The intelligent use of visual and concrete mathematics materials can enable young students to grasp mathematical concepts and thus not have to rely upon memorization. I would like to illustrate this principle with a few fraction activities, especially since fractions are a topic that students tend to find particularly confusing. These activities are appropriate for average students of ages six and seven, and for gifted students of age five.
Understanding the Meaning of a Unit Fraction
The first and most basic concept students need to know about fractions is the concept of a unit fraction, 1/n. We need to communicate to our students this principle: if a whole is partitioned into “n” equal parts, each part is a unit fraction and has the name of 1/n. The best approach for introducing the meaning of a unit fraction is through the use of pattern blocks. We consider the yellow block as the whole. The child places the equal-sized red blocks on the yellow block and notices that two of them form the whole, and therefore each one is called a half. We can abbreviate this result by writing if Y = 1, than R = 1/2. Similar reasoning leads to the conclusion that if Y = 1, then B = 1/3 and G = 1/6.
Figure 1. Each colored block represents a unit fraction.
It does not take more than two or three minutes for children to associate the values of one, one half, one third and one sixth with the corresponding colored block.
Understanding the Meaning of a Fraction
Once students understand the meaning of a unit fraction, say one sixth, they are ready to understand the meaning of a non-unit fraction such as five sixths. We ask them to place down one sixth on the table; they do so by placing one green block. Next, we ask them to show us two sixths; they place another green block down. In this manner they represent five sixths by five green blocks and come to realize that five sixths are five copies of the unit fraction one sixth. Likewise, they display three halves by three red blocks to represent three copies of the unit fraction one half.
Figure 2: Pattern block representation of the fractions 5/6 and 3/2.
Finding Sums and Differences of Fractions With the Same Denominator
With the above knowledge under their belts, students are ready to use their blocks to find the answers to examples such as 3/6 + 2/6 – 1/6, without being provided with any rules for adding or subtracting fractions. Rather, students place down three of the green blocks to represent three sixths, place another two of the green blocks to represent the addition of two sixths, and then take away one of the green blocks to represent the subtraction of one sixth. In total there remain four green blocks and so the answer is four sixths.
Young students who work with these pattern blocks tend to avoid the common error of adding two sixths to three sixths and obtaining five twelfths. After all, adding more green blocks only affects the number of blocks we have, but not their size or their color.
One of the key concepts of mathematics is the relational meaning of equality, which compares two expressions, such as 4 + 1 = 3 + 2. Before we provide young students with a relational equivalence problem involving fractions, we provide them with a relational equivalence problem involving whole numbers. I have found that even children of age five can grasp this concept using the approach discussed here, where a circle separates the two expressions, and the correct symbol needs to be inserted inside the circle.
Figure 3. Student uses chips to
learn about equality of expressions.
Using chips, the student begins with the left side by placing down five chips and then taking two of them away, and finally adding one more; she follows a similar process on the right side, then compares both sides to see if they have the same number of chips. With this type of experience, students quickly learn that the equal sign may be used to compare two expressions involving the operations of addition or subtraction.
Equivalence Problems Involving Expressions Containing Fractions
Using the above instructional steps, the teacher may now present a problem involving equivalence between expressions containing sums and differences of fractions having a common denominator, such as 2/3 – 1/3 + 3/3 and 2/3 + 2/3 + 1/3.
Students use the pattern blocks, in this case the green blocks, to determine if both sides are the same. Young students (age six and up) LOVE doing this kind of problem because it LOOKS important. In the process, they are reinforcing their knowledge of addition and subtraction of fractions with the same denominator and their knowledge of the relational meaning of the equal sign. And just as important, they are developing a positive attitude about their ability to do mathematics.
|Batya Chava, age six, shows off her work.|
Fractions Which are Equal to 1
Early on in their work with the pattern blocks, students learned that two red blocks make up the whole (see Figure 1). We now show how to express in writing the relationship that two halves are equal to a whole, namely, 2/2 = 1. We proceed in a similar manner with the blue and green blocks and show that 3/3 =1 and 6/6 = 1. Through these examples, students learn that the equal sign may be used to show that a fraction is equal to one.
Subtracting a Proper Fraction From 1
A nice mental exercise to provide students is to ask, “What do we have left if we take away one-third from a whole?” At first, we encourage them to use their pattern blocks. Students quickly see that since the whole consists of three blue blocks, taking one away leaves two of them behind, and hence the answer is two thirds. After a short while, young students can visualize the blocks and are able to mentally provide the answer to problems such as 1 – 1/2, 1 – 2/3 and 1 – 1/6.
Considering Other Blocks As the Whole
To further consolidate the idea of a unit fraction, we ask the question, “If the red block is the whole, how much is the green block?” Since three green blocks make up the whole, students readily see that each green block is now one third. Likewise, if the blue block is the whole, the green is one half.
Comparing the Larger Block to the Smaller One
We now ask, “If it costs five dollars to make the red block, what does it cost to make the yellow block?” Since the yellow block is twice the size of the red block, students will respond with $10. Problems such as this one introduce students to proportional thinking, although we do not introduce this term. A similar problem would be, “If it costs two dollars to make the green block, how much does the yellow block cost?”
Summary visual and concrete materials, used intelligently by the teacher, can enable young students to grasp foundational concepts related to fractions and equality. Furthermore students learn that mathematics is not a mysterious subject, but one by which they can enjoy and understand.