Equivalent Fractions is a concept that is generally introduced in the 3rd grade. In the US 3rd grade Common Core (CCSS.Math.3.NF.3):
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
This is further extended in the 4th grade (CCSS.Math.4.NF):
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
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.
Logic Reasoning and Induction
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”. So,
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.
Another better way is to use bar models or fraction strips. The fraction strip paper folding exercise which we first wrote about after a lesson study at a local school, is what we generally use to introduce the topic of equivalent fractions.
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.
Let the students have fun exploring equivalent fractions by deriving their own “rules” through induction. To read more about our experience using the fraction strip exercise, see our previous post here.
Equivalent Fractions
Equivalent Fractions on the Number Line
Generally, students first learn about equivalent fraction using an area model, e.g.
Through area models, student observed “how the number and size of the parts differ even though the two fractions themselves are the same size” (CCSS.Math.4.NF.1).
The next step is to transfer their knowledge to the number line (see our previous post on the importance of understanding fractions on 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/parents can present the fraction strips along with the number lines.
In this way, it is very intuitive to see how the concepts of equivalent fractions can be transferred to the number line.
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
Special Notes
Lastly, some special notes to be mindful about when teaching equivalent fractions.
Fraction Simplification
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.
Don’t say ‘cancel’ or ‘reduce’
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’.
TeachableApp
Want to see a fun way to teach Equivalent Fractions using an interactive manipulative? Check out the Fraction Wheel App here.
Conclusion
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.
This article is part of a series of blog posts on Fractions:
- >> Read the next post on Fractions: On Comparing Fractions
- << Read the previous post on Fractions: Fractions on the Number Line
- Or start from the beginning: Understanding Fractions as Equal Parts
This was just what I needed as I move on to fractions for my students!