Understanding the concept of fraction computation is one of the most important areas of elementary school mathematics education. Basic math skills learned in the previous grades are applied to fraction computation with addition and subtraction. These skills lay the foundation for the future study of fraction computation with multiplication and division. The student must already have mastered addition and subtraction of whole numbers in order to use these skills with fractions. A knowledge of multiplication and division is also necessary to find the lowest common denominator. The most important concept for the student to understand is that fractions are numbers. This must be stressed in every grade in which fractions are studied.
There are certain skills that all students should fully grasp in the third grade. If these skills are not acquired, the student will struggle with learning fraction computation in the fourth grade. The third grader will be introduced to fractions and realize that fractions are numbers. The student will also learn that fractions are a result of dividing a whole into parts of the same size. In addition to this, the child will also gain an understanding of a fraction as a number on a number line. The interval from zero to 100 represents the whole and the number line should be partitioned into equal parts. The student will then learn how to locate each number on the number line. The next concept a third grader must learn is that two fractions are equivalent if they are the same size or are on the same point of the number line. For example, 1/2 = 2/4. A whole number can also be expressed as a fraction (4 = 4/1) and will be on the same point on the number line. Finally, fractions can only be compared when they are part of the same whole.
Various instructional methods can be used at the third grade level. First, the use of discrete objects such as pencils should be introduced. If the whole is three pencils, then one pencil is ⅓ of the whole. The problem with discrete objects is that it focuses on how many rather than how much and can give children the wrong idea of what a fraction is. The use of contiguous models give the student the ideas of area, length and volume. It is a good idea to begin with the number line. Addition and subtraction of fractions are easier to understand on the number line because only length is involved. Finally the symbols of greater than and less than are introduced and used to compare fractions. If the child fully understands these concept then he or she is ready to move on to fourth grade math and fraction computation with addition and subtraction.
Fourth graders will build on what they have learned in the third grade as they develop an understanding of fraction equivalence. The teacher should use visual fraction models to explain why fractions with different numbers are the same. For example, ¼ = 2/8. This concept was introduced in the third grade but now it will be explored more fully. Next, the students will learn how to compute the lowest common denominator in order to add or subtract fractions. They will compare the results of the problem by using > or < symbols and then justify the results by using a visual fraction model. The teacher should now teach the students to apply their previous understanding of whole numbers to fractions. How to build fractions from unit fractions will be discussed. For example, ¾ = ¼ + ¼ + ¼. Students will begin to add and subtract mixed numbers with like denominators. They will also begin to solve word problems with fractions. The language level of each student should be assessed. The children’s previously learned understanding of multiplication will now be extended to multiply a fraction by a whole number. Finally, the students must understand decimal notations for fractions and be able to compare them in order to be prepared for fifth grade.
Instructional practices for the fourth grade include introducing the use of symbols. For example, a/b = n x a/ nx b or a/b na/ba. In other words, if the numerator and denominator are both multiplied by the same number, the fractions are equal. The teacher can use fraction strips or the area model to explain this to children. More complicated fractions can be taught by the number line. The next task the teacher faces is to explain to children how to reduce fractions to their simplest form. For example, 24/42 can be reduced to 4/7. The teacher will use previously learned division and multiplication of whole numbers to introduce this new idea. Adding and subtracting fractions can be taught by using segments on the number line. Adding fractions should be taught as a continuation of the earlier concept of adding whole numbers. For instance, the students will learn that if you add fractions with the same denominator, you will get an answer with the same denominator. The students will also learn that the same can be said of subtracting fractions with the same denominator. The acquisition of these skills is necessary to be successful in fifth grade.
At this grade level, the students will begin to move past addition and subtraction of fractions. However, what they have learned will be a foundation for more complicated computations. The concept of a common denominator, which was learned in the fourth grade, will now be used to add and subtract equivalent fractions. The fifth grade students will perform operations with decimals to the hundreth. They will apply previously learned multiplication and division skills to multiply and divide fractions. Students will also learn to divide the numerator by the denominator. Their knowledge of adding and subtracting fractions will be extended to adding and subtracting decimals. The multiplication algorithm for decimals will be introduced by the teacher.
Fifth grade teachers provide an in depth study of addition, subtraction, multiplication and division of fractions and decimals by building on what was learned in the previous grades. Instructional practices include using algorithms, formulas and the number line to give students a thorough understanding of fraction computation. At this point, teachers can use common errors as examples but they must be sure to present them in a positive way.
The developmental stages for acquisition of the concepts of fraction computation can vary with the individual student. The Common Core State Standards for Mathematics (CCSSM) offers guidelines for teachers as to what is developmentally appropriate for various age groups. However, these are only guidelines and each student should be assessed as an individual. The language level and ability of each child should be taken into account in order to accurately assess the child’s progress. Every teacher should understand how students acquire a concept. Most children will be able to develop fraction concepts in the fourth and fifth grades. If they understand the ideas of order and equivalence at this stage, they will be more successful later on. The job of 3rd, 4th and 5th grade teachers is to be sure they understand the concepts which is more important than understanding the procedures. It is very important that students understand concepts with whole numbers before they can understand fraction computation. In other words, children should be trained to think rather than just memorize procedures.
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