Practice Maths

Developing Mathematical Models

Mathematical Models and Variables

Key Terms

independent variable
the value you choose or control; plotted on the x-axis
dependent variable
the value that results from the independent variable; plotted on the y-axis
mathematical model
a formula or equation that represents a real-world situation
formula
an algebraic rule showing how variables are related

Defining Dependent and Independent Variables

When we create tables and graphs, we need to know which number goes where. We organise them by deciding which value relies on the other.

Independent Variable (x) Dependent Variable (y)
This is the value you choose or control. It changes naturally (like time). It is usually the top row of a table. This is the result. Its value depends on what 'x' is. It is usually the bottom row of a table.

Example:

If you are buying apples, the cost depends on how many apples you pick up.

  • Independent (x): Number of apples (You decide how many to grab).
  • Dependent (y): Total cost (This is calculated based on your choice).

From Sequences to Formulas

We can look at a number sequence and find the "Gap" (what we add each time) to build a formula.

Sequence: 7, 12, 17, 22...

  • The Gap is +5. This means the formula starts with 5x.
  • Check: 5 × 1 = 5. But our first term is 7. We need 2 more.
  • Formula: y = 5x + 2
Hot Tip Always check your formula against more than one term. Substitute x = 1 and x = 2 to make sure both give the correct values from the sequence.

Worked Example

Problem: Concert tickets cost $x each. Mia buys 4 tickets and pays $60 in total. Find the cost of one ticket.

Step 1 — Identify the variables and write an equation.
The total cost depends on the price per ticket. Write: 4x = 60

Step 2 — Solve the equation using an inverse operation.
Divide both sides by 4: x = 60 ÷ 4 = 15

Step 3 — State the answer with units.
Each ticket costs $15. Check: 4 × 15 = 60. ✓

What Is a Mathematical Model?

A mathematical model is a formula or equation that describes a real-world situation. Scientists model the spread of diseases. Engineers model the strength of bridges. Meteorologists model tomorrow's weather. Even the algorithm your streaming service uses to recommend videos is a mathematical model. It all starts with the skills you're learning right now.

Independent vs Dependent: Who Controls Whom?

These terms are easier to remember if you think of a cause-and-effect story. The independent variable is the cause — it's the thing you control or that naturally changes. The dependent variable is the effect — it reacts to the independent variable.

Think about driving a car: the further you drive (you control that), the more fuel you use (it depends on the distance). So distance is independent, fuel used is dependent.

  • Ask: "Which variable CAUSES the change?" — that's the independent variable (x).
  • Ask: "Which variable REACTS or results?" — that's the dependent variable (y).

Finding the Rule from a Sequence: The Gap Method

The Key Ideas tab showed the sequence 7, 12, 17, 22… and explained the gap method. Here's a deeper look at why it works:

  • The gap (+5) tells you the rate of change — how much y grows per step. This is always the coefficient of x in the formula.
  • Multiply that rate by the term number (x) to get a "base" value. For x = 1: 5 × 1 = 5. But the actual value is 7. So the extra amount is 7 − 5 = 2. That's the constant!
  • Formula: y = 5x + 2. Check: x = 2 → y = 12 ✓, x = 3 → y = 17 ✓.
Remember: The gap in the sequence = coefficient of x. Then substitute x = 1 to find what constant you need to add to make it work. Always verify with x = 2!

Reading Tables: Setting Up the Variables

When you see a table of values in a real-world problem, always label your variables before writing the formula. Don't just jump to the numbers!

  1. Identify what's changing — that's x (independent variable). Put it in the top row.
  2. Identify what results — that's y (dependent variable). Put it in the bottom row.
  3. Look for the pattern (gap), then build the formula.

Example: A typing student types 40 words per minute. x = minutes, y = words typed. y = 40x. After 5 minutes: y = 40 × 5 = 200 words. Clean, simple, powerful.

Common Mistake: When writing "formula to table" questions, students sometimes reverse x and y, substituting the y-value instead of the x-value. Always substitute the independent variable (x) into the formula to find y.

Working Backwards: The Formula as a Reverse Machine

Once you have a formula, you can go both directions. Forward: given x, find y. Backward: given y, find x. The backwards direction is really solving an equation — and that's the next big topic! For now, practise both directions so they feel natural.

Practice Questions

  1. Identifying Variables Fluency

    For each situation, identify the independent variable (x) and the dependent variable (y).

    1. The further you drive, the more petrol your car uses. Variables: distance driven and petrol used.
    2. At a fruit stall, apples cost $2.50 each. Variables: number of apples and total cost.

  2. Reading a Table of Values Fluency

    A sprinkler waters 12 square metres of lawn per minute.

    Minutes (x)1234
    Area (m²) (y)12
    1. Copy and complete the table.
    2. How many square metres are watered in 8 minutes?
    3. Write the algebraic formula for this relationship.

  3. Formula Evaluation Fluency

    Use the formula y = 4x + 3 to find the value of y for each input.

    1. x = 2
    2. x = 5
    3. x = 0

  4. Writing a Formula from Words Fluency

    A taxi charges a $3 flag fall plus $2.50 per kilometre.

    1. What is the independent variable?
    2. What is the dependent variable?
    3. Write the formula for total cost (C) in terms of kilometres (k).
    4. Find the cost of a 6 km trip.

  5. Speed and Distance Understanding

    A cyclist rides at a constant speed of 15 km/h.

    Hours (x)1234
    Distance (km) (y)15
    1. Copy and complete the table.
    2. Write the algebraic formula.
    3. How far does the cyclist travel in 5.5 hours?

  6. Two-Step Model: Delivery Fee Understanding

    A courier service charges a $5 booking fee plus $2.50 per kilometre.

    1. Write the formula for cost (C) in terms of kilometres (k).
    2. Copy and complete the table:
      Kilometres (k)0246
      Cost (C)
    3. How much does a 10 km delivery cost?

  7. Decreasing Model: Water Tank Understanding

    A water tank holds 200 litres. Water drains out at 8 litres per minute.

    1. Copy and complete the table:
      Minutes (x)0123
      Volume (L) (y)200
    2. Write the formula for volume (y) in terms of minutes (x).
    3. How much water is left after 15 minutes?

  8. Finding a Formula from a Sequence Understanding

    Consider the number sequence: 5, 9, 13, 17, ...

    1. Create a table of values for the 1st, 2nd, 3rd, and 4th terms.
    2. What is the common difference (the "gap")?
    3. Write the algebraic formula for the n-th term (y in terms of x).
    4. Use your formula to find the 25th term.

  9. Solving Backwards: Electrician's Bill Problem Solving

    An electrician charges a $75 call-out fee plus $55 per hour.

    1. Write the formula for total cost (C) in terms of hours worked (h).
    2. How much does a 4-hour job cost?
    3. A customer receives a bill of $405. How many hours did the electrician work?
    4. A family can afford at most $600. What is the maximum number of whole hours they can hire the electrician?

  10. Comparing Two Plans Problem Solving

    Two maths tutors offer the following rates:

    • Tutor A: No booking fee, $45 per hour.
    • Tutor B: $30 booking fee + $30 per hour.
    1. Write a cost formula for each tutor.
    2. How much does each tutor charge for a 3-hour session?
    3. For how many hours of tutoring do the two tutors charge the same amount?
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