How to use this lesson
1. Complete the warm-up questions from memory
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
🔥
Warm-Up: Retrieval Practice⏱ Before You Start
Answer from memory — no notes! Get all 3 correct before moving on.Q1 What are the coordinates of a point 3 right and 4 up from the origin?
Q2 Find the gradient from (0, 0) to (2, 6).
Q3 True or False: Parallel lines have the same gradient.
Q4 🔄 RECALL Find the nth term of: 5, 8, 11, 14
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
The Big Idea: The Journey MapEvery straight line can be described by one simple formula:
y = mx + c
Think of it like giving directions: m tells you how steep the hill is, c tells you where you start.
m = gradient (how steep)
c = y-intercept (where you start on the y-axis)
The gradient tells you how steep the line is. A bigger number means steeper. Negative means it slopes downhill.
📖
Core DefinitionsCoordinate
A point written (x, y). Memory trick: "along the corridor (x), up the stairs (y)"
Gradient
How steep a line is. Calculate: rise ÷ run (change in y ÷ change in x)
y-intercept
Where the line crosses the y-axis. It's the y-value when x = 0. In y = mx + c, it's the c
Parallel Lines
Lines with the same gradient — they never meet because they have identical steepness
🎯 Why Gradient Works
If a line goes up 6 and across 3:
Gradient = 6 ÷ 3 = 2
This means for every 1 step you take to the right, the line goes up 2.
✏️
Worked ExamplesWorked Example 01Plotting Points
Plot (2, 5) and (5, 11).
1
For (2, 5): move right 2 along the x-axis, then up 5. Mark the point.
2
For (5, 11): move right 5, then up 11. Mark the point.
3
Join both points with a straight line using a ruler.
Go along the corridor (x = 2), then up the stairs (y = 5)
✓ Key Rule
Always read x first (across), then y (up or down).
Worked Example 02Reading Coordinates from a GraphNEW
Point A is 4 units right and 3 units up from the origin. What is its coordinate?
1
Read across first (x-value): A is 4 units to the right → x =
42
Read up next (y-value): A is 3 units up → y =
3Memory trick: "Along the corridor, up the stairs" = (x, y)
✓ Coordinate
(4, 3)
⚠️ The most common mistake is writing (y, x) instead of (x, y). Always go across FIRST.
Worked Example 03Finding Gradient from Two Points
Find the gradient from (1, 3) to (4, 12).
1
Find change in y (rise):
12 − 3 = 92
Find change in x (run):
4 − 1 = 33
Divide:
9 ÷ 3 = 3Rise (green) ÷ run (amber) = gradient. Always do y-change first, then x-change.
✓ Gradient
3
Worked Example 04Identifying m and c
Given: y = 4x + 7. Identify the gradient and y-intercept.
1
Compare with
y = mx + c. The number in front of x is m (gradient).2
The number on its own is c (y-intercept).
✓ Result
Gradient m = 4 | y-intercept c = 7
Worked Example 05Drawing a Line from y = mx + cNEW
Draw the line: y = 2x + 1
1
Find the y-intercept: c = 1, so plot the point
(0, 1) on the y-axis. This is your starting point.2
Use the gradient: m = 2 means "go right 1, go up 2." From (0, 1), move right 1 and up 2 → new point:
(1, 3)3
Draw: join the points with a straight line using a ruler. Extend both ways.
Always plot c first (orange), then use m to find the next point (green), then draw the line
✓ Key Rule
Always plot the y-intercept first, then use the gradient to find a second point.
Worked Example 06Writing the Equation from a GraphNEW
A line crosses the y-axis at 3 and goes up 2 for every 1 across. Write its equation.
1
Read the y-intercept: the line crosses the y-axis at 3, so
c = 32
Find the gradient: goes up 2 for every 1 across, so
m = 23
Substitute into y = mx + c: put the numbers in place of the letters
Read c from the y-axis, calculate m from rise ÷ run, then plug into y = mx + c
✓ Equation
y = 2x + 3
Worked Example 07Negative Gradient
A line passes through (0, 10) and (5, 0). Find its gradient.
1
Change in y (rise):
0 − 10 = −10 (it goes DOWN, so it's negative)2
Change in x (run):
5 − 0 = 53
Divide:
−10 ÷ 5 = −2Negative gradient = line goes downhill from left to right. The minus sign tells you the direction!
✓ Gradient
−2
⚠️ If the answer is negative, it means the line goes downhill. Positive = uphill, negative = downhill.
✅
Method Checklists📋 Finding Gradient
✓
Identify two points✓
Change in y first (rise)✓
Change in x second (run)✓
Divide rise by run📋 Drawing from y = mx + c
✓
Plot y-intercept first✓
Use gradient (rise / run)✓
Draw straight line through📋 Writing Equation from Graph
✓
Find y-intercept (x = 0)✓
Calculate gradient✓
Write as y = mx + c🧮
Arithmetic Support StripNegative ÷ positive = negative
Always subtract in same order
Rise = y2 − y1
Run = x2 − x1
Gradient = rise ÷ run
c = value of y when x = 0
🎯
Interactive Practice ZoneHow to use
Attempt each question first. Then tap a card to reveal the full working and answer.🟢 Green — Core Skills
✍️ Type Your Answer
State the gradient and y-intercept of: y = 3x − 5
In y = mx + c: gradient m = 3, y-intercept c = −5
✓ m = 3, c = −5
Problem 1
Plot the point (2, 4)
Move right 2 along x-axis
Move up 4
Mark the point
✓ Point at (2, 4)
👆 Tap to reveal
Problem 2
Gradient from (0, 0) to (4, 8)
Rise = 8 − 0 = 8
Run = 4 − 0 = 4
Gradient = 8 ÷ 4
✓ Answer: 2
👆 Tap to reveal
Problem 3
Identify m and c in y = 5x − 2
Compare with y = mx + c
m = number before x
c = number on its own
✓ m = 5, c = −2
👆 Tap to reveal
Problem 4
A point is 3 right, 7 up from origin. Coordinates?
x = 3 (across first)
y = 7 (up second)
✓ Answer: (3, 7)
👆 Tap to reveal
🟠 Amber — Multi-Step
🟡 Amber · TYPE YOUR ANSWER
State m and c in y = 7x − 3
Problem 5
Draw the line y = 3x + 2
c = 2 → plot (0, 2)
m = 3 → rise 3, run 1
From (0,2): right 1, up 3 → (1, 5)
Join with straight line
✓ Line through (0, 2) and (1, 5)
👆 Tap to reveal
Problem 6
Gradient from (2, 3) to (6, 15)
Rise = 15 − 3 = 12
Run = 6 − 2 = 4
Gradient = 12 ÷ 4
✓ Answer: 3
👆 Tap to reveal
Problem 7
Write the equation: gradient 4, y-intercept −1
m = 4, c = −1
Substitute into y = mx + c
✓ Answer: y = 4x − 1
👆 Tap to reveal
Problem 8
Gradient from (0, −4) to (4, 4)
Rise = 4 − (−4) = 4 + 4 = 8
Run = 4 − 0 = 4
Gradient = 8 ÷ 4
✓ Answer: 2
👆 Tap to reveal
🔴 Red — Exam Style (Show Full Working)
🔴 Red · TYPE YOUR ANSWER
Find the gradient from (1, 2) to (5, 10)
Problem 9 · 2 marks
Draw y = −2x + 5
c = 5 → plot (0, 5)
m = −2 → rise −2, run 1
From (0,5): right 1, DOWN 2 → (1, 3)
Join with straight line
✓ Line through (0, 5) and (1, 3) — slopes downward
👆 Tap to reveal
Problem 10 · 2 marks
Gradient between (1, −2) and (5, 6)
Rise = 6 − (−2) = 6 + 2 = 8
Run = 5 − 1 = 4
Gradient = 8 ÷ 4
✓ Answer: 2
👆 Tap to reveal
Problem 11 · 2 marks
A line crosses y-axis at −3 and has gradient ½. Write its equation.
m = ½, c = −3
Substitute into y = mx + c
✓ Answer: y = ½x − 3
👆 Tap to reveal
0 / 11
Problems revealed
Tap problems to reveal working and answers
🔍
Mistake Detective⚠️ These mistakes cost marks
Study each one carefully — then look for them in your own work.❌ Wrong
Gradient = run ÷ rise
Formula reversed — run and rise are swapped
✅ Correct
Gradient = rise ÷ run
(change in y ÷ change in x)
❌ Wrong
y = 2x + 5 drawn without
plotting (0, 5) first
y-intercept not plotted — line starts from wrong position
✅ Correct
Step 1: Plot (0, 5) first
Step 2: Use gradient m = 2
Step 3: Draw line through
🎓
Examiner Lens🎓 To Gain Full Marks
- Show rise and run clearly — write the subtraction
- Plot y-intercept first before using gradient
- Write the full equation in the form y = mx + c
- Draw a straight line accurately — use a ruler
- Include units when reading values from real-life graphs
🏆
Mastery Checklist🏆 Unit 6 Mastery
✓
I can plot coordinates accurately
✓
I can calculate gradient from two points
✓
I can draw a line from y = mx + c
✓
I can write the equation from a graph
✓
I show full method clearly
Nice work — you’ve completed Unit 6!
4 units remaining. Consistent practice is what turns Grade 3 into Grade 4/5. Keep going.
Ready for Unit 7?
If you can do these 3 things confidently, you're ready to move on:
👨👩👧 For Parents
Your child has completed Unit 6 of a structured GCSE Foundation Maths revision programme covering Algebra and Graphs — key topics on every Foundation paper.
✔ AQA, Edexcel & OCR aligned
✔ 190+ questions & 70 worked examples
✔ Step-by-step explanations
Up Next
Unit 7: Non-Linear Graphs →
Your Guru Academy is an independent resource and is not affiliated with, endorsed by, or connected to AQA, Edexcel (Pearson), OCR, or any exam board. All exam board names are the property of their respective owners. Content is designed to support Foundation GCSE revision but does not guarantee any specific grade outcome.