B1.3 Represent equivalent fractions from halves to twelfths, including improper fractions and mixed numbers, using appropriate tools, in various contexts.
Activity 1: Determining Equivalent Fractions Using Various Models
The following are representations that allow the student to develop a sense of what equivalent fractions are.
Linear Model
The linear model is very useful for representing fractions, finding equivalent fractions and performing operations with fractions. It is a very effective visual reference for students to better understand the quantity represented by a fraction.
Interactive activity: L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
The same thing can be represented with strips of metric grid paper (the whole is 36 squares long). The length of the whole must be the same.
Source: L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
Area Model
Think about how students should solve this problem.
The red part represents the part of the crepe that Mona ate.
If this crepe had been cut into 4, 6 or 8 equal parts, would Mona have been able to eat the same amount?
Interactive activity: Click on "Solution" to see a suggested problem-solving strategy.
Source: L’@telier — Ressources pédagogiques en ligne (atelier.on.ca).
Set Model
Consider how students should solve this type of problem using counters.
Janika states that the 3 twelfths of the months of the year begin with the letter "j". She shows her assertions using counters: green tokens to represent months that start with a "j" and red tokens to represent months that start with another letter. After looking at the representation, she even says that \(\frac{1}{4}\) of the months start with a "j". She places the counters in groups of 3 to show the 4 groups, 1 of which represents the 3 months that begin with the letter “j”.
Interactive activity: Click on the Solution button to see a suggested solution.
Source: L’@telier — Ressources pédagogiques en ligne (atelier.on.ca).
Proportionality Relationship
A proportionality relationship is present in any situation of equivalent fractions.
Interactive Activity: L’@telier — Ressources pédagogiques en ligne (atelier.on.ca).
Note: \(\frac{3}{7}\; = \;\frac{{21}}{{49}}\) are 2 equivalent fractions. This notation is also called a proportion.
Source: L’@telier — Ressources pédagogiques en ligne (atelier.on.ca).
Activity 2: Equivalent Fractions
Directions
Hand out the worksheet to the students.
To the right of the 1st rectangle, ask students to write the fraction that represents the shaded area.
Ask students to shade an area on the other rectangles equal to the shaded area on the first rectangle.
To the right of each rectangle, ask students to write the fraction that represents the shaded part of each rectangle.
On the board, write the following equalities:
Ask students to explain the relationship between the numerators and denominators.
Note the relationships using arrows and operations.
Ask students for a rule for creating sets of equivalent fractions.
Have students use manipulatives to demonstrate the rule they found.
Source: L’@telier — Ressources pédagogiques en ligne (atelier.on.ca)
Activity 3: Mixed Numbers and Improper Fractions
Group the students into 2 or 3 and give them pattern blocks. Introduce the blocks by identifying the hexagon as the whole. Have the students recognize that the red trapezoid is equivalent to \(\frac{1}{2}\) of the hexagon, the blue diamond to \(\frac{1}{3}\) of the hexagon and the green triangle \(\frac{1}{6}\) from the hexagon.
First, ask students to represent \(\frac{{16}}{3}\) hexagons using pattern blocks. Since the improper fraction is in thirds and the diamond represents \(\frac{1}{3}\) of the whole, students should choose 16 diamonds.
Invite students to group the diamonds to form wholes. In this way, students can determine the mixed number that corresponds to the given improper fraction.
Conclude by pointing out that the given improper fraction (\(\frac{{16}}{3}\)) and the determined mixed number (\(5\frac{1}{3}\)) represent the same quantity.
Following the same approach, explore other equivalences (for example, \(\frac{{13}}{2}\; = \;6\frac{1}{2},\;\frac{27}{6}\;= \;4\frac{3}{6}\)) with the students. After a few examples, ask them to carefully consider the reasoning used to determine a strategy that does not require concrete materials.
Second, ask the students to represent \(3\frac{5}{6}\) hexagons using pattern blocks. To represent the given mixed number, most students should choose 3 hexagons and 5 triangles.
Invite students to group the diamonds to form wholes. In this way, students can then determine the improper fraction corresponding to the given mixed number.
Conclude by pointing out that the given mixed number (\(3\frac{5}{6}\)) and the determined improper fraction (\(\frac{{23}}{6}\)) represent the same quantity. Following the same approach, explore other equivalences (for example, \(7\frac{1}{2}\; = \;\frac{{15}}{2};\;3\frac{2} {3}\;=\;\frac{{11}}{3}\)) with the students. After a few examples, ask them to carefully consider the reasoning used to determine a strategy that does not require concrete materials.
Note: This activity can be done with other types of manipulatives (for example, Cuisenaire rods, interlocking cubes). In addition, to deepen student understanding of the relationship between a fraction and its whole, the whole can be modified; for example, if the whole is 2 hexagons, then the green triangle is equivalent to \(\frac{1}{{12}}\) of the whole, the blue diamond is equivalent to \(\frac{1}{6}\) of the whole, and the red trapezoid is equivalent to \(\frac{1}{4}\) of the whole.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 157-159.
Activity 4: Flags
Suggest that the students create a class flag on which will appear 4 elements of equal importance; thus, it will be separated into 4 equivalent sections.
Provide students with a sheet of paper showing several congruent rectangles. First, ask them to divide the rectangles into fourths in various ways.
Then invite students to draw the rectangle on the board. Discuss the various possible ways it could be divided. Generally, the whole is divided into 2 parts and each of these parts is then divided into 2 parts. Each part is therefore 1 half of 1 half, which is one fourth of the whole.
Point out to students that although the shapes of the parts differ, each part covers the same area, one fourth of the rectangle.
Invite students to choose one of the divided rectangles and create a flag respecting the criterion of the four elements having equal importance.
Source: Guide d’enseignement efficace des mathématiques de la 4e à la 6e année,p. 135.