B2.5 Add and subtract fractions with like denominators, in various contexts.
Skill: Adding and Subtracting Fractions With Like Denominators, in Various Contexts
Making Connections Between Operations on Whole Numbers and Operations on Fractions
Students have already developed a solid understanding of addition and subtraction. It is important to make connections between operations on whole numbers and operations on fractions. For example, adding 3 eighths and 2 eighths is the same as adding 3 candies and 2 candies. Only the notation is different.
Valuing Informal Procedures to Develop Strategies
It is important to value informal procedures, as they contribute to the development of number sense and operation sense. In situations that involve operations, many students use personal algorithms rather than procedures. For example, a student with good number sense might approach the addition of \(\frac{7}{8}\) of a sandwich and \(\frac{4}{8}\) of another identical sandwich in this way: "I know that with \(\frac{7}{8}\) of a sandwich, I am missing \(\frac{1}{8}\) of a sandwich to have a complete sandwich. So if I add \(\frac{1}{8}\) to \(\frac{7}{8}\), I have a whole and I'm left with \(\frac{3}{8 }\). So \(\frac{7}{8}\; + \;\frac{4}{{8\;}}\; = \;1\frac{3}{8}\)".
Nature of the Fundamental Operations
Applying an operation on numbers has the effect of reorganizing the quantities involved. It is very important that students understand this facet of quantity when using any of the four operations.
Junior students have had the opportunity to learn this relationship in relation to the 4 operations on whole numbers. In addition, two quantities are put together to form a new quantity while in subtraction, one quantity is taken away from another. We can also recognize when we are looking for a quantity by which 2 given quantities differ. However, whether one is joining, comparing, combining, or separating, it is important to know and understand that the operation has an effect on the quantities.
When fractions are involved, it becomes even more important to focus on the meaning of the operation, choose an appropriate model, and think about the quantities. Building a sense of operations on fractions takes a lot of time, as it requires thinking about the numerators, denominators, and wholes involved. Students should be given the opportunity to work with concrete and semi-concrete models and to develop a sense of the result of performing an operation before moving on to operations involving symbolic representations.
Addition and Subtraction
According to the curriculum, Grade 5 students see the addition and subtraction of fractions with like denominators. This is essentially the addition and subtraction of objects or quantities of the same nature. For example, in \(\frac{3}{7}\; + \;\frac{2}{7}\; = \;\frac{5}{7}\), the addition simply represents 3 sevenths + 2 sevenths = 5 sevenths, just like 3 apples + 2 apples = 5 apples. In this example, sevenths are counted the same way as apples. We have 3 pieces of a certain size and 2 pieces of the same size, which gives 5 pieces of this size. Adding fractions becomes more complex when the fractions have different denominators, since the parts are not the same size and are not of the same nature. These operations are studied in grade 6 .
Subtraction is treated the same way. For example, if I take 3 marbles out of a bag that contains 5 marbles, I have 2 marbles left. Similarly, if I subtract 3 eighths from 5 eighths, I have 2 eighths left \(\frac{5}{8}\; - \frac{3}{8}\; = \frac{2}{8}\).
Adding two whole numbers has the effect of increasing the original quantity, while subtracting two whole numbers has the effect of decreasing the original quantity. It is important for students to understand that the same is true for adding and subtracting fractions. This allows them to understand the reasonableness of some answers obtained from incorrect procedures. For example, a student who adds the numerators and denominators of \(\frac{2}{3}\; + \;\frac{1}{3}\) to obtain \(\frac{3}{6}\) should see that this answer, which is equal to \(\frac{1}{2}\), is less than one of the original fractions, \(\frac{2}{3}\).
Exploring Operations Using Multiple Models
It is important for students to learn fraction-related concepts in a variety of situations using a variety of models such as area models, linear models, and set models. The same is true for learning operations on fractions.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 75-77.
When working with fractions, the most important and sometimes the most difficult thing is to represent the whole correctly. In Grade 5, students add and subtract fractions with like denominators. With concrete or semi-concrete material, the visual effect of addition and subtraction is enhanced. Drawing on their experiences with whole numbers and developing a sense of operation, students may recognize, for example, that adding 2 thirds (\(\frac{2}{3}\)) to 1 third (\(\frac{1}{3}\)) gives 3 thirds (\(\frac{3}{3}\)), or the whole.
However, it is not always clear to students that addition is in relation to the numerator. In a situation where there are 3 eighths left of a pizza and 2 eighths left of another pizza of the same size, we look at how many are left in total. Are we counting eighths or sixteenths, since the pizzas had a total of 16 pieces?
Students should also explore situations involving improper fractions. For example, the operation \(\frac{{10}}{4}\; + \;\frac{3}{4}\) could be represented using the following model.
This model allows the result to be expressed in the form \(\frac{{13}}{4}\) or \(3\frac{1}{4}\).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 92-94.
Examples
Joining Problems
Mila ate \(\frac{2}{8}\) of a submarine sandwich, while Peter ate \(\frac{3}{8}\). What fraction of the submarine sandwich was eaten by the 2 children?
Comparing Problem
Mila took \(\frac{2}{8}\) from a pitcher of water to water her plant. Pierre took \(\frac{5}{8}\) from the pitcher to water his own. What fraction represents the difference between the amounts of water used by Pierre and Mila?
This situation translates to \(\frac{5}{8}\; - \;\frac{2}{8}\; = \;\frac{3}{8}\).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 78.
There is a difference of \(\frac{3}{8}\) between the amount of water used by Pierre and Mila.
Knowledge: Numerator
Number of equal parts of the whole being considered.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 34.
The numerator is the number at the top of the fraction. It tells us the number of equal parts or groups involved.
Knowledge: Denominator
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.
The denominator is the number at the bottom of the fraction. This number allows us to know how many equivalent parts there are in the whole. When there are like denominators, this means that the 2 denominators are identical.