B2.6 Multiply and divide fractions by fractions, as well as by whole numbers and mixed numbers, in various contexts.
Skill: Multiplying and Dividing Fractions by Fractions, by Whole Numbers and Mixed Numbers in Various Contexts
Multiplication
When learning to multiply fractions, students have moved through a progression since the primary grades, they have explored concepts related to multiplication using concrete materials, calculators, visuals and symbols.
By the time students are in Grade 8, they should flexibly be able to multiply and divide fractions by fractions, as well as by whole numbers and mixed numbers, in various contexts.
For example, they should be able to use an area model to solve the following problem:
In a rectangular field with an area of 100 m2, Mr. Siby has planted cucumbers on
Source: adapted from Les mathématiques... un peu, beaucoup, à la folie, Guide pédagogique, Numération et sens du nombre/Mesure, 6e année, Module 2, Série 2, Activité 9, Activités à la carte, p. 309.
- Multiplication using an area model
I decomposed 100 into
I added up the partial products to arrive at 40.

The area of the field dedicated to the cultivation of cucumbers is 40 m2.
- Multiplication using a personal algorithm
I decomposed
I multiplied
Multiplying by
I get 40.
The area of the field dedicated to the cultivation of cucumbers is 40 m2.
Multiplying a Fraction by a Fraction
The problem-solver uses manipulatives and pictures to represent fractions and to simulate the action that results from the statement. It is from these visual representations that they construct the meaning of the operations (×, ÷).
Source: inspired by Les mathématiques… un peu, beaucoup, à la folie, Guide pédagogique, Numération et Sens du nombre/Mesure,8e année, Module 1, Série 2, p. 143-144.
Multiplication Without Splitting
- Multiplication performed with an area model
The student can use the area model, either the rectangle or the square to multiply a fraction by another fraction.
Example
Since I am looking for

Then I horizontally divided the same rectangle into 4 equal parts and coloured 1 part, which is

The fraction that represents
The fraction
- Multiplication performed with a symbolic representation
The student multiplies the numerators together and the denominators together.
Source: inspired by Les mathématiques… un peu, beaucoup, à la folie, Guide pédagogique, Numération et Sens du nombre/Mesure, 8e année, Module 1, Série 2, p. 253-254.
- Multiplication performed by finding the LCM
The lowest common multiple (LCM) of 4 and 3 is 12.
I divided a rectangle into 12 equal parts (3 units by 4 units) and shaded
Then I found

Then,
Multiplication With Splitting
- Multiplication performed with an area model
Since I am looking for

Then I decomposed
I divided the same rectangle horizontally into 4 equal parts and coloured 1 row, which corresponds to

I'm looking for
Multiplying a Fraction by a Whole Number
In Grade 8, students will multiply fractions by whole numbers (for example,

In order to multiply a fraction by a whole number, students develop personal strategies using various models. Consider the following situation:
On an activity day, we want the students to experience six different activities, each lasting three fourths of an hour. How long will all the activities be?
To solve this problem, we can recognize that we can perform the operation
- perform the repeated addition;
;
- use a concrete representation;

- use a visual representation;

- use an array;

Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 95-96.
- perform calculations.
As a result of using various models, some students often notice that they can determine the product using calculations. For example, in order to calculate
This last strategy, although effective, should not be applied without understanding. It is important for students to realize that when they multiply

Subsequently, when they perform

Multiplying a Fraction by a Mixed Number
Examples
Note: To calculate the area of a rectangle, multiply the two dimensions, which is equivalent to the area formula.
A = length × width
- Multiplication using an area model
I decomposed

I did the partial multiplications.
I know that
To do

Now I have to add up the partial products. I found like denominators, which are tenths.
SO,
- Multiplication performed by transforming a mixed number into an improper fraction
I first changed
Now I can multiply the two fractions by multiplying the numerators together and multiplying the denominators together.
SO,
- Multiplication using an area model
I decomposed the mixed number
I did the partial multiplications.
I added up all the partial products.
- Multiplication performed by transforming mixed numbers into improper fractions
I transform the two mixed numbers into improper fractions.
Now I can multiply the two improper fractions by multiplying the numerators together and multiplying the denominators together.
Division
When dividing fractions, there is a certain progression to follow. The exploration of division, like that of other operations, should focus on concrete and visual (semi-concrete) representations and not on algorithms. Students can then reactivate their prior knowledge and grasp the meaning of the operation. In order to understand division, it is essential to examine the meaning of the division and the nature of the numbers that make up the division. Division is partitive when we look for the size of the groups; it is quotative when we look for the number of groups.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 79-80.
Division of a Fraction by a Whole Number
The division of a fraction by a whole number (for example,
Examples of representations of
Area model | Linear Model | Set Model |
---|---|---|
Three friends want to share |
Three sisters are on their way to school and there is |
In a bag, there were candies. Peter has eaten some and there are |
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Students can then tackle more complex division situations, where the numerator is not divisible by the divisor (for example,
The complexity comes from the fact that the situation is more difficult to represent. To be successful, it must be understood that partitioning in these situations involves subdividing the parts. The operation is easier to understand when it is presented in context.
Example
The 6 members of a family want to share
The question clarifies what one is looking for, that is a fraction of a full pie. Recognizing that division is associated with the concept of partitioning, the situation can be represented symbolically by

How do we divide these two thirds? You can divide each third into three equal pieces for a total of six equal pieces (Figure 1). You can also divide the first third into six pieces that will be shared, then do the same with the second third (Figure 2).

In division problems, the main difficulty experienced by students is to find the quantity in relation to the whole. In the previous situation (

Thus, according to the first split, each receives a piece, or
Consider the same division (
Example
We want to cut
When we understand the meaning of the problem, we can recognize that we must perform

Thus, we can determine that each of the 6 sections corresponds to
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 97-100.
Division of a Fraction by a Fraction
Division Without Splitting
Example
- Division performed with an area model
I represented

There are 5 one sixths in
- Division performed with a symbolic representation
Since the dividend and divisor have like denominators, I can divide the numerators and divide the denominators.
Division With Splitting
Example
- Division performed using a number line
I represented
Knowing that

On the number line, I can see that there is one group of
- Division performed using like denominators
Division of a Fraction by a Mixed Number
Example
- Division performed with a symbolic representation
I changed the mixed number into an improper fraction.
Possible answer 1: "I divided the numerators and denominators.
So,
Possible answer 2: "I found like denominators and then divided.
So,
Knowledge: Fractions
The word fraction comes from the Latin fractio which means "break". In order to determine a fraction of a whole that has been divided into parts, the parts must be of equal or equivalent size.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 33.
Knowledge: Numerator
Number of equal parts into which a whole or set is divided.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 34.
Knowledge: Denominator
Number of equivalent parts by which the whole is divided.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 34.
Knowledge: Fractional Notation
The fractional notation
Example
- I gave 1 fourth (
) of my sandwich to Alex.

- A fourth (
) of my marbles are blue.

However, fractional notation can also be associated with other concepts such as division, ratio and operator.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 36.