Tools
There are several tools that I plan to use to teach my unit. Paper folding, partitioning and number lines are the tools I have chosen to help my students develop a visual image of fractions in accordance with the Singaporean method.
Paper Folding
Singapore Math uses paper folding as a concrete manipulative to teach fraction concepts. The technique is kinesthetic and visual. Square and rectangular pieces of paper are the preferred shapes. An example is folding a square piece of paper into four equal parts with the same area. One part would represent one fourth. Students could share the different ways that they folded the square into equal parts. Sharing the variations helps students to deepen their knowledge. Then the students can name two parts as two fourths. Paper folding is one tool that will be used to teach the objectives of my unit.
Partitioning
Partitioning is another tool I will incorporate. Partitioning is the process of dividing an object or objects into more parts. The parts should not overlap. All of the object or objects should be contained within the parts, meaning no part should be left over or unused. When working with fractions, these parts must be equal in area. Developing the skill of partitioning will take time to develop in children. Students will begin with simple partitioning activities with concrete objects such as paper folding. The second step in the CPA (concrete, pictorial, algorithm) Approach is for students to partition pictorial representations. For example, students may subdivide each half of a picture into three equal parts. Then the smaller part would represent sixths. Susan Lamon describes the importance of partitioning in this way, "Partitioning is fundamental to the production of quantity, to mathematical concepts, and to reasoning and operations." 3 Over time, more complex partitioning activities will be taught. Two goals of the partitioning activities are 1) to develop a deep understanding of the concept of equivalence and 2) to develop flexible thinking about units. These strategies will build the students' fundamental concepts in a meaningful way, as opposed to memorizing algorithms and rules that are easily forgotten.
Number Lines
I plan to provide my students the opportunity to work with number lines. A number line is built by prescribing an origin, a unit and a positive direction. It is important for the teacher and the students to understand that a number line is a metric entity, visually showing the distance or length from the origin. All of the numbers on the number line refer to a unit and as you move away from the origin you work with multiples of the unit distance. Negative numbers are placed to the left of the origin, although we will restrict ourselves to positive numbers and fractions in this unit. For example, 2/4 represents two copies of the distance 1/4 and it is twice the distance from the origin, zero. Likewise, 3/4 would represent three copies of 1/4 and would be three times the distance from zero.
I would begin by having the students "count" by one fourth on the number line. Using the number line above, I would have the students count aloud, "one fourth, two fourths, three fourths..." I would build this skill using different unit fractions such as ½, 1/3, and 1/5. In addition to counting orally, it is important for students to label number lines in the same fractional increments. These exercises help students to understand that the numerator "counts" the multiples of the unit. It is important to have the students go past one whole, so that they develop the conceptual understanding that fractions can be greater than one whole.
Another important use of number lines is to develop a rich understanding of the size of unit fractions and general fractions. It is very important to keep the size of the whole unit constant when students compare the sizes of different fractions. A common problem is that textbooks represent fractions with so many different pictures that it is difficult for students to build a clear concept of the comparable size of fractions. Therefore, the students must develop a meaningful sense of the relative size of the fractions in order to create a strong mathematical foundation.
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