S2 Maths | Linear Graphs and Slope: Tutor Prep Guide | y = mx + c, Two Points, Parallel and Perpendicular

2026-11-13

Tutor lesson-prep guides · Junior Secondary Maths

Students can draw a straight line but answer “what does the slope mean” with “it is slanted”. This lesson turns the slope into “how much y rises per step of x”, and uses a filling water tank so that m and c each carry a sentence a student can say out loud.

Objectives — Students can: (1) state that in y = mx + c the m is how much y rises when x increases by one and the c is the value at x = 0, then use A(1, 5) and B(4, 17) to get m = 4 and c = 1, giving y = 4x + 1; (2) read information back off the equation, namely the x-intercept at minus one divided by four, that is minus 0.25, the point (0, 1), and a rise of 12 for every 3 steps right; (3) use slopes to decide parallel lines, where a parallel through (2, minus 3) is y = 4x minus 11, and perpendicular lines, where the product of slopes is minus one and so the perpendicular slope is minus 0.25.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Substitution	What is 4x + 1 when x = 4?	Answers 17; anything else means substitution or order of operations is not secure
Coordinates	In the point (3, 5), which number is x and which is y?	Says x = 3 and y = 5; swapping them inverts every slope in the chapter
Directed number division	What is (17 minus 5) divided by (4 minus 1)?	Answers 4; minus 4 or 12 means the subtraction order or the divisor is wrong

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Tank opening	Describe a tank holding 5 litres that fills at 4 litres a minute and ask how much is in it after 12 minutes	Work it out and name the starting amount and the amount added each minute
5-13 min	Slope as change per unit	Plot A(1, 5) and B(4, 17) and show the change in y divided by the change in x	Compute m = 4 and say that y rises 4 when x rises 1
13-22 min	Recovering the intercept	Show y minus mx to find c and point out that c is the value at x = 0	Get c = 1, write y = 4x + 1 and mark the crossing on the y-axis
22-31 min	Three reading questions	Ask for the x-intercept, the change in y when x rises by 3, and whether (2, 9) lies on the line	Answer minus 0.25, 12, and 9 does not equal 9, each with a reason
31-40 min	Parallel and perpendicular	Write the slopes of two parallel lines and one perpendicular pair and ask for the pattern	State equal slopes for parallel and a product of minus one for perpendicular, then compute minus 0.25
40-45 min	Exit check	Collect three answers and mark correct, partly correct or wrong on the spot	Hand in three answers with one reason each

03 Teaching script (read it out)

Scene 1: Act one: m is the rise per step

The m is not an adjective, it is a ratio. In y = 4x + 1, every time x goes up by 1, y goes up by 4. From A(1, 5) to B(4, 17): x rises by 3, y rises by 12, and 12 divided by 3 is 4, so the slope is 4. If x rose by 3 and y only rose by 3, the slope would be 1 and the line would be much flatter. When an exam asks for the slope, it wants a ratio, not the word steep.

Students restate: slope is the change in y divided by the change in x, that is the rise per step.

Scene 2: Act two: c is the starting point

Once m is known, substitute either point into y = mx + c. With A(1, 5): 5 equals 4 times 1 plus c, so c is 1. The meaning of c is the value of y at x = 0, the point where the line crosses the y-axis. On the graph you start at (0, 1), go right 1 and up 4, then right 1 and up 4 again, and join the points. For the x-intercept ask when y = 0: 0 equals 4x plus 1, so x is minus 0.25. A negative intercept is normal and does not mean the working is wrong.

Students restate: c is the value at x = 0, and every equation should be checked against a point.

Scene 3: Act three: two points, three steps

Given (2, 7) and (6, 19) there are only three steps. First the slope: 19 minus 7 is 12, 6 minus 2 is 4, and 12 over 4 is 3. Second substitute (2, 7): 7 equals 6 plus c, so c is 1. Third write y = 3x + 1 and check with the other point: 3 times 6 plus 1 is 19, correct. Parallel lines keep the slope, so the parallel through (2, minus 3) is y = 4x minus 11 because minus 3 equals 8 plus c. Perpendicular slopes multiply to minus one, so the perpendicular to slope 4 has slope minus 0.25.

Students restate: slope first, then substitute one point for c, then check with the other point.

04 Question types and worked steps

Type	Example	Steps	Answer
Equation from two points	Find the equation of the line through A(1, 5) and B(4, 17).	(1) change in y and change in x (2) divide to get m (3) substitute one point for c (4) check with the other point	m = 4 and c = 1, so y = 4x + 1, and substituting (4, 17) gives 17 as required.
Reading an equation	For y = 4x + 1 find the x-intercept and the change in y when x rises by 3.	(1) set y = 0 and solve (2) multiply m by the increase in x (3) state what the sign means	x-intercept minus 0.25; y rises by 12.
Parallel line	Find the line parallel to y = 4x + 1 that passes through (2, minus 3).	(1) keep the slope (2) substitute the new point for c (3) write the equation (4) confirm the two c values differ	y = 4x minus 11, since minus 3 equals 8 plus c.
Perpendicular line	Find the slope perpendicular to y = 4x + 1 and give one example line.	(1) take the negative reciprocal (2) verify the product is minus one (3) choose a point and write the full equation	Slope minus 0.25, because 4 times minus 0.25 is minus one, for example y = minus 0.25x plus 2.

Sanity check (teach this): Reasonableness check: a positive slope must rise from lower left to upper right, so a negative answer means the change in y was reversed; the change in x must not be zero except for a vertical line, so a zero denominator means the two points share an x value; when x rises by 3 the rise in y must be 3 times m, here 12, and an answer of 3 treats the slope as 1; an x-intercept of minus 0.25 is negative and perfectly normal, so “no intercept” means y was never set to zero; parallel lines must have different c values, and two identical c values describe the same line.

05 Independent practice (3 questions)

Question	Answer	What to watch
Find the equation of the line through (2, 3) and (5, 12).	m = 3 and c = minus 3, so y = 3x minus 3.	Was y subtracted first, and was c found by substitution
A line has slope minus 2 and passes through (4, 1). Find its equation and x-intercept.	y = minus 2x plus 9, x-intercept 4.5.	Is c equal to 9, and was the intercept solved correctly with y = 0
A tank holds V litres after t minutes with V = 4t + 5. How much after 12 minutes, and when is there 65 litres?	53 litres after 12 minutes; 65 litres at 15 minutes.	Was t substituted rather than multiplied, and was the equation reversed to find time

06 Common mistakes and what to say

Mistake	Student writes	Say this
Slope inverted	“The slope is 3 divided by 12, that is 0.25.”	Say: “Slope is the change in y over the change in x, so 12 over 3 is 4. Inverted, you are claiming 4 steps across for 1 up, which flattens the line.”
Intercept read as slope	“In y = 4x + 1 the slope is 1.”	Say: “The number in front of x is the slope; the number added on is the intercept. Slope 4 means up 4 per step, while 1 only fixes where the line starts.”
Parallel given as the same line	“The parallel line is y = 4x + 1.”	Say: “A line is not parallel to itself. Parallel keeps the slope but must change c, for example y = 4x minus 11 through (2, minus 3).”
Reciprocal without the sign	“The perpendicular slope is 0.25.”	Say: “It is the negative reciprocal, so 4 becomes minus 0.25. Check the product: 4 times 0.25 is 1, not minus one.”

07 Homework (paper, section and question numbers)

Maths index: redo practice questions 1 to 3 and plot all three equations on one grid to compare their slopes — This chapter, practice 1-3
Maths index: write your own tank problem with a chosen rate and starting volume, give the equation and answer two questions about it — This chapter, section 5 application question 1

08 Exit check (last 5 minutes)

Question	Answer	Decision
A line passes through (0, 7) with slope minus 3. Write its equation.	y = minus 3x plus 7.	Writing y = 7x minus 3 means intercept and slope were swapped
What is the slope through (1, 2) and (3, 10)?	4, from 8 divided by 2.	Answering 0.25 means the fraction was inverted
How are the slopes of two parallel lines related?	They are equal, with different c values.	Answering that the product is minus one confuses this with perpendicular lines

09 Tutor FAQ

Which year is this for?

S2. S3 reuses the slope when testing regions in inequalities and linear programming, and S4 functions build directly on it.

Is 45 minutes enough for three acts?

Yes. About nine minutes each, leaving time for the parallel and perpendicular matching and the exit check; draw the graph once only.

Is there practice material?

Yes. The maths index links this chapter practice with answers, and it pairs with the simultaneous equations chapter on where two lines meet.

Find a tutor — free Tutor Registration →

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.