Practice Maths

Describing Patterns Using Algebraic Rules

Full Lesson

Key terms: variable, algebraic expression

• A variable is a letter or symbol that represents a number e.g. x
• An algebraic expression is a mathematical phrase containing numbers, symbols ( + − × ÷ ) and variables e.g. 2x + 3

Creating Algebraic Expressions Using Numbers and Variables

Consider this statement: Sarah has 5 apples. Then, she buys a bag containing an unknown number of apples.

Let the number of apples in the bag be represented by the letter: a
Sarah had 5 apples, and now she has a more, so Sarah's total number of apples can be represented by the expression: a + 5

Hot Tip You can use any letter to represent an unknown quantity, but it is common to use a letter that relates to the object (like a for apples or c for cost).

Worked Example

Question: A parking station charges a $3 entry fee plus $2 for every hour parked. Let h be the number of hours. Write an algebraic expression for the total cost, then find the cost for 4 hours.

Step 1 — Identify the fixed amount and the variable amount.
Fixed: $3 entry fee (always paid regardless of time)
Variable: $2 per hour, so for h hours the cost is 2h

Step 2 — Write the expression.
Total cost = 2h + 3

Step 3 — Substitute to find the cost for 4 hours.
Total cost = 2 × 4 + 3 = 8 + 3 = $11

What Is Algebra, Really?

Algebra sounds scary, but you already use it every day without realising it. When you say "if I have 5 apples and someone gives me some more, I'll have __ apples," you are doing algebra! The unknown number is just waiting for a letter to represent it.

Mathematicians use letters like x, n, or a as placeholders for "the number we don't know yet." Think of a letter as a mystery box — it holds a number, we just haven't opened it yet.

Two Ways to Describe a Pattern

Suppose you're building fences from posts. One fence post, two fence posts, three fence posts… there are two ways to describe what comes next:

  • Term-to-term rule: look at what you do to get from one term to the next. Example: "add 1 each time."
  • Position-to-term rule: look at how the term number connects to its value. Example: "the number of posts equals the term number." This lets you jump straight to the 100th term without listing them all.

The position-to-term rule is far more powerful — that's the one we turn into algebra.

A Real-Life Example: Parking Costs

The Key Ideas tab showed a parking station: $3 entry fee plus $2 per hour. Let's think about WHY we write 2h + 3:

  • No matter how long you park, you always pay $3 — that's the fixed part (the constant).
  • Every hour adds $2 — that's the variable part (it changes with h).
  • After 1 hour: 2(1) + 3 = $5. After 3 hours: 2(3) + 3 = $9. See how substituting works?

Spreadsheet programs like Excel use exactly this idea — you type a formula like =2*A1+3 and it calculates for any number you put in column A. That's algebra running the world's businesses!

Expressions vs Equations

There's an important difference worth knowing early:

  • An expression is like a recipe without a total: 2h + 3. No equals sign.
  • An equation says two things are equal: 2h + 3 = 11. This lets us solve for h.
Remember: An expression has no equals sign. An equation does. This is the same difference as a shopping list (no total yet) versus a receipt (total included).

Patterns in Nature and Architecture

Patterns are everywhere. Sunflower seeds spiral in sequences following mathematical rules. Architects use algebraic rules to calculate how many bricks they need per row. Computer game designers use expressions to set player scores. Once you can spot a pattern and write it algebraically, you can answer any question about it — even the 1000th term — in seconds.

Common Mistake: Students often write the variable part before thinking about the fixed part. Always ask two questions: (1) "What never changes?" and (2) "What grows with the variable?" Then write: variable part + fixed part.

Practice Questions

  1. Fill in the tables for the following scenarios: Fluency

    1. I have 8 blue marbles in a jar and some white marbles. Let w be the number of white marbles.
      Number of white marblesw
      Total number of marbles
    2. What is the total number of marbles if there are 12 white marbles in question 1a? (Let w = 12)
    3. A player scores some points in a game and then receives a bonus of 15 points. Let p be the initial score.
      Initial scorep
      Total score
    4. If the player's total score was 40 in question 1c, what was their initial score? (Find the value of p)
    5. A skyscraper is three times taller than a nearby tree. Let t be the height of the tree.
      Tree heightt
      Skyscraper height
    6. How tall is the skyscraper if the tree is 25 meters tall? (Let t = 25)
    7. A sports car drives 20 km/h faster than a truck. Let s be the speed of the sports car.
      Sports car speeds
      Truck speed
    8. How fast is the sports car going if the truck is travelling at 80 km/h? (Find s)

  2. Writing equivalent algebraic expressions: Fluency

    1. Show two alternatives for writing this algebraic expression:
      I had a gift card with value v and bought two books that cost $15 each.
    2. If the gift card originally had $50 on it, how much is left?

  3. Number puzzles: Fluency

    1. Write an algebraic expression for this situation:
      Think of a number, triple it, and then subtract 4.
    2. If the answer is 26, what was the original number you thought of?

  4. Below are some algebraic expressions. Create a word story that might represent the expression. Understanding

    1. m + m + 5
    2. 3 × b − 2
    3. 100 − 4 × k

  5. Go back to the scenarios you wrote in question 4. Write the algebraic expressions in a more compact mathematical way (e.g., 4y instead of 4 × y). Fluency

  6. Is the expression 5 × m + 2 the same as 2 + 5 × m? Understanding

    Substitute any number for m to find out.

  7. Is the expression 2a + 3a the same as 5a? Understanding

    Substitute any number for a to find out.

  8. Look at the pattern of square dining tables below. Understanding

    One square table seats 4 people. If you push two tables together to make a long rectangle, they seat 6 people. Three tables pushed together seat 8 people.

    1. Draw or visualize what 4 tables pushed together would look like. How many people could be seated?
    2. Let t be the number of tables. Copy and complete the table below:
      Number of tables (t) 1 2 3 4
      People seated (p) 4 6 8
    3. Write an algebraic rule connecting the number of tables (t) to the number of people (p).

  9. Real-world Algebra: The Taxi Fare Problem Solving

    A taxi company charges a $5.00 "flat rate" fee (just for getting in the car) and then charges $2.00 for every kilometre travelled.

    1. Let k represent the number of kilometres travelled. Write an expression for the total cost of the trip.
    2. Use your expression to calculate the cost of a 12 km trip.
    3. If a passenger has $25.00, what is the maximum number of whole kilometres they can travel?

  10. Find the Rule Problem Solving

    Look at the table of Input and Output numbers below. The same rule is applied to every Input number to get the Output.

    Input (x) 1 2 3 4 10
    Output (y) 7 12 17 22
    1. Describe the pattern of the Output numbers in words (e.g., "The numbers go up by...").
    2. Find the value of the Output when the Input is 10.
    3. Write the algebraic rule for y in terms of x.
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