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{{T}}
[[Category:English]]
[[Category:Factoring Simple Expressions]]

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= Introduction =

'''Factoring Simple Expressions''' is about rewriting an algebraic expression as a product of simpler factors. For example, <math>12x+18</math> can be rewritten as <math>6(2x+3)</math>. The two forms are [[English:Equivalent expressions|equivalent]] because they have the same value for every allowed value of <math>x</math>.

Factoring is the reverse of using the [[English:Distributive property|distributive property]]. It helps you see structure in an expression, simplify later algebra, solve problems with areas, and prepare for more advanced [[English:Factorization|factorization]].

In this aiMOOC you will learn how to identify common factors, find the [[English:Greatest common divisor|greatest common factor]], factor simple expressions, and check your result by expanding.

[[File:Factor tree of 24.png|500px|frameless|center]]

A factor tree reminds you that whole numbers can be broken into factors. That same idea helps when you factor the numerical coefficients in algebraic terms.

{{#ev:youtube|https://www.youtube.com/watch?v=CUEOL3_Wm3Y|500|center}}

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= Learning Goals =

By the end of this course, you should be able to explain what a factor is, identify the greatest common factor of two or more terms, rewrite a sum or difference as a product, include common variable factors when appropriate, and verify a factored expression by expanding it.

You should already be comfortable with [[English:Arithmetic|whole-number arithmetic]], [[English:Multiplication|multiplication]], [[English:Division|division]], [[English:Variables|variables]], [[English:Terms|terms]], and the basic [[English:Distributive property|distributive property]].

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= Core Ideas =

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== Factors, Terms, and Coefficients ==

A '''factor''' is a number or expression multiplied by another number or expression. In <math>5x</math>, both <math>5</math> and <math>x</math> are factors.

A '''term''' is a part of an expression separated by addition or subtraction signs. In <math>8x+12</math>, the terms are <math>8x</math> and <math>12</math>.

A '''coefficient''' is the numerical factor multiplying a variable. In <math>8x</math>, the coefficient is <math>8</math>.

When you factor an expression, you look for something that is a factor of '''every''' term.

[[File:Algebra tile physical attributes.jpg|500px|frameless|center]]

Algebra tiles can represent constants, variables, and squared variables. Physical or paper tiles can make the structure of an expression visible before you write the symbolic factorization.

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== The Greatest Common Factor ==

The '''greatest common factor''', often called the GCF, is the greatest factor shared by all relevant numbers or terms.

For <math>12x+18</math>, the numerical coefficients are <math>12</math> and <math>18</math>. Their greatest common factor is <math>6</math>. Therefore:

<math>12x+18=6(2x+3)</math>

For <math>10x+15x^2</math>, both terms contain a factor of <math>5x</math>. Therefore:

<math>10x+15x^2=5x(2+3x)</math>

For Grades 7–8, it is useful to separate the process into two questions: '''What number divides every coefficient?''' and '''What variable factor appears in every term?'''

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== Factoring Reverses the Distributive Property ==

The distributive property says:

<math>a(b+c)=ab+ac</math>

Factoring uses the same relationship in reverse:

<math>ab+ac=a(b+c)</math>

For example:

<math>4x+20=4(x+5)</math>

The common factor <math>4</math> is placed outside the parentheses. Inside the parentheses, you write what remains after dividing each original term by <math>4</math>.

[[File:Distributive property with rectangles.svg|500px|frameless|center]]

The rectangle model shows why distribution and factoring are linked: one large area can be viewed as the sum of smaller areas, or the sum can be recombined into a product of side lengths.

{{#ev:youtube|https://www.youtube.com/watch?v=v-6MShC82ow|500|center}}

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= Factoring Step by Step =

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== Method for Simple Expressions ==

Use this routine when all terms share a common factor.

# [[English:Identify terms|Identify terms]]: Separate the expression into terms using addition and subtraction signs.
# [[English:Find the GCF|Find the GCF]]: Find the greatest numerical factor common to all coefficients.
# [[English:Find common variables|Find common variables]]: Include any variable factor present in every term.
# [[English:Divide each term|Divide each term]]: Divide every original term by the complete common factor.
# [[English:Write the product|Write the product]]: Place the common factor outside parentheses and the quotients inside.
# [[English:Check by expanding|Check by expanding]]: Distribute the outside factor and confirm that you recover the original expression.

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== Worked Example: Numerical Common Factor ==

Factor <math>18x+24</math> completely.

The terms are <math>18x</math> and <math>24</math>. The greatest common factor of <math>18</math> and <math>24</math> is <math>6</math>. Divide each term by <math>6</math>:

<math>18x\div6=3x</math>

<math>24\div6=4</math>

So:

<math>18x+24=6(3x+4)</math>

Check by distributing:

<math>6(3x+4)=18x+24</math>

Because the expanded form matches the original expression, the factorization is correct.

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== Worked Example: Common Variable Factor ==

Factor <math>12x^2+18x</math>.

The coefficients <math>12</math> and <math>18</math> share a greatest common factor of <math>6</math>. Both terms also contain at least one factor of <math>x</math>. The complete common factor is therefore <math>6x</math>.

<math>12x^2+18x=6x(2x+3)</math>

Check:

<math>6x(2x+3)=12x^2+18x</math>

This example is especially useful in Grade 8 because it combines number factors with variable factors.

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== Worked Example: A Negative Term ==

Factor <math>-8y+12</math>.

The greatest positive common factor is <math>4</math>, so one correct factorization is:

<math>-8y+12=4(-2y+3)</math>

You may also factor out <math>-4</math>:

<math>-8y+12=-4(2y-3)</math>

Both forms are equivalent. In later algebra, factoring out a negative factor is sometimes useful because it can make the first term inside the parentheses positive.

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= Area Models and Visual Thinking =

Factoring can describe dimensions. Imagine two adjacent rectangular strips that both have a width of <math>3</math> metres. One strip has length <math>x</math> metres and the other has length <math>4</math> metres. Their combined area is:

<math>3x+12</math>

Factoring gives:

<math>3x+12=3(x+4)</math>

The factored form shows the dimensions of the whole rectangle: width <math>3</math> and total length <math>x+4</math>.

[[File:Algebra tile factoring.jpg|500px|frameless|center]]

This algebra-tile image shows a more advanced area-model example. You do not need to master quadratic factoring in this course, but the image gives you a preview of how the same product-and-area idea extends to later algebra.

{{#ev:youtube|https://www.youtube.com/watch?v=RHnWD8yRh2E|500|center}}

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= Worked Practice =

Try each expression before reading the explanation.

{| class="wikitable"
! Expression
! Common factor
! Factored form
! Quick check
|-
| <math>6x+9</math>
| <math>3</math>
| <math>3(2x+3)</math>
| <math>3\cdot2x+3\cdot3=6x+9</math>
|-
| <math>14a-21</math>
| <math>7</math>
| <math>7(2a-3)</math>
| <math>7\cdot2a-7\cdot3=14a-21</math>
|-
| <math>20m+30</math>
| <math>10</math>
| <math>10(2m+3)</math>
| <math>10\cdot2m+10\cdot3=20m+30</math>
|-
| <math>9x^2+12x</math>
| <math>3x</math>
| <math>3x(3x+4)</math>
| <math>3x\cdot3x+3x\cdot4=9x^2+12x</math>
|-
| <math>16p-24q</math>
| <math>8</math>
| <math>8(2p-3q)</math>
| <math>8\cdot2p-8\cdot3q=16p-24q</math>
|}

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= Common Mistakes and Self-Check =

'''Mistake 1: Taking a common factor that is not the greatest.''' For example, <math>12x+18=3(4x+6)</math> is correct, but it is not fully factored because <math>4x+6</math> still has a common factor of <math>2</math>. The fully factored form is <math>6(2x+3)</math>.

'''Mistake 2: Forgetting to divide every term.''' If you factor <math>5</math> from <math>10x+15</math>, the inside terms must be <math>2x</math> and <math>3</math>.

'''Mistake 3: Losing a minus sign.''' In <math>14a-21=7(2a-3)</math>, the negative sign must remain with the second inside term.

'''Mistake 4: Missing a variable factor.''' In <math>12x^2+18x</math>, both terms contain <math>x</math>, so factoring only <math>6</math> does not produce the most complete common-factor form.

'''Best self-check: expand.''' If distribution returns the exact original expression, your factorization is equivalent.

{{#ev:youtube|https://www.youtube.com/watch?v=I6TBBzIvgB8|500|center}}

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= Interactive Tasks =

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== Quiz: Test Your Knowledge ==

{{MC}}

'''What does it mean to factor an algebraic expression?'''
(Rewrite it as a product)
(!Rewrite it as a fraction)
(!Replace every variable)
(!Add all coefficients)

{{E}}
<br>

{{MC}}

'''What is the greatest common factor of 12 and 18?'''
(6)
(!2)
(!3)
(!36)

{{E}}
<br>

{{MC}}

'''Which property connects expanding and factoring?'''
(Distributive property)
(!Commutative property only)
(!Identity property only)
(!Zero product property only)

{{E}}
<br>

{{MC}}

'''What is the greatest common numerical factor of 15x and 20?'''
(5)
(!10)
(!15)
(!20)

{{E}}
<br>

{{MC}}

'''If every term in an expression contains x, what can be true?'''
(x can be a common factor)
(!x must equal zero)
(!x must be removed)
(!x becomes a coefficient)

{{E}}
<br>

{{MC}}

'''After factoring 4 from 12x plus 20, what is the coefficient of x inside the group?'''
(3)
(!4)
(!5)
(!12)

{{E}}
<br>

{{MC}}

'''Which common factor should you take first to factor 18x plus 24 completely?'''
(6)
(!2)
(!3)
(!9)

{{E}}
<br>

{{MC}}

'''What is the most reliable way to check a simple factorization?'''
(Expand the factored form)
(!Change every sign)
(!Square each term)
(!Delete the common factor)

{{E}}
<br>

{{MC}}

'''What is a coefficient?'''
(A number multiplying a variable)
(!A variable with no number)
(!A sign between terms)
(!A pair of parentheses)

{{E}}
<br>

{{MC}}

'''What is the greatest positive common numerical factor of negative 6x and positive 9?'''
(3)
(!2)
(!6)
(!9)

{{E}}
<br>

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== Memory Game ==

<div class="memo-quiz">
{|
|-
| Factor || A quantity multiplied by another quantity to form a product
|-
| Term || A part of an expression separated by addition or subtraction
|-
| Coefficient || The numerical factor multiplying a variable
|-
| GCF || The greatest factor shared by all selected terms
|-
| Distributive property || A rule connecting a product with a sum or difference
|-
| Equivalent expressions || Different forms that have the same value
|}
{{E}}
<br>

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== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Greatest common factor'''
| Largest shared factor of all terms
|-
| '''Common variable factor'''
| Variable part present in every term
|-
| '''Reverse distribution'''
| Rewriting a sum or difference as a product
|-
| '''Factored form'''
| Expression written as multiplication of factors
|-
| '''Expanded form'''
| Expression produced after distributing multiplication
|}
{{E}}

<br>
<br />

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== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Factor || What do you call a quantity multiplied by another quantity in a product?
|-
| Variable || What letter or symbol can represent a changing or unknown value?
|-
| Coefficient || What is the numerical factor multiplying a variable called?
|-
| Distributive || Which property name describes multiplying across a sum or difference?
|-
| Equivalent || What word describes two expressions with the same value?
|-
| Rectangle || Which shape is often used in an area model for factoring?
|}
{{E}}
<br>

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== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Factoring+Simple+Expressions </iframe>

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== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
Factoring rewrites an expression as a { product }. The process reverses the { distributive property }. When factoring, begin by looking for the { greatest common factor } of all terms. A number multiplying a variable is called a { coefficient }. If the same variable appears as a factor in every term, it can be a { common factor }. After factoring, you can check your result by { expanding }. Correct original and factored forms are { equivalent }. Factoring out a negative common factor can make the leading term inside the parentheses { positive }.
</quiz>

<br>

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= Open-Ended Tasks =

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=== Easy ===
# [[English:Factor Card Sort|Factor Card Sort]]: Create twelve cards showing simple expanded expressions and twelve matching cards showing factored forms; mix them, match the pairs, and explain two of your choices to a partner.
# [[English:GCF Interview|GCF Interview]]: Interview a classmate about how they find the greatest common factor of two numbers, record their method in clear English, and compare it with your own method.
# [[English:Algebra Poster|Algebra Poster]]: Design a one-page poster that explains factoring as reverse distribution and includes at least three correct examples and one self-check.
# [[English:Distributive Property Video|Distributive Property Video]]: Record a short video in which you expand one expression and then reverse the process to factor it again.

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=== Standard ===
# [[English:Algebra Tile Model|Algebra Tile Model]]: Build paper or physical algebra tiles, model a simple expression with a common factor, photograph or draw the model, and explain how the groups show the factorization.
# [[English:Rectangle Area Investigation|Rectangle Area Investigation]]: Draw two adjacent rectangles with one shared side length, write the total area as a sum, factor the expression, and explain what the factored form says about the whole rectangle.
# [[English:Error Analysis|Error Analysis]]: Invent three realistic factoring mistakes, solve each one correctly, and write feedback that would help another learner understand the error.
# [[English:Factoring Mini-Lesson|Factoring Mini-Lesson]]: Prepare and teach a five-minute mini-lesson on common-factor factoring to a small group, then collect one question from your audience and answer it.

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=== Advanced ===
# [[English:Factoring Strategy Guide|Factoring Strategy Guide]]: Create a decision guide that helps a learner decide whether to factor out a number, a variable, or both, and test the guide on at least eight expressions.
# [[English:School Space Modeling|School Space Modeling]]: Visit a classroom, corridor, sports area, or other suitable school space, create a simplified rectangle-based area model, and show how a factored expression can represent combined dimensions or repeated widths.
# [[English:Expression Experiment|Expression Experiment]]: Use a spreadsheet or a short computer program to generate expressions with known common factors, factor them by hand, and verify the results by expanding.
# [[English:Peer Teaching Project|Peer Teaching Project]]: Design a short practice session for younger learners that combines explanation, visual media, guided practice, and an exit question about factoring simple expressions.

{{:Open Task - Create a MOOC}}

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= Learning Assessment =

# [[English:Explain and verify|Explain and verify]]: Factor <math>14x+21</math>, explain why your common factor is greatest, and verify your answer by expanding.
# [[English:Compare factorizations|Compare factorizations]]: Decide whether <math>2(6x+9)</math> and <math>6(2x+3)</math> represent the same expanded expression, then explain which form is more completely factored and why.
# [[English:Diagnose an error|Diagnose an error]]: A learner writes <math>10x+15=5(2x+15)</math>; identify the exact error, correct it, and show a distribution check.
# [[English:Create an example|Create an example]]: Write an expression whose complete common factor is <math>4x</math>, factor it, and justify why no larger common factor is possible.
# [[English:Transfer to geometry|Transfer to geometry]]: Create a two-part rectangle area problem whose total area can be represented by a simple expression, then factor the expression and interpret both factors as dimensions or shared measures.
# [[English:Reason about signs|Reason about signs]]: Compare factoring <math>-8y+12</math> by <math>4</math> and by <math>-4</math>; explain why both results are equivalent and when one form may be easier to use.

<br>
<br>

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= Evidence of Learning =

# '''Knowledge''': You can define factor, term, coefficient, greatest common factor, distributive property, factored form, and expanded form.
# '''Skills''': You can identify common numerical and variable factors, factor simple sums and differences, and verify results by expansion.
# '''Reasoning''': You can explain why factoring is reverse distribution and justify why a chosen factor is the greatest common factor.
# '''Products''': You can produce correct worked examples, diagrams, card sorts, short explanations, models, or videos that communicate the factoring process.
# '''Transfer''': You can use factoring in new contexts such as rectangle-area models, error analysis, peer teaching, and generated expressions.
# '''Communication''': You can describe your method in clear mathematical English and respond to questions about each step.

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= OERs on the Topic =

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

<br>

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= Linked Learning Areas =

{| align=center
{{:D-Tab}}
'''[[English:Factoring Simple Expressions|Factoring Simple Expressions]]'''
# [[English:Factorization|Factorization]]
# [[English:Greatest common divisor|Greatest common factor]]
# [[English:Distributive property|Distributive property]]
# [[English:Algebraic expression|Algebraic expression]]
# [[English:Equivalent expressions|Equivalent expressions]]
# [[English:Monomial|Monomial]]
# [[English:Area|Area]]
|}

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= aiMOOC Projects =
[[Category:English]]
[[Category:Factoring Simple Expressions]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Grades 7-8]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Factoring Simple Expressions - Versionsgeschichte

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