📌 What is Variation?

Variation is all around us—it’s the reason things change and react to one another! Imagine you're filling a balloon with air. The more air you pump in, the bigger it gets. That’s variation in action! In simple terms, variation describes how one thing depends on another. If you walk faster, you cover more distance in less time. If you save more money, your bank account grows.

Some changes happen in a steady, predictable way—like earning the same amount for every hour you work. Others may be more unpredictable, like how the weather changes. Understanding variation helps us make sense of patterns in the world, from how prices rise with demand to how shadows grow longer as the sun sets. Now, let’s zoom in on linear variation, where change happens at a constant, steady rate!

📌 Modelling Linear Variation With Maths

Use the PHET applet to investigate the amount by which a spring is stretched when different weights are attached to it.

• Check the 'Unstretched Length' and 'Resting Position' options.
• Attach, in turn, the 50g, 100g and 250g weights to a spring.
• Use the ruler to measure the distance between the 'Unstretched Length' and 'Resting Position' lines.
• Record the measurements in a table of values.

mass(g)	0	50	100	250
dist (mm)	0

Credit: University of Colorado Boulder: https://phet.colorado.edu/en/simulations

The collected data can be graphed in Desmos to examine the relationship between the size of the mass and the stretch in the spring.

• Enter the data as points and tick 'Label'. An axes can be rescaled by pointing to it with the mouse, holding down the 'Shift' key and dragging. The number plane can be repositioned using 'click and drag'.
• The data points appear to have a linear relationship.
  Enter 'y=kx' into Desmos and click 'Add Slider'.
• Click on the Slider and add these values: 0 < k < 1 Step: 0.01
• Now use the mouse to adjust the slider until the line passes as close to all 3 points as possible.
  Write down this value of k.

📌 Linear Variation

The equation that you have found is called a linear model. It describes how a variation in mass (independent variable) is related to a change in the distance the spring is stretched (dependent variable). Linear variation occurs when one quantity increases or decreases at a constant rate when the other changes.

For example, if an uber charges $2 per kilometre, the total cost increases is a straight-line pattern:

• 1 km = $2
• 2 km = $4
• 3 km = $6 etc.

If the independent variable (distance) changes be 1 km, there is a corresponding change in the dependent variable (cost) of $2.