The gradient from two points
(−1, −1) and (2, 5)
up: 5 − (−1) = 6, across: 2 − (−1) = 3
m = 6 ÷ 3 = 2
5 = 2 × 2 + c, so c = 1
y = 2x + 1
Maths · Foundation and Higher · AQA · 45 minutes
Straight-line graphs What do m and c do to a straight line?
Students find a line's equation from three points, then move m and c to see what each one does. Slides to present, a vote that shows who's stuck, practice and homework.
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Move m. One point never moves. Which one, and why?
y = 2x + 1
The idea in one minute
In y = mx + c, c is where the line crosses the y-axis, because there x = 0 and y = c. m is the gradient: how far the line goes up for each one step to the right. A negative m means the line goes down; m = 0 means it is flat.
The lesson goes picture first. Students write an equation for three points before anyone explains, then move c and m on the graph and say what each does before they are told. Only then do they find m from two points and rearrange equations into y = mx + c. The core is for Foundation and Higher; the stretch finds a line through two points with a negative gradient.
By the end, students can
- read the gradient and y-intercept from y = mx + c
- find the gradient from two points as up ÷ across
- write the equation of a line through two points
- rearrange an equation into y = mx + c, and spot parallel lines
Key words
- gradient
- y-intercept
- parallel
- coordinates
- y = mx + c
Before this lesson
Students should plot coordinates in all four quadrants and substitute a number into an expression such as 2x + 1. Gradient is taught here, from the graph.
The lesson, step by step
30 minutes in total. Each step says what changes for this length.
3 min 1 · Hook · students think
Guess my line
Students write an equation for three points before anyone explains.
For this length: Show the three points and y = 2x + 1 together; skip the guessing round
Say “These three points are on one straight line: (0, 1), (1, 3) and (2, 5). Write an equation it could have. Don't show anyone yet.”
Ask- What happens to y each time x goes up by 1?
- Where does it cross the y-axis?
Watch for Writing y = x + 2 because it works for (1, 3): ask them to try (2, 5).
For the teacher
Give two minutes of silent writing, then gather three or four equations. Test each one with all three points: does every point work?
Don't explain yet. The question is the point: which numbers in the equation set the line's slope and where it sits?
Gather a few guesses with Ask the class. Keep y = 2x + 1 for the next step, and a wrong one to test.
Shorter: Show the three points and y = 2x + 1 together; skip the guessing round (−2 min).
8 min 2 · Explain · you explain
What c does, then what m does
Move c, then m: predict first. Then the gradient from two points.
For this length: Move c and m without the votes
Say “Before I move anything: what will change if I make c bigger? Write one word.”
Ask- c is now 3. Where does the line cross the y-axis?
- What will the line look like when m is 0? When m is −1?
Watch for Thinking a bigger c makes the line steeper: c moves the whole line up, the slope stays.
For the teacher
On the first slide keep m = 2 and move c. At x = 0 the equation says y = c, so the line crosses the y-axis at c. The slope never changes: every line stays parallel to the first.
On the second slide keep c = 1 and move m. For each step right the line goes up m. A negative m goes down; m = 0 is flat. The point (0, 1) never moves, which answers the hook's question.
Then the gradient from two points: up ÷ across, both measured the same way round. Put one point in to find c.
Vote before each move (Ask the class): c to 3, then −2; m to 1, 0, then −1.
Shorter: Move c and m without the votes (−4 min).
5 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
For this length: Do the first pair only
Say “Watch me do one. Then you do its twin, which is almost the same.”
Ask- What is the same in the twin? What changed?
Watch for Dividing across by up: the gradient is up ÷ across.
A line through two points
(1, 4) and (3, 8)
m = (8 − 4) ÷ (3 − 1) = 2
4 = 2 × 1 + c, so c = 2
y = 2x + 2
Then theirs
(1, 1) and (3, 7)
The number with x first
y = 5 − 3x
m = −3, c = 5
Then theirs
y = 7 − x
Rearranged first
2y = 4x + 6
y = 2x + 3
m = 2, c = 3
Then theirs
3y = 6x − 9
For the teacher
Work the first example aloud, writing each line. Then students solve its twin while you watch who writes up ÷ across.
Do the same for each pair. The second pair catches students who read the first number as the gradient; the third, those who forget to divide c too.
Open the answers slide once most have finished their twins.
Shorter: Do the first pair only (−3 min).
9 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: After the vote, questions 1–4 only
Say “Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- How do you know which number is the gradient?
Watch for Reading the number written first as the gradient.
Show your working on paper. Write negative numbers with a minus sign. The stretch is optional.
The question that shows who's stuck
- y = 3 − 2x. What are its gradient m and its y-intercept c?
In your Classroom, as they answer (an example) - A11
- B8
- C4
- D2
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- y = 4x − 3 has gradient ___ and y-intercept ___
- y = 6 − x. What is its gradient?
- A line goes through (0, 2) and (2, 8). What is its gradient?
- 2y = 8x + 2. What is the gradient of this line?
- Which line is parallel to y = 2x + 1?
- Is the point (3, 7) on the line y = 2x + 1?
- y = 2x + 1 meets the x-axis where y = 0, at x = ___
- The line y = mx + 1 goes through (2, 7). What is m?
- Which line is steeper: y = −4x or y = 3x?
- Stretch: the line through (1, 5) and (3, 1) is y = ___x + ___
Support · For question 3: count up from y = 2 to y = 8, then across from x = 0 to x = 2. The gradient is up ÷ across. The line crosses the y-axis where x = 0.
For the teacher
Everyone chooses before anyone explains. If they read the first number as the gradient, ask which number is multiplied by x. If they lost the sign, ask what the minus belongs to.
Questions 1–2 build fluency; 3–4 change the look; 5–7 ask for reasons; 8 works backwards; 9 compares steepness. The stretch has a negative gradient.
Students show their working on paper or the board; the quiz keeps their final answers.
If many read the first number as the gradient, rewrite the equation together with the x term first, before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: why are lines with the same m parallel? (+10 min).
5 min 5 · Exit · students show
Sketch three lines
Three sketches and one sentence show who sees what m and c each do.
Say “Sketch three lines on one pair of axes. Then answer the exit question in one or two sentences.”
Watch for A parallel line drawn through the same y-intercept: that is the same line.
One or two sentences are enough.
- Write the equation of a line that is steeper than y = 2x + 1 and crosses the y-axis at the same point. Say how you know.
For the teacher
Sketch y = 2x + 1, then a line parallel to it, then a line that goes down and crosses the y-axis at 1. Label where each crosses the y-axis.
Then answer the exit question.
The sentences show who still mixes up m and c: start the next lesson with the most common mistake. Set the homework.
Longer: Compare sketches: how do you know two lines are parallel? (+5 min).
The answers, their slides and the mistake behind each wrong answer come with your copy. The worked examples above are open to everyone.
One-to-one: let the student move m and c first and say what they see before you name anything; ask them to check each equation by putting a point back in.
Homework: read, find and sketch
Show your working. For question 3, find the gradient as up ÷ across, then put a point in to find c. Check each equation with a point.
- Write the gradient and y-intercept of y = 5x + 2.
- Write the gradient and y-intercept of y = 4 − 3x.
- Find the equation of the line through (0, −1) and (2, 5).
- Write 2y = 6x + 8 in the form y = mx + c.
- Is the point (4, 9) on the line y = 2x + 1? Show how you know.
- Find where y = 3x − 6 meets the x-axis. Then sketch the line, labelling where it crosses both axes.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching straight-line graphs: common questions
How do you teach y = mx + c so that it sticks?
Let students see it before they are told it. Give three points on a line and ask for its equation, then move c and m on a graph and ask what each does. c moves the line up and down; m tilts it. Then name them: the y-intercept and the gradient.
How do you find the gradient of a line from two points?
Gradient is up ÷ across: the change in y divided by the change in x, both worked out in the same order. Through (−1, −1) and (2, 5) the line goes up 6 and across 3, so the gradient is 2. A line going down has a negative gradient.
Is this straight-line graphs lesson suitable for Foundation GCSE?
Yes. The core reads m and c, finds a gradient from two points and spots parallel lines, for students who can plot coordinates and substitute into an expression. The stretch, the same with a negative gradient, is a step up at Foundation.
Can I use the lesson without creating an account?
Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.
Curriculum reference: AQA GCSE Mathematics 8300: A9 (straight-line graphs, y = mx + c, parallel lines, the line through two points) and A10 (gradients and intercepts).
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