How to introduce the concept of Equivalent Fractions
Conceptual Understanding:
Equivalent Fractions via Paper Folding
The fraction strip paper folding exercise is generally used to introduce the topic of equivalent fractions.
Another better way is to use bar models or fraction strips.
First, have the students fold a paper strip in half and note that there are two equal parts. Then fold it again and note that now we have 4 equal parts.
Without folding it again, ask the students how many equal parts do they think they will have if we were to fold the paper strip a third time. Some might guess 6, a natural progression from 2 and 4, while others might reason that the pattern is multiplying by two, not adding.
Next, have the students shade the fraction 1/2, 2/4, 4/8 etc on different strips of paper and paste them on the same blank paper, on top of each other. An example is shown below.
Ask the following questions, and have a discussion with the class:
What do you notice?
Is there a pattern?
Is there a rule?
Extend the exercise to other fractions pictorially, using different shapes.
Exploring Equivalent Fractions
Equivalent Fractions using Area Models
After students have grasp the concept of paper folding to derive various equivalent fractions, you can now transfer that knowledge to the use of area models:
Let the students have fun exploring equivalent fractions by deriving their own “rules” through induction.
Through area models, students will observe “how the number and size of the parts differ even though the two fractions themselves are the same size” .
Example 1
Each fraction in example 1 represents the same number. These fractions are equivalent.
Let's look at some more examples of equivalent fractions.
Example 2
Two-thirds is equivalent to four-sixths.
Example 3
The fractions three-fourths, six-eighths, and nine-twelfths are equivalent.
In this way, it is very intuitive to see how the concepts of equivalent fractions can be transferred to the number line.
Equivalent Fractions using the Number Line
The next step is to transfer their knowledge to the number line However, many students have problem visualizing equivalent fractions on the number line. For example, it is not intuitive to see that 2/3 and 4/6 are the same point on the number line.
Here again, the bar model or fraction strips will come in very handy. To make it easier to visualize, teachers can present the fraction strips along with the number lines.
Procedural Understanding of Equivalent Fractions
Multiply the numerator and denominator by the same factor to get another fraction that is equivalent to the origin.
“Whatever you do to the numerator, you’d do for the denominator”. So,
The case of 1
The case of 1 is often overlooked by teachers, but the concept is so important. This simply refers to the fact that the whole number 1 is also made up of equivalent fractions, e.g.
1 = 3/3
This is extended to other whole numbers, e.g.
3 = 3/1
and even further to
3 = 9/3
The concept is important when the students start to apply their knowledge of fractions in addition and subtraction and other fraction manipulations. For example, in fraction subtraction, many students resort to converting the mixed fraction to an improper fraction before proceeding to subtract, and finally convert the resulting improper fraction back to mixed.
5 1/3 – 2/3 = 16/3 – 2/3 = 14/3 = 4 2/3
If the students understood the concept that whole numbers also have equivalent fractions, they can do a “re-grouping” as follows
5 1/3 – 2/3 = 4 4/3 – 2/3 = 4 2/3
Lastly, some special notes to be mindful about when teaching equivalent fractions.
Equivalent fractions is not always about multiply up. It is also important to learn that simplifying fractions to lower terms is also finding equivalent fractions. Fraction simplification is very important when it comes to fraction arithmetic (add, subtract), algebra and general word problems.
Terms like ‘cancel’ or ‘reduce’ give the impression that the “size” is somehow reduced and can be confusing to young students who have not fully understood equivalence yet. Instead, use the universal term ‘simplify’.
Want to see a fun way to teach Equivalent Fractions using an interactive manipulative? Check out the Fraction Wheel App here.
Equivalent fractions is such an important concept for students to understand, however under the pressure of time, it is sometimes tempting for teachers and parents to skip to procedural methods and not emphasize on conceptual understanding. However with some thoughts and design, the topic can be a fun way for students to discover more facts about fractions that they have not realized before and at the same time strengthen their confidence in fraction manipulations for the future.
In this post, we are going to understand what equivalent fractions are and how you can find out what fractions are equivalent.
Equivalent fractions are fractions that represent the same quantity. For example, which of the following do you believe will be the biggest?
What have you figured out? We are going to see it with an example, sharing this pizza in as many slices as the fraction indicates.
In order to represent 1/2, we will divide the pizza in 2 slices and keep 1 slice for ourselves:
In order to represent 3/6, we will divide the pizza into 6 slices and keep 3 slices for ourselves:
In order to represent 4/8, we will divide the pizza into 8 slices and keep 4 slices for ourselves:
Is there a piece of pizza that is bigger? No! Notice, the three fractions represent the same quantity of pizza, one half; therefore those are equivalent fractions:
How do we know if two fractions are equivalent?
Two fractions are equivalent if they represent the same decimal number.
For example, the three previous fractions represent the same decimal: 0.5.
Common mistakes made in introducing the concept of equivalent fractions
When teaching equivalence of fractions, teachers often start by stating the procedural rules. “Whatever you do to the numerator, you’d do for the denominator”.
Not only is this not helping in the conceptual understanding of equivalent fractions, but introducing the topic in this way wastes a perfectly good opportunity for the students to exercise their logic reasoning and induction muscles and discover for themselves what equivalence means, which fractions are equivalent and how to find them.
The concept of equivalent fractions seems simple – just multiply the numerator and denominator by the same factor to get another fraction that is equivalent to the origin. However it is not trivial at all, and with intentional design in the instruction delivery, the topic can be introduced in a way that strengthens the students’ reasoning and inductive skills and at the same time, lay a stronger foundation for the future, especially in fraction arithmetic and algebraic manipulations.
