B1.6 Use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 10 items among 2, 3, 4, and 6 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts.
Skill: Representing and Solving Fair-Share Problems
The development of understanding of the quantity represented by a fraction relies on the student's experience with concrete materials and on instruction that focuses more on the meaning of the fraction than on procedures.
(Bezuk and Cramer, "Teaching about Fractions: What, When, and How?" in P. Trafton (Ed.), National Council of Teachers of Mathematics 1989 Yearbook: New Directions For Elementary School Mathematics, 1989, p. 157)
In the primary grades, students have the opportunity to explore fractions by sharing sets of objects (for example, if 3 friends want to share 18 apples equally, each will receive 1/3 of the 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. Note that in Grade 2 , the curriculum limits the study of fractions to halves, thirds, fourths and sixths.
By continuing to build on the principle of division, students create connections between the action of division, the whole and the parts of the whole. They are then able to better understand fractionation and develop a sense of fraction. Students 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 one fourth of a cupcake).
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 28.
Exploration of Fractions
Baroody and Coslick(Fostering Children's Mathematical Power: An Investigative Approach to K-8 Mathematics Instruction, 1998, pp. 9-14 and 9-15) advocate a "meaningful approach" to learning fractions. They advocate that learning fractions follow a progression that moves students from informal, concrete representations to formal, abstract representation. The first step in this progression is the sharing of quantities.
Sharing
Any activity that engages students in a meaningful experience of sharing, without explicit reference to terminology or symbolism, provides them with a concrete foundation for understanding the concept of fraction. Beginning with a common everyday task (for example, sharing felt pens or crayons for an art project), students can develop the ability to divide a set of items into equivalent parts. Later, they may have experiences that involve more sophisticated thinking in which a single item is split (for example, dividing a cardboard box into three equivalent parts to make a craft).
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 and 3/4).
Example
If you have 2 pies, you can cut the 2 pies into fourths, making 8 fourths. Adding the other 3 fourths gives a total of 11 fourths, or \frac{11}{4}}.
If you have 11 fourths of pie, you can group the fourths, 4 at a time, to form whole pies. This way you can form 2 pies with 8 pieces. This will leave 3 fourths. So we have \(2\frac{3}{4}\) pies.
Source: Guide d'enseignement efficace des mathématiques de la 4e à la 6e année, p. 53 et 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.
Example
Sacha has 12 apples. He wants to share them equally among 4 friends. How many apples will each friend get?
Note: Several students begin by distributing whole objects, then separate the remaining ones. Others separate each element and distribute the parts. When the number of people is greater than the number of objects to be shared, the objects must be divided from the start of the sharing process.
Half: When a whole is divided into 2 equivalent parts, each part is half of the original quantity. Two halves make a whole.
One fourth: When a whole is divided into 4 equivalent parts, each part represents 1 fourth of the original quantity. Four fourth make a whole.
One third: When a whole is divided equally into 3 parts, each part represents 1 third of the whole. Three thirds form a whole.
One sixth: When a whole is divided equally into 6 parts, each part represents 1 sixth of the whole. Six sixths form a whole.