problem solving find equivalent fractions lesson 6 5

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Unit 7: Equivalent fractions and comparing fractions

About this unit.

You already know that fractions can be broken into parts, but did you know that different fractions can be equivalent? In this lesson, you'll learn all about equivalent fractions and how to compare them. With the help of models, number lines, and benchmark fractions, you'll be a fraction master in no time!

Equivalent fractions

• Equivalent fractions and comparing fractions: FAQ (Opens a modal)
• Equivalent fractions with models (Opens a modal)
• Equivalent fractions on number lines (Opens a modal)
• Visualizing equivalent fractions review (Opens a modal)
• Equivalent fractions (Opens a modal)
• More on equivalent fractions (Opens a modal)
• Equivalent fractions and different wholes (Opens a modal)
• Comparing fractions of different wholes (Opens a modal)
• Equivalent fractions (fraction models) Get 3 of 4 questions to level up!
• Equivalent fractions (number lines) Get 3 of 4 questions to level up!
• Equivalent fractions Get 5 of 7 questions to level up!
• Fractions of different wholes Get 5 of 7 questions to level up!

Common denominators

• Finding common denominators (Opens a modal)
• Common denominators: 1/4 and 5/6 (Opens a modal)
• Common denominators: 3/5 and 7/2 (Opens a modal)
• Common denominators review (Opens a modal)
• Common denominators Get 3 of 4 questions to level up!

Comparing fractions with unlike denominators visually

• Comparing fractions: tape diagram (Opens a modal)
• Comparing fractions: number line (Opens a modal)
• Comparing fractions: fraction models (Opens a modal)
• Visually comparing fractions review (Opens a modal)
• Visually compare fractions with unlike denominators Get 5 of 7 questions to level up!

Comparing fractions with unlike denominators

• Comparing fractions 1 (unlike denominators) (Opens a modal)
• Comparing fractions word problems (Opens a modal)
• Compare fractions using benchmarks Get 3 of 4 questions to level up!
• Compare fractions with different numerators and denominators Get 5 of 7 questions to level up!
• Compare fractions word problems Get 3 of 4 questions to level up!

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1. Connect the pictures that show equivalent fractions. Write the name of each fraction beside its picture.

2. Make a chain of equivalent fractions.

3. Split the pieces by drawing the new pieces in the right-hand picture. Write the equivalent     fractions.

b.   Split each piece into three .

c.   Split each piece in two .

d.   Split each piece in two .

e.   Split each piece into three .

f.   Split each piece in two .

g.   Split each piece in two .

h.   Split each piece in two .

i.   Split each piece into five .

4. Write the equivalent fraction. Imagine or draw the helping arrows.

b. Split each piece in two .

c.  Split each piece into six .

d. Split each piece into four .

e. Split each piece into five .

f. Split each piece into three .

g. Split each piece into ten .

h. Split each piece into eight .

i. Split each piece into seven .

j. Split each piece into eight .

5. Figure out how many ways the pieces were split and write the missing numerator or denominator.

b.  Pieces were     split into ____ .

c.  Pieces were     split into ____ .

d.  Pieces were     split into ____ .

e.  Pieces were     split into ____ .

6. Mark the equivalent fractions on the number lines.

    c. Find and mark two fractions on the 12th parts number line that do not         have an equivalent fraction on the 3rd parts number line. Write them here →

    d. Find and mark two fractions on the 24th parts number line that do not         have an equivalent fraction on the 12th parts number line. Write them here →

Math Mammoth Fractions 1

A self-teaching worktext for 5th grade that teaches fractions and their operations with visual models. The book covers fractions, mixed numbers, adding and subtracting like fractions, adding and subtracting mixed numbers, adding and subtracting unlike fractions, and comparing fractions.

Download ($3.50) . Also available as a printed copy.

Learn more and see the free samples!

See more topical Math Mammoth books

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• Fractions - converting

Equivalent fractions

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Math worksheets: Finding equivalent fractions

Below are six versions of our grade 5 math worksheet on equivalent fractions. Students must write in the missing numerator or denominator to make the fractions in each problem equal.  These fractions worksheets are pdf files .

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Explore all of our fractions worksheets , from dividing shapes into "equal parts" to multiplying and dividing improper fractions and mixed numbers.

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Teaching Equivalent Fractions

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.
• 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.

This is extended to other whole numbers, e.g.

and even further to

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 .

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

More Fraction Resources

For more fraction resources, refer to our main fractions page .

1 thought on “Teaching Equivalent Fractions”

This was just what I needed as I move on to fractions for my students!

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Published 2013 Revised 2019

Exploring Fractions

• The first group  gives you some starting points to explore with your class, which are applicable to a wide range of ages.  The tasks in this first group will build on children's current understanding of fractions and will help them get to grips with the concept of the part-whole relationship. 
• The second group of tasks  focuses on the progression of ideas associated with fractions, through a problem-solving lens.  So, the tasks in this second group are curriculum-linked but crucially also offer opportunities for learners to develop their problem-solving and reasoning skills.   

• are applicable to a range of ages;
• provide contexts in which to explore the part-whole relationship in depth;
• offer opportunities to develop conceptual understanding through talk.

Go Math Interactive Mimio Lesson 6.5 Find Equivalent Fractions

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Equivalent Fractions Challenge

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

21 January 2016

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Thank you for this resource. Great for my Year 6 children!

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Fantastic activity - perfect for my more able Year 6s. Thank you very much!

great break down of concepts and worksheet to reflect the main fraction skils

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