At a Glance...

B2.8 Multiply and divide fractions by fractions, using tools in various contexts.

Skill: Multiplying and Dividing Fractions by Other Fractions, Using Tools, in a Variety of Contexts

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Multiplication

When multiplying fractions, there is a certain progression to follow. In the junior grades, students already have a background of knowledge about multiplication. Since the primary grades, students have been exploring concepts related to multiplication using concrete materials, the calculator, pictures and symbols. In Grade⁴ , fraction multiplication is limited to the multiplication of a unit fraction by a whole number. This type of multiplication can be understood by relating it to repeated addition. For example, students easily grasp that \(3\; \times \;\frac{1}{2}\) which can be read as "3 times 1 half," is a multiplication that can be represented by repeated addition, that is, \(\frac{1}{2}\; + \;\frac{1}{2}\; + \;\frac{1}{2}\). These should be explored to help students understand fraction multiplication.

Example

How many cards will the student who is to distribute half a card to 4 classmates need?

By understanding the situation, students recognize that there is a multiplication of a quantity, either \(4\; \times \;\frac{1}{2}\), which can be represented by repeated addition, or \(\frac{1}{2}\; + \;\frac{1}{2}\; + \;\frac{1}{2}\; + \;\frac{1}{2}\). To find the answer, the students represent the situation in a mental or semi-concrete way or use their knowledge of the addition of the fractions in question. Some students can visualize that 2 equal parts of a 1^st card are distributed to 2 comrades and that 2 equal parts of a 2^nd card are distributed to the 2 other comrades. They therefore needs 2 boxes.

Others may think of the following abstract representation: "I need 4 times a half-carton. I know that \(4; \times \;\frac{1}{2}\) is equal to 2, because 2 halves make 1. Therefore, I need 2 boxes." Others may illustrate the problem as follows, then mentally combine the pieces 2 by 2 to see that there are the equivalent of 2 full boxes.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 86-87.

However, it is more difficult to make sense of multiplying a whole number by a fraction (for example, \(\frac{1}{2}; \times \;3\)). These situations are explored starting in grade 5 by multiplying a whole number by a unit fraction.

Example

In a Grade 6 classroom, students are wearing toques.

If there are 24 students in the classroom, how many students are wearing toques?

• Multiplication using a double number line

To find \(\frac{1}{2}\) of 24, I divide 24 by 2, which gives me 12.

I represent the 24 students below the line and half of 24 on the top of the line.

There are 12 students wearing toques.

• Multiplication performed using the area model

I decompose 24 into \(20\; + \;4\). I determine half of 20 and half of 4, \(10\; + \;2\; = \;12\).

There are 12 students wearing toques.

In grade 6 , as they examine the concept of multiplying a whole number by a proper fraction, they will learn that the fraction of a set \(\frac{2}{3}\) of 6 is related to multiplication and that this situation can be represented by multiplication \(\frac{2}{3}\; \times \;6\). However, there is a danger in introducing this multiplication in grade 6 , as there is a danger of forcing obedience to rules and rote learning without conceptual understanding. It takes a good degree of abstraction to accept a situation like \(\frac{1}{2}\) of 12 as multiplication.

Example

In a rectangular field with an area of 100^m2, Mr. Longpré has planted cucumbers on \(\frac{2}{5}\)of the field . What area of the field, in m2, is devoted to cucumbers?

Source: Les mathématiques… , Guide pédagogique, Numération et Sens du nombre/Mesure,^6e année, Module 2, Série 2, p. 309.

• Multiplication performed using the area model

I have decomposed 100 into \(25\; + \;25\; + \;25\; + \;25\). I also decomposed \(\frac{2}{5}\) into \(\frac{1}{5}\; + \;\frac{1}{5}\). I know that \(\frac{1}{5}\) of 25 is 5 since \(5\; \times \;5\; = \;25\).

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

\[\begin{array}{l}\frac{2}{5}\; \times\;100\; = \;2\; \times\;\frac{1}{5}\; \times\;100\; = \;\frac{1}{5}\; \times\;2\; \times \;100\\\quad \quad \quad \quad \quad \quad \quad \quad \,\; = \;\frac{1}{5}\; \times 200\\\quad \quad \quad \quad \quad \quad \quad \quad \,\; = \;200\; \div \;5\\\quad \quad \quad \quad \quad \quad \quad \quad \,\; = \;40\end{array}\]

The area of the field dedicated to the cultivation of cucumbers is 40^m2.

Multiplying a Fraction by a Fraction

In grade 7 , students multiply and divide fractions by other fractions. Thus, the teacher needs to get students to understand the meaning of operations and to represent them visually using various representations.

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

\(\frac{1}{4}\; \times \;\frac{2}{3}\;{\rm{ou}}\;\frac{1}{4}\;{\rm{de}}\;\frac{2}{3}\)

Since I am looking for \(\frac{1}{4}\) of \(\frac{2}{3}\), I vertically divide a rectangle into 3 equal parts and colour 2 parts, which corresponds to \( \frac{2}{3}\) of the rectangle.

Then I horizontally divide the same rectangle into 4 equal parts and colour 1 part, which is \(\frac{1}{4}\) of the rectangle.

The fraction that represents \(\frac{1}{4}\) corresponds to the number of parts coloured twice, that is \(\frac{2}{{12}}\). The numerator corresponds to the darker coloured parts and the denominator corresponds to the number of parts of the same size in the whole, that is 12 parts.

The fraction \(\frac{2}{{12}}\) can be simplified to an equivalent fraction of \(\frac{1}{6}\) by dividing the numerator and denominator by 2.

• Multiplication performed with a symbolic representation

The student multiplies the numerators together and the denominators together.

\(\frac{1}{4}\; \times \;\frac{2}{3}\; = \;\frac{{1\; \times \;2}}{{4\; \times \;3}}\;= \frac{2}{12}\)

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 divide a rectangle into 12 equal parts (3 units by 4 units based on the denominators) and represent \(\frac{2}{3}\)of the rectangle.

Then I find \(\frac{1}{4}\) of 8 squares or \(\frac{1}{4}\) of 2 columns = 2 squares by 12.

Then, \(\frac{1}{4}\)of \(\frac{2}{3}\)of the rectangle is equal to 2 squares over the set of squares (12).

\(\frac{1}{4}\;{\rm{of}}\;\frac{2}{3}\;{\rm{of rectangle}}\;{\rm{ = }}\; \frac{2}{{12}}\;{\rm{ou}}\;\frac{1}{6}\)

Multiplication With Splitting

• Multiplication performed with an area model

\(\frac{3}{4}\; \times \;\frac{2}{3}\;{\rm{or}}\;\frac{3}{4}\;{\rm{of}}\;\frac{2}{3}\)

Since I am looking for \(\frac{3}{4}\) of \(\frac{2}{3}\), I vertically divide a rectangle into 3 equal parts and colour 2 parts, which corresponds to \( \frac{2}{3}\) of the rectangle.

Then I can decompose \(\frac{3}{4}\) into \(\frac{1}{4}\; + \;\frac{1}{4} + \;\frac{1}{ 4}\).

I divide the same rectangle horizontally into 4 equal parts and first colour 1 part, which corresponds to \(\frac{1}{4}\) of the rectangle, that is 3 squares among 12. The fraction that represents \(\frac{1}{4}\) of \(\frac{2}{3}\) corresponds to the number of parts coloured twice, that is \(\frac{2}{{12}}\). The numerator corresponds to the darker coloured parts and the denominator corresponds to the number of parts of the same size in the whole, that is 12 parts (figure 1).

I'm looking for \(\frac{3}{4}\), then I can do \(2\; + \;2\; + \;2\) since \(\frac{1}{4}\) represents 2 squares. So, \(\frac{3}{4}\) represents 6 squares. \(\frac{3}{4}\) of \(\frac{2}{3}\) represents \(\frac{6}{{12}}\) at all (Figure 2). If I move 2 of the squares (Figure 3), I can see that \(\frac{6}{{12}}\) is also \(\frac{1}{2}\) at all.

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 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 has the meaning of sharing when we look for the size of the groups; it has the meaning of grouping when we look for the number of groups.

Thus, in grade 5 , dividing a whole number by a unit fraction (for example, \(2\; \div \;\frac{1}{3}\)) is well represented using grouping. In this case, the fraction is the divisor.

For example, if we have 2 licorices and we want to give each child \(\frac{1}{3}\) licorice, we proceed to a division since we have to separate a quantity (2 licorices ) in equal amounts of (\(\frac{1}{3}\) of licorice) to determine how many equal amounts or groups can be created (6 children will receive \(\frac{1}{3}\) of licorice each). In this case, it is important to recognize that the quotient expresses a number of sections, or thirds, and not a quantity of objects (liquorice).

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 80-82.

In the case of a division of a whole number by a fraction, division usually takes on the meaning of grouping. Thus, the analogy of repeated subtraction is appropriate since it involves separating parts.

For example, in dividing 2 by \(\frac{1}{4}\) (\(2\; \div \;\frac{1}{4}\)), doing \(2\; - \;\frac{1}{4}\; - \;\frac{1}{4}\; - \;\frac{1}{4}\; - \;\frac{1}{4} \; - \;\frac{1}{4}\; - \;\frac{1}{4}\; - \;\frac{1}{4}\; - \;\frac{1}{ 4}\), we can create 8 groups of \(\frac{1}{4}\). However, the group created is rather abstract since it is a group that is a fraction of a whole. The questions "How many \(\frac{1}{4}\) can be created with 2 wholes?" and "How many times does \(\frac{1}{4}\) go into 2?" can help visualize the operation.

In grade 6 , students divide a whole number by a proper fraction.

Example

\(6\; \div \;\frac{3}{5}\; = \;10\)

Area Model

In order to create a tessellation, each team needs the equivalent of \(\frac{3}{5}\;\)squares on a sheet. How many teams can complete the task if we have 6 sheets?

Linear Model

A teacher has a 6-metre rope and wants to cut it into sections of one metre each. How many sections can she create?

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 100-102.

In grade⁷ , students will divide fractions by other fractions.

Division of a Fraction by a Fraction

Division Without Splitting

Example

\(\frac{5}{6}\; \div \;\frac{1}{6}\)

• Division performed with an area model

I represent \(\frac{5}{6}\) of a rectangle by dividing the rectangle into 6 equal parts of which 5 parts are shaded. I ask myself the question "How many \(\frac{1}{6}\) are there in \(\frac{5}{6}\)?" »

There are 5 one-sixths in \(\frac{5}{6}\). So, \(\frac{5}{6}\; \div \;\frac{1}{6}\; = \;5\).

• Division performed with a symbolic representation

Since the dividend and divisor have a common denominator, I can divide the numerators and divide the denominators.

\(\frac{5}{6}\; \div \;\frac{1}{6}\; = \frac{{5\; \div \;1}}{{6\; \div \; 6}}\;=\frac{5}{1}\;=5\)

Division With Splitting

Example

\(\frac{3}{4}\; \div \;\frac{1}{2}\)

• Division performed using a number line

I represent \(\frac{3}{4}\)on the number line (red line). I ask myself the question "How many groups of \(\frac{1}{2}\) are there in \(\frac{3}{4}\)?" »

Knowing that \(\frac{2}{4}\; = \;\frac{1}{2}\), I determine how many \(\frac{2}{4}\) there are in \(\frac{3}{4}\).

On the number line above, I can see that there is a group of \(\frac{1}{2}\) or \(\frac{2}{4}\) and \(\frac {2}{4}\)to \(\frac{3}{4}\), there is half of \(\frac{1}{2}\).

(\(\frac{1}{2}\)) from another, so in total there are 1\(\frac{1}{2}\) groups of \(\frac{1}{2 }\).

• Division Using a Common Denominator

\(\begin{array}{l}\frac{3}{4}\; \div \;\frac{1}{2}\; = \;\frac{3}{4}\; \div \ ;\frac{2}{4}\\\quad \quad \;\; = \;\frac{{3\; \div \;2}}{{4\; \div \;4}}\\ \quad \quad \;\; = \;\frac{{\frac{3}{2}}}{1}\; = \;\frac{3}{2}\\\quad \quad \;\ ;= \;1\frac{1}{2}\end{array}\)

Knowledge: Fractions

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The word fraction comes from the Latin fractio which means "break". A part of a broken object can therefore represent a fraction, because it is a part of a whole. However, to determine a fraction of an object divided into parts, the parts must be equivalent.

Knowledge: Fractional Notation

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The fractional notation \(\frac{a}{b}\ is usually associated with the concept of part of a whole. The whole can be an element or a set of elements.

For example:

• I gave 1 fourth (\(\frac{1}{4}\)) of my sandwich to Alex.

• A fourth (\(\frac{1}{4}\)) of my marbles are blue.

However, fractional notation can also be associated with other concepts such as division, ratio and operator.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 33-36.