fraction multiplication into eight types of problems, in progressively challenging order:

1. Unit fraction x whole number, e.g. ⅓ x 24
2. Proper fraction x whole number, e.g. ⅔ x 24
3. Unit fraction x unit fraction, e.g. ¼ x ⅓
4. Proper fraction x proper fraction, e.g. ⅔ x ¾
5. Improper fraction x proper fraction, e.g. 4/3 x ⅔
6. Improper fraction x improper fraction, e.g. 4/3 x 3/2
7. Mixed number x whole number, e.g. 1 ½ x 2
8. Mixed number x mixed number, e.g. 2 ⅓ x 3 ¼

Below, we outline some teaching strategies we employ in our own classroom for each type.

### Unit fraction x whole number, e.g. ⅓ x 24

Fraction of a set is the first step in introducing fraction multiplication. Here we start with unit fractions because of its direct relationship to whole number division, e.g. ⅓ x 24 can be interpreted as 24÷3. The goal is to use this familiarity to introduce multiplication with fractional values instead of whole numbers. In general, we would use paper folding and bar models to let the students “see” the fraction in pictorial form, e.g.

### Proper fraction x whole number, e.g. ⅔ x 24

This is a major step because now students do not have a direct way of interpreting fraction multiplication in terms of whole number operations. The illustration we use is similar to the previous case and we use the same bar models, but with more units shaded, e.g. (See our previous post on fraction of a set for more details)

### Unit fraction x unit fraction, e.g. ¼x⅓

This is the point where we introduce the area model for fraction multiplication. Unit fraction multiplication is introduced as a first step because we need to get students familiar with the meaning of one unit area in the area model, i.e. the result of multiplying two unit fractions. Here, we would start with a paper folding activity to introduce the area model of fraction multiplication. We ask students to:

1. Fold a paper along one edge into fourths.
2. Unfold the paper and shade one of the columns (1/4).
3. Fold along the other edge into thirds.
4. Unfold the paper and shade (a different color) one of the row (1/3).
5. Discuss, e.g.
• How many pieces/units have we divided the paper into (12)?
• What is the fraction of each piece (1/12)?
• How many units have both shaded colors (1 unit of 1/12)?
• Hence the answer for ¼x⅓ is? (1/12).

### Proper fraction x proper fraction, e.g. ⅔ x ¾

Now, we are ready to use the full area model to understand multiplication of two proper fractions. We extend the same paper folding activity in the previous step to shade more column/rows/units for non-unit proper fraction multiplication.

After several examples and discussion, we let the students come up with their own rules about fraction multiplication. Most of the time, they would be able to deduce that ⅔ x ¾ can be solved using (2×3)/(3×4).

### Improper fraction x proper fraction, e.g. 1/2 x 4/3

After students grasp the concept from the paper folding exercises, we replace the paper with a pictorial area model. This makes it easier to show improper fractions and mixed numbers. As a first step, we only have one of the fraction as an improper fraction. For example, for 1/2 x 4/3, the area model would be: ### Improper fraction x improper fraction, e.g. 3/2 x 4/3

When both fractions are improper, we continue using area model in pictorial form, but now, we have to draw another set since we are multiplying by a fraction that is greater than 1. New: Check out our Fraction Multiplication Area Model App! ### Mixed number x whole number, e.g. 1 ½ x 2

Students should be comfortable expressing improper fractions as mixed numbers by this time. Before introducing multiplication with two mixed numbers, we use only one mixed number to illustrate the “distributive property” of fraction multiplication, e.g. 1 ½ x 2 can be expressed as 1 x 2 + ½ x 2 because, This is a precursor to the next step, where we try to highlight a common misconception with students when it comes to multiplying mixed numbers.

### Mixed number x mixed number, e.g. 2 ⅓ x 3 ¼

Here, we want to highlight a very common confusion with the “distributive” approach to multiplying two mixed numbers, and the area model is an excellent tool to clarify the misconception. For example, some students think that 2 ⅓ x 3 ¼ = 6 1/12, since 2 x 3=6 and ⅓ x ¼ is 1/12. However, this is wrong since we missed out the two other factors, i.e.

(2 + ⅓) x (3 + ¼) = (2×3) + (⅓ x 3) + (2 x¼) + (⅓x¼)

Using the area model, we can clearly see the missing pieces, Here, both areas represented by 2 x ¼ and 3 x ⅓ were left out, resulting in a product that is less than the actual answer.

## Common Misconception / Tips

Here are some further misconceptions with regards to fraction multiplication:

• Multiplying always increase the size

Some students have the misconception that multiplying always result in a bigger value. This is not the case with fractions, as multiplying by a proper fraction always reduces a number. Use bar models/area models to illustrate why this is not always the case.

• Multiplying a fraction by a fraction always result in a fraction

Again, this is a common question asked by many students. Using counter examples, we can easily show this is not the case, e.g. 12/5 x 5/4 = 3.

• Misinterpreting fraction word problems

Students should associate the word “of” with multiplication, e.g. ⅔ of ¾ should be translated mathematically to ⅔ x ¾. In fact, this starts way before fractions, e.g. “3 groups of 2″ is 3 x 2.

## Conclusion

Fraction multiplication is a very important topic, but teachers and parents often have difficulty understanding what to teach, since the procedure is much easier than the fraction addition and subtraction – “just multiply the numerators and denominators separately”. However, without a firm grounding in their conceptual understanding, students may find that while they are able to solve the fraction questions easily using the straightforward procedures, they are not able to interpret fraction word problems and other more advanced applications.