Unlocking Fractions: A Beginner's Guide to Mastering Parts of a Whole
Fractions are a fundamental concept in mathematics, serving as the basis for more advanced topics like algebra. While they may seem daunting at first, understanding fractions is achievable with the right approach. This article provides a comprehensive guide for beginners, covering the basics of fractions and effective learning strategies.
What is a Fraction?
A fraction, in its simplest form, represents a part of a whole. Imagine a whole pizza; a fraction would be a single slice of that pizza. When examining a fraction, you'll notice two numbers.
Basic Fraction Concepts
- Numerator: The top number of a fraction represents how many parts of the whole you have (e.g., the number of pizza slices you have).
- Denominator: The bottom number of a fraction indicates how many equal parts the whole has been divided into (e.g., the total number of pizza slices).
Early Introduction to Fractions
Typically, students begin their journey with fractions in the 1st grade. At this stage, they focus on recognizing basic fractions such as ½, ⅓, and ¼. They learn to identify these fractions by dividing shapes into wholes, halves, thirds, and fourths.
Visual Learning: A Key to Understanding
When teaching fractions, it's beneficial to make learning visual. This approach helps students grasp the concept more easily.
Simplified Fractions
Simplified fractions are fractions that have been reduced to their simplest, lowest form. For example, 4/16 can be simplified to 1/4. To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common factor.
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Let’s consider 5/15. Obviously, 5 is divisible by 5, right? So 5÷5 = 1 And 15 is also divisible by 5.
Fractions as Division
Fractions can also be interpreted as division problems. The numerator is divided by the denominator to get a decimal equivalent of the fraction.
Improper Fractions
An improper fraction is one in which the numerator is larger than the denominator. For example, 5/4 is an improper fraction. Improper fractions can be converted into mixed numbers.
Prime Numbers and Greatest Common Factor
A prime number is a number that can only be divided by itself or by one. Examples of prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc. The greatest common factor is the largest factor that divides two numbers. We find this by figuring out what prime numbers are multiplied together to make up each number.
Operations with Fractions
Adding and Subtracting Fractions
To add or subtract two fractions, the two fractions must have the same denominator. Therefore, you will have to find the lowest common denominator and change each fraction to an equivalent fraction.
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Multiplying Fractions
Multiplying fractions is fairly simple. You may want to put each fraction in its lowest terms to start. Then multiply the numerators to get a numerator and multiply the denominators to get a new denominator. And reduce or simplify the fraction to put it in its lowest terms. If you are multiplying a fraction by an integer, put the integer over one to make a fraction of it. Five times 4/5: Put the five over one, then multiply 5 x 4 to get 20 and 1 x 5 to get 5. Twenty can be divided evenly by 5 to give us an answer of 4.
Negative Fractions
If either the denominator or numerator is negative, the fraction is considered a negative fraction. Adding two fractions of the same sign (either positive or negative) gives an answer with the same sign.
Challenges in Learning Fractions
Fractions can be challenging for students. Even if students get the basic concepts right-that fractions don’t behave the same way that whole numbers do-the rules of how to operate on fractions can stump learners. Their abstract nature also makes fractions notoriously hard to teach.
Common Mistakes
Students often treat fractions like whole numbers well into the upper-elementary grades. A common mistake is to confuse the size of fractions based on which has the larger denominator: They may think that 1/6 is a bigger number than 1/4, for instance. Another common struggle: the parts-of-a-whole problem. Fractions are introduced through the “pizza model,” but students’ thinking often gets stuck there. Taking two slices from an eight-slice pizza is easy to comprehend, but taking two slices out of two different pizzas-one with six slices and another with seven-is difficult to grasp. If students believe that a fraction is always less than 1, they will have a hard time understanding ratios or proportions written in fraction form. Another sticking point for many students is adding fractions with different denominators-for example, coming up with 11/12 when asked to add 3/5 and 8/7. This conceptual problem can crop up in later grades, too, if it isn’t tackled earlier.
Effective Teaching Strategies
Teachers need to focus on getting students’ basic knowledge of fractions right. But they don’t have to start from scratch. Experts suggest that teachers should tap into the information students already have.
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Leveraging Prior Knowledge
Students come to school with an informal understanding of proportionality-they are familiar with the idea of sharing a candy bar or a cookie in equal proportions. Teachers can use this basic knowledge as a ramp to introduce fractions by asking students to divide one object into equal parts or several objects equally among themselves.
Using the Number Line
The number line is a math teacher’s most trusted friend when it comes to fractions and can be used well beyond elementary classrooms. It forms the conceptual basis for students to relate fractions to each other and to whole numbers and to grasp that there are an infinite number of fractions between any two numbers.
Measurement is a good place to start. The use of fraction strips-manipulatives that provide a visual representation-can help young students measure and compare different items’ lengths. Teachers can use these strips to introduce the concept of improper fractions (11/2; 6/4; 23/16) and show that fractions can be more than one whole (1 and 1/2). These strips can be stacked up vertically to see how 1/2=2/4=4/8 or 3/6 (equivalence) or how close or far a fraction is to zero compared with other fractions. In higher grades, experts suggest that students mark these notations on the number line to understand their relationship.
Research suggests “fading” from concrete representations, like pies or pizzas, to fraction strips, and finally to abstract notations of x/y on a number line. This means it’s not a strict start-and-stop between concepts but more of an overlap. Experts also suggest using representations together-like the pizza model and the number line in one lesson-so that students can compare the different ways of expressing a fraction.
Conceptual Understanding
The jump from learning about fractions to adding and then multiplying them happens fast. At every stage, students need to understand the operation conceptually and pair that with learning how to solve. Otherwise, students could make common mistakes like incorrectly adding the numerators and denominators.
Here, using a visual aid or a manipulative can help. Students may initially think 1/2+1/2 =2/4, but if they saw a picture, they would know 1/2 and 1/2 make 1. Before students begin adding fractions, teachers can stress the example of sharing or dividing candy bars equally, so they know to compare like denominators. Let students guess the answers and explain their thinking before they solve a fraction problem. Teachers can use the student’s own thinking to explain the solution. For instance, a student may estimate that 1/2+1/5 is greater than 1/2 but smaller than 3/4 but still calculate the answer at 5/7. Now, the teacher can show why that’s wrong: 5/7 is smaller than 1/2 in size. Research indicates that when teachers start with fraction arithmetic, they should use smaller values to reduce the cognitive load of adding or subtracting big numbers. Also, teachers should use the same unit-like a chocolate bar-when comparing and adding fractions initially. Another potential pitfall that teachers encounter when teaching fractions is an over-reliance on cross-multiplication to identify equivalent fractions or solve for an unknown. Once students have learned it as a trick, the specter of cross-multiplication is difficult to exorcise. Students tend to try it on every fraction problem, even when it’s not appropriate, which can leave them with incorrect answers. This is where real-world applications come in handy. Teachers can get students to think about ratio and proportions in terms of adjusting recipes or calculating a car’s mileage. They can also boost multiplicative reasoning by using the “build up” method- “If two dishes are needed for five people, how many dishes do we need for 15?” Students will add on both sides of 2:5 ratio to get the answer of six dishes.
Teaching Fractions Using the Singapore Maths Method
Many children (and adults for that matter) find fractions difficult to understand. This is often because fraction notation (writing a fraction as a number, e.g. 1⁄2) is very confusing. Children therefore struggle to relate the symbol to the ‘thing’ and end up guessing. In Singapore, the understanding of fractions is rooted in the Concrete, Pictorial, Abstract (CPA) model, where children use paper squares and strips to learn the link between the concrete and the abstract.
At the heart of understanding fractions is the ability to understand that we’re giving an equal part a name. It is simply a naming activity!
4 Steps to Develop Understanding of Fractions
- Finding equal parts: Children need to understand what a fraction is. When we divide a whole into equal parts we create fractions. A fraction is just an equal part. In this lesson, children are encouraged to create 4 equal parts. Tip: Ask the children to also show you 4 unequal parts (if haven’t done it already!).
- Naming equal parts: Once the children can make/identify equal parts (fractions), they need to give them a name. In this lesson, children cut a pizza into a different number of equal parts. To show that the equal parts are different, the children give them different names. This is the denominator (name). Tip: Call the denominator the ‘namer’ to start with. Write out the denominator as a word as well as a number.
- Operations involving equal parts: If children can name a fraction, they are ready to do calculations using like fractions (they have the same name). In this lesson, children can use strips of paper to model the problem and see how it links to the written and symbolic notation. Tip: Prepare 3 apples (or any other item). Show the items to the children and ask them what is 2 apples + 1 apple? Now show them 3 equal pieces of paper. Write a ‘quarter’ on each. Then ask the children what is 2 quarters + 1 quarter? Is this any harder than adding apples?
- What if the parts aren’t equal? Can we add 3 apples and 2 oranges? Is it 5 apples? Is it 5 oranges? It is neither because we cannot add things with different names. We have to give them the same name, and in this case we could rename them as ‘fruit’. They now all have the same name and so we can do the calculation (5 pieces of fruits). The same is true for fractions. We can’t add 2 quarters and 1 eighth because they have different names, however, if we can give them the same name (equivalent) it is possible. In this lesson, children cut up a quarter to show more equal parts and name the parts. Tip: Ask the children to shade in their piece.
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