Properties and Operations

Chapter 2: Solving Equations

Lesson 2.1: Properties and Operations

Use properties to simplify expressions, evaluate formulas, and convert measurements.

Learning Objectives

By the end of this lesson, you should be able to:

Use commutative, associative, and identity properties to simplify and evaluate expressions.
Use operations and conversion factors to solve measurement and real-life problems.

Christ Holds All Things Together

Mathematical properties help us recognise order, consistency, and relationships. The same God who gives order to mathematics created and sustains the whole universe. The Bible teaches that all things were created through Jesus Christ and that everything holds together in Him.

Jesus is more than a wise teacher or an example of good living. He is the Son of God, the Creator and Lord. Because sin separates us from God, we cannot save ourselves by our own effort or good works. God sent Jesus to live a sinless life, die on the cross for our sins, and rise again. Everyone who turns from sin and trusts in Jesus receives forgiveness and new life through Him. As you study order and relationships in mathematics, let them remind you that your life also finds its true purpose and peace in Christ.

Memory verse: “For in him all things were created: things in heaven and on earth, visible and invisible.” — Colossians 1:16

1. Commutative Properties

In English, commute means to change locations. A commutative property allows you to change the order of numbers without changing the result.

Property	Words	Algebra	Example
Commutative Property of Addition	Numbers may be added in any order.	$a+b=b+a$	$4+(-7)=(-7)+4$
Commutative Property of Multiplication	Numbers may be multiplied in any order.	$ab=ba$	$8(-5)=(-5)8$

Worked Example 1: Changing the Order

Evaluate $12+35+8$ using a helpful order.

Reorder the addends to combine the numbers that are easiest to add:

$12+35+8=12+8+35=20+35=55$

The order changed, but the sum did not. This is the commutative property of addition.

2. Associative Properties

In mathematics, associate means to group numbers together. An associative property allows you to change the grouping of numbers without changing the result.

Property	Algebra	Example
Associative Property of Addition	$(a+b)+c=a+(b+c)$	$(9+6)+2=9+(6+2)$
Associative Property of Multiplication	$(ab)c=a(bc)$	$(3\cdot10)4=3(10\cdot4)$

Worked Example 2: Regrouping for Mental Math

Evaluate $25+17+3$ by grouping numbers that make a multiple of 10.

$25+17+3=25+(17+3)=25+20=45$

The associative property changed the grouping from $(25+17)+3$ to $25+(17+3)$ .

Picture space for adding integers on a number line — Changing order or grouping is allowed for addition and multiplication.

3. Identity Properties

An identity is a number that leaves another number unchanged when an operation is performed.

Property	Words	Algebra	Example
Identity Property of Addition	The sum of a number and 0 is the number.	$a+0=a$	$-6+0=-6$
Identity Property of Multiplication	The product of a number and 1 is the number.	$a\cdot1=a$	$4\cdot1=4$

Worked Example 3: Simplifying with Identities

Identify the property shown by each statement.

$mn+0=mn$ shows the Identity Property of Addition.

$(-7u)(1)=-7u$ shows the Identity Property of Multiplication.

4. Properties and Operations

Commutative and associative properties work for addition and multiplication. They do not work for subtraction and division.

Subtraction is not commutative: $8-3=5$ , but $3-8=-5$ .

Division is not commutative: $10\div5=2$ , but $5\div10=0.5$ .

Subtraction is not associative: $(15-9)-4=2$ , but $15-(9-4)=10$ .

Division is not associative: $(48\div6)\div2=4$ , but $48\div(6\div2)=16$ .

Always check whether changing order or grouping is allowed before using a property.

5. Conversion Factors

A conversion factor is a ratio equal to 1. Multiplying by a conversion factor changes the unit while keeping the measurement equivalent.

For example, $1\text{ foot}=12\text{ inches}$ , so either ratio below equals 1:

$\frac{1\text{ foot}}{12\text{ inches}}=1 \qquad \text{and} \qquad \frac{12\text{ inches}}{1\text{ foot}}=1$

Worked Example 4: Converting Yards to Inches

Convert 5 yards to inches. Use $1\text{ yard}=3\text{ feet}$ and $1\text{ foot}=12\text{ inches}$ .

$5\text{ yd}\cdot\frac{3\text{ ft}}{1\text{ yd}}\cdot\frac{12\text{ in}}{1\text{ ft}}$

$=5\cdot3\cdot12\text{ in}=180\text{ in}$

The units cancel from left to right, leaving inches.

Common Mistakes to Avoid

Do not use a commutative or associative property with subtraction or division.
Do not confuse changing order with changing grouping.
When converting units, choose a factor that cancels the original unit.
Remember that an identity leaves the original number unchanged: add 0 or multiply by 1.
Show the property or conversion factor used so your reasoning can be checked.

Practice Questions

Show each step and name the property when requested.

1. Skill Practice

A. Evaluate each expression. Justify the steps using a property when appropriate.

$32+16+8$
$15(-9)(2)$
$7\cdot1+0$
$45+29+55$

B. Evaluate each expression when $a=9$ and $b=-4$ .

$5ab$
$b(25a^2)$
$11+4b+a$
$3a+b^2+13$

C. Simplify each expression.

$x+17+12$
$3+j+(-9)$
$-8(6c)$
$(5y)(26)$

D. Identify the property shown.

$mn+0=mn$
$19\cdot5^3=5^3\cdot19$
$(2x+3y)+z=2x+(3y+z)$
$(-7u)(1)=-7u$

E. Use a conversion factor.

Convert 4 miles to feet. Use $1\text{ mile}=5{,}280\text{ feet}$ .
Convert 7.5 kilograms to grams. Use $1\text{ kilogram}=1{,}000\text{ grams}$ .
Convert 360 seconds to minutes. Use $1\text{ minute}=60\text{ seconds}$ .
Convert 432 square inches to square feet. Use $1\text{ square foot}=144\text{ square inches}$ .

2. Applications

Surveying: A surveyor measures the depth of a river at three points: meters, meters, and meters.
1. Use properties of addition to find the total of the measurements.
2. What is the mean depth?
Fish length: A swordfish can grow to about 5 yards. An adult mackerel is about 18 inches long.
1. Convert 5 yards to inches using conversion factors.
2. How many times as long as a mackerel is the swordfish?
Logical reasoning: Calculate each pair and decide whether the operation is commutative.
1. $8-3$ and $3-8$
2. $10\div5$ and $5\div10$
3. $6+11$ and $11+6$
4. $4\cdot9$ and $9\cdot4$
Algebra: Evaluate each expression when , , and .
1. $x^2yz^2$
2. $15xyz$
3. $2x+9y+5z$
Critical thinking: When you divide any number $a$ by 1, what is the result? Write an algebraic statement that expresses this property.
Real life: A recipe uses 2 cups of flour for one batch. A baker makes 6 batches and then divides the flour equally among 3 large bowls. Write an expression and find how many cups of flour go into each bowl. Explain how the associative property helps you calculate.
Challenge: Find two different ways to evaluate $18+25+2+5$ using commutative and associative properties. Explain why both methods give the same result.

Answers

Show answers: Skill Practice

$56$ ; for example, $32+8+16=40+16=56$ .
$-270$ ; $15(-9)(2)=15(-18)=-270$ .
$7$ ; the identity properties leave 7 unchanged.
$129$ ; regroup as $(45+55)+29=100+29$ .
$-180$ .
$-8{,}100$ .
$4$ .
$56$ .
$x+29$ .
$j-6$ .
$-48c$ .
$130y$ .
Identity Property of Addition.
Commutative Property of Multiplication.
Associative Property of Addition.
Identity Property of Multiplication.
$21{,}120$ feet.
$7{,}500$ grams.
$6$ minutes.
$3$ square feet.

Show answers: Applications

a. $4.7+8.5+6.3=19.5$ meters. b. $19.5\div3=6.5$ meters.
a. $5\text{ yd}\cdot\frac{3\text{ ft}}{1\text{ yd}}\cdot\frac{12\text{ in}}{1\text{ ft}}=180$ inches. b. $180\div18=10$ ; the swordfish is 10 times as long.
a. $5$ and $-5$ : not commutative. b. $2$ and $0.5$ : not commutative. c. $17$ and $17$ : commutative. d. $36$ and $36$ : commutative.
a. $300$ . b. $-450$ . c. $27$ .
$a\div1=a$ ; this is the Identity Property of Division.
$(2\cdot6)\div3=12\div3=4$ cups per bowl. Another grouping is $2(6\div3)=2\cdot2=4$ .
Answers may vary. For example, $(18+2)+(25+5)=20+30=50$ , or $18+(25+5)+2=18+30+2=50$ .

Show the key ideas to remember

Commutative means change the order.
Associative means change the grouping.
These properties apply to addition and multiplication.
The additive identity is 0: $a+0=a$ .
The multiplicative identity is 1: $a\cdot1=a$ .
Subtraction and division are neither commutative nor associative.
Use conversion factors so unwanted units cancel.

Lesson Summary

Properties make calculations easier and help us explain why expressions are equivalent. Commutative properties change order, associative properties change grouping, and identity properties leave a number unchanged. These ideas support mental math, algebraic simplification, and unit conversion. As you continue into solving equations, remember that clear steps and faithful reasoning matter.

Jesus Christ is the One through whom all things were created. Seek Him for wisdom, forgiveness, and new life.

Lesson 2.1: Properties and Operations

Learning Objectives

Christ Holds All Things Together

1. Commutative Properties

Worked Example 1: Changing the Order

2. Associative Properties

Worked Example 2: Regrouping for Mental Math

Commutative and Associative Properties

3. Identity Properties

Worked Example 3: Simplifying with Identities

4. Properties and Operations

5. Conversion Factors

Worked Example 4: Converting Yards to Inches

Unit Conversion Chain

Common Mistakes to Avoid

Practice Questions

1. Skill Practice

2. Applications

Answers

Lesson Summary

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Pre-Algebra

Properties and Operations

Lesson 2.1: Properties and Operations

Learning Objectives

Christ Holds All Things Together

1. Commutative Properties

Worked Example 1: Changing the Order

2. Associative Properties

Worked Example 2: Regrouping for Mental Math

Commutative and Associative Properties

3. Identity Properties

Worked Example 3: Simplifying with Identities

4. Properties and Operations

5. Conversion Factors

Worked Example 4: Converting Yards to Inches

Unit Conversion Chain

Common Mistakes to Avoid

Practice Questions

1. Skill Practice

2. Applications

Answers

Lesson Summary

Related

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