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!Q1 Identify m and c in: y = 3x + 4
Q2 What is the gradient from (0, 2) to (2, 6)?
Q3 What does it mean if a line passes through the origin?
Q4 🔄 RECALL Simplify: 34 × 32
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
The Big Idea: Graphs as Real-Life ModelsLinear graphs can model real-life relationships using y = mx + c
m = gradient
=
rate of change
c = y-intercept
=
starting value
Examples: cost & quantity, distance & time, temperature conversion, taxi fares
Direct proportion (solid green) passes through the origin. Non-direct (dashed violet) has c = 4.
📖
Core DefinitionsModel
A mathematical representation of a real situation
Direct Proportion
A relationship that passes through (0, 0)
Intercept (c)
Where the graph crosses the y-axis — the starting value
Gradient (m)
Rate of change — rise ÷ run
🔍 Direct Proportion Test
A line shows direct proportion if it is straight AND passes through (0, 0). Always check the intercept first!✏️
Worked ExamplesWorked Example 01Recognising Direct Proportion
A graph passes through (0, 0) and (2, 6)
1
Check origin: passes through
(0, 0) ✓ → Direct proportion2
Gradient = rise ÷ run =
6 ÷ 2 = 3✓ Answer
y = 3x
💡 No +c term because c = 0 (passes through origin)
Worked Example 02Equation from Two Points
A line passes through (0, 5) and (4, 13)
1
y-intercept: when x = 0, y = 5 →
c = 52
Gradient = (13 − 5) ÷ (4 − 0) =
8 ÷ 4 = 2✓ Answer
y = 2x + 5
Worked Example 03Equation from a GraphNEW
Graph shows: line crosses y-axis at −2, passes through (3, 4)
1
From graph:
c = −22
Gradient = (4 − (−2)) ÷ (3 − 0) =
6 ÷ 3 = 2✓ Answer
y = 2x − 2
⚠️ Always read the scale carefully when identifying the intercept!
Worked Example 04Cost Model
A taxi charges: £4 starting fee + £3 per mile
1
Starting fee = y-intercept:
c = 42
Cost per mile = gradient:
m = 33
x = miles, y = total cost (£)
✓ Answer
y = 3x + 4
Worked Example 05Conversion Graph
Temperature conversion through (0, 32) and (10, 50)
1
Intercept:
c = 32 (when °C = 0, °F = 32)2
Gradient = (50 − 32) ÷ (10 − 0) =
18 ÷ 10 = 1.8✓ Answer
°F = 1.8 × °C + 32
The y-intercept (£4) is the starting fee. The gradient (£3/mile) is the rate.
✅
Process Checklists📋 Writing Equation from Graph
✓
Identify y-intercept first (c)✓
Calculate gradient (rise ÷ run)✓
Use correct subtraction order✓
Write in form y = mx + c📋 Direct Proportion
✓
Line must pass through origin✓
Equation has no +c term✓
Check visually before calculating🧮
Arithmetic SupportRise = y2 − y1
Run = x2 − x1
If (0,0) is on the line → direct proportion
Always check graph scale
💪
Tiered Practice🔓 How It Works
Tap any card to reveal the full working and answer.0 / 11
Problems revealed
Tap problems to reveal working and answers
🟢 Green — Core Skills
🟢 Green · TYPE YOUR ANSWER
Does the line through (0,0) and (2,6) show direct proportion?
✍️ Type Your Answer
A line passes through (0, 3) with gradient 2. Write the equation.
y = mx + c → y = 2x + 3
✓ y = 2x + 3
Green · Q1
Does the line through (0,0) and (3,9) represent direct proportion?
Passes through (0,0) ✓
Line is straight ✓
✓ Yes — direct proportion
Tap to reveal answer
Green · Q2
Find equation: line through (0, 7) and (3, 13)
c = 7 (y-intercept)
Gradient = (13−7)÷(3−0) = 6÷3 = 2
✓ y = 2x + 7
Tap to reveal answer
Green · Q3
Identify intercept from graph crossing y-axis at 5
y-intercept = where x = 0
✓ c = 5
Tap to reveal answer
🟡 Amber — Multi-Step
🟡 Amber · TYPE YOUR ANSWER
Write equation: gradient 3, y-intercept 5
Amber · Q4
Write equation: gradient 5, y-intercept −3
m = 5, c = −3
Use y = mx + c
✓ y = 5x − 3
Tap to reveal answer
Amber · Q5
Graph through (0, 4) and (2, 10). Find equation
c = 4
Gradient = (10−4)÷(2−0) = 6÷2 = 3
✓ y = 3x + 4
Tap to reveal answer
Amber · Q6
Does graph through (0,5) and (2,10) show direct proportion?
Does NOT pass through (0,0)
Passes through (0,5) instead
✓ No — not direct proportion
Tap to reveal answer
Amber · Q7
Gym costs £20 joining fee + £5/month. Write equation
Joining fee = c = 20
Per month = m = 5
x = months, y = total cost
✓ y = 5x + 20
Tap to reveal answer
Amber · Q5b
A plumber charges £40 call-out + £25 per hour. Write an equation for total cost (y) in terms of hours (x).
Call-out fee = intercept = 40
Hourly rate = gradient = 25
y = mx + c
Hourly rate = gradient = 25
y = mx + c
✓ y = 25x + 40
Tap to reveal answer
Amber · Q5c
A graph passes through (0, 0) and (5, 15). Is this direct proportion? Find the equation.
Passes through origin → yes, direct proportion
Gradient = 15 ÷ 5 = 3
c = 0 (through origin)
Gradient = 15 ÷ 5 = 3
c = 0 (through origin)
✓ Yes, direct proportion. y = 3x
Tap to reveal answer
🔴 Red — Exam Style
🔴 Red · TYPE YOUR ANSWER
Taxi costs £4 start + £2.50/mile. Write equation for cost C and miles m
Red · 3 marks · Q8
Line through (0, −1) and (4, 7). Write equation
c = −1
Gradient = (7−(−1))÷(4−0) = 8÷4 = 2
✓ y = 2x − 1
Tap to reveal answer
Red · 3 marks · Q9
Taxi: £5 starting + £2.50/mile. Write cost equation
Starting fee = c = 5
Per mile = m = 2.5
x = miles, y = cost
✓ y = 2.5x + 5
Tap to reveal answer
Red · 3 marks · Q10
Conversion graph: (0, 32) and (100, 212). Find equation & convert 25°C
c = 32
Gradient = (212−32)÷(100−0) = 180÷100 = 1.8
y = 1.8(25)+32 = 45+32 = 77
✓ y = 1.8x + 32; 25°C = 77°F
Tap to reveal answer
Red · 3 marks · Q11
Two phone plans: Plan A costs £10/month + £0.05/text. Plan B costs £15/month with unlimited texts. For how many texts are they equal?
Plan A: y = 0.05x + 10
Plan B: y = 15
Set equal: 0.05x + 10 = 15
0.05x = 5
x = 100
Plan B: y = 15
Set equal: 0.05x + 10 = 15
0.05x = 5
x = 100
✓ 100 texts per month
Tap to reveal answer
Red · 4 marks · Q12
A conversion graph for £ to € passes through (0, 0) and (50, 60). Find the equation and convert £125 to euros.
Gradient = 60 ÷ 50 = 1.2
c = 0 (through origin)
y = 1.2x
£125: y = 1.2 × 125 = 150
c = 0 (through origin)
y = 1.2x
£125: y = 1.2 × 125 = 150
✓ y = 1.2x; £125 = €150
Tap to reveal answer
🔍
Mistake Detective⚠️ These mistakes cost marks
Study each one carefully.❌ Wrong
Writing y = 2x + 5
for graph through (0, 0)
Direct proportion should have c = 0
✅ Correct
y = 2x
(no +c term — passes through origin)
❌ Wrong
Gradient = (x2 − x1) ÷ (y2 − y1)
Rise and run are reversed
✅ Correct
Gradient = (y2 − y1) ÷ (x2 − x1)
Rise over run
🎓
Examiner Lens🎓 To Gain Full Marks
- Show gradient calculation clearly (rise ÷ run)
- Identify intercept correctly from graph or points
- Write full equation in correct form y = mx + c
- Interpret gradient and intercept in context when asked
- Check if line passes through origin for proportion questions
🏆
Mastery Checklist🏆 Unit 9 Mastery
✓
I can identify direct proportion from a graph
✓
I can write an equation from two points or a graph
✓
I can interpret gradient in real-life context
✓
I can interpret intercept in real-life context
✓
I can use conversion graphs and cost models
Nice work — you’ve completed Unit 9!
1 unit remaining. Consistent practice is what turns Grade 3 into Grade 4/5. Keep going.
Ready for Unit 10?
If you can do these 3 things confidently, you're ready to move on:
👨👩👧 For Parents
Your child has completed Unit 9 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
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Unit 10: Coordinate Geometry →
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