A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Functions in Modelling — mark scheme explained
The short answer
In AQA A-Level Mathematics, the use of functions in modelling is a crucial skill that helps you apply mathematical concepts to real-world scenarios. This involves creating and interpreting mathematical models using functions, understanding their limitations, and refining them as necessary. ### What is Mathematical Modelling?
The question
A city's population grows exponentially. The initial population is 50,000 and the growth rate is 2% per year. Write an exponential function to model this growth and find the population after 10 years.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
Marks are typically awarded for correctly identifying the problem, formulating an appropriate model, solving the equations accurately, interpreting the results in context, and validating the model. Partial marks may be given for correct steps even if the final answer is incorrect.
What the command words demand
- Identify
- Clearly define the problem or context.
- Formulate
- Choose appropriate functions to represent the relationships in the problem.
- Solve
- Use algebraic techniques to solve the equations derived from the model.
- Interpret
- Translate the mathematical results back into the context of the original problem.
- Validate
- Check the model against real-world data to ensure it fits well.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: For a 6-mark question on this topic, allocate approximately 7-8 minutes to ensure you have enough time to carefully formulate, solve, interpret, and validate your model.
- Identify the given values: P 0 = 50,000, k = 0.02, t = 10.0 marks
- Write the exponential growth function: P(t) = P 0 × e kt .1 mark
- Substitute the values into the function: P(10) = 50,000 × e 0.02 × 10 .1 mark
- Calculate the exponent: 0.02 × 10 = 0.2.1 mark
- Evaluate the exponential term: e 0.2 ≈ 1.2214.1 mark
- Multiply to find the population: P(10) = 50,000 × 1.2214 ≈ 61,070.1 mark
Final answer: The population after 10 years is approximately 61,070.
Work through every step correctly and you earn all 5 marks.
Another worked example
A logistic growth model for a city's population is given by P(t) = 100,000 / (1 + 9e -0.1t ) . Find the carrying capacity and the initial population.
- Identify the logistic growth function: P(t) = K / (1 + Ae -kt ) .0 marks
- From the given function, K = 100,000 (carrying capacity).1 mark
- To find the initial population, set t = 0: P(0) = 100,000 / (1 + 9e -0.1 × 0 ) .1 mark
- Simplify the exponent: e 0 = 1.0 marks
- Substitute back into the function: P(0) = 100,000 / (1 + 9) = 100,000 / 10 = 10,000.2 marks
Final answer: The carrying capacity is 100,000 and the initial population is 10,000.
Work through every step correctly and you earn all 4 marks.
Common mistakes
Using an inappropriate function for the problem
Why it happens: Choosing a linear function for exponential growth or decay can lead to inaccurate models.
Fix: Identify the nature of the problem and select the appropriate type of function (e.g., linear, quadratic, exponential).
Ignoring domain restrictions
Why it happens: Extrapolating beyond the valid range of input values can lead to unrealistic results.
Fix: Always consider the domain of the function and ensure it is appropriate for the context of the problem.
Failing to validate the model
Why it happens: Not checking the model against real-world data can result in a model that does not accurately represent the situation.
Fix: Compare the model's predictions with actual data and refine the model if necessary.
Overlooking simplifying assumptions
Why it happens: Making unrealistic or overly simplistic assumptions can lead to a model that does not accurately reflect reality.
Fix: Re-evaluate the assumptions and consider more realistic factors if necessary.
Incorrectly interpreting results
Why it happens: Translating mathematical results back into the context of the problem can be challenging, leading to misinterpretations.
Fix: Ensure that the interpretation makes sense in the real world and is consistent with the data and assumptions.
Using incorrect parameters
Why it happens: Using inaccurate or outdated parameters can lead to a model that does not fit the data well.
Fix: Use the most accurate and up-to-date data available to determine the parameters of the model.
Failing to refine the model
Why it happens: Not adjusting the model based on feedback or new information can result in a less accurate representation of the problem.
Fix: Continuously refine the model by incorporating new data, re-evaluating assumptions, and considering additional factors.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Exponential Growth Model | Formulate an exponential growth model and calculate the population after a given time. | 5 |
| Logistic Growth Model | Read the carrying capacity from the model and evaluate the initial population at t=0. | 4 |
| Linear Function Modelling | Form a linear equation from a real-world context and evaluate it at a given value. | 4 |
| Maximum Of Quadratic | Find the maximum height of a ball modelled by a quadratic height-time function. | 6 |
| Total across these question types | 19 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.