Why is the gradient of a straight line considered 'constant'?

A straight line does not curve, meaning its steepness never changes. Therefore, the ratio of vertical change to horizontal change is the same between any two points on that line.

How do you convert an integer gradient like $-4$ into a 'rise' and 'run' for drawing?

Express the integer as a fraction: $\frac{-4}{1}$. This means the 'rise' is $-4$ (move $4$ units down) and the 'run' is $1$ (move $1$ unit right).

What is the most common error when using the gradient formula $m = \frac{y_2 - y_1}{x_2 - x_1}$?

The most common error is swapping the numerator and denominator, resulting in 'run over rise' ($\frac{\Delta x}{\Delta y}$). Always ensure the change in $y$ is on top.

Why might a student get the wrong sign when calculating the gradient of a downhill line?

This usually happens if they only calculate the lengths of the triangle sides without considering the direction, or if they subtract the $x$ and $y$ coordinates in inconsistent orders.

What mistake is made if a student counts grid squares to find the gradient on a graph with different axis scales?

Counting squares ignores the actual numerical values of the units. If the $y$-axis goes up in $2$s and the $x$-axis in $1$s, one square vertically represents a different change than one square horizontally.

Define the term 'Rise' in the context of coordinate geometry.

The 'Rise' is the vertical change between two points on a line, calculated as the difference between their $y$-coordinates ($y_2 - y_1$).

Define the term 'Run' in the context of coordinate geometry.

The 'Run' is the horizontal change between two points on a line, calculated as the difference between their $x$-coordinates ($x_2 - x_1$).

What does a gradient of $0$ represent visually?

A gradient of $0$ represents a perfectly horizontal line. This occurs because the 'rise' is zero, meaning the $y$-coordinate never changes regardless of the $x$-value.

If a line has a gradient of $-2$, what happens to the $y$-value as the $x$-value increases by $1$?

The $y$-value decreases by $2$ units. The negative sign indicates a downward movement for every step taken to the right.

What is the difference between a gradient of $4$ and a gradient of $\frac{1}{4}$?

A gradient of $4$ is much steeper, rising $4$ units for every $1$ unit right. A gradient of $\frac{1}{4}$ is shallower, rising only $1$ unit for every $4$ units right.

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Gradient of a Line

Summary

The gradient is a fundamental measure of the steepness and direction of a straight line. It quantifies the rate at which the vertical coordinate changes relative to the horizontal coordinate, providing a single numerical value that defines the slope's character.

1. Definition & Core Concepts

Gradient is a numerical measure of the steepness of a straight line on a Cartesian plane. It represents the constant rate of change between the $x$ and $y$ variables along that line.
A positive gradient indicates an 'uphill' slope where the line rises from the bottom-left to the top-right. Conversely, a negative gradient indicates a 'downhill' slope where the line falls from the top-left to the bottom-right.
The magnitude (absolute value) of the gradient determines the steepness. A gradient of $-5$ is steeper than a gradient of $3$ because the vertical change per horizontal unit is greater, regardless of the direction.

A coordinate grid showing a red line with a blue right-angled triangle underneath it. The horizontal side is labeled 'Run' and the vertical side is labeled 'Rise', illustrating the components of a gradient.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

The Sign Check: Always perform a visual 'sanity check' before finalizing your answer. If the line goes downhill but your calculated gradient is positive, you have likely made a subtraction error.
Scale Awareness: Examiners often use different scales for the $x$ and $y$ axes. Do not simply count grid squares; instead, use the actual values on the axes to determine the rise and run.
Point Selection: When choosing points from a graph, pick 'clean' points where the line crosses the grid corners exactly. This minimizes estimation errors and ensures your fraction simplifies correctly.

6. Common Pitfalls & Misconceptions