From the time students start Kindergarten until they finish their schooling they will learn various mathematical concepts. Math is an enormous subject to be such a small word and students can feel overwhelmed if they do not get the basics as they go along each math level. Fractions are no exception and for fourth graders it can be intimidating and confusing learning fractions but the teacher will make sure they have learned their prerequisite skills in order to help them master fractions. Teachers will use prerequisite skills to enable students to enter the beginning of fractions and give students the confidence to use their abilities to connect new knowledge with previous knowledge to succeed with fractions with unlike denominators.
Prerequisite Skills
a. It is essential for the student to be fluent when working with multiplication facts. Articulacy is crucial in dealing with multiplication facts and the student must possess this ability in order for them to understand how to find a common denominator with fractions when trying to perform a mathematical function, (addition, subtraction, multiplication, and division). It may be relevant for the student to multiply the numerator and denominator in order to find the common denominator. This enables the students to find fractions that are equivalent. This process will help students in dealing with fractions in greater terms.
b. It is essential for the student to fluently know division facts. It is imperative for students to possess the ability to use division to assist them in finding the common numerator and denominator to avoid difficulties in finding equivalent fractions. This process is as imperative as multiplication when dealing with fractions. This will help the students find fractions in reduced terms.
c. It is crucial for students to retain simple terminology of fractions, (numerator and denominator). Students need to understand principal math terminology of fractions and fractional procedures in order to understand the basis of fractions. This is the basis of the foundation of fractions and ensures the student can proceed with fractions when they understand what it is they are doing.
d. It is critical for students to be able to divide whole items into equal portions to help them understand the concepts of fractions. This process will enable them to identify with equivalent fractions.
Using Manipulatives
An exceptional instrument students may use to launch the concept of equivalent fractions is using manipulatives. Manipulatives bestow students with the solid standards to assist them in linking the concept of equivalent fractions to the figurative mode which is the desired outcome. The students have a variety of manipulatives to choose from to help them with equivalent fractions.
I can implement the concept of equivalent fractions to a fourth grade class by using colored triangle blocks that fit in the shape of a pie, each student will have 2 pies. In the first pie we have 8 pieces that are red and in the second pie we have 4 pieces that are blue. I will ask the students how many pieces are in the red pie and they will answer 8. I will ask them what is the value of each piece and they will answer 1/8. When I ask how many pieces make a whole, they will answer 8 1/8 pieces. Now we look at the second pie and I will ask how many pieces are in the blue pie and they will answer 4. I will then ask them what is the value of the fraction of each piece of pie and they will answer 1/4. I will ask them how many pieces make a whole pie and they will answer 4 1/4 pieces. I will then implement the concept of equivalent fractions by asking how many 1/8 pieces are in 1/4 piece and they will answer 2 which is 2/8. How many 1/8 pieces are in 2/4 and they will answer 4 which is 4/8. Last I will ask them how many 1/4 pieces and 1/8 pieces are in 1/2 of the pie and they will answer 2/4 and 4/8. This will let them see how 1/2, 2/4 and 4/8 are all equal fractions. By using different colors for the fractions it will help the students see how the fractions are equivalent.
The same exercise can be used with thirds and sixths. Each student will have 2 pies, one is yellow and one is green. The first pie is yellow and it is divided in thirds. The second pie is green and it is divided in sixths. I will ask the students how many pieces are in the yellow pie and they will answer 3. I will ask them what fraction does each piece of pie equal and they will answer 1/3. I will then ask them how many 1/3 pieces equal a whole pie and they will answer 3. Next I will ask them how many pieces are in the green pie and they will answer 6. I will ask them what fraction does each piece of pie equal and they will say 1/6. I will then ask them how many 1/6 pieces equal a whole and they will say 6. At this point I will implement the concept of equivalent fractions by asking how many 1/6 are in 1/3 and 2/3 of the pie. The students will answer 2/6 for 1/3 and 4/6 for 2/3. I will then ask if 3/3 equals 6/6 and they will answer yes. This will show them how thirds and sixths can be equal.
When teaching students about equivalent fractions we need to make sure that we correctly inform the student about the appropriate way to find equivalent fractions. We first will look at fractions with no specific denominator. If we have 1/8 and we want to find an equivalent fraction of bigger terms we must multiply 1/8 by a bigger number: 1/8 x 3/3 = 3/24. Therefore we know that 3/24 is equivalent to 1/8. If we have a specific denominator that we want to find an equivalent fraction we must look at the denominators and see what number we must multiply by to get the fraction. We have the problem of 2/6=?/18; we look at the 18 and see how many times 6 goes into 18, the answer is 3. So we know that 3 x 2 = 6 therefore 2/6 = 6/18. Those are equivalent fractions. We can have instances where fractions are not equivalent. If you have 2/3 and 3/5 you could work all day and never have those two equivalent because 3 multiplied by any number will never equal 5. Therefore we know these two fractions are not equivalent.
We can then teach the students if they are looking for equivalent fractions in smaller terms by division. If we have a denominator that is not specified we could have a problem like 4/8 = 1/2. If you divide 4/8 by 4/4 you get 1/2. Therefore we know the 4/8 is equivalent to 1/2. Then if we have a specific denominator we could have a problem of 4/16 = ?/4. We need to figure out what common number goes in to 16 and 4. The answer is 4. When we divide 4 by 4, the answer is 1, therefore the answer to the equivalent fraction would by 4/16 = 1/4. Sometimes a fraction cannot be reduced, if a fraction is 7/12 and you want to see if it is in simplest form look at the numerator and see what times what equals it. In this case 7 times 1 is the only factors and we know 7 will not go into 12 so this fraction is in its simplest form. Sometimes a fraction can be reduced for instance you have 15/25; you can divide by numerator and denominator by 5 and get 3/5. Therefore we know 15/25 = 3/5.
Transitioning Students From Concrete Manipulatives to Paper-and-Pencil Problems
At this point the students should have made the link between equivalent fractions by working with manipulatives. Using the fraction pies with different colors have helped the students go from solid interpretations to the symbolic portion of equivalent fractions. The students will be given paper versions of pies that are different colors according to the fraction that they represent. The pies will represent halves, fourths, sixths and eighths.
Starting with halves the students will be told to take 1/2 of the pie and see how many 1/4 pieces will fit in the 1/2, and the answer is 2. You would have 2/4. At this point the students would record this answer in their math journal. Have the students explain how 1/2 relates to 2/4, they will find it is twice as many pieces but it equals the same amount. Next take the 1/2 and see how many 1/6 pieces will fit and the answer is 3. Have the students figure out how 1/2 relates to 3/6 and they will find they equal the same but it takes 3 times as many 1/6 pieces to make the 1/2. The students would record in the math journal that 3/6 is equal to 1/2.
The students would now take the 2/3 of the third pie and see how many 1/6 pieces fit into that section. The answer would be 4 and this is what they would record in their journals that 2/3 = 4/6 and the relationship between the two fractions. Have the students identify that 2/3 and 4/6 are the symbols for the fractions to minimize any confusion the students may have between symbolic forms and written out forms.
Students must realize that you will have fractions that are not equivalent. For instance you have 5/6=?/7 and the first thing they look at is 5/6 is in the simplest form and 6 cannot go into 7 therefore these fractions are not equivalent. The students can use this example in their journal to refresh their memories if they have problems.
Quiz On Fractions
To make sure that the students have mastered the lesson on equivalent fractions I will give them a 9 question quiz. This will help me decide if the students have retained the information needed to go on to the next lesson.
1. 4/8 = x/16 What is x? 2. 5/20 = y/4 What is y? 3. 6/36 = 1/z What is z? 4. 2/3 = 6/a What is a? 5. 15/30 = b/10 What is b? 6. 6/18 = 1/c What is c? 7. 9/27 = d/3 What is d? 8. 1/2 = 8/e What is e? 9. 2/3 = 6/f What is f?