In this topic of IBDP Mathematics, we will be discussing mathematical models, how they are constructed, and different types of mathematical models.

**Mathematical Modelling**

In IBDP Mathematics, we may come across mathematical problems that are more complex than basic counting.

- For example, you can create a model to determine how much time it takes for a laptop to run out of battery.
- Whilst you could simply measure the time it takes for a laptop to run out of battery, this would be very time consuming and not very efficient.
- Instead, you could take smaller measurements, such as how much time it took a laptop to get down to 95%, and use this information to generate a mathematical model.
- You can then use this model to predict when your laptop will run out.

The example above shows the benefits to mathematical modelling. Modelling allows us to not only make predictions, but it can also help explain the relationship between two variables. Thus, it can be extremely useful in helping us understand real-world problems.

In mathematical modelling, we often have to make assumptions in order to simplify the problem. By making assumptions and removing less important details, it allows us to construct a mathematical description of the problem which is simple enough for us to work with. This enables the creation of a simpler model that is easier to understand, and we can use the model to approximate the real-world situation.

- However, it is important to note that these models are not completely accurate - given the assumptions made, there will be some error to the model. But the assumptions have to be made in order for us to construct a model that can be understood.

**The Modelling Cycle**

In IBDP Mathematics, mathematical models are developed using the **modelling cycle**:

- Pose a real world problem. Make assumptions which simplify the problem without missing key features.
- Develop a model which represents the problem with mathematics. This may involve a formula or an equation. The model should consider constraints such as the range of possible values each variable can take in the real world.
- Test the model by comparing its predictions with known data. If the model is unsatisfactory, return to Step 2.
- Reflect on your model and apply it to your original problem, interpreting the solution in a real world context.
- If appropriate, extend your model to make it more general or accurate as needed.

Revisiting the example above, we can follow the steps in the modelling cycle to generate a model for that problem.

**Step 1**- the problem, as mentioned above, is to construct a model to describe the battery level after t minutes in order to help determine the time it takes for the battery to run out.- In order to simplify the problem, we can make some assumptions, such as 'the battery is used at a constant rate for all laptops'.
- When we are considering a relationship between two variables, one variable is known as the independent variable (which is placed on the horizontal axis) and the other is known as the dependent variable (which is placed on the vertical axis).
- The independent variable is the variable that is used to predict the dependent variable. In this example, given that we want to use time to predict battery level (because we want to determine the battery level after t minutes), the independent variable is the time, and the dependent variable is battery level.
**Step 2**- in order to develop a model for this problem, we will need some data. Given the assumption of the model, we can simply record the time it takes for the laptop to lose 1% of battery.- For the sake of this example, lets say a laptop loses 1% of battery every minute.
- Therefore, at 0 minutes, the laptop will have 100%. At 1 minute, the laptop will have 99%. At 2 minutes, the laptop will have 98%. This can be summarised into the following equation:
*battery level = - t + 100.* - Given this rate, we can construct a linear model (a linear model is used since we assume a constant rate), with battery level on the y-axis and the time on the x-axis.
- It is important to note that this model will have constraints on the y-axis. Since battery levels only consist of values between 0 and 100, the model should only produce values that are between 0 and 100.
**Step 3**- to test the model, you can collect some data on the rate of laptop battery use.- If the data does not match the model, use this set of data to refine to model.
- For the sake of this example, we can assume that the model is satisfactory.
**Step 4**- given our equation for the linear model, we can substitute the values to determine the time it takes for the battery level to become 0%.- We can substitute 0 for battery level, and solve for t. This should give us t = 100.
- Interpreting this value, we can say that the laptop will run out of battery after 100 minutes of use.
**Step 5**- In order to extend, we can add in more factors which affect battery level, such as how old the laptop is, the type of activity used on laptop etc.

In IBDP Mathematics, not only will you need to be familiar with constructing models, you are required to be able to use these models, state the assumptions made when constructing the model, and whether these assumptions are reasonable. You may also be asked to describe whether you think the actual value is larger or smaller than the value predicted by the model. Thus, you must be able to think critically about these models.

**Types of Mathematical Models**

In IBDP Mathematics, there are 3 types of mathematical models:

- Linear models - these are the models where two variables are linearly related (i.e. they form a straight line on the graph).
- Piecewise models - a model made up of several different straight line segments.
- Non-lines piecewise models - model made of up several segments, some of which are not linear.

To learn more about these types of models, click __here__!

This is the end of this topic.