Equation of perpendicular lines
GCSE Mathematics – Numeracy and GCSE Mathematics
Possible learning objectives
 Explore properties of linear graphs
GCSE Mathematics
Understanding and using functional relationships
 Identifying the equation of lines parallel or perpendicular to a given line, to satisfy given conditions.
Possible learning outcomes
 Find the gradient of a line given two points
 Find the equation of a line given two points, one of which is the y-intercept
 Find the equation of a line given two points
 Know that parallel lines have the same gradient
 Know that product of the gradients of perpendicular lines equals -1
 Identify parallel lines of the form y = mx + c
 Identify perpendicular lines of the form y = mx + c
 Rearrange the equation of a line into the form y = mx + c
 Identify parallel lines not of the form y = mx + c
 Identify perpendicular lines not of the form y = mx + c
 Find the equation of a line perpendicular to a given line and through a given y-intercept
 Find the equation of a line perpendicular to a given line and through a given point
Prerequisites
Mathematical language
Pedagogical notes
 Plot the graphs of lines of the form y = mx + c by generating and Equation
 Students should be familiar with the word ‘perpendicular’ as a result of earlier
Formula
plotting points
work in geometry. Intermediate and higher tier students will also encounter the
Variable
concept when working with circle theorems: Understanding that the tangent at any
 State the gradient and y-intercept of a line y = mx + c
Coefficient
point on a circle is perpendicular to the radius at that point.
 Find the gradient of a line drawn on a unit grid
Line
 The gradient of a line is a measure of its steepness. Informally it can be thought of
 Find the equation of a line drawn on a unit grid
Linear
as ‘the change in y divided by the change in x’, but care should be taken to ensure
 Plot the graphs of lines of the form y = mx + c by using
Parallel
that the gradient is negative if the line slopes down from left to right. Formally, if
knowledge of the gradient and y-intercept
Perpendicular
two points are (x1, y1) and (x2, y2) then the gradient is (y2 – y1) ÷ (x2 – x1)
 Change the subject of a formula
Product
 If the equation of a straight line is written in the form y = mx + c, then ‘m’ is the
Gradient
gradient of the line. If two lines are given by y = m1x + c1 and y = m2x + c2, and m1 ×
Slope
m2, = –1, then the two lines are perpendicular.
y-intercept
 Students need to be able to rearrange equations into the form y = mx + c
Rearrange
 It is particularly important to write fractions with a horizontal fraction bar when
Subject
working in contexts such as this: 1/2x or 1/2x is misleading when handwritten.
 WJEC: New content guidance includes worked examination questions and
Notation
annotated candidates’ responses.
y = mx + c is typically used in the UK. Elsewhere, a common
form is y = mx + b
Reasoning opportunities and probing questions
Possible activities
Potential misconceptions
 Show me the equation of a line parallel (perpendicular) to this one. Hwb: Question 40: Parallel and perpendicular lines
 Some students may use ‘change in y divided by change in x’ to
Hwb: Question 57: Hidden square
And another, and another, …
calculate the gradient of a line and neglect to consider the sign of the
Hwb:
Question
58:
How
do
we
know?
gradient
 What is the same and what is different: y = 2x + 1, y + 2x + 1 = 0, 2y =
Hwb: Question 82: Perpendicular pairs
1 + x, 2y = 1 – x ?
 When multiplying fractions some students may think that a common
Kangaroo
Maths:
The
gradients
of
perpendicular
lines
–
using
denominator is required
 Always / Sometimes / Never: Two perpendicular lines have
Autograph
gradients with a product of -1
 When changing the subject of a formula such as 2y = x + 1 some
NRICH: Perpendicular lines
students may not divide all three terms by 2, and produce y = x/2 + 1.
In this case, if they were looking to identify perpendicular lines, the
error would not prevent them finding the correct solution.
Nonetheless, the error should be pursued.