B1.4 Compare and order fractions from halves to twelfths, including improper fractions and mixed numbers, in various contexts.
Activity 1: Comparing Fractions
Write on the board a set of fractions between 0 and 2 (for example, \(\frac{5}{6},\;\frac{5}{8},\;\frac{4}{4},\; \frac{5}{3},\;\frac{9}{{10}},\;1\frac{2}{3},\;\frac{8}{5},\;\frac{ 3}{5}\)).
Invite students to sort these numbers as follows: those that are close to 0, those that are close to \(\frac{1}{2}\), those that are close to 1, and those that are close to 2. Then, focusing on fractions that are close to \(\frac{1}{2}\), ask them to name those that are greater than \(\frac{1}{2}\) and those that are less than \(\frac{1}{2}\).
Then draw a number line, locating the numbers \(0,\;\frac{1}{2},\;1,\;1\frac{1}{2}\;\rm{and} \ 2\), and invite students to come and locate the numbers written on the board on this number line, explaining their decision.
Repeat the exercise using other classification criteria (for example, between 0 and 1, between 1 and 2) or using other numbers.
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. 163.
Activity 2: Which Key to Choose?
This activity allows students to examine the use of fractions to define the measurements of tools (wrenches, bolts, nuts).
Materials
- bolts and nuts of various sizes previously arranged;
- set of wrenches identified in imperial measurement (examples of the most common measurements: \(\frac{1}{4}\) inch, \(\frac{3}{8}\) inch, \(\frac{1} {2}\) inch, \(\frac{5}{8}\) inch, \(\frac{3}{4}\) inch).
Directions
Invite students to sit in a circle around the materials on the floor. Explain to the students that the metric system is the system of measurement generally used in Canada, but that in some contexts such as construction, the imperial system is still commonly used.
Show the students a wrench with a fraction on it and explains that the measurement of the opening of the wrench is a fraction of an inch. For example, the wrench with the fraction \(\frac{3}{4}\) on it has an opening of \(\frac{3}{4}\) of an inch. It would be helpful to have a ruler or tape measure to show that an inch is a measurement representing approximately 2.5 cm. Next, explain that the wrench is the tool used to tighten a nut and to be useful, it is important to ensure that the measurement of the wrench opening matches the width of the nut.
Present students with fractions corresponding to all available wrenches and nuts. Fractions can be recorded on the board or on a large piece of paper.
Next, choose a nut (partially threaded onto a bolt), and pass it among the students, asking them to estimate which wrench should be used to screw or tighten it.
Invite a student to try to tighten the nut in question using the wrench of their choice. If the wrench does not fit the nut precisely, ask whether the chosen wrench is too big or too small. For example, the student (pictured below) explains that he chose the \(\frac{5}{8}\) inch key, but it is a bit too big.
If the first attempt is unsuccessful, ask the student to try again and pick a different wrench. Continue by inviting other students to try until the correct wrench for the nut is found, then repeat the process with other nuts.
Throughout the activity, encourage students to compare fractions (for example, the opening of the \(\frac{3}{4}\) inch wrench is larger than that of the \(\frac{3}{8}\) inch wrench) and to justify their choice of wrench (for example, "During the last test, the \(\frac{1}{2}\) inch wrench was too big; so I'm going to try the \(\frac{3}{8}\) inch wrench, since \(\frac{3}{8}\) of an inch is a little less than \(\frac{1} {2}\) of an inch”).
To complete the activity, ask students to place the fractions in ascending order (for example, \(\frac{1}{4}\), \(\frac{3}{8}\), \(\frac{1}{2}\), \(\frac{5}{8}\), \(\frac{3}{4}\)) and verify their answer by placing the corresponding wrenches in order of size.
Note: To make the task more authentic, instead of loose nuts and bolts, a bicycle (or other object that includes several nuts) can be presented and students can be asked, for example, to identify which wrench to use to loosen the nut to mount the seat or to remove the wheel.
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. 105-106.
Activity 3: In Order
Materials
- Appendix 5.4 (Fractions)
Group students into groups of four. Give each group a copy of Appendix 5.4 and point out that the shaded area represents a fraction of the strip. Ask them to cut out each strip to make a set of strips, each with the corresponding fraction.
Invite students to combine strips to create mixed numbers or improper fractions. Students may also create their own strips.
Example
The student creates the mixed number \(1\frac{4}{6}\) or the improper fraction \(\frac{{10}}{6}\).
Now ask students to place all the fractions in their group in ascending order (smallest part to largest part).
Circulate and monitor for understanding, by asking questions such as:
- Why did you place this fraction in this location?
- How do the numbers in the numerator and denominator help you place fractions in the correct order?
- Some fractions represent the same quantity. What does this mean? How did you know where to place them? How can you justify that they are equivalent?
When the task is complete, have a mathematical discussion to emphasize the strategies used to determine the ordering of the fractions. Next, have students observe unit fractions (for example, \(\frac{1}{2},\;\frac{1}{3},\;\frac{1}{4},\;\frac{1}{6},\;\frac{1}{8},\;\frac{1}{9},\;\frac{1}{{12}}\)), more particularly their numerator and their denominator, to emphasize that when fractions have the same numerator, one having a larger denominator represents a smaller part. Then, ask them to compare all the fractions with the same denominator (for example, \(\frac{1}{8},\;\frac{2}{8},\;\frac{3}{8} \ldots \)) to emphasize that when fractions have the same denominator, the one with a larger numerator represents a larger part. Finally, invite them to analyze equivalent fractions in order to recognize the relationship of proportionality.
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. 160.