At a Glance...

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

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Multiplication

When multiplying fractions, there is a certain progression to follow. In the junior division, students already have knowledge about multiplication. In fact, since the primary division, they have explored concepts related to multiplication using concrete materials, calculators, illustrations and symbols. In^Grade 4, multiplying fractions is limited to multiplying a unit fraction by a whole number. This type of multiplication can be understood by relating it to repeated addition. Thus, students easily grasp that \(3\; \times \;\frac{1}{2}\), which can be read as "3 times a half", is a multiplication which can be represented by the repeated addition, that is \(\frac{1}{2}\; + \;\frac{1}{2}\; + \;\frac{1}{2}\). They should be explored to help students understand the multiplication of fractions.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 78.

Example

"How many cardboards will the student need if he is to distribute half a cardboard to four classmates?"

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 may visualize that two equal parts of a first cardboard are distributed to two classmates and that two equal parts of a second cardboard are distributed to the other two classmates. So he needs two cardboards.

Others may think of the following abstract representation: "I need 4 times a half-cardboard. I know that \(\ 4 \times \frac{1}{2}\) is equal to 2, because two halves are 1. Therefore, I need 2 cardboards." Others may illustrate the problem as follows, then mentally group the pieces two at a time to see that they are equal to two full cardboards.

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

Multiplying a Whole Number by a Fraction

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 from the5th grade by multiplying a whole number by a unit fraction.

Example 1

In a^Grade 8 classroom, \(\frac{1}{2}\) students are wearing toques.

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

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 7, p. 281.

• Multiplication using a double number line

To find \(\frac{1}{2}\;{\rm{de}}\;{\rm{24}}\), 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 using an 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.

Example 2

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: 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 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

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

Multiplying a Fraction by a Fraction

In grade⁷ , 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

\(\frac{1}{4}\; \times \;\frac{2}{3}\) or \(\frac{1}{4}\) of \(\frac{2}{3} \)

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.

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

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

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

Multiplying a Fraction by a Whole Number

In^Grade 8, students learn about multiplying fractions by whole numbers (for example, \(7\; \times \;\frac{3}{4}\)). In the primary and junior divisions, students recognized the relationship between multiplication and repeated addition. This relationship can also be applied in the case of multiplications of fractions by whole numbers. However, to emphasize multiplication as an operation, it is useful to represent it in various ways. For example, the operation \(4\; \times \;\frac{1}{2}\) which is read "four times one half" can be represented by 4 groups of \(\frac{1}{2 }\)(Figure 1) or using an array (Figure 2).

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 \(6\; \times \;\frac{3}{4}\). In order to determine the result, various computational strategies are possible such as:

• perform the repeated addition; \(\frac{3}{4}\; + \;\frac{3}{4}\; + \;\frac{3}{4}\; + \;\frac{3}{4}\ ; + \;\frac{3}{4}\; + \;\frac{3}{4}\; = \;\frac{18}{4}\;{\rm{so}}\; 4\frac{2}{4}\;{\rm{ou}}\;4\frac{1}{2}\);

• use a concrete representation;

• use a semi-concrete representation;

• use an array;

Subtract the missing fourths (either \(\frac{6}{4}\) or \(1\frac{2}{4}\)) from the 6 units and determine that it is \(4\frac{2} {4}\left( {6\; - \;1\; - \;\frac{2}{4}} \right)\) or \(4\frac{1}{2}\).

Source: 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 \(6\; \times \;\frac{3}{4}\), I first multiply 6 by 3, then divide that result by 4 and get \(\frac {{18}}{4}\;{\rm{ou}}\;{\rm{4}}\frac{2}{4}\).

This last strategy, although effective, should not be applied without understanding. It is important for students to realize that when they multiply \(6\; \times \;3\), they are determining the number of pieces. Since these pieces are fourths, they then determined that there are a total of 18 fourths (\(\frac{{18}}{4}\)).

Subsequently, when they perform \(18\; \div \;4\), they determine the number of groups of 4 fourths. Thus, they determine that there are 4 wholes and two fourths remain, hence the result \(4\frac{2}{4}\).

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

\(\frac{2}{5}\; \times \;2\frac{1}{4}\;\)

• Multiplication using an area model

I decompose the number \(2\frac{1}{4}\) into \(2\; + \;\frac{1}{4}\).

I perform partial multiplications.

I know that \(2\; \times \;\frac{2}{5}\) is the same as \(\frac{2}{5}\; + \;\frac{2}{5} \), which gives me \(\frac{4}{5}\).

To perform \(\frac{2}{5}\; \times \;\frac{1}{4}\), I multiply the numerators together and I multiply the denominators together. So, \(\frac{2}{5}\; \times \;\frac{1}{4}\; = \;\frac{2}{{20}}\;{\rm{ou}} \;\frac{1}{{10}}\).

Now I have to add up the partial products. I find like denominators, which are tenths.

\(\frac{4}{5}\; + \;\frac{1}{{10}}\; = \;\frac{8}{{10}}\; + \frac{1}{{ 10}}\;=\;\frac{9}{{10}}\)

SO, \(\frac{2}{5}\; \times \;2\frac{1}{4}\; = \;\frac{9}{{10}}\).

• Multiplication performed by transforming a mixed number into an improper fraction

I first transform \(2\frac{1}{4}\) into an improper fraction, that is \(\frac{9}{4}\) since \(\frac{4}{4}\; + \;\frac{4}{4}\; + \;\frac{1}{4}\; = \;\frac{9}{4}\) (distributive property).

Now I can multiply the two fractions by multiplying the numerators together and multiplying the denominators together.

\(\begin{array}{c}\frac{2}{5}\; \times \;2\frac{1}{4}\; = \;\frac{2}{5}\; \times \;\frac{9}{4}\\ = \;\frac{{18}}{{20}}\;{\rm{ou}}\;\frac{9}{{10}}\end {array}\)

SO, \(\frac{2}{5}\; \times \;2\frac{1}{4}\; = \;\frac{9}{{10}}\).

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

• Multiplication using an area model

I decompose the mixed number \(1\frac{1}{3}\;{\rm{en}}\;{\rm{1}}\;{\rm{ + }}\;\frac{1} {3}\) and \(2\frac{3}{4}\;{\rm{and}}\;{\rm{2}}\;{\rm{ + }}\;\frac{3 }{4}\).

I perform partial multiplications.

\(\begin{align}{c}\left( {1\; + \;\frac{1}{3}} \right) \left( {2\; + \;\frac{3}{4} } \right)\; &= \;\left( {1\; \times \;2} \right)\; + \;\left( {1\; \times \;\frac{3}{4} } \right)\; + \;\left( {2\; \times \;\frac{1}{3}} \right)\; + \;\left( {\frac{1}{3}\ ; \times \;\frac{3}{4}} \right)\\ &= \;2\; + \;\frac{3}{4}\; + \;\frac{2}{3} \; + \;\frac{3}{{12}}\end{align}\)

I add up all the partial products.

\(\begin{align}2\; + \;\frac{3}{4}\; + \;\frac{2}{3}\; + \;\frac{3}{{12}}\ ; &= \;2\; + \;\frac{9}{{12}}\; + \;\frac{8}{{12}}\; + \;\frac{3}{12}\\ &= \;2\; + \;\frac{{20}}{{12}}\\ &= \;2\; + \;1\frac{8}{{12}}\\ &= \;3\frac{8}{{12}}\;{\rm{ou}}\;{\rm{3}}\frac{2}{3} \\ {\rm{So,} }\; \;1\frac{1}{3} \times \; \;2\frac{3}{4} \; &= \;3\frac{2}{3}\end{align} \)

• Multiplication performed by transforming mixed numbers into improper fractions

I transform the two mixed numbers into improper fractions.

\(1\frac{1}{3}\; = \;\frac{3}{3}\; + \;\frac{1}{3}\; = \;\frac{4}{3} \) \(2\frac{3}{4}\; = \;\frac{4}{4}\; + \;\frac{4}{4}\; + \;\frac{3}{4} \;=\;\frac{11}{4}\)

Now I can multiply the two improper fractions by multiplying the numerators together and multiplying the denominators together.

\(\frac{4}{3}\; \times \;\frac{{11}}{4}\; = \;\frac{44}{12}\; = \;3 \frac{8}{{12}}\;{\rm{ou}}\;{\rm{3}}\frac{2}{3}\)

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 is partitive when we look for the size of the groups; it quotative when we look for the number of groups.

Source : 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, \(\frac{3}{4}\; \div \;5\)) is more often found in partitive situations. We can start with situations where the numerator is divisible by the divisor (for example, \(\frac{6}{9}\; \div \;3,\;\frac{3}{4}\; \ div \;3,\;\frac{4}{9}\; \div \;2,\;\frac{{12}}{{20}}\; \div \;4\)). The examples below show different ways to get the result of dividing \(\frac{6}{9}\;{\rm{par}}\;{\rm{3}}\).

Examples of representations of \(\frac{6}{9}\; \div \;3\; = \;\frac{2}{9}\)

Students can then tackle more complex division situations, where the numerator is not divisible by the divisor (for example, \(\frac{2}{3}\; \div \;6, \;\frac{3}{5}\; \div \;12\)).

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 \(\frac{2}{3}\) of a pie. What fraction of pie will each have?

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 \(\frac{2}{3}\; \div \;6\). In order to represent the operation, we can illustrate \(\frac{2}{3}\) of a pie.

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 (\(\frac{2}{3}\; \div \;6\)), the fraction \(\frac{2}{3}\) temporarily acts as a whole, because it must be divided into 6. However, the answer must be expressed in relation to the whole to which \(\frac{2}{3}\) refers (the pie). Thus, from the representations above, the student determines that the solution is 1 piece or 2 pieces depending on the division made, but has difficulty in recognizing that it is 1 or 2 pieces of the whole. But how do we know what fraction of the pie each gets? To find out, we must also divide the missing third.

Thus, according to the first split, each receives a piece, or \(\frac{1}{9}\) of a pie (Figure 3). According to the second, each receives two pieces, or \(\frac{2}{{18}}\) of a pie (Figure 4). However, they are the same quantity since \(\frac{1}{9}\) and \(\frac{2}{{18}}\) are equivalent fractions.

Consider the same division (\(\frac{2}{3}\; \div \;6\)) from a situation that relates more to the linear model.

Example

We want to cut \(\frac{2}{3}\) from a roll of rope into 6 sections. What fraction of the roll of rope will each section be?

When we understand the meaning of the problem, we can recognize that we must perform \(\frac{2}{3}\; \div \;6\). In order to determine the quotient, as in the previous example, one can proceed in two ways:

Thus, we can determine that each of the 6 sections corresponds to \(\frac{1}{9}\) or \(\frac{2}{{18}}\) of the roll of rope.

Source : Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, p. 97-100.

Division of a Whole Number by a Fraction

In^Grade 5, the division of a whole number by a unit fraction (for example, \(2\; \div \;\frac{1}{3}\)) is well represented using the quotative division. In this case, the fraction is the divisor.

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

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 81-82.

In the case of a division of a whole number by a fraction, division usually takes on the quotative meaning. Thus, the analogy of repeated subtraction is in order 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 in 2?" can help visualize the operation.

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

In Grade⁶ , 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 6m rope and wants to cut it into sections of \(\frac{3}{5}\;\)meters each. How many sections will she be able to create?

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

By grade⁷ , students 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 like denominators, 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 (\(\frac{1}{2}\)) of another, so in total there are 1\(\frac{1}{2}\) groups of \(\frac{1}{2}\).

• Division performed using like denominators

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

Division of a Fraction by a Mixed Number

Example

\(\frac{7}{{12}}\; \div \;1\frac{3}{4}\)

• Division performed with a symbolic representation

I transform the mixed number into an improper fraction.

\(1\frac{3}{4}\; = \;\frac{4}{4}\; + \;\frac{3}{4}\; = \;\frac{7}{4} \)

Possible answer 1: I divide the numerators and denominators.

\(\frac{7}{{12}}\; \div \;\frac{7}{4}\; = \;\frac{{7\; \div \;7}}{{12\; \div \;4}}\;=\;\frac{1}{3}\)

So, \(\frac{7}{{12}}\; \div \;1\frac{3}{4}\; = \;\frac{1}{3}\)

Possible answer 2: I can find like denominators and then do the division.

\(\frac{7}{{12}}\; \div \;\frac{7}{4}\; = \;\frac{7}{{12}}\; \div \;\frac{ {21}}{{12}}\;=\;\frac{{\frac{7}{{21}}}}{1}\;=\;\frac{7}{{21}}\; {\rm{ou}}\;\frac{1}{3}\)

So, \(\frac{7}{{12}}\; \div \;1\frac{3}{4}\; = \;\frac{1}{3}\)

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, in order to determine a fraction of an object divided into parts, the parts must be equivalent.

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

Knowledge: Numerator

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Number of equal parts into which a whole or set is divided.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 34.

Knowledge: Denominator

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Number of equivalent parts by which the whole is divided.

Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 34.

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.

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. 36.