The learning and teaching of fractions can be a problem since fractions are perceived to be complex and challenging. Fractions are introduced as early as the second grade but most are in fourth grade. The effective introduction of equivalence in working with fractions that have unlike denominators depends on the learner’s understanding of prior knowledge. First, the learner needs to be able to correctly add and subtract real numbers. This is to help them differentiate the numbers in terms of their size so that they can comprehend what happens to the number as a fraction. The learner also needs to be able to multiply numbers of sets up to twelve. This is to help him/her in division of fractions and relating fractions upon simplification such as ½ is the same as 5/10, 6/12 and so on. The learner also needs to be able to comprehend the concept of whole. Further, the learner needs to have knowledge that whole numbers represented as fractions have one as the denominator.
Manipulatives help learners form valid mental pictures that enable them to apply them hence performing well in fractions. The concept of fractions can be introduced through models such as fractional parts of a circle, pattern blocks, counters and paper folding activities among others. The manipulatives are introduced in different forms such as having a full black circle, white circle cut into two, blue circle cut into six and so on but the numbers must not exceed twelve. The students are then engaged in understanding the relationship of the circles by first understanding the full circle as a whole. They are then demonstrated on determining the fractional parts of the whole. For instance, for the blue circle divided into six pieces, each piece is a fraction of the total number of pieces that form the complete circle. Students are then allowed and assisted in creating their own sets of colored circles and dividing them into a number of pieces then determining the fractional part of the whole.
The process of finding equivalent fractions involves learners first being introduced to the basic manipulatives. They are then involved in activities that allow them to use physical models and diagrams while naming symbols of fractions through picture drawing. In this step words are used and symbols introduced later, for example, three-fourths is then written as ¾. The concept of variation is then introduced. For example, in using circles you let one piece of the blue circle be 1 instead of having the whole circle as the unit. The students are then assisted to value the other pieces in order to understand the unit concept. Learning is then extended through other models such as Cuisenaire rods or number lines while the interpretation models are expanded to include such as the quotient model. As the learner progresses, the ordering of fractions is done with comparisons to ½ and 1. Later on the applications of addition and subtraction operations are introduced.
The transition process from concrete manipulatives to more representative paper and pencil problems is made efficient by following the steps provided in equivalent fractions and focusing on helping the learner to understand. This transition can be made easier by making use of manipulatives and later diagrams to reinforce the understanding of the concept of unit. In addition, the fractions commonly used may be ½, 1/3, ¼ and ¾ with more emphasis on ½. It requires that during the introduction stages, the fraction used either as unit or non-unit should have denominators less than 8. This is expanded to 12 as the learner’s scope and understanding increases. In addition, the interpretation models such as the quotient and other models of things that students relate to daily form a good foundation for transition. Discussions and active engagement of students is also important.
Examples of fractions that can be used to gauge if students have transitioned to paper and pencil representatives from concrete manipulatives include:
Use circular pieces to show:
a) ¼ b) ¾ c) 2/3 d) 8/12 e) 3/12 f) ½ g) 2/8
Solve these problems using fraction circles. The blue circle is 1. What is the value of?
3 blacks 2 whites 6 Reds
Use Cuisenaire rods to solve these problems.
Find the value of each of the following rods given that the value of the yellow rod is 1:
a) White b) Red
c) Green
The Cuisenaire rod of green color is 1; find the value of each of the following rods:
a) Red b) White c) Yellow