Practice Maths

Developing Algebraic Rules to Describe Patterns

Full Lesson

Key terms: variable, term, expression

• A variable is a letter representing an unknown number (e.g., x, n)
• A term is one position in a sequence (e.g., the 3rd term)
• An expression is a rule written in symbols (e.g., y = 2x + 2)

Creating algebraic expressions

We have previously created algebraic expressions to describe number patterns. For example:

Term (x)123
No. of counters (y)345

The algebraic expression to describe the relationship:

  • Words: number of counters = term + 2
  • Symbols: y = x + 2

Creating expressions from "connected" patterns

Consider a garden walkway. 1 concrete slab needs 4 pieces of wood framing. When we connect 2 slabs, they share a side, so we only need 6 pieces of wood.

No. of Slabs (x)123
Wood Pieces (y)468

We are adding two pieces of wood for every new slab. This means the rule involves multiplying by 2.

Rule: y = 2x + 2

Worked Example

Question: A bike rental shop charges a $15 fixed fee plus $10 per hour. Write the algebraic rule for Cost (C) in terms of hours (h), then find the cost for 6 hours.

Step 1 — Identify the fixed part and the variable part.
Fixed: $15 (always charged regardless of time)
Variable: $10 per hour = 10h

Step 2 — Write the rule.
C = 10h + 15

Step 3 — Substitute h = 6.
C = 10 × 6 + 15 = 60 + 15 = $75

Connected Patterns Are Trickier — Here's Why

Simple patterns like "3 matchsticks per triangle" are straightforward — the formula is just a multiplication. But connected patterns (like the garden walkway from the Key Ideas tab) are more interesting. When you join two slabs, one side is shared, so you don't need as many pieces of wood. This is why the rule involves a multiplier and an extra constant.

The Four-Step Method for Finding Any Rule

Here is a reliable method you can use on any table of values:

  1. Find the gap: look at how much the output changes each time the input increases by 1. This gap becomes your multiplier.
  2. Write the multiplier × x: e.g. if the gap is 2, write y = 2x + ?
  3. Find the constant: substitute x = 1 into your partial rule and compare with the actual value. The difference tells you what to add or subtract.
  4. Verify: check with x = 2 and x = 3. If all values match, your rule is correct.

Example using the garden walkway (y = 2x + 2):

  • Gap = +2 each time. So start with y = 2x + ?
  • At x = 1, y should be 4. But 2(1) = 2. So we need to add 2. Rule: y = 2x + 2.
  • Check x = 2: y = 2(2) + 2 = 6. Correct! Check x = 3: y = 2(3) + 2 = 8. Correct!

The Bike Rental: Fixed and Variable Costs

The worked example in the Key Ideas tab (bike rental: $15 fixed + $10/hour) is a very common type of real-world formula. Let's think about the structure:

  • Fixed cost: This is what you pay no matter what — like an entry fee, call-out fee, or sign-up charge. It doesn't change. This is your constant.
  • Variable cost: This grows with usage — hours, kilometres, items. It's the multiplier × your variable.

Plumbers, electricians, taxi drivers, and caterers all charge this way. Once you can write the formula, you can work out any bill or reverse-calculate to find how long something took.

Remember: When you test your rule, always check at least two values — not just one. A rule might accidentally work for x = 1 but fail for x = 2 if you've made an error in the constant.

Descending Patterns: Subtraction Rules

Not all patterns grow — some shrink! A candle burning down, a car using fuel, a water tank draining. These rules subtract each time. The technique is the same: find the gap (this time it's negative), write the multiplier, then find the constant.

Example: height = 30 − 2h means the candle starts at 30 cm and loses 2 cm per hour. After 5 hours: 30 − 2(5) = 20 cm. Simple!

Common Mistake: When the pattern decreases, students sometimes write the rule as y = 2x instead of y = 30 − 2x. Always check whether the output goes up or down as x increases. Down means you subtract.

Practice Questions

  1. Baking Muffins (Trays vs Total Muffins) Fluency

    A baker uses large trays that hold a specific number of muffins. 1 tray holds 6 muffins. 2 trays hold 12 muffins.

    1. Complete the table:
      Number of trays (t)123
      Total muffins (m)612
    2. Write the algebraic relationship between trays and muffins in both words and symbols.
    3. How many muffins would there be if the baker used 35 trays?
    4. How many trays would be needed to hold 168 muffins?

  2. Square Tables at a Party (Visual Pattern) Fluency

    One square table seats 4 people. Two square tables pushed together seat 6 people (because the side where they join cannot be used).

    1. Put the information for this seating pattern in the table:
      Number of tables (t)123
      Number of people (p)4

  3. Using the Party Tables Pattern Understanding

    This question uses the table you completed in Q2.

    1. What would be the equation in symbols relating the number of tables (t) to the number of people (p)? (Hint: Refer to the "Garden Walkway" example in the Full Lesson.)
    2. How many people could be seated if 15 tables were pushed together in a line?
    3. If there were 42 people seated, how many tables were pushed together?

  4. Building a Lego Tower Understanding

    A child is building a tower pattern. Level 1 uses 5 blocks. Level 2 uses 8 blocks. Level 3 uses 11 blocks.

    1. Complete the table:
      Level Number (n)123
      Total Blocks (b)
    2. Write the algebraic relationship in words.
    3. What would be the equation in symbols?
    4. How many blocks would be needed to make a tower with 20 levels?
    5. If a tower used 95 blocks, what level number is it?

  5. Bead Necklace Pattern Understanding

    A jeweller creates a necklace pattern using Blue and Silver beads. 1 Blue bead requires 5 Silver beads around it. 2 Blue beads (linked) require 8 Silver beads.

    1. Put the information for this pattern in the table:
      Blue Beads (x)123
      Silver Beads (y)
    2. Write the algebraic relationship in symbols.
    3. How many Silver beads are needed for 20 Blue beads?
    4. How many Blue beads are there if 107 Silver beads were used?

  6. Stadium Seating Sections Understanding

    A stadium is built in curved sections. Row 1 has 10 seats. Row 2 has 12 seats. Row 3 has 14 seats.

    1. Put the information for this seating pattern in the table:
      Row (r)12345
      Number of Seats (s)
    2. Write the algebraic relationship in symbols (s = ...).
    3. How many seats would be in the 10th row?
    4. Could a row have exactly 81 seats? Explain why or why not.

  7. Bike Rental (Fixed and Variable Costs) Fluency

    A bike rental shop charges a $15 fixed fee (insurance) plus $10 for every hour you rent the bike.

    1. Complete the table of costs:
      Hours (h)123
      Total Cost (C)$25
    2. Write the algebraic rule for Cost (C) in terms of hours (h).
    3. How much would it cost to rent the bike for 8 hours?

  8. The Burning Candle (Descending Pattern) Understanding

    A 30 cm tall candle is lit. It burns down at a rate of 2 cm per hour.

    Hours Lit (h)0123
    Height (y)302826
    1. Describe the pattern of the height numbers in words.
    2. Write an algebraic rule for the height (y) based on hours (h).
      Hint: Start with 30 and subtract...
    3. How tall will the candle be after 5 hours?

  9. Abstract Number Machine Understanding

    Find the rule for this input/output table.

    Input (n)12310
    Output (p)91317?
    1. What is the "gap" or difference between consecutive Output numbers?
    2. Write the algebraic rule (p = ...).
    3. Calculate the Output when the Input is 10.

  10. Which Rule is Correct? Problem Solving

    A plumber charges $50 to come to your house, plus $40 per hour for labour.

    1. Which of the following equations represents the Total Cost (C) for (h) hours?
      A) C = 90h
      B) C = 50h + 40
      C) C = 40h + 50
    2. Use the correct formula to calculate the cost of a 3-hour job.
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