To determine the slope between the points $(1, 1)$ and $(4, 5)$, we can utilize the formula:

The slope, also referred to as the gradient, quantifies the steepness of a line. To compute the slope between two points, it is essential to know their coordinates. Here, we have the points $(1, 1)$ and $(4, 5)$. We will label these points as $(x_1, y_1)$ and $(x_2, y_2)$, respectively. Thus, we identify:

• $x_1 = 1$
• $y_1 = 1$
• $x_2 = 4$
• $y_2 = 5$

Now, we can substitute these values into the slope formula:

Next, we simplify the expression:

Therefore, the slope between the points $(1, 1)$ and $(4, 5)$ is $\frac{4}{3}$. This indicates that for every 3 units moved horizontally to the right, there is a corresponding rise of 4 units vertically upward. Understanding how to calculate the slope is essential in GCSE Maths, as it enables you to analyze the rate of change between two points on a graph.