Practice Maths

Creating Algebraic Expressions

Full Lesson

Key terms: term, sequence, relationship

• A term is one number in a sequence (e.g., the 3rd term of 3, 6, 9, 12 is 9)
• A sequence is an ordered list of numbers that follow a rule (e.g., 3, 6, 9, 12 ...)
• A relationship describes how two quantities are connected (e.g., y = 3x)

In this lesson, you will continue your work on algebra to create expressions that describe patterns of numbers.

Describing number patterns as algebraic equations

Consider a pattern made with triangles where each separate triangle is made of 3 matchsticks.

Term (x) 1 2 3 x
No. of sticks (y) 3 6 9 ?

It's easy to see that you 'add 3' to each number to continue the pattern. But how could you easily find the 30th shape? You could just keep 'adding 3' 30 times, but that would be slow.

This is where algebra helps.

You can see that the number of matchsticks is three times the term number:

  • Term 1: 3 × 1 = 3
  • Term 2: 3 × 2 = 6
  • Term 3: 3 × 3 = 9

The relationship in words is: number of matchsticks = 3 × term

The algebraic expression is: y = 3x

Hot Tip To be sure of which sequence is used, you usually need at least three numbers to start.

Worked Example

Question: Sarah has $50 in her bank and saves $5 every week. Write an algebraic rule for her total savings (S) after w weeks, then find her savings after 8 weeks.

Step 1 — Find the starting amount and the rate of change.
Starting amount: $50 (Week 0)
Added each week: $5

Step 2 — Build the rule.
S = 50 + 5 × w    or    S = 5w + 50

Step 3 — Substitute w = 8.
S = 5 × 8 + 50 = 40 + 50 = $90

From Words to Symbols

In the previous lesson, we used letters to represent unknown quantities. Now we're going a step further — using a table of numbers to discover a rule, and writing that rule in algebra.

Here's the big idea: if a pattern adds the same amount each time (like 3, 6, 9, 12…), that amount is your multiplier. The letter x represents which term you're at, and y is the value at that term.

Matchsticks: Building the Insight

The matchstick triangles from the Key Ideas tab are a classic. Let's think about WHY y = 3x works:

  • Each new triangle you add requires exactly 3 new matchsticks — one for each side.
  • Term 5 would need 5 × 3 = 15 matchsticks. Term 30? Just 30 × 3 = 90. No counting required!

This is the power of algebra — you write a rule once, and it answers every question about that pattern instantly.

Two-Step Patterns: Starting Amount + Growth

Real life is often more complex. Sarah starts with $50 and adds $5 each week. Let's build this carefully:

  • Week 0 (start): $50. Week 1: $55. Week 2: $60. Week 3: $65.
  • The growth rate is $5 per week, so the first part of the rule is 5w.
  • But at week 0, she already has $50. So we add that: S = 5w + 50.
  • Check week 3: S = 5(3) + 50 = 15 + 50 = $65. Matches our table!
Remember: Always check your formula with at least two values from the table. If both work, you can be confident. If one doesn't, look again at either the multiplier or the starting number.

Term-to-Term vs Position-to-Term

You could describe the savings sequence as "add $5 each week" — that's a term-to-term rule. But what if you want to know Sarah's savings after 52 weeks? You'd have to add $5 fifty-two times! The position-to-term rule S = 5w + 50 lets you substitute w = 52 and get the answer immediately: S = 5(52) + 50 = $310.

That's why position-to-term rules (algebraic formulas) are so important — they are shortcuts to any answer, any time.

Where Is This Used in the Real World?

Phone plans work exactly like this: a fixed monthly fee (the constant) plus a rate per GB of data used (the variable part). Every time a phone company sets a price, an algebraic formula is running behind the scenes. Spreadsheets, coding, physics, economics — all of these rely on the same idea you're learning right now.

Did You Know? Scientists use formulas to predict how fast bacteria multiply, how far a rocket travels, and how much medicine a patient needs based on their body weight. All of it starts with finding a rule from a table of data — exactly what you're practising here!

Practice Questions

  1. Answer the following questions using algebraic expressions: Fluency

    1. The length of a rectangular swimming pool is 15 meters longer than its width. Write an expression for the length of the pool (let width = w).
    2. You are running a 10 km race. Write an expression for how many kilometres are left to run (let k be the distance you have already run).

  2. Finishing number patterns. Fill in the missing numbers in these sequences: Fluency

    1. 20, 21, 22, 23, ___, ___, ___
    2. 100, 85, 70, ___, ___, ___
    3. 10, 6, 2, ___, ___, ___
    4. 5.0, 4.5, 4.0, ___, ___, ___

  3. There can often be more than one sequence beginning with the same numbers. Understanding

    Consider the sequence: 3, 6, ...

    1. What could the next numbers be if the rule is "Add 3"?
    2. What could the next numbers be if the rule is "Multiply by 2"?

  4. Refer to the "Matchsticks" example in the Full Lesson (y = 3x). Understanding

    1. Use the relationship y = 3x to work out how many sticks you'd need to make the 30th term.
    2. What term would use 150 sticks?

  5. You are creating separate Pentagon shapes using toothpicks. Each pentagon uses 5 toothpicks. Understanding

    1. Complete the table below based on your pattern:
      Term (Number of Pentagons) 1 2 3
      No. of toothpicks
    2. How many toothpicks would be needed for the 4th term?
    3. See if you can find a relationship between the term (x) and the number of toothpicks (y). Write the relationship in words.
    4. Write the algebraic expression (e.g., y = ...).
    5. How many toothpicks would you need to make the 100th term?

  6. Sarah is collecting trading cards. She started with a few, and buys one new card every day. Here is her pattern: Understanding

    Day number (d) 1 2 3
    Total cards (c) 12 13 14
    1. Write the relationship in words.
    2. Write the relationship as an algebraic expression (c = ...).
    3. How many cards would she have on the 50th day?
    4. On what day number would she have 92 cards?

  7. Movie Tickets (Multiplication Rules) Fluency

    A cinema sells tickets for $12 each.

    1. Create a table showing the cost for 1, 2, 3, and 4 tickets.
    2. Let t be the number of tickets and C be the total cost. Write an algebraic expression for C in terms of t.
    3. Use your formula to calculate the cost of buying 8 tickets.

  8. Geometry Rules (Formulas) Understanding

    We can use algebra to create formulas for shapes.

    1. An equilateral triangle has side length s. Write an expression for the Perimeter (P).
    2. A regular hexagon has side length h. Write an expression for the Perimeter (P).
    3. If the perimeter of the regular hexagon is 60 cm, what is the side length h?

  9. The Number Machine (Subtraction Rules) Understanding

    Look at the Input and Output of this number machine.

    Input (x) 10 20 50 100
    Output (y) 5 15 45 95
    1. Describe the rule in words (e.g., "The machine subtracts...").
    2. Write the rule as an algebraic equation (y = ...).
    3. If the Input is 150, what is the Output?

  10. The Savings Pattern (Two-Step Scenarios) Problem Solving

    Liam has $50 in his bank account. He gets a job and decides to save $5 every week.

    Week (w) 0 (Start) 1 2 3
    Total Savings (S) $50 $55 $60 ?
    1. How much money will he have after 3 weeks?
    2. Write an algebraic rule for his Savings (S) based on the number of weeks (w).
      Hint: Think about the starting amount + the amount added each week.
    3. Use your rule to find out how much money he will have after 10 weeks.
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