Introduction to Creating Models

In mathematics, creating models, especially using linear functions, is an essential skill that finds applications in diverse fields such as economics, biology, and engineering. It involves establishing a mathematical relationship, often using data, to predict future trends or understand underlying patterns. This section will delve deeper into the process of using data and predicting trends with linear models, providing a comprehensive guide for IB Mathematics students. To gain a deeper understanding, students can explore the basics of linear regression, which plays a critical role in creating predictive models.

Using Data

Data Collection

• Importance: The initial step in creating a model involves gathering relevant data. This data serves as the foundation upon which our model will be built, providing the raw information that will be analysed and interpreted. Understanding the importance of data representation is crucial for effective model creation.
  • Example: Suppose we have data representing the cost of production (in pounds) for different quantities of a product: (10, 200), (20, 350), (30, 500).
• Types of Data: Data can be continuous or discrete, and understanding the nature of the data is crucial in determining the type of model to be developed.

Data Representation

• Graphical Representation: Once collected, data is often plotted on a graph to visualise the relationship between variables. In the context of linear functions, we’re typically looking for a straight-line relationship. Graphs play a significant role in interpreting correlation between variables.
  • Example: Plotting the above data on a graph with quantity on the x-axis and cost on the y-axis.
• Scatter Plots: These are particularly useful in initially visualising data and can often indicate whether a linear model might be appropriate. For a deeper understanding, see the section on calculating correlation.

Identifying Variables

• Dependent and Independent Variables: Determine the dependent and independent variables. The independent variable is the input (often denoted as x), while the dependent variable (often denoted as y) is what we are trying to predict or understand.
  • Example: In our production cost scenario, quantity might be the independent variable, while cost is the dependent variable.

Predicting Trends

Establishing a Linear Relationship

• Linear Equation: Using the data, we derive a linear equation of the form y = mx + c, where m is the slope and c is the y-intercept. This equation becomes the model, predicting y for any given x. The process of finding the slope and intercept can be further explored in sections on slope basics and intercept basics.
  • Example: Using two points from our data, say (10, 200) and (20, 350), we can find the slope m using the formula m = (y2 - y1) / (x2 - x1). Substituting these values, we get m = (350 - 200) / (20 - 10) = 15.
• Intercept: The y-intercept c is the value of y when x is zero. It can be found by rearranging the equation to c = y - mx and substituting a known point.

Using the Model for Predictions

• Predictive Analysis: Once the linear model is established, it can be used to predict the dependent variable for any given independent variable. This is a crucial step in making predictions using the model.
  • Example: To predict the cost for producing 25 units, we substitute x = 25 into our model (once we have found c) and calculate the corresponding y value.
• Model Validation: It's crucial to assess the validity of the model by comparing predicted values with actual data and ensuring the model is applicable within the relevant context.
  • Example: If our model predicts a cost of 375 pounds for 25 units, but the actual cost is significantly different, we may need to reassess our model.

Application of Linear Models

Case Study: Lemonade Stand

Imagine running a lemonade stand where we observe the following: selling 10 cups costs 5 pounds and selling 20 cups costs 8 pounds. We wish to create a model to predict the cost for any number of cups.

Step 1: Find the Slope

Using the two points (10, 5) and (20, 8), we find the slope m as follows:

m = (8 - 5) / (20 - 10) = 0.3

Step 2: Find the Y-Intercept

Using the slope and one point, say (10, 5), we find the y-intercept c using the rearranged linear equation:

c = y - mx

Substituting in our values:

c = 5 - (0.3 * 10) = 2

Step 3: Formulate the Model

Our linear model becomes:

y = 0.3x + 2

Step 4: Use the Model

To predict the cost of selling 15 cups of lemonade, substitute x = 15 into the model:

y = (0.3 * 15) + 2 = 6.5

Thus, it would cost 6.5 pounds to sell 15 cups of lemonade.

Considerations in Modelling

• Accuracy: Ensure the model accurately represents the available data and is reliable for predictions within a relevant range.
• Simplicity vs Complexity: While a simple model (like a linear model) is easier to use and understand, it may not always accurately represent data, especially if the true relationship is non-linear.
• Context: Always consider the real-world implications and applicability of the model. Ensure it makes logical sense in the given context.

Challenges and Practice Questions

Question 1

Given the data points (5, 120) and (10, 200) representing the cost of producing certain quantities of a product:

• Find the linear model representing the data.
• Use the model to predict the cost of producing 7 units.

Question 2

You observe that your weekly spending is 50 pounds in week 1 and 70 pounds in week 3:

• Create a linear model to represent your spending over time.
• Predict your spending in week 5 using the model.

Question 3

A car rental company charges 20 pounds per day to rent a car, and an additional one-time fee of 10 pounds for cleaning:

• Formulate a linear model representing the total cost, y, to rent a car for x days.
• Use the model to find the cost of renting the car for 6 days.