Operation M

# Operations with Fractions: Multiplication and Division

## Divide a fraction by a smaller friendly denominator fraction with a whole number result (e.g., 6⁄4 × 3⁄8 = 4)

The denominator of a fraction is the fractional unit. Although division is often instructed using the invert and multiply algorithm, division can be completed a number of ways. Consider 64 × 38. This could be interpreted as 'how many 38 are in 64?' Students could count up to 64 by 38. With a recognition that 64 = 128, students can skip count 38, 68, 98, 128 to obtain the correct answer of 4. Students may also use their understanding of equivalent fractions to reorganize the question to 128 ÷ 38. Then, they could use the same thinking as 12 cm ÷ 3 cm to obtain an answer of 4. The whole number result helps reduce the cognitive load somewhat and helps students to connect this new learning to their prior experience with whole numbers.

### Background

Multiplication and division involving fractions is widely recognized to be more comple than multiplication and division with whole numbers. Often algorithms are introduced with little emphasis on understanding the mathematics behind the algorithm. There is some research evidence that suggests that early emphasis on procedures over concepts can actually impede students' fractions understanding in the long term (Brown and Quinn, 2007). However, with precise instruction, students can develop both conceptual procedural knowledge.

Division with fractions may increase or decrease a quantity, or leave it unchanged. Division with whole numbers is often connected to either repeated subtraction (Partitive: how many threes are in six?) or making groups (Quotative: how many groups of three are in six?); however, with fractions, other interpretations, such as the unit rate and the inverse of multiplication, support increased understanding.

The strategies for whole number division of quotative (measurement), partitive (fair share) and Cartesian product (array/area) also apply to fractions. The determination of a unit rate as well as connecting division to the inverse of multiplication are strategies that apply to fractions uniquely. A measurement, or linear, model is very useful for understanding division of fractions.

Consider 3 ÷ 23. We can use hops on a number line to answer this question by considering 'how many two-thirds are in 3?'

It is clear that there are four full hops and one-half a hop, so the answer to 3 ÷ 23 is

There are several effective strategies for teaching division of fractions. See Fractions Operations: Multiplication and Division Literature Review for more details.

Best practices include:

1. Increase the focus on conceptual understanding;
Instruction should start with contextual problems which allow students to focus on what they are solving rather than how they are solving. They also should have hands-on experiences with materials such as relational rods, paper folding and visual models like number lines.
2.
3. Draw on student familiarity with whole number operations;
Specific connections to division with whole numbers should be made as the properties hold true for division with fractions as well.

1. The Identity Property: a ÷ 1 = a
2. The Zero Property: 0 ÷ a = 0; a ÷ 0 is undefined
3.
As well, students should understand that multiplication is the inverse of division and vise versa. That is to say that if a × b = c, then a = c ÷ b and b = c ÷ a. This is true for a, b, c ≠ 0. For example, although 3 × 0 = 0, 0 ÷ 0 ≠ 3.
4.
5. Recognize and draw on students' informal knowledge and prior experiences;
Students have prior experience with fractions, including partitioning different models (which is helpful when solving division problems) and an understanding of the unit fraction. Since students undestand division as the inverse of multiplication, they may solve division questions using multiplication. For example, rather than determine, a student may instead consider. As well, students can draw upon their understanding of the various constructs of fractions (i.e., part-whole, part-part, quotient, operator and linear measure) and thier work with equivalent fractions.
6.
7. Include carefully selected and multiple representations to convey meaning;
There are a few representations across fractions learning allows students to understand the structure of the representations and use them effectively to solve questions. Models help students understand the mathematics better - both conceptually and procedurally - and enhance their retention of their learning.

### References:

Brown, G., & Quinn, R. J. (2007). Investigating the relationship between fraction proficiency and success in algebra. Australian Mathematics Teacher, 63(4): 8- 15.