Practice Questions: Number Plane, Slope, Rise, Run and Distance
How to use this quiz: Open one topic at a time, type your answer, then check your working. The questions cover rise and run, slope, distance, coordinates, intercepts, equations and mixed number plane skills.
1. Find the rise and run
2. Find the slope or gradient
3. Distance when the points have the same x-value or y-value
4. Distance between two points using the distance formula
5. Coordinates, axes and quadrants
6. Substitute into y = mx + c and find missing values
7. Intercepts and equations of straight lines
8. Mixed number plane questions
Number Plane Straight Lines: Gradient and Y-Intercept
Understanding straight lines on the number plane is a key maths skill. It helps you read graphs, find how steep a line is, and write the rule (equation) for a line.
What Is a Number Plane?
A number plane (Cartesian plane) is made using two axes:
- x-axis → horizontal
- y-axis → vertical
The point where they meet is the origin: (0,0)
Every point on the plane is written as: (x,y)
The Equation of a Straight Line
The general equation of a straight line is:
y = mx + c
Where:
- y → the y-coordinate (vertical value) of a point on the line
- x → the x-coordinate (horizontal value) of a point on the line
- m → the gradient (slope) of the line
- c → the y-intercept (where the line crosses the y-axis)
Example: If the equation is y = 2x + 3
- Gradient m = 2 → the line rises 2 units for every 1 unit across
- Y-intercept c = 3 → the line crosses the y-axis at (0,3)
Rise and Run
Gradient is based on two simple movements:
- Rise → vertical change between two points
- Run → horizontal change between two points
The gradient can be found using:
$$ m = \frac{\text{Rise}}{\text{Run}} $$
Or using coordinates:
$$ m = \frac{y_2 – y_1}{x_2 – x_1} $$
- If the line moves upwards, the gradient is positive.
- If the line moves downwards, the gradient is negative.
Finding Gradient and the Equation from Two Points
Suppose two points are:
\( A(x_1,y_1) \) and \( B(x_2,y_2) \)
Step 1: Find the Gradient
$$ m = \frac{y_2 – y_1}{x_2 – x_1} $$
Step 2: Find the Y-Intercept
Substitute the gradient \( m \) and the coordinates of one point into:
$$ y = mx + c $$
Step 3: Write the Equation
Once you know \( m \) and \( c \), write:
$$ y = mx + c $$
Example: Points \( A(2,3) \) and \( B(6,11) \)
Slope (Gradient):
$$ m = \frac{11 – 3}{6 – 2} = \frac{8}{4} = 2 $$
Find the y-intercept: Use point \( (2,3) \) in \( y = mx + c \)
$$ 3 = 2(2) + c $$
$$ 3 = 4 + c $$
$$ c = -1 $$
Equation of the line:
$$ y = 2x – 1 $$
Finding Gradient and Y-Intercept from an Equation
If the equation is already in the form \( y = mx + c \), you can read \( m \) and \( c \) directly.
Example 1:
$$ y = 3x – 4 $$
- \( m = 3 \)
- \( c = -4 \)
Example 2:
$$ 2x + 3y = 12 $$
Rearrange into \( y = mx + c \):
$$ 3y = -2x + 12 $$
$$ y = -\frac{2}{3}x + 4 $$
- \( m = -\frac{2}{3} \)
- \( c = 4 \)
Distance Formula (Extension)
To find the distance between two points on the number plane, use:
$$ \text{Distance} = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $$
Example: Distance between \( A(2,3) \) and \( B(4,5) \)
$$ \text{Distance} = \sqrt{(4-2)^2 + (5-3)^2} $$
$$ = \sqrt{2^2 + 2^2} $$
$$ = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 $$
Quick Summary
- Straight lines can be written as \( y = mx + c \)
- Gradient tells how steep a line is: \( m = \frac{y_2-y_1}{x_2-x_1} \)
- Y-intercept is where the line crosses the y-axis at \( (0,c) \)
- You can find the equation of a line using two points
Tip: The best way to get confident is to practise drawing lines and calculating gradient using rise and run.
🔗 Related Learning Resources
Practise and strengthen your understanding of straight lines, gradients, and the number plane with these helpful resources:
- Number Plane – The Complete Beginner Guide
- Interactive Number Line Maths Game
- Simple Linear Equations Practice Tests
- Algebra Made Easy – Student Friendly Guide
- Parabola and Graphing – Complete Guide
- NSW Year 9 Maths Syllabus Overview
- NSW Year 10 Maths Syllabus Overview
Tip: The best way to master gradients and straight lines is to combine theory with practice. Try the interactive games and practice tests above to build confidence quickly!
📘 Trusted External Resources
These high-quality external resources can help you explore the topic further:
- MathsIsFun – Introduction to Linear Equations
- Khan Academy – Linear Equations and Graphs
- BBC Bitesize – Straight Line Graphs and Gradients
- Math Planet – Understanding Gradient and Slope
- Purplemath – Equation of a Straight Line Explained
Tip: The best way to master gradients and straight lines is to combine theory with practice. Use the interactive games and trusted learning sites above to build strong confidence quickly!
