In a model y = mx + c, the gradient m is the change in y for each increase of 1 in x, and the intercept c is the value of y when x is 0. Interpreting them means writing one sentence for each that names the quantity and gives its unit.
This lesson comes after fitting a model to data in International Mathematics modelling. It is where you turn numbers back into a real-world statement.
How do you write the interpretation?
Use a sentence frame and fill it in.
- Gradient: “For each extra 1 [input unit], the [output quantity] increases (or decreases) by [m] [output unit].”
- Intercept: “When [input] is 0, the [output quantity] is [c] [output unit].”
- Unit check: the unit of m is the output unit divided by the input unit, written “per”.
Then ask whether the intercept makes sense. If zero input is outside the situation, say that the intercept is only a starting value of the model.
Worked example
A water tank is being emptied. Its volume V litres after t minutes is modelled by V = 800 − 25t. Interpret both numbers and find when the tank is empty.
Step 1, gradient: the number multiplying t is −25. Output unit is litres and input unit is minutes, so the unit is litres per minute. The volume decreases by 25 litres each minute.
Step 2, intercept: at t = 0, V = 800. The tank holds 800 litres at the start.
Step 3, empty tank: set V = 0. Then 800 − 25t = 0, so 25t = 800 and t = 800 ÷ 25 = 32.
The tank is empty after 32 minutes, if the flow stays constant. Check: 25 × 32 = 800, so 800 − 800 = 0.
The mistake to watch for
A common slip is to state the gradient without a unit, or with the unit upside down.
Mistaken statement: “The gradient is −25, so the tank loses 25 minutes per litre.”
That reverses the unit.
The correction is to build the unit from the axes: output over input, which is litres over minutes, or litres per minute. One check is to test the sentence with a small input. In 2 minutes the volume falls by 50 litres, so “25 litres each minute” is right, while “25 minutes per litre” would mean one litre takes 25 minutes to drain, which does not match the model.
Check yourself
Try these on paper, then open each answer.
1. The cost model for the gym is C = 50 + 35m, where C is in RM and m is the number of months. Interpret the 35 and the 50.
Show answer
35 is the gradient: the cost rises by RM35 for each extra month. 50 is the intercept: when m = 0 the cost is RM50, which is the joining fee.
2. A prepaid balance B (RM) after d days is modelled by B = 40 − 2.5d. Interpret the gradient and find when the balance reaches zero.
Show answer
The gradient −2.5 means the balance decreases by RM2.5 per day. Set B = 0: 2.5d = 40, so d = 40 ÷ 2.5 = 16 days.
Check: 2.5 × 16 = 40.
3. A model for a car journey gives distance d (km) = 60t, where t is in hours. State the unit of the gradient and what it represents.
Show answer
The unit is kilometres per hour (km/h), and the gradient 60 represents the average speed, 60 km/h, assuming the car moves at a constant speed.
Where this leads next
Once you can say what the numbers mean, you can judge how well a model predicts: compare a prediction with an observed value. Go back to define variables and assumptions if the units keep slipping. The modelling practice set has mixed questions.
Some students know the algebra but lose marks because the final sentence has no unit. That is a habit our teachers can fix in online one-to-one Mathematics tuition. The percentage-base explorer also helps if you need to practise stating what a percentage is a percentage of.