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Year 7 Maths

Patterns & Algebra

Generalise number patterns using pronumerals, build algebraic rules, and practise substituting values into formulas.

Recognising and Extending Patterns

A number pattern (or sequence) is a list of numbers that follow a rule. To find the rule, look at what operation connects each term to the next.

Example: Pattern of squares made from matchsticks

1 square

4 sticks

2 squares

7 sticks

Squares (n)1234
Matchsticks471013

The pattern adds 3 each time. We can write this as: Matchsticks = 3n + 1

Using Pronumerals

A pronumeral is a letter used to represent an unknown or changing number in an algebraic rule. In Australia, we use the term pronumeral; in other countries, it is often called a variable.

Writing Algebraic Rules

Words "The total cost is $5 for each ticket plus a $2 booking fee."
Rule Let n = number of tickets. Total cost = 5n + 2

Common Translations

"5 more than n"n + 5
"3 times a number"3n
"half of m"m ÷ 2 or m/2
"4 less than p"p − 4

Substitution

Substitution means replacing a pronumeral with a given number and then calculating the result. This lets us evaluate formulas for specific inputs.

Step-by-step process

Formula: T = 4n + 2

Step 1: Write the formula: T = 4n + 2

Step 2: Replace n with the given value, e.g., n = 6

Step 3: T = 4(6) + 2 = 24 + 2 = 26

Note: Always write the letter being substituted and the value so your working is clear. e.g., "When n = 6..."

Tables of Values

A table of values shows input values and their corresponding outputs for a rule. It helps us see the pattern and can also be used to plot graphs.

Rule: y = 2x + 1
x (input) 0 1 2 3
y = 2x + 1 1 3 5 7

Notice the output increases by 2 each time x increases by 1 — this matches the coefficient of x in the rule.

Key Vocabulary

Pronumeral

A letter that stands for an unknown or changing number in an algebraic expression or formula.

Substitution

Replacing a pronumeral with a specific number to find the value of an expression.

Generalise

To write a rule that describes a pattern for all possible values, not just specific ones.

Formula

A rule written using pronumerals that expresses a mathematical relationship, e.g., A = l × w.

Worked Examples

1

Find the next two terms and write a rule for: 5, 8, 11, 14, ...

Pattern: Each term increases by 3.

Next two terms: 14 + 3 = 17, 17 + 3 = 20

Rule: Term = 3n + 2, where n is the position (n = 1, 2, 3, ...)

Check: When n = 1: 3(1) + 2 = 5 ✓

2

The area of a rectangle is A = l × w. Find A when l = 8 cm and w = 5 cm.

Substitute: A = 8 × 5

Answer: A = 40 cm²

3

Write an algebraic rule: a taxi charges a $3 flagfall plus $2.50 per kilometre.

Let d = distance in kilometres.

Cost = 2.5d + 3

For 10 km: Cost = 2.5(10) + 3 = 25 + 3 = $28

Knowledge Check

Select the correct answer for each question.

Question 1

What is the next term in the pattern: 3, 7, 11, 15, ...?

Question 2

Evaluate 4n − 1 when n = 5.

Question 3

Which algebraic rule represents "8 more than twice a number m"?

Question 4

A rule is T = 5n − 3. What is T when n = 4?

Question 5

A table of values for y = 3x − 1 shows x = 0, 1, 2, 3. What is y when x = 3?

Key Concepts Summary

Year 7: Circles Year 7: Financial Maths