S3 Maths | Equations of Straight Lines Lesson Prep Guide (for tutors) | Slope, intercept form, parallel and perpendicular

2026-11-09

Tutor lesson-prep guides · Secondary Maths (DSE)

Slope is not y divided by x; it is the change in y divided by the change in x, and parallel and perpendicular rules are only consequences of that one sentence.

Objectives — Students can: (1) find the slope from two points or a graph and separate positive, negative, zero and undefined slopes; (2) write the slope-intercept form and convert between it and the general form; (3) use equal slopes for parallel lines and a product of minus one for perpendicular lines, scoring at least 9 of 12 in the diagnostic.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Negative numbers	What is 17 minus 5?	12
Fractions	What is 12 divided by 4?	3
Substitution	When x is 4, what is 3x minus 1?	11.0

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-6 min	Slope is a ratio of two changes	From (2,5) to (6,17) y rises by 12 while x rises by 4, so the slope is 3	Explain why the slope is 3
6-18 min	Slope-intercept form	y equals 3x minus 1; at x equal to 0, y is -1, so the y-intercept is -1	Write the equation and name the intercept
18-32 min	Intercepts and the graph	The x-intercept sets y to 0, giving x equal to one third; the two intercepts are where the line meets the axes	Read both intercepts off a graph
32-44 min	Parallel and perpendicular	Parallel means equal slopes; perpendicular slopes multiply to minus one, so the perpendicular to 3 is minus one over 3	Classify three pairs of lines
44-56 min	12-question diagnostic	Four question types, marked on the spot: 9 right, 3 wrong	Independent work
56-60 min	Exit check	Three oral questions; any miss points to the stage to reteach	Oral answers

03 Teaching script (read it out)

Scene 1: Slope is the ratio of two changes

The slope is the change in y over the change in x. From (2,5) to (6,17), y goes from 5 to 17, a change of 12, while x goes from 2 to 6, a change of 4; 12 over 4 is 3. Both points must be taken in the same direction; reversing only one of the two changes turns the slope from positive to negative.
Teaching move: have students write a small arrow beside each point, saying change in y then change in x before dividing.

Students do: find the slope through (1,2) and (5,10), answer 2, and say whether swapping the order changes it.

Scene 2: One slope plus one intercept

The form is y equals mx plus c, with m the slope and c the y-intercept. With slope 3 through (2,5), 5 equals 3 times 2 plus c, so c is -1 and the equation is y equals 3x minus 1. Check: x equal to 4 gives y equal to 11.0, matching the point (4,11).
Teaching move: substitute one point back every time; that single line is the cheapest check available.

Students do: write the line with slope minus 2 through (1, minus 2), which is y equals minus 2x, and explain a y-intercept of 0.

Scene 3: Converting between forms

The general form is ax plus by plus c equals 0. Rearranging y equals 3x minus 1 gives 3x minus y minus 1 equals 0, so a is 3, b is minus 1 and c is minus 1. Going back, keep y on the left, move the rest right and divide.
Teaching move: point out that b cannot be 0, because that is a vertical line with no slope, which is why vertical lines have no slope-intercept form.

Students do: convert 2x plus 3y minus 6 equals 0 into y equals minus two thirds x plus 2 and read off slope and intercept.

Scene 4: Parallel, perpendicular and two special cases

Parallel lines have equal slopes: y equals 3x plus 7 is parallel to y equals 3x minus 1. Perpendicular lines have slopes multiplying to minus one, so the perpendicular slope to 3 is minus one over 3. A horizontal line y equals 4 has slope 0, while a vertical line x equals minus 3 has no slope at all and must never be written as 0.
Teaching move: keep a three-line table (equal, product minus one, undefined) and ask “is this a vertical line” before every classification.

Students do: classify y equals 2x plus 1 against y equals minus one half x plus 3 as perpendicular and give the reason.

04 Question types and worked steps

Type	Example	Steps	Answer
Slope from two points	(2,5) and (6,17)	Change in y over change in x, same direction for both	3
Write the equation	Slope 3 through (2,5)	Substitute to find c, then check with a point	y equals 3x minus 1
Slope from the general form	2x plus 3y minus 6 equals 0	Keep y left, move the rest right and divide	slope minus two thirds, y-intercept 2
Parallel and perpendicular	y equals 3x minus 1 against another line	Equal slopes for parallel, product minus one for perpendicular	perpendicular slope minus one over 3

Sanity check (teach this): Sanity checks: a rising line has positive slope and a falling line negative; horizontal means 0 and vertical means undefined; the order of numerator and denominator must match; perpendicular slopes multiply to minus one; 9 out of 12 is 75 percent, not 9 percent.

05 Independent practice (3 questions)

Question	Answer	What to watch
Three slopes from points	Include one negative and one zero slope	Is the order consistent
Two equations	One from a slope and a point, one from two points	Was a point substituted back
Three classifications	Parallel, perpendicular and a form conversion	Was the vertical case checked first

06 Common mistakes and what to say

Mistake	Student writes	Say this
Slope inverted	Writing 4 over 12 instead of 12 over 4	The numerator is the change in y and the denominator the change in x; inverting them describes the steepness backwards.
Inconsistent order	Using later minus earlier for one point and the reverse for the other	Take both changes in the same direction, write them down first, then divide.
Vertical line given slope 0	Writing 0 as the slope of x equals minus 3	A vertical line has no slope; 0 belongs to a horizontal line, the opposite extreme.
Mixing the two conditions	Using equal slopes to claim perpendicular	Parallel is equal and perpendicular is product minus one; write both sentences in the margin and read them before answering.

07 Homework (paper, section and question numbers)

Secondary maths chapter index — All chapters in one place
Master prep guide index — Compare related chapters
Redo this chapter — Redo diagnostic questions 1 to 6, writing the two changes first

08 Exit check (last 5 minutes)

Question	Answer	Decision
Slope through (2,5) and (6,17)	3	Answering one third means the ratio is inverted; reteach stage one
Perpendicular slope to 3	minus one over 3	Answering 3 means parallel and perpendicular are confused; reteach stage four
Slope of x equals minus 3	Undefined	Answering 0 means vertical and horizontal are confused; reteach stage four

09 Tutor FAQ

Must the slope be a fraction?

An integer or a fraction in lowest terms is fine; if a decimal is required, state the accuracy and never convert a fraction into a recurring decimal midway.

Can one point and the slope give the equation?

Yes: point-slope form is y minus y1 equals m times (x minus x1), which expands into the slope-intercept form.

Why does a vertical line have no slope?

The denominator is the change in x, which is zero for a vertical line; division by zero is undefined, so we say the slope does not exist.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.