B1.6 Use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts.
Skill: Representing and Solving Fair-Share Problems
The development of quantitative understanding of a fraction is based on student’s experiences with physical models that emphasize meaning rather than procedure.
(Bezuk and Cramer, 1989, p. 157)
In the primary grades, students have the opportunity to explore fractions by partitioning sets of objects (for example, if 3 friends want to share 18 apples equally, each will receive 6 apples) and by examining whole parts separated into equivalent parts (for example, a rectangle separated into fourths). Partitioning can then serve as a springboard for the study of fractions in the junior grades.
As students continue to build on the principle of sharing, they create connections between the action of sharing, the whole, and the parts of the whole. They are able to better understand partitioning and develop a sense of fraction. They understand that the fraction is also used to illustrate a remainder after sharing (for example, if 4 students want to share 5 cupcakes, each gets 1 cupcake and 1 cake).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 28.
Exploration of Fractions
Baroody and Coslick (1998, pp. 9-14 and 9-15) emphasize a "meaningful approach" to learning fractions. They advocate that learning fractions follows a progression that moves students from informal, concrete representations to formal, abstract representation. The first 2 steps of this progression are inherent in this content.
- Partitioning : Any activity that engages students in a meaningful experience of sharing, without explicit reference to terminology or symbolism, provides a concrete foundation for understanding the concept of fraction. Beginning with a common everyday task (for example, partitioning felt pens or coloured pencils for an art project), students can develop the skill of dividing a set of items into equivalent parts. Later, students may have experiences that involve more elaborate thinking patterns in which they must split a single item (for example, dividing a cardboard box into 3 equivalent parts to make a craft).
- Naming fractions represented by models: When they name the fraction associated with a situation, students make a connection between the fraction, the representation used, and the situation, which solidifies their sense of the fraction. This step is therefore important. For example, given a problem in which four friends must share twelve (12) crackers, students are asked to explicitly name the fraction that determines each friend's share of one-quarter (1/4) of the total crackers.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 58.
Knowledge: Mixed Numbers
Learning about fractions usually begins with the use of proper fractions, that is, fractions less than 1. In Grade 2, students encounter fractions that represent quantities greater than 1 (for example, counting pieces of pie cut into fourths, there are 11 fourths left). Such situations lead students to mixed numbers (for example, \(2\frac{3}{4}\)).
Example
If there is 2 and three fourths of pies, the 2 pies can be cut into fourths, making 8 fourths. Adding the other 3 fourths make a total of 11 fourths, or \(\frac{11}{4}\).
If there is 11 fourths of pie, the fourths can be grouped, 4 at a time, to form whole pies. This way, 2 pies can be formed with 8 pieces. This will leave 3 fourths. There is 2 and three fourths pies.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 53-54.
Knowledge: Fraction
Understanding the representation of quantities by fractions deepens the concept of quantity. It is important to understand that fractions (for example, one-half, one-third, one-fourth) represent different quantities depending on whether they refer to a part of a whole (a length, an area, or a solid) or to a part of a set. For example, one third of a chocolate bar (part of a whole) represents a quantity of chocolate based on the size of the original bar. However, one third of a dozen eggs (part of a set) represents four eggs.
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 48.
Provide opportunities for students to discover that:
- when the fraction represents an area (area model), the area of each part must be of equivalent size;
- when fractions are used to describe sets (set model), the objects making up the sets may be different sizes (for example, if the fruit bowl is said to be made up of apples, the other fruit bowl may be made up of grapes that are smaller than the apples);
- the fraction represents a relationship rather than a particular number. It is important for students to know that half of a small amount can be much smaller than one third of a large amount;
- the fraction represents a part of a whole.
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 30.
Knowledge: Fair-Sharing
A situation where a set is shared or distributed among a known number of people or groups.