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{{T}}
[[Category:English]]
[[Category:Equivalent Fractions]]

{{BR}}
= Introduction =

'''Equivalent fractions''' are fractions that look different but name the same amount. For example, 1/2, 2/4, 4/8, and 8/16 all represent one half of the same whole. In this aiMOOC, you will learn to recognize, create, simplify, explain, and use equivalent fractions. The course is designed for learners in Grades 5–6.

[[File:Fraction Equivalents.jpg|500px|frameless|center]]

Look closely at the image. The number of pieces changes, but the shaded amount stays the same. That is the key idea of [[English:Fraction|fractions]] that are equivalent: '''the value does not change'''.

{{#ev:youtube|https://www.youtube.com/watch?v=U2ovEuEUxXQ|500|center}}

{{BR}}
== What You Will Learn ==

By the end of the course, you should be able to explain why two fractions are equivalent, create equivalent fractions by multiplying or dividing, simplify a fraction, locate equivalent fractions on a [[English:Number line|number line]], solve missing-number problems, and use equivalent fractions in everyday situations.

{{BR}}
= Fraction Basics =

A fraction describes equal parts of a whole, a group, or a distance. In the fraction 3/5, the '''numerator''' is 3 and the '''denominator''' is 5.

The [[English:Numerator|numerator]] tells how many equal parts are being considered. The [[English:Denominator|denominator]] tells how many equal parts make one whole. The denominator cannot be zero.

{| class="wikitable"
! Fraction word
! Meaning
! Example
|-
| Numerator
| Number above the fraction bar
| In 3/5, it is 3
|-
| Denominator
| Number below the fraction bar
| In 3/5, it is 5
|-
| Whole
| The complete object, set, or length being divided
| One full strip or one full cake
|-
| Equal parts
| Parts with the same size
| Four equal quarters
|}

[[File:Four quarters circle.svg|500px|frameless|center]]

Four quarters make one whole. If two of the four equal parts are selected, the amount is 2/4, which is the same amount as 1/2.

{{BR}}
== The Same Whole Matters ==

When you compare fraction pictures, they must refer to the same-sized whole. Half of a small pizza is not the same amount of food as half of a much larger pizza. Equivalent fraction models work only when the whole stays the same and each whole is divided into equal-sized parts.

[[File:Cake-quarters.jpg|500px|frameless|center]]

The cake model shows that two quarters cover the same amount as one half. This gives the equivalence 2/4 = 1/2.

{{BR}}
= Understanding Equivalent Fractions =

Two fractions are equivalent when they represent exactly the same value. You can see this with shaded models, fraction strips, number lines, multiplication, division, or a check using cross products.

[[File:Equal Fractions 123.svg|500px|frameless|center]]

The image shows that one whole can be partitioned in different ways. Changing the number of equal parts changes the fraction name, but it does not have to change the amount.

{{BR}}
== Visual Models and Fraction Strips ==

Fraction strips are useful because strips of the same total length can be divided into different numbers of equal parts. If the endpoint of 1/2 lines up with the endpoint of 2/4 and 4/8, the three fractions represent the same distance.

[[File:FractionStrips.PNG|500px|frameless|center]]

Try covering a 1/2 strip with fourths. You need two fourths. Cover the same 1/2 strip with eighths. You need four eighths. Therefore:

'''1/2 = 2/4 = 4/8'''

[[File:Fraction Circles Shaded.png|500px|frameless|center]]

Fraction circles are another way to compare equal parts of the same-sized whole. They help you see how the size of one part changes as the denominator changes.

{{#ev:youtube|https://www.youtube.com/watch?v=W72eW83rrC8|500|center}}

{{BR}}
== The Multiplication Rule ==

To create an equivalent fraction, multiply the numerator and denominator by the '''same nonzero whole number'''.

For example:

'''2/3 × 2/2 = 4/6'''

The fraction 2/2 is equal to 1, so multiplying by 2/2 does not change the value. It only changes how the same amount is named.

{| class="wikitable"
! Start
! Same factor
! Equivalent fraction
|-
| 1/2
| Multiply both by 2
| 2/4
|-
| 2/3
| Multiply both by 3
| 6/9
|-
| 3/5
| Multiply both by 4
| 12/20
|-
| 5/8
| Multiply both by 2
| 10/16
|}

A useful sentence to remember is: '''Whatever you do to the numerator, do the same to the denominator.'''

{{BR}}
== The Division Rule and Simplifying ==

You can also create an equivalent fraction by dividing the numerator and denominator by the same common factor. This is called [[English:Simplifying fractions|simplifying]] or reducing a fraction.

Example:

'''12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12/18 = 2/3.'''

Both 12 and 18 are divisible by 6, so dividing both by 6 keeps the value unchanged.

A fraction is in '''simplest form''' when the numerator and denominator have no common factor greater than 1. For example, 2/3 is in simplest form.

{{BR}}
== Why the Rule Works ==

When you multiply a fraction by 3/3, you are really multiplying it by 1 because 3/3 = 1. Multiplying any number by 1 keeps its value. When you divide the numerator and denominator by the same factor, you reverse that process.

This is why 3/4, 6/8, and 9/12 can all name the same number.

{{BR}}
= Finding Missing Numbers =

Equivalent-fraction problems often hide one numerator or denominator. Look for the factor that changes one known number into another, then apply that same factor to the other part of the fraction.

Example:

'''3/4 = ?/20'''

The denominator changes from 4 to 20 by multiplying by 5. Multiply the numerator by 5 too:

'''3 × 5 = 15'''

So:

'''3/4 = 15/20'''

Another example:

'''5/6 = 20/?'''

The numerator changes from 5 to 20 by multiplying by 4. Multiply the denominator by 4:

'''6 × 4 = 24'''

So:

'''5/6 = 20/24'''

{{BR}}
== A Reliable Strategy ==

# [[English:Read the fraction|Read the fraction]]: Decide which numerator or denominator is missing.
# [[English:Find the scale factor|Find the scale factor]]: Ask what multiplication or division changes the known number into its partner.
# [[English:Use the same factor|Use the same factor]]: Apply that factor to the other numerator or denominator.
# [[English:Check the value|Check the value]]: Use a visual model, simplification, or cross products to confirm your result.

{{BR}}
= Checking Whether Fractions Are Equivalent =

There are several reliable ways to check whether two fractions are equivalent.

{{BR}}
== Method One: Simplify ==

Simplify each fraction and compare the results.

Example:

12/16 simplifies to 3/4 because both numbers can be divided by 4.

9/12 also simplifies to 3/4 because both numbers can be divided by 3.

Therefore, 12/16 and 9/12 are equivalent.

{{BR}}
== Method Two: Use a Common Denominator ==

Rewrite both fractions with the same denominator.

Example:

2/3 and 8/12 can be compared by changing 2/3 to twelfths.

2/3 × 4/4 = 8/12

Both fractions become 8/12, so they are equivalent.

{{BR}}
== Method Three: Cross Products ==

For two fractions a/b and c/d, you can compare the cross products. If a × d and b × c are equal, the fractions are equivalent.

Example:

3/4 and 9/12

3 × 12 = 36

4 × 9 = 36

The cross products match, so the fractions are equivalent.

This method is useful as a check, but you should also understand the visual and multiplication ideas behind equivalence.

{{BR}}
= Equivalent Fractions on a Number Line =

Equivalent fractions land on the same point of a number line. On a line from 0 to 1, 1/2 is halfway. If the same line is divided into fourths, 2/4 lands at exactly the same location. If it is divided into eighths, 4/8 lands there too.

So:

'''1/2 = 2/4 = 4/8'''

A number line proves that equivalent fractions are not just similar pictures. They are the same number.

{{BR}}
= Common Mistakes and How to Fix Them =

'''Mistake 1: Changing only the numerator.''' Turning 1/2 into 2/2 changes one half into one whole. Fix it by multiplying both numerator and denominator by the same factor.

'''Mistake 2: Adding the same number to the numerator and denominator.''' Changing 1/2 to 2/3 does not keep the value. Equivalent fractions are normally created by multiplying or dividing both parts by the same factor.

'''Mistake 3: Using unequal visual wholes.''' Two shaded regions may look similar even when their wholes have different sizes. Compare models that use the same-sized whole.

'''Mistake 4: Dividing when the numbers do not divide evenly.''' To simplify using whole-number factors, choose a number that divides both numerator and denominator exactly.

'''Mistake 5: Stopping before simplest form.''' For example, 8/12 can be simplified to 4/6, but 4/6 can still be simplified to 2/3.

{{BR}}
= Real-Life Uses =

Equivalent fractions appear in cooking, measurement, sharing, maps, probability, music, and many other settings.

In a recipe, 2/4 cup is the same amount as 1/2 cup. In measurement, 6/12 of a foot is 1/2 of a foot. When a group shares food equally, equivalent fractions can describe the same share using different numbers of pieces.

{{#ev:youtube|https://www.youtube.com/watch?v=vKXqzpz-G0s|500|center}}

{{BR}}
== Connection to Percents ==

A [[English:Percent|percent]] is a fraction with a denominator of 100. Equivalent fractions can help you connect common fractions with percents.

1/2 = 50/100 = 50 percent

1/4 = 25/100 = 25 percent

3/4 = 75/100 = 75 percent

This connection becomes especially useful when you compare discounts, survey results, test scores, and probabilities.

{{#ev:youtube|https://www.youtube.com/watch?v=kmVfZ9o-2gg|500|center}}

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''Which fraction is equivalent to 1/2?'''
(Two fourths)
(!Two thirds)
(!Three fifths)
(!Four fifths)

{{E}}
<br>

{{MC}}

'''What must you do to create an equivalent fraction by multiplication?'''
(Multiply both parts by the same factor)
(!Multiply only the numerator)
(!Multiply only the denominator)
(!Add the same number to both parts)

{{E}}
<br>

{{MC}}

'''Which fraction is equivalent to 2/3?'''
(Four sixths)
(!Three sixths)
(!Four fifths)
(!Five sixths)

{{E}}
<br>

{{MC}}

'''Which is the simplest form of 12/18?'''
(Two thirds)
(!Three fourths)
(!Four fifths)
(!Five sixths)

{{E}}
<br>

{{MC}}

'''What does the denominator tell you?'''
(Number of equal parts in one whole)
(!Number of selected parts)
(!Number of whole objects)
(!Number of operations to perform)

{{E}}
<br>

{{MC}}

'''Where do equivalent fractions appear on a number line?'''
(At the same point)
(!At opposite ends)
(!At neighboring points)
(!At random points)

{{E}}
<br>

{{MC}}

'''What factor changes 3/4 into 15/20?'''
(Five)
(!Two)
(!Three)
(!Four)

{{E}}
<br>

{{MC}}

'''Which action can simplify a fraction without changing its value?'''
(Divide both parts by a common factor)
(!Subtract from the numerator only)
(!Add to the denominator only)
(!Swap the numerator and denominator)

{{E}}
<br>

{{MC}}

'''Which statement about equivalent fractions is true?'''
(They have the same value)
(!They always have the same numerator)
(!They always have the same denominator)
(!They must use consecutive numbers)

{{E}}
<br>

{{MC}}

'''Why does multiplying by 3/3 keep a fraction equivalent?'''
(Because three thirds equals one)
(!Because the numerator becomes larger)
(!Because every denominator must be three)
(!Because multiplication always makes equal fractions)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Numerator || Number written above the fraction bar
|-
| Denominator || Number written below the fraction bar
|-
| Equivalent || Describes fractions that have the same value
|-
| Factor || Whole number that divides another whole number exactly
|-
| Simplify || Write an equal fraction with smaller numbers
|-
| Fraction strip || Bar model divided into equal parts
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Multiply both parts'''
| Create an equivalent fraction with larger numbers
|-
| '''Divide both parts'''
| Simplify a fraction using a shared factor
|-
| '''Same point'''
| Number-line evidence of equal value
|-
| '''Equal shaded area'''
| Visual evidence of equal value
|-
| '''Common factor'''
| Divides the numerator and denominator exactly
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Numerator || What is the top number of a fraction called?
|-
| Denominator || What is the bottom number of a fraction called?
|-
| Equivalent || Which word describes different fractions with the same value?
|-
| Simplify || What verb means to write a fraction in an equal form with smaller numbers?
|-
| Factor || What word names a whole number that divides another whole number exactly?
|-
| Fraction || What kind of number can represent part of a whole?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Equivalent+Fractions </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
Equivalent fractions have the same { value } even when their numerators and denominators look different. The top number of a fraction is the { numerator }. The bottom number is the { denominator }. To create an equivalent fraction by multiplication, use the same { factor } on both parts. A fraction can be simplified by dividing both parts by a common { divisor }. Equivalent fractions name the same { point } on a number line. Visual fraction models must use the same-sized { whole }. Changing only one part of a fraction usually changes its { value }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Fraction Collage|Fraction Collage]]: Cut or draw equal-sized shapes, shade pairs of equivalent fractions, and label each pair with a short sentence explaining why the amounts match.
# [[English:Fraction Strip Lab|Fraction Strip Lab]]: Make paper fraction strips for halves, fourths, eighths, and sixteenths, then record at least four equivalence statements that you can prove by lining up the strips.
# [[English:Equivalent Fraction Story|Equivalent Fraction Story]]: Write a short story about sharing food or objects in which two different fraction names describe the same amount.
# [[English:Kitchen Fraction Hunt|Kitchen Fraction Hunt]]: With an adult or teacher, look for measuring tools or recipe amounts and make a small illustrated list of equivalent measurements you can find or create.

{{BR}}
=== Standard ===
# [[English:Classmate Interview|Classmate Interview]]: Interview a classmate about how they recognize equivalent fractions, compare your strategies, and write a short report explaining one method you learned from each other.
# [[English:Number Line Poster|Number Line Poster]]: Design a poster that places several equivalent fraction families on number lines and explains why fractions in each family land on the same point.
# [[English:Recipe Scaling|Recipe Scaling]]: Choose a simple recipe, rewrite at least four ingredient amounts using equivalent fractions, and explain why the amount of each ingredient stays unchanged.
# [[English:Photo Fraction Hunt|Photo Fraction Hunt]]: Photograph or sketch examples of equal parts in your classroom, home, playground, or local area, then annotate the images with possible equivalent fraction descriptions.

{{BR}}
=== Advanced ===
# [[English:Proof Video|Proof Video]]: Create a two- to three-minute video that proves one equivalence in three ways using a model, multiplication or division, and a number line or cross-product check.
# [[English:Fraction Game Design|Fraction Game Design]]: Design and test a card or board game in which players must match, generate, or simplify equivalent fractions, then revise the rules after observing a test round.
# [[English:Data Display Investigation|Data Display Investigation]]: Collect simple class survey data, express selected results as fractions, generate equivalent forms, and explain which form makes each comparison easiest to understand.
# [[English:Fraction Field Visit|Fraction Field Visit]]: Visit a supermarket, cafeteria, kitchen, workshop, or other suitable place with an adult or teacher, document where fractional quantities appear, and present at least three examples in equivalent forms.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Visual Proof Assessment|Visual Proof Assessment]]: Prove that 3/4 and 6/8 are equivalent by drawing equal-sized models and explaining how the partitions are related.
# [[English:Missing Number Reasoning|Missing Number Reasoning]]: Solve 5/7 = 20/? and explain how the same scale factor must be used on numerator and denominator.
# [[English:Error Analysis|Error Analysis]]: A learner says that 2/3 becomes 3/4 by adding one to both numbers; explain why this does not preserve value and show a correct equivalent fraction.
# [[English:Method Comparison|Method Comparison]]: Decide whether 8/12 and 10/15 are equivalent, using two different checking methods, and compare which method is more efficient.
# [[English:Real-Life Transfer|Real-Life Transfer]]: Create a short real-life problem in which two equivalent fractions describe the same quantity, solve it, and explain why equivalence matters in the situation.
# [[English:Fraction Connection|Fraction Connection]]: Explain how 3/5 can be rewritten as a fraction with denominator 100 and use that result to connect the fraction to a percent.

<br>
<br>

{{BR}}
= Evidence of Learning =

A strong collection of evidence should show what you know, what you can do, what you can create, and how you can transfer your learning.

* '''Knowledge:''' You can explain that equivalent fractions are different names for the same value and identify the roles of numerator and denominator.
* '''Skills:''' You can generate, simplify, compare, and check equivalent fractions using multiplication, division, models, number lines, and reasoning.
* '''Products:''' You can produce accurate fraction strips, posters, explanations, games, photos, reports, or videos that show equivalence clearly.
* '''Reasoning:''' You can justify why the same factor must be applied to both numerator and denominator and identify mistakes in incorrect methods.
* '''Transfer:''' You can recognize and use equivalent fractions in recipes, measurement, surveys, percents, sharing situations, and other new contexts.
* '''Communication:''' You can use correct mathematical vocabulary and explain your strategy so another learner can follow it.

<br>
<br>

{{BR}}
= OERs on the Topic =

The English Wikipedia article on [[English:Fraction|fractions]] includes a section about equivalent fractions and simplifying.

<iframe> https://en.m.wikipedia.org/wiki/Fraction </iframe>

<br>

{{BR}}
= Linked Learning Areas =

Equivalent fractions connect fraction models with multiplication, division, factors, number lines, simplest form, ratios, and percents. These links help you move from visual understanding to calculation and real-life problem solving.

{| align=center
{{:D-Tab}}
'''[[English:Equivalent Fractions|Equivalent Fractions]]'''
# [[English:Fraction|Fraction]]
# [[English:Numerator|Numerator]]
# [[English:Denominator|Denominator]]
# [[English:Multiplication|Multiplication]]
# [[English:Division|Division]]
# [[English:Simplifying fractions|Simplifying fractions]]
# [[English:Number line|Number line]]
# [[English:Factor|Factor]]
# [[English:Ratio|Ratio]]
# [[English:Percent|Percent]]
|}

[[Category:English]]
[[Category:Equivalent Fractions]]
[[Category:Mathematics]]
[[Category:Fractions]]
[[Category:Grades 5-6]]
[[Category:Elementary mathematics]]

{{BR}}
= aiMOOC Projects =
[[Category:AI_MOOC]] [[Category:GPT aiMOOC]]
{{MT}}

English:Equivalent Fractions - Versionsgeschichte

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