Part A: Numeracy
4

Algebra & Patterns

Master linear equations, sequences, algebraic expressions, and function rules

Learning Objectives

  • ✓ Identify arithmetic and geometric number patterns in real-world teaching data
  • ✓ Simplify algebraic expressions by combining like terms correctly
  • ✓ Apply the distributive law to expand brackets
  • ✓ Solve linear equations using inverse operations step-by-step
  • ✓ Formulate algebraic equations from written school scenarios
Chapter Progress: 0%

1. Number Patterns & Sequences

Arithmetic vs Geometric Sequences

A number sequence is an ordered list of numbers following a mathematical rule. An arithmetic sequence adds or subtracts a constant common difference (d) between terms. A geometric sequence multiplies or divides by a constant common ratio (r).

Example 1

ACER Authentic Problem: Spreadsheet Formula for Average (July 2017 Q20)

❓ Question: Which formula could Ms Bell have used in cell E2 to calculate Trixie’s average quiz score across Quiz 1 (B2), Quiz 2 (C2), and Quiz 3 (D2)?
📄 Source Passage / Reference Data:

SPREADSHEET FORMULA

A (Student)B (Quiz 1)C (Quiz 2)D (Quiz 3)E (Average)
2Trixie13141514

About spreadsheets: * stands for multiplication; / stands for division.

📝 Step-by-Step Method:
1
Recall formula for arithmetic mean: Mean = (Sum of scores) ÷ Number of scores.
2
Substitute cell references for Trixie: B2, C2, D2.
3
Apply parentheses to enforce order of operations before division.
🔍 Evidence Extracted: "Sum enclosed in brackets before dividing: =(B2+C2+D2)/3"
🎯 Final Answer: =(B2+C2+D2)/3
💡 Why It Is Correct: Without brackets, =B2+C2+D2/3 would only divide D2 by 3 due to standard operator precedence. Brackets ensure the three scores are summed before dividing by 3.
⚡ LANTITE Exam Strategy: LANTITE Spreadsheet Strategy: In spreadsheets, always enclose the numerator sum in brackets: =(Cell1 + Cell2 + Cell3) / n.

2. Simplifying Algebraic Expressions

Distributive Law

a(b + c) = ab + ac
a = Multiplier outside brackets
b, c = Terms inside brackets

Example

3(2x + 4) = 3(2x) + 3(4) = 6x + 12
Example 2

ACER Authentic Problem: Setting Up a Savings Equation (July 2017 Q13)

❓ Question: Kerry earns $225 a week in a part-time job and saves half of the money she earns each week. Which equation calculates how much money Kerry saves from her earnings?
📄 Source Passage / Reference Data:

SAVING MONEY

Kerry earns $225 a week. Kerry saves half (0.5) of the money she earns each week.

📝 Step-by-Step Method:
1
Express half savings as a decimal multiplier.
2
Calculate weekly savings amount.
3
Extend across an arbitrary number of weeks (n).
🔍 Evidence Extracted: "Savings rate = $225 × 0.5 per week"
🎯 Final Answer: savings = 225 × 0.5 × number of weeks
💡 Why It Is Correct: Saving half of $225 weekly means $225 × 0.5 is saved each week. Multiplying by the number of weeks gives total accumulated savings.
⚡ LANTITE Exam Strategy: Algebraic Modelling: Identify the fixed weekly rate (225 × 0.5) and multiply by the independent variable (number of weeks).

3. Solving Linear Equations

Example 3

ACER Authentic Problem: Simple Interest Substitution (July 2017 Q11)

❓ Question: What is the simple interest on a principal of $1000 at 5% annual interest rate over 3 years using the formula I = PRT / 100?
📄 Source Passage / Reference Data:

SIMPLE INTEREST

A rule for calculating simple interest on a loan is: I = PRT / 100, where I = interest ($), P = principal ($), R = annual rate (%), T = time (years).

📝 Step-by-Step Method:
1
Identify the parameters: P = 1000, R = 5, T = 3.
2
Substitute into formula.
3
Compute numerator and divide.
🔍 Evidence Extracted: "I = (1000 × 5 × 3) / 100 = $150"
🎯 Final Answer: $150
💡 Why It Is Correct: Substituting P = 1000, R = 5, and T = 3 into I = PRT / 100 gives $150 total interest.
⚡ LANTITE Exam Strategy: Formula Substitution: When R is already divided by 100 in the formula (PRT/100), enter R as the whole percentage number (5, not 0.05).

4. Setting Up Equations from Word Problems

Translating Words to Algebraic Symbols

Key translation rules: "sum" or "more than" = addition (+); "difference" or "less than" = subtraction (-); "times" or "product" = multiplication (×); "per" or "divided equally" = division (÷); "is" or "equals" = =.

Example 4

Setting Up Equations for Resource Quantities

❓ Question: A school bus fuel tank contains an unknown volume of diesel (x litres). After adding 15 litres, the tank holds 40 litres. Find the initial fuel volume.
📝 Step-by-Step Method:
1
Define variable and write equation based on scenario.
2
Solve for x by subtracting 15 from both sides.
🎯 Final Answer: 25 litres
💡 Why It Is Correct: Subtracting the 15 added litres from the total 40 litres reveals an initial volume of 25 litres.

Chapter 4 Practice Quiz