What is an Algebraic Expression?

An algebraic expression is a mathematical phrase made up of:

• Numbers
• Variables (letters)
• Mathematical operations

Unlike an equation, an expression does not contain an equals sign (=).

Examples include:

$x+5$x+5$3y$3y$4a-7$4a−7$2m+3n$2m+3n

Algebraic expressions describe quantities and relationships.

---

Parts of an Algebraic Expression

Every algebraic expression is made up of different parts.

Consider the expression:

$5x+3$5x+3

It contains:

• Term:$5x$5x
• Term:$3$3
• Coefficient: 5
• Variable:$x$x
• Constant: 3

Understanding these parts makes expressions easier to work with.

---

Terms

A term is a single part of an expression separated by addition (+) or subtraction (−) signs.

Example:

$4x+7-2y$4x+7−2y

The three terms are:

• $4x$4x
• $7$7
• $-2y$−2y

Each term is treated as one unit.

---

Variables

A variable is a letter that represents a number.

Examples include:

• $x$x
• $y$y
• $a$a
• $n$n

Variables may represent:

• Unknown numbers.
• Quantities that can change.

Example:

$6x$6x

The variable is:

$x$x

---

Coefficients

A coefficient is the number multiplied by a variable.

Example:

$8y$8y

The coefficient is:

$8$8

Another example:

$-3a$−3a

The coefficient is:

$-3$−3

If no number is written, the coefficient is 1.

Example:

$x=1x$x=1x

---

Constants

A constant is a number without a variable.

Examples:

• 7
• −5
• 12

Constants have fixed values.

Example:

$4x+9$4x+9

The constant is:

$9$9

---

Identifying the Parts

Example:

$3a-5+2b$3a−5+2b

---

Writing Algebraic Expressions

Words can be translated into algebraic expressions.

Examples:

A number plus 6

$x+6$x+6

---

Five more than a number

$x+5$x+5

---

Seven times a number

$7n$7n

---

A number divided by 4

$\frac{x}{4}$4x​

---

Three less than a number

$x-3$x−3

Carefully reading mathematical language helps produce the correct expression.

---

Translating Mathematical Phrases

Certain words represent specific mathematical operations.

Learning these words makes it easier to write algebraic expressions.

---

Be Careful with "Less Than"

The phrase less than reverses the order.

Example:

5 less than a number

means:

$x-5$x−5

However:

5 is less than a number

means:

$x>5$x>5

And:

5 less than 20

means:

$20-5$20−5

Always read the phrase carefully to determine the correct order.

---

Expressions vs Equations

Students often confuse expressions and equations.

Expression

Contains:

• Numbers.
• Variables.
• Operations.

No equals sign.

Example:

$4x+2$4x+2

---

Equation

Contains:

• Two expressions.
• An equals sign.

Example:

$4x+2=18$4x+2=18

Expressions describe quantities.

Equations state that two quantities are equal.

---

Interpreting Expressions

Expressions often describe real situations.

Example:

$5n$5n

If:

$n$

n represents the number of notebooks,

then:

$5n$5n

means:

"The total cost if each notebook costs $5."

---

Another example:

$2a+6$2a+6

If:

$a$

a represents age,

then:

$2a+6$2a+6

means:

"Twice the person's age plus six."

The meaning depends on what the variable represents.

---

Real-World Models

Expressions help model many situations.

Shopping

Each apple costs $2.

Buying

$a$

a apples costs:

$2a$2a

---

Taxi Fare

Starting fee:

$4

Plus:

$3 per kilometre.

Expression:

$4+3k$4+3k

---

Perimeter

Rectangle:

Length =

$l$

l

Width =

$w$

w

Perimeter:

$2l+2w$2l+2w

Expressions make mathematical models concise and flexible.

---

Why Algebraic Expressions are Important

Expressions allow mathematicians and scientists to:

• Describe patterns.
• Write formulas.
• Solve problems.
• Model real situations.
• Communicate mathematical ideas efficiently.

They are one of the building blocks of algebra.

---

Worked Example

Write an algebraic expression for:

"Eight more than three times a number."

Solution

Step 1:

Let the number be:

$x$x

Step 2:

Three times the number:

$3x$3x

Step 3:

Add eight:

$3x+8$3x+8

The expression is:

$\boxed{3x+8}$3x+8​

---

Key Terms

Algebraic Expression — A mathematical phrase containing numbers, variables, and operations, but no equals sign.

Term — A single part of an expression separated by addition or subtraction signs.

Variable — A letter representing a number.

Coefficient — The numerical factor multiplied by a variable.

Constant — A number without a variable.

Expression — A mathematical phrase without an equals sign.

Equation — A mathematical statement showing that two expressions are equal.

---

Key Takeaways

• An algebraic expression contains numbers, variables, and operations but does not contain an equals sign.
• Expressions are made up of terms, coefficients, variables, and sometimes constants.
• Mathematical phrases can be translated into algebraic notation using the correct operations.
• Expressions describe quantities, while equations state that two quantities are equal.
• The meaning of an expression depends on what its variables represent.
• Algebraic expressions are used to model patterns, formulas, and real-world situations.

---

Suggested Images

Suggested placement:

1. After "Parts of an Algebraic Expression" – Labelled diagram of an expression such as 5x + 3, identifying the terms, coefficient, variable, and constant.
2. After "Writing Algebraic Expressions" – Infographic translating common verbal phrases (e.g., "five more than a number") into algebraic notation.
3. After "Expressions vs Equations" – Side-by-side comparison showing an expression and an equation, highlighting the presence or absence of an equals sign.
4. After "Real-World Models" – Everyday examples such as shopping costs, taxi fares, and rectangle perimeter represented with algebraic expressions.
5. After "Translating Mathematical Phrases" – Reference chart linking words like sum, difference, product, and quotient to their corresponding mathematical operations.
6. Near "Why Algebraic Expressions are Important" – Illustration of a student using algebraic expressions to solve a real-world problem or identify a number pattern.

Why Do We Need an Order of Operations?

Many mathematical expressions contain more than one operation.

For example:

$6+4\times5$6+4×5

Should we:

• Add first?
• Multiply first?

If everyone chose a different order, they would get different answers.

To avoid confusion, mathematicians around the world use a standard order of operations.

This ensures that everyone obtains the same answer.

---

PEMDAS / BEDMAS

The order of operations is commonly remembered using:

PEMDAS

• P – Parentheses
• E – Exponents
• M – Multiplication
• D – Division
• A – Addition
• S – Subtraction

or

BEDMAS

• B – Brackets
• E – Exponents
• D – Division
• M – Multiplication
• A – Addition
• S – Subtraction

Both acronyms describe the same mathematical rules.

Important: Multiplication and division have equal priority. Work from left to right.

Addition and subtraction also have equal priority. Work from left to right.

---

Step 1: Parentheses (Brackets)

Always evaluate expressions inside parentheses first.

Example

$8+(6-2)$8+(6−2)

Step 1:

$6-2=4$6−2=4

Step 2:

$8+4=12$8+4=12

Answer:

$\boxed{12}$12​

---

Step 2: Exponents

Evaluate powers after parentheses.

Example

$4+3^2$4+32

Calculate the exponent first.

$3^2=9$32=9

Then:

$4+9=13$4+9=13

Answer:

$\boxed{13}$13​

---

Step 3: Multiplication and Division

Perform multiplication and division from left to right.

Example

$24\div6\times2$24÷6×2

First:

$24\div6=4$24÷6=4

Then:

$4\times2=8$4×2=8

Answer:

$\boxed{8}$8​

Notice that multiplication is not always done before division.

---

Step 4: Addition and Subtraction

Finally, perform addition and subtraction from left to right.

Example

$18-5+2$18−5+2

First:

$18-5=13$18−5=13

Then:

$13+2=15$13+2=15

Answer:

$\boxed{15}$15​

Addition is not always done before subtraction.

---

Worked Examples

Example 1

Evaluate:

$7+3\times4$7+3×4

Multiply first:

$3\times4=12$3×4=12

Then:

$7+12=19$7+12=19

Answer:

$\boxed{19}$19​

---

Example 2

Evaluate:

$(9-4)\times6$(9−4)×6

Parentheses first:

$9-4=5$9−4=5

Multiply:

$5\times6=30$5×6=30

Answer:

$\boxed{30}$30​

---

Example 3

Evaluate:

$18\div3+7$18÷3+7

Division first:

$18\div3=6$18÷3=6

Then:

$6+7=13$6+7=13

Answer:

$\boxed{13}$13​

---

Example 4

Evaluate:

$2^3+5$23+5

Exponent first:

$2^3=8$23=8

Then:

$8+5=13$8+5=13

Answer:

$\boxed{13}$13​

---

Evaluating Algebraic Expressions

When an expression contains variables, first substitute the given values.

Example

Evaluate:

$3x+4$3x+4

when:

$x=5$x=5

Step 1:

Substitute:

$3(5)+4$3(5)+4

Step 2:

Multiply:

$15+4$15+4

Step 3:

Add:

$19$19

Answer:

$\boxed{19}$19​

---

Another Example

Evaluate:

$2a^2-3$2a2−3

when:

$a=4$a=4

Step 1:

Substitute:

$2(4)^2-3$2(4)2−3

Step 2:

Exponent:

$4^2=16$42=16

Step 3:

Multiply:

$2\times16=32$2×16=32

Step 4:

Subtract:

$32-3=29$32−3=29

Answer:

$\boxed{29}$29​

---

Multi-Step Calculations

Many expressions require several steps.

Example

$4+2(7-3)^2$4+2(7−3)2

Step 1:

Parentheses:

$7-3=4$7−3=4

Step 2:

Exponent:

$4^2=16$42=16

Step 3:

Multiplication:

$2\times16=32$2×16=32

Step 4:

Addition:

$4+32=36$4+32=36

Answer:

$\boxed{36}$36​

Working carefully through each step reduces mistakes.

---

Common Mistakes

Mistake 1

Ignoring multiplication.

Incorrect:

$5+4\times2=(5+4)\times2=18$5+4×2=(5+4)×2=18

Correct:

$5+8=13$5+8=13

---

Mistake 2

Ignoring parentheses.

Incorrect:

$10-3\times2=14$10−3×2=14

Correct:

$10-6=4$10−6=4

---

Mistake 3

Forgetting exponents.

Incorrect:

$3+2^3=(3+2)^3=125$3+23=(3+2)3=125

Correct:

$3+8=11$3+8=11

---

Mistake 4

Doing multiplication before division (or addition before subtraction) regardless of order.

Incorrect:

$24\div6\times2=2$24÷6×2=2

Correct:

$24\div6=4,\qquad4\times2=8$24÷6=4,4×2=8

Always work from left to right when operations have equal priority.

---

Why the Order of Operations Matters

The order of operations allows mathematicians, scientists, engineers, and computer programmers to communicate calculations clearly.

It ensures that:

• Everyone follows the same rules.
• Expressions have only one correct value.
• Mathematical formulas work consistently.

Without these rules, mathematics would be unreliable.

---

Real-World Applications

The order of operations is used in many fields.

Science

Calculating:

• Speed.
• Force.
• Energy.
• Chemical formulas.

---

Engineering

Design calculations require accurate multi-step mathematics.

---

Finance

Calculating:

• Interest.
• Taxes.
• Discounts.
• Investment returns.

---

Computing

Computer programs follow the order of operations exactly when evaluating expressions.

---

Worked Example

Evaluate:

$5+2(9-5)^2\div4$5+2(9−5)2÷4

Solution

Step 1:

Parentheses:

$9-5=4$9−5=4

Step 2:

Exponent:

$4^2=16$42=16

Step 3:

Multiply and divide from left to right:

$2\times16=32$2×16=32$32\div4=8$32÷4=8

Step 4:

Addition:

$5+8=13$5+8=13

Answer:

$\boxed{13}$13​

---

Key Terms

Order of Operations — The agreed sequence for evaluating mathematical expressions.

Parentheses (Brackets) — Symbols used to group parts of an expression that should be evaluated first.

Exponent — A number indicating repeated multiplication.

Substitution — Replacing a variable with a known value.

Expression — A mathematical phrase containing numbers, variables, and operations.

Evaluate — To calculate the value of an expression.

---

Key Takeaways

• The order of operations ensures that mathematical expressions are evaluated consistently.
• Follow PEMDAS or BEDMAS:
  1. Parentheses (Brackets)
  2. Exponents
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)
• Multiplication and division have equal priority, as do addition and subtraction.
• When evaluating algebraic expressions, substitute the given values before applying the order of operations.
• Solving multi-step calculations carefully helps avoid common mistakes and ensures accurate results.

---

Suggested Images

Suggested placement:

1. After "PEMDAS / BEDMAS" – Colourful infographic illustrating the sequence of operations, emphasizing that multiplication/division and addition/subtraction are performed left to right.
2. After "Worked Examples" – Step-by-step visual solution showing each stage of evaluating a multi-operation numerical expression.
3. After "Evaluating Algebraic Expressions" – Diagram demonstrating how to substitute a value for a variable before applying the order of operations.
4. After "Common Mistakes" – Educational poster highlighting frequent errors (such as ignoring brackets or performing multiplication before division regardless of order) alongside the correct solutions.
5. After "Real-World Applications" – Images of a calculator, engineering calculations, and scientific formulas illustrating where the order of operations is used in everyday life and STEM fields.

What is Algebra Used For?

Many people think algebra only belongs in mathematics classrooms.

In reality, algebra is used every day to solve problems, make predictions, and describe relationships between quantities.

Algebra allows us to:

• Represent unknown values.
• Describe patterns.
• Calculate costs.
• Plan journeys.
• Design buildings.
• Predict future outcomes.

Whenever one quantity depends on another, algebra can help describe the relationship.

---

Variables Represent Real Quantities

A variable is a letter or symbol that represents a number.

In real-life situations, variables often represent quantities such as:

• Time
• Distance
• Money
• Age
• Temperature
• Number of people
• Weight

For example:

$d=60t$d=60t

where:

• $d$d = distance (km)
• $t$t = time (hours)

The variable helps describe how the quantities are related.

---

Modelling Real-World Relationships

A mathematical model is an equation or expression that represents a real situation.

Models allow us to calculate values quickly without starting from scratch each time.

Example

Each notebook costs $4.

If you buy

$n$

n notebooks:

$\text{Cost}=4n$Cost=4n

This model works for:

• 1 notebook
• 5 notebooks
• 100 notebooks

Simply substitute the value of

$n$

n.

---

Shopping

Algebra is useful whenever we buy goods or services.

Example

Each apple costs $2.

Buying

$a$

a apples costs:

$2a$2a

If:

$a=6$a=6

then:

$2(6)=12$2(6)=12

The apples cost $12.

---

Taxi Fares

Taxi companies often charge:

• A fixed starting fee.
• An additional cost for each kilometre travelled.

Suppose:

• Starting fee = $5
• Cost per kilometre = $2

If the trip is

$k$

k kilometres:

$C=5+2k$C=5+2k

If:

$k=8$k=8

then:

$C=5+2(8)=21$C=5+2(8)=21

The fare is $21.

---

Mobile Phone Plans

Many phone plans include:

• A monthly fee.
• An additional cost for extra usage.

Example:

Monthly fee:

$20

Extra data:

$3 per GB

If

$g$

g GB of extra data are used:

$C=20+3g$C=20+3g

This allows customers to predict future bills.

---

Saving Money

Suppose you save $15 each week.

If you save for

$w$

w weeks:

$S=15w$S=15w

After:

10 weeks:

$15(10)=150$15(10)=150

You have saved $150.

---

Travelling

Distance depends on speed and time.

The relationship is:

$d=st$d=st

where:

• $d$d = distance
• $s$s = speed
• $t$t = time

Example:

Travelling at:

80 km/h

for:

3 hours

gives:

$d=80\times3=240\text{ km}$d=80×3=240 km

---

Geometry

Algebra is used to calculate measurements.

Perimeter of a Rectangle

$P=2l+2w$P=2l+2w

where:

• $l$l = length
• $w$w = width

---

Area of a Rectangle

$A=lw$A=lw

One formula works for rectangles of every size.

---

Science

Scientists use algebra constantly.

Examples include:

Physics

Speed:

$v=\frac{d}{t}$v=td​

---

Chemistry

Density:

$\rho=\frac{m}{V}$ρ=Vm​

---

Biology

Scientists model:

• Population growth.
• Disease spread.
• Enzyme activity.

Algebra allows scientists to predict how systems behave.

---

Predicting Future Values

One of the greatest strengths of algebra is prediction.

Example:

A tree grows 25 cm each year.

If:

• Current height = 120 cm

After

$y$

y years:

$H=120+25y$H=120+25y

After:

4 years:

$120+25(4)=220\text{ cm}$120+25(4)=220 cm

Algebra predicts future growth without measuring the tree every year.

---

Making Decisions

Algebra helps compare different choices.

Example

Two internet plans:

Plan A

$25 per month.

Plan B

$10 per month plus $2 per day used.

If you use the service for

$d$

d days:

Plan A:

$25$25

Plan B:

$10+2d$10+2d

Using algebra allows you to determine which plan is cheaper for different values of

$d$

d.

Businesses and consumers use algebra to make informed financial decisions.

---

Solving Everyday Problems

Example

Emma buys:

• 3 sandwiches.
• Each costs $6.

Expression:

$3(6)$3(6)

Total:

$18$18

---

Example

A movie ticket costs $12.

A group buys

$n$

n tickets.

Expression:

$12n$12n

If:

$n=7$n=7

then:

$12(7)=84$12(7)=84

The total cost is $84.

---

Why Algebra is Important

Algebra helps us:

• Describe relationships.
• Solve unknowns.
• Predict future values.
• Compare choices.
• Solve practical problems.

It is one of the most powerful tools used in mathematics, science, engineering, economics, and technology.

---

Real-World Applications

Algebra is used in:

Engineering

Designing bridges, buildings, and machines.

---

Medicine

Calculating medicine doses and interpreting medical data.

---

Business

Managing costs, profits, and budgets.

---

Computer Science

Programming algorithms and computer graphics.

---

Environmental Science

Predicting population growth and climate change.

---

Worked Example

A gym charges:

• $25 membership fee.
• $8 per visit.

Question

Write an expression for the total cost after

$v$

v visits.

Calculate the cost for 6 visits.

Solution

Expression:

$C=25+8v$C=25+8v

Substitute:

$v=6$v=6$C=25+8(6)$C=25+8(6)$C=25+48=73$C=25+48=73

The total cost is:

$\boxed{\$73}$$73​

---

Key Terms

Variable — A letter representing a number or quantity.

Model — A mathematical representation of a real-world situation.

Expression — A mathematical phrase containing numbers, variables, and operations.

Equation — A mathematical statement showing two expressions are equal.

Substitution — Replacing a variable with a known value.

Prediction — Using mathematics to estimate future values or outcomes.

---

Key Takeaways

• Algebra is used to represent and solve real-world problems.
• Variables can represent quantities such as money, time, distance, age, or temperature.
• Algebraic expressions and equations provide mathematical models of everyday situations.
• Algebra helps calculate costs, compare options, solve practical problems, and make predictions.
• Mathematical models are widely used in science, engineering, business, technology, and many other fields.
• Learning algebra develops problem-solving skills that are useful throughout life.

---

Suggested Images

Suggested placement:

1. After "What is Algebra Used For?" – Infographic showing common everyday applications of algebra, including shopping, travel, science, engineering, and finance.
2. After "Shopping" and "Taxi Fares" – Illustrations of shopping receipts and taxi fare calculations represented by algebraic expressions.
3. After "Travelling" – Diagram showing the relationship between distance, speed, and time, including the formula$d = st$d=st.
4. After "Geometry" – Diagram of a rectangle labelled with length, width, perimeter, and area formulas.
5. After "Making Decisions" – Comparison chart of two pricing plans demonstrating how algebra helps compare costs.
6. Near "Real-World Applications" – Images of engineers, scientists, and business professionals using mathematical models to solve practical problems.