Algebraic Expressions and Equations

MYP Unit Framework

Key Concept: RELATIONSHIPS Related Concepts: Equivalence, Simplification, Generalization Global Context: Scientific and Technical Innovation (How does algebra enable us to model and solve problems in science and technology?) Statement of Inquiry: Algebra generalises arithmetic relationships and enables powerful problem-solving strategies.

Inquiry Questions

ATL Skills

• Thinking: Recognise patterns and generalise relationships; translate between words and algebraic symbols
• Communication: Use algebraic notation correctly; explain reasoning step by step
• Research: Apply algebra to solve real-world problems
• Self-Management: Organise solutions systematically; check work for errors

1. Algebraic Expressions

What Is Algebra?

Algebra is the branch of mathematics that uses letters (variables) to represent unknown or changing numbers. It allows us to generalise arithmetic relationships.

Key Components

• Variable: A letter representing an unknown number (x, y, a, b)
• Constant: A fixed number (3, -7, 1/2)
• Term: A number, variable, or product of both (3x, -2y², 5)
• Coefficient: The number multiplying a variable (in 3x, the coefficient is 3)
• Expression: A combination of terms without an equals sign (3x + 5)

Writing Algebraic Expressions

Simplifying Expressions by Collecting Like Terms

Like terms have the same variable(s) raised to the same power.

Example: 3x + 2y + 5x - y = 3x + 5x + 2y - y = 8x + y

Expanding Brackets

Multiply each term inside the bracket by the term outside.

Example: 3(x + 4) = 3x + 12

Example with two brackets (FOIL method): (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

2. Equations

What Is an Equation?

An equation is a statement that two expressions are equal. It contains an equals sign.

Example: 2x + 3 = 11

Solving Linear Equations

The goal is to isolate the variable on one side of the equation. What you do to one side, you must do to the other.

One-Step Equations:

• x + 5 = 12 → x = 7
• 3x = 15 → x = 5
• x/4 = 6 → x = 24

Two-Step Equations:

• 2x + 3 = 11 → 2x = 8 → x = 4
• 5x - 7 = 13 → 5x = 20 → x = 4

Equations with Variables on Both Sides:

• 3x + 4 = 2x + 9 → 3x - 2x = 9 - 4 → x = 5

Equations with Brackets:

• 2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4

3. Factorisation

What Is Factorisation?

Factorisation is the reverse of expanding brackets — expressing an expression as a product of its factors.

Common Factor Factorisation: Identify the highest common factor (HCF) of all terms.

Example: 6x + 9 = 3(2x + 3)

Example: 12x² - 8x = 4x(3x - 2)

Factorising Quadratic Expressions

For quadratics of the form x² + bx + c, find two numbers that:

• Multiply to give c
• Add to give b

Example: x² + 7x + 12 = (x + 3)(x + 4)

• 3 x 4 = 12 (check!)
• 3 + 4 = 7 (check!)

Example: x² - 5x + 6 = (x - 2)(x - 3)

• (-2) x (-3) = 6
• (-2) + (-3) = -5

Difference of Two Squares

a² - b² = (a - b)(a + b)

Example: x² - 9 = (x - 3)(x + 3)

4. Forming and Solving Equations from Word Problems

Problem-Solving Strategy

1. Read the problem carefully — what are you being asked to find?
2. Define a variable — let x represent the unknown quantity
3. Form an equation — translate the words into algebraic symbols
4. Solve the equation using algebraic methods
5. Check — does the answer make sense in context?
6. State the answer in a sentence

Examples

Age Problem: Sarah is 5 years older than twice her brother's age. The sum of their ages is 23. Find each age.

Let brother's age = x. Then Sarah's age = 2x + 5. x + (2x + 5) = 23 → 3x + 5 = 23 → 3x = 18 → x = 6 Brother is 6. Sarah is 2(6) + 5 = 17. Check: 6 + 17 = 23.

Perimeter Problem: A rectangle has length 4 cm more than its width. The perimeter is 40 cm. Find the dimensions.

Let width = w. Then length = w + 4. 2(w + (w + 4)) = 40 → 2(2w + 4) = 40 → 4w + 8 = 40 → 4w = 32 → w = 8 Width = 8 cm. Length = 12 cm. Check: 2(8 + 12) = 40.

5. Patterns and Sequences

Linear Sequences

A sequence where each term increases or decreases by a constant amount (common difference).

Example: 3, 7, 11, 15, 19... (common difference = 4)

• Term 1: 3 = 4(1) - 1
• Term 2: 7 = 4(2) - 1
• Term n: 4n - 1

Finding the nth Term

1. Find the common difference (d)
2. Write: nth term = d x n + c
3. Substitute a known term to find c

Quadratic Sequences

A sequence where the second differences are constant.

Example: 1, 4, 9, 16, 25... (second difference = 2)

• nth term = n²

6. Inequalities

What Is an Inequality?

An inequality compares two quantities using <, >, less than or equal to, or greater than or equal to.

Solving Inequalities: Similar to equations, but:

• Multiplying or dividing by a negative number REVERSES the inequality sign

Example: -2x < 8 → x > -4

Representing Solutions

Inequality solutions can be shown on number lines.

Summative Assessment

Task: Mathematical investigation (800-1000 words equivalent) using algebra to solve a real-world problem.

Criteria:

• A: Knowing and Understanding — Select and apply algebraic procedures correctly
• B: Investigating Patterns — Identify patterns and generalise relationships
• C: Communicating — Use algebraic notation clearly; explain reasoning
• D: Applying Mathematics in Real-World Contexts — Apply algebra to authentic situations

Option 1: Investigate number patterns. Find a general rule. Prove it algebraically.

Option 2: Create and solve word problems based on real situations (age, money, geometry). Show all steps and check answers.

Option 3: Model a simple financial situation (saving money, phone plans, taxi fares) using algebraic expressions. Compare different options.

Formative Assessment

• Algebraic expression translation (words to symbols)
• Simplifying and expanding expressions (skills practice)
• Solving equations (varied difficulty levels)
• Factorisation drills
• Word problem creation and exchange with peers
• Pattern investigation worksheets
• Quick quizzes on key skills

Interdisciplinary Connections

• Science: Using formulas in science — speed = distance/time, density = mass/volume
• Design: Calculating material quantities for projects
• Economics: Simple financial models — profit, cost, revenue
• Computer Science: Variables and expressions in programming

Service as Action

• Tutoring: Help younger students with algebra through a peer tutoring programme. Create practice worksheets.
• Budgeting Project: Use algebra to create a budget for a school event. Determine costs, ticket prices, and profit.

IB Learner Profile

• Thinkers: Apply logical reasoning to solve problems
• Communicators: Express mathematical ideas using algebraic language
• Inquirers: Investigate patterns and relationships in the world
• Reflective: Check work for errors and reflect on different problem-solving strategies

Self-Test

1. Simplify: 4x + 3y - 2x + 5y
2. Expand: 3(2x - 5)
3. Expand and simplify: (x + 3)(x + 4)
4. Solve: 2x + 7 = 19
5. Solve: 5x - 3 = 2x + 12
6. Factorise: 8x + 12
7. Factorise: x² + 8x + 15
8. Factorise: x² - 16
9. Form and solve: The sum of a number and three times the number is 28. Find the number.
10. Find the nth term of the sequence: 5, 9, 13, 17, 21...

This unit aligns with IB MYP Mathematics guide, developed for Year 3 (Class 8) students.