B2.5 Add and subtract fractions, including by creating equivalent fractions, in various contexts.
Skill: Adding and Subtracting Fractions in Various Contexts
Make connections between operations on whole numbers and operations on fractions
By Grade 7, students have already developed a solid understanding of addition and subtraction. It is important to make connections between operations with whole numbers and operations with fractions. For example, adding 3 eighths and 2 eighths is the same as adding 3 candies and 2 candies. The unit fraction can help to bridge student understanding about operations with whole numbers to fractions. For example, saying "3 one-eighths and 2 one-eighths". Only the notation is different and more complex.
Valuing informal procedures to develop strategies
It is important to value informal procedures, as they contribute to the development of number and operation 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 \(\frac{4}{8}\) of another identical sandwich in this way: "I know that with \(\frac{7}{8}\) of a sandwich, I have \(\frac{1}{8}\) less than a whole sandwich. So if I add \(\frac{1}{8}\) to \(\frac{7}{8}\), I have a whole sandwich and I'm left with \(\frac{3}{8 }\) or the second sandwich. So altogether I have \(\frac{7}{8}\; + \;\frac{4}{8}\; = \;1\frac{3}{8}\) sandwiches.
Fundamental Operations
Applying an operation on numbers has the effect of changing the quantities involved. It is very important that students understand this aspect of quantity when using addition, subtraction, multiplication and division. Junior students have had the opportunity to learn about these relationships in relation to the four operations on whole numbers. For addition, two quantities are put together and result in a new quantity. For subtraction, one quantity is taken away from another quantity to result in a new quantity. We can also recognize that we are looking for the difference between two quantities.
It takes a long time to build a sense of operations with fractions, as one must think about the numerators, denominators, and wholes involved. Students should be given the opportunity to work with concrete and visual (semi-concrete) models before moving on to operations involving symbolic representations. A key concept for students to understand is that when adding or subtracting fractions, we can only add and subtract like things; in the same way that adding and subtracting whole numbers is essentially adding the unit “ones”.
According to the Ontario curriculum, the expectation is that Grade 6 students add and subtract fractions with and without common denominators.
For addition and subtraction with fractions having like denominators, it is essentially the addition and subtraction of units, objects or quantities of the same kind.
For example, in \(\frac{3}{7}\; + \;\frac{2}{7}\; = \;\frac{5}{7}\), the addition simply represents 3 one seventh + 2 one sevenths = 5 one sevenths, just as 3 apples + 2 apples = 5 apples. In this example, one sevenths (the unit fraction) are counted in the same way as apples. We have 3 parts of a certain size and 2 parts of the same size, which gives 5 parts of that 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 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, \(\frac{2}{3}\).
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 75-77.
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 with fractions.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 76.
When working with fractions, the most important and sometimes the most difficult part is correctly representing the whole. 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 2 thirds (\(\frac{2}{3}\)) to 1 third \(\frac{1}{3}\)) results in 3 thirds (\(\frac{3}{3}\)) which is the whole.
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However, it is not always clear to students that addition is related to the numerator. In a situation where there are 3 eighths (\(\frac{3}{8}\)) left of a pizza and 2 eighths (\(\frac{2}{8}\)) left of another pizza of the same size, when determining how much is left, we need to think about how many eights are left in total. We are counting eighths, not sixteenths, even though the pizzas had a total of 16 pieces.
Students progress to problems involving fractions with different denominators. However, students can explore this type of problem in context using concrete and visual (semi-concrete) representations.
For example, Alexis and his brother Mycolas each have a granola bar. The bars are identical. Alexis ate one-third (\(\frac{1}{3}\)) of her granola bar and Mycolas ate three fourths (\(\frac{3}{4}\)) of his. How many granola bars do they have left altogether? While this is a challenging situation for students, if they use benchmarks, a visualization of the situation, or a semi-concrete representation, they are able to conclude that there is almost a whole granola bar left.
This situation is represented with the help of an area model. The rectangular shape is rather natural, since it looks like a granola bar. The granola bar is also a good choice of context because it is not already pre-divided into any number of fractional parts, so students can divide it into thirds or fourths, as needed to solve the problem. However, other representations, such as the linear model, should not be overlooked.
The following example is about distance and uses mixed numbers:
As part of his training for a race, William must run at least \(6\frac{1}{{10}}\) km a day. This morning, before going to school, he ran \(3\frac{3}{{10}}\) km. How many kilometres does he have to run after school?
Since the situation deals with a linear measurement, students can use a linear model such as a number line. The first step is to locate \(6\frac{1}{{10}}\) on a number line (below left) or an open number line (below right),
then subtract 3 from \(\ 6 \frac{1}{10}\).
then subtract the fractional part, \(\frac{3}{{10}}\).
Thus, we can conclude that William must run \(2\frac{8}{{10}}\) km. We therefore have \(6\frac{1}{{10}}\; - \;3\frac{3}{{10}}\; = \;2\frac{8}{{10}}\).
Although the linear model accurately represents the situation, students could represent the operation using an area model as shown below.
This situation can also be solved by decomposing \(6\frac{1}{{10}}\) into \(5\; + \;\frac{{10}}{{10}}\; + \;\frac{1}{{10}}\).
Now we can subtract \(5\frac{{11}}{{10}}\; - \;3\frac{3}{{10}}\; = \;2\frac{8}{{10 }}\).
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 the result to be expressed as \(\frac{{13}}{4}\), if the fourths are counted, or\(3\frac{1}{4}\), as a mixed number.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 92-94.
The following are examples of the different types of problems relating to addition and subtraction (joining problems, separating problems, comparing problems and part-part-whole problems):
Examples
Joining Problems
Mila ate \(\frac{2}{8}\) of a submarine sandwich, while Peter ate \(\frac{1}{2}\). a) What fraction of the submarine sandwich was eaten by the 2 children?
\(\frac{2}{8}\; + \;\frac{1}{2}\; = \;?\)
Representation using fraction strips:
\[\begin{array}{l}\frac{2}{8}\; + \;\frac{1}{2}\; = \;\frac{2}{8}\; + \frac{4}{8}\; = \;\frac{6}{8}\\\frac{6}{8}\; = \;\frac{3}{4}\end{array}\]
\(\frac{6}{8}\;{\rm{or}}\;\frac{3}{4}\)of the submarine sandwich was eaten by the children.
Source: modified problem translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 78.
The red ant is \(\frac{3}{{10}}\) cm long, the cicada is \(1\frac{1}{5}\) cm long, the beetle is \(\frac{2} {{10}}\) cm long and the ladybug is \(\frac{3}{{10}}\) cm long.
- Which improper fraction represents the total length in centimetres of the insects?
- What is this length in metres?
To add fractions, one may choose to represent them using a common denominator. So, for the length of the cicada, one could find an equivalent fraction that has the same denominator as the other fractions.
The total length of all the insects is \(\frac{{20}}{{10}}\) cm or 2 cm.
Source: adapted and translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
Comparing Problem
Mila used \(\frac{1}{4}\) of a pitcher of water to water her plant. Peter used \(\frac{5}{8}\) of a pitcher to water his plant. What fraction represents the difference between the amounts of water used by Peter and Mila?
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 78.
To subtract fractions, one may choose to represent them using a common denominator. So, for Mila's quantity of water, one could find the equivalent fraction that has the same denominator as the other fraction. For example, "I know that \(\frac{1}{4}\; = \;\frac{2}{8}\) since 8 is a multiple of 4. So I multiply the denominator by 2 to get eighths, and I then multiply the numerator by 2 to find an equivalent fraction, which is \(\frac{2}{8}\).
There is a difference of \(\frac{3}{8}\) water from the pitchers.
Mental Math
In everyday life, students often encounter situations involving fraction operations, such as a store sale when items are offered at half price, or the headline in the newspaper that indicates enrollment in a college program has increased by one-third. In such situations, a mental calculation is often required. Depending on the context, either an estimation or an exact answer is sought.
Mental math is based on the use of relationships between numbers and operations. It requires flexibility in reasoning about the calculations to be performed. In mental math, numbers are often decomposed (for example, \(\frac{3}{4}\; = \;\frac{2}{4}\; + \;\frac{1}{4}\) or \(\frac{1}{2}\; + \;\frac{1}{4}\)) or recomposed (for example, for \(\frac{1}{2}\; + \;\frac{3}{4}\; = \;1\frac{1}{2}\)) knowing that the 3 fourths are composed of 1 half and 1 fourth, the 2 halves are combined to make a whole, then 1 fourth is added.
Examples
I know that \(\frac{3}{6}\) is equal to \(\frac{1}{2}\), so I decompose \(\frac{5}{6}\) into \(\frac{3}{6}\;+\;\frac{2}{6}\). I eliminate the \(2\frac{1}{2}\). I still have \(\frac{2}{6}\).
The fraction \(\frac{2}{6}\) can be simplified to \(\frac{1}{3}\).
Clearly, in order to mentally perform an operation with fractions, students must have a good understanding of fractions, operations, and operations with fractions.
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 87.
Knowledge: Denominator
The number below the line in a fraction that represents the 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, Numération et sens du nombre, Fascicule 2, Fractions, p. 34.
For example, the whole is a length of \(\frac{3}{3}\).
\(\frac{2}{3}\) of the segment is red.
The denominator "3" of the fraction \(\frac{2}{3}\), represents the 3 equal 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 of \(\frac{3}{3}\).
\(\frac{2}{3}\) of the segment is red.
The numerator "2" of the fraction \(\frac{2}{3}\), represents the 2 red parts of the 3 equal sections of the number line.
Knowledge: Equivalent Fractions
To recognize the equality of two fractions is to recognize that the two fractions represent the same quantity. According to Van de Walle and Folk (2005, pp. 236-237), we must distinguish between the concept of equivalent fractions and the algorithm for determining equivalent fractions.
Concept: Two fractions are equivalent if they represent the same quantity.
Algorithm: To obtain a fraction equivalent to any fraction, multiply or divide its numerator and its denominator by the same number (other than 0).
Determining equivalent fractions means finding fractions that represent the same quantity. Specifically, we’re looking for a number of “small parts” that correspond to a particular number of “large parts” of the same whole. For example, find the number of sixteenths that corresponds to 1 and 1 fourth (\(1\frac{1}{4}\; = \;\frac{?}{{16}}\)):
Source: translated from Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, Numération et sens du nombre, Fascicule 2, Fractions, p. 50-51.