B2.5 add and subtract fractions with like and unlike denominators, using appropriate tools, in various contexts
Skill: Adding and Subtracting Fractions with Fractions With and Without Common Denominators, Using Appropriate Tools and Strategies
Make 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.
Source: A Guide to
Effective Instruction in Mathematics, Grades 4 to 6, p. 75.
Valuing Informal Procedures to Develop Strategies
It is important to value informal procedures, as they contribute to the development of number sense and operational
sense. In situations that involve operations, many students use personal algorithms rather than procedures. For
example, a student with good number sense could tackle the addition of \(\frac{7}{8}\) from a sandwich and
\(\frac{4}{8}\) from 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, \(\frac{7}{8}\; + \;\frac{4}{8}\; = \;1\frac{3}{8}\).
»
Source: A Guide to
Effective Instruction in Mathematics, Grades 4 to 6, p. 76.
Nature of the Fundamental Operations
Applying an operation on numbers has the effect of rearranging the quantities involved. It is very important that students understand this aspect of quantity when using any of the four operations. Junior students have had the opportunity to learn this relationship in relation to the four operations on whole numbers. In addition, two quantities are put together to form a new quantity. In subtraction, one quantity is taken away from another. We can also recognize that we are looking for a quantity by which two given quantities differ.
It takes a long time to build a sense of operations on fractions, as one must think about the numerators, denominators, and all involved. Students should be given the opportunity to work with concrete and semi-concrete models and develop a sense of the order of magnitude of results before moving on to operations involving symbolic representations.
According to the curriculum, 6th graders see addition and subtraction of fractions with and without common denominators.
For addition and subtraction with like denominators, it is essentially the addition and subtraction of objects or quantities of the same kind.
For example, in \(\frac{3}{7}\; + \;\frac{2}{7}\; = \;\frac{5}{7}\), the addition simply represents \( 3\;{\rm{one \ sevenths}}\; + \;2\;{\rm{one \ sevenths}}\; = \;5\;{\rm{one \ sevenths}}\), any 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.
Subtraction is treated the same way. For example, if I take 3 marbles out of a bag that contains 5 marbles, 2 marbles remain. Similarly, if I subtract 3 one-eighths from 5 one-eighths, I have 2 one-eighths left (\(\frac{5}{8}\; - \;\frac{3}{8}\; = \;\frac{2}{8}\)).
Adding two whole numbers increases both initial quantities, while subtracting two whole numbers decreases the initial quantity. It is important for students to understand that the same is true for adding and subtracting fractions.
This allows them to understand the implausibility of certain answers obtained from erroneous procedures. For example,
the student who, to calculate \(\frac{2}{3}\; + \;\frac{1}{3}\), adds the numerators and denominators and obtains
\(\frac{3 }{6}\), should see that this answer, which is equal to \(\frac{1}{2}\), is less than one of the initial
fractions, i.e. \(\frac{2}{3}\).
Source: A Guide to Effective
Mathematics Teaching Grades 4-6, p. 76-77.
Explore 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: A Guide to
Effective Instruction in Mathematics, Grades 4 to 6, p. 76.
When working with fractions, the most important and sometimes the most difficult part is getting the whole thing
right. With concrete or semi-concrete materials, the visual effect of addition and subtraction is enhanced. Drawing on
their experiences with whole numbers and developing a sense of operation, students can recognize, for example, that
adding two-thirds to one-third yields three-thirds, 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 three eighths left of a pizza and two 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 sixteen pieces in all?
Students progress to problems involving fractions with different denominators. However, students can explore this
type of problem in context using concrete and semi-concrete representations.
For example, Alexis and his brother Mycolas each have a soft bar. The bars are identical. Alexis ate one-third of her
soft bar and Mycolas ate three-quarters of his. How much of the soft bar do they have left? While this is a
challenging situation for students, if they use cues, a visualization of the situation, or a semi-concrete
representation, they are able to conclude that there is almost a whole soft bar left.
This is represented using a surface model. The rectangular shape is rather natural, since it looks like a soft bar shelf. However, other representations, such as the length model, should not be overlooked.
The following example is about distance and uses fractional numbers:
As part of his training for a race, William must run at least 6 km a day. This morning, before going to school, he ran 3 km. How many miles does he have to run after school?
Since the situation deals with a linear measurement, students can use a length model such as a number line. The first
step is to locate 6 on a graded number line (left) or a non graded number line (right),
then subtract 3 from 6
then subtract the fractional part, i.e. 3 times \(\frac{3}{10}\) or 3 times \(\frac{1}{10}\).
Thus, we can conclude that William must travel 2 km. We have therefore \(6\frac{1}{10} - 3\frac{3}{10} = \;2\frac{8}{10}\).
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 to express the result as \(\frac{13}{4}\), if we count the quarters, or \(\frac{1}{4}\), if we take into account the whole.
Source: A Guide to
Effective Mathematics Teaching Grades 4-6, p. 92-94.
Here are different types of problems relating to
addition and subtraction (addition problems, subtraction problems, comparison problems and union problems).
Examples:
Joining Problems
- Mila ate a submarine, while Peter ate it. What fraction of the submarine was eaten by the two children?
\(\frac{2}{8}\; + \frac{1}{2}\; = \frac{1}{2}{8}\)
Representation using fractional bands
\(\frac{6}{8}\;{\rm{ou}}\;\frac{3}{4}\) of the submarine was eaten by the two children.
Source: A Guide
to Effective Instruction in Mathematics, Grades 4 to 6, p. 78.
- The red ant measures \(\frac{3}{{10}}\) centimetres, the cicada measures \(1\frac{1}{5}\) centimetres, the beetle measures \(\frac{2}{{10}}\) of a centimetre and the ladybug measures \(\frac{3}{{10}}\) of a centimetre. What improper fraction represents the length in cm of the sequence of insects? What is this length in metres?
To add fractions, you need to represent them with a common denominator.
\(\frac{12}{10} + \frac{3}{{10}} + \frac{2}{10} + \frac{3}{10} = \frac{20}{10} = 2\)
Its length is centimetres or 2 cm. Its length in metres is metres or 0.02 m.
Source: Adapted from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
Comparing Problem
Mila took \(\frac{1}{4}\) from a pitcher of water to water her plant. Peter took \(\frac{5}{8}\) from the pitcher to water his own. What fraction represents the difference between the amounts of water used by Peter and Mila?
To subtract fractions, they must be represented using a common denominator.
There is a difference in the amount of water in the pitcher.
Source: A Guide to Effective
Instruction in Mathematics for Grades 4-6, p. 78.
Knowledge: Denominator
Number of equal parts into which a whole or set is divided.
For example, the whole is a length.
The denominator of the segment is red. The denominator "3" of the fraction \(\frac{2}{3}\), represents
the 3 equivalent sectioned parts of the number line.
Knowledge: Numerator
The number above the line in a fraction that represents the number of equal parts being considered.
For example, the whole is a length.
\(\frac{2}{3}\) of the segment is red. The numerator "2" of the fraction \(\frac{2}{3}\), represents the 2 red parts of the 3 equivalent sections of the number line.
Source: A Guide
to Effective Instruction in Mathematics, Grades 4 to 6, p 32 and 34.