Learning objectives
- Recognise, explain, and apply “Coordinates”.
- Recognise, explain, and apply “Linear graphs”.
- Recognise, explain, and apply “Gradient”.
- Recognise, explain, and apply “Intercept”.
- Recognise, explain, and apply “`y = mx + c`”.
KS3 Year 8 / Mathematics / Curriculum
Graphs
Graphs: structured theory, worked examples, answered practice, and a mastery checklist for KS3 Year 8.
CHAPTER PLAN
Learn, practise, and check your progress
Follow the steps in order or jump directly to the part you need.
—/5
completed
Estimated active study time
209 minutes
1. Understand
Objectives, key ideas, and structured theory.
Open →2. Follow the method
Worked examples that explain every step.
Open →3. Practise
Graded tasks with hints and answer guidance.
Open →4. Consolidate
Assignments from core work to challenge.
Open →5. Check
A timed, marked test with solution guidance.
Open →
Progress is stored only in this browser and does not require AI credits.
Key ideas and checks
- I move between formula, table, and graph.
- I interpret slope, intercepts, and key features.
- Every solution or explanation for “Graphs” should include a method, justification, and final check.
Unit
Core theory
The essential chapter ideas in a clear sequence before practice.
Input-output relationship
A function maps each allowed input to one output. Formula, table, and graph are different representations of the same relationship.
Parameters and graph
Parameters change slope, position, amplitude, or rate of change. Read the key features before plotting individual points.
Domain and interpretation
The domain states which inputs are allowed. In a real model, also check whether the output is physically meaningful.
Coordinates
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Linear graphs
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Gradient
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Intercept
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
`y = mx + c`
Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.
Mathematics
Worked examples
Follow the method step by step and check why every step is valid.
Worked function example
For f(x) = 2x - 6, find f(5) and the x-intercept.
- 1Substitute x = 5: f(5) = 2·5 - 6.
- 2For the intercept set f(x) = 0: 2x - 6 = 0.
- 3Solve for x and check the point on the graph.
f(5) = 4, x-intercept = 6/2
Reasoning example
Before calculating, explain the key idea from “Coordinates” and which conditions must be checked.
- 1Define the idea in one clear sentence.
- 2Connect it to a representation, law, or formula.
- 3State a restriction, unit, or final check that makes the solution valid.
The answer should show not only which rule is used for “Coordinates”, but also why it is valid here.
Graphs
Practice with answers
Eight graded tasks from core fluency to exam-style application. Work independently before opening a hint or answer.
Core fluency4 minutes
Solve 4x + 5 = 25.
Hint
Subtract 5 from both sides, then divide by 4.
Answer guide
x = 5.
Core fluency5 minutes
Expand and simplify 7(x + 2) - 2x.
Hint
Multiply every term inside the bracket.
Answer guide
7x + 14 - 2x = 5x + 14.
Application6 minutes
Factorise 3x + 12.
Hint
Take out the common factor 3.
Answer guide
3(x + 4).
Application7 minutes
For f(x) = 6x + 6, find f(12) and solve f(x) = 42.
Hint
Substitute, then solve the equation.
Answer guide
f(12) = 78; x = 6.
Reasoning8 minutes
A sequence starts 3, 12, 21. Find its nth term and 10th term.
Hint
The common difference is 9.
Answer guide
9n + -6; term 10 = 84.
Reasoning9 minutes
Solve 5x + 5 > 30.
Hint
Use equation steps; the coefficient of x is positive.
Answer guide
x > 5.
Exam style10 minutes
Solve x + y = 10, x - y = 6.
Hint
Add the two equations.
Answer guide
x = 8, y = 2.
Exam style12 minutes
A learner writes 4(x + 4) = 4x + 4. Identify and correct the error.
Hint
The outside factor multiplies every term.
Answer guide
4(x + 4) = 4x + 16.
Common mistakes
- Confusing f(x) with the product f times x.
- Plotting without a scale or key points.
- Ignoring domain and model restrictions.
Mastery check
- I move between formula, table, and graph.
- I interpret slope, intercepts, and key features.
- I state domain and relevant restrictions.
Graphs
Chapter homework
Six distinct assignments from core fluency to challenge, each with an estimated time, hint, and answer guide.
Core8 minutes
Homework 1: Solve 7x + 6 = 48.
Hint
Subtract 6 from both sides, then divide by 7.
Answer guide
x = 6.
Core10 minutes
Homework 2: Expand and simplify 3(x + 3) - 3x.
Hint
Multiply every term inside the bracket.
Answer guide
3x + 9 - 3x = 0x + 9.
Stretch12 minutes
Homework 3: Factorise 6x + 30.
Hint
Take out the common factor 6.
Answer guide
6(x + 5).
Stretch15 minutes
Homework 4: For f(x) = 9x + 2, find f(11) and solve f(x) = 20.
Hint
Substitute, then solve the equation.
Answer guide
f(11) = 101; x = 2.
Challenge18 minutes
Homework 5: A sequence starts 4, 9, 14. Find its nth term and 10th term.
Hint
The common difference is 5.
Answer guide
5n + -1; term 10 = 49.
Challenge20 minutes
Homework 6: Solve 8x + 6 > 54.
Hint
Use equation steps; the coefficient of x is positive.
Answer guide
x > 6.
50 minutes / 40 marks
Full chapter test
A timed, full-mark self-assessment with model-answer guidance.
Start the timer when ready, work without notes, show every step, and open model answers only after finishing.
1. Solve 5x + 6 = 36.
2 marksAnswer guide
x = 6.
2. Expand and simplify 8(x + 3) - 3x.
3 marksAnswer guide
8x + 24 - 3x = 5x + 24.
3. Factorise 4x + 20.
3 marksAnswer guide
4(x + 5).
4. For f(x) = 7x + 2, find f(9) and solve f(x) = 16.
4 marksAnswer guide
f(9) = 65; x = 2.
5. A sequence starts 4, 7, 10. Find its nth term and 10th term.
4 marksAnswer guide
3n + 1; term 10 = 31.
6. Solve 6x + 6 > 42.
4 marksAnswer guide
x > 6.
7. Solve x + y = 12, x - y = 6.
5 marksAnswer guide
x = 9, y = 3.
8. A learner writes 5(x + 5) = 5x + 5. Identify and correct the error.
5 marksAnswer guide
5(x + 5) = 5x + 25.
9. Model this: 2 tickets cost x euros each plus a fixed 8 euros. Write the cost and find it for x = 10.
5 marksAnswer guide
C = 2x + 8; C(10) = 28.
10. Represent “Gradient” as an equation, a graph, and a verbal description of the same model.
5 marksAnswer guide
Example y = 4x + 4: a line with gradient 4, intercept 4, meaning “starts at 4 and rises by 4 per unit”.
Unit
Official sources and verification
Curriculum reference sources. Always confirm the teaching sequence with the school and tutor.
Mathematics
Start from the board specification and work topic by topic before full papers.
What this chapter covers
The structure follows the official textbook layout and is used to organise study.
Coordinates
Linear graphs
Gradient
Intercept
`y = mx + c`
Real-life graphs
Conversion graphs
Where to focus
The areas that usually create mistakes or need extra revision.
I move between formula, table, and graph.
I interpret slope, intercepts, and key features.
I state domain and relevant restrictions.
Sources, daily material, and resources
Where to start: textbook, daily material, PDFs, videos, and worked examples.
Start from the official textbook or specification referenced on the subject page.
Use the notes and examples as support, not as a replacement for the official syllabus.
Check the current syllabus version before exam preparation.
Practice by subtopic
Targeted practice before full tests so coverage is clear.
Solve 4x + 5 = 25.
Expand and simplify 7(x + 2) - 2x.
Factorise 3x + 12.
For f(x) = 6x + 6, find f(12) and solve f(x) = 42.
A sequence starts 3, 12, 21. Find its nth term and 10th term.
Solve 5x + 5 > 30.
Solve x + y = 10, x - y = 6.
A learner writes 4(x + 4) = 4x + 4. Identify and correct the error.
Mocks and progress checks
How to measure progress in this chapter and when it enters a cumulative mock.
Start with untimed practice by subtopic.
Move to a short timed checkpoint only after completing the mastery checklist.
Record each error with the correct method and revisit it after 48 hours.
Next step
What to do after finishing the chapter and how it connects to the next unit.
Complete the practice without support.
Explain the core method aloud in under two minutes.
Continue to the next chapter or request targeted tutor support.
Note: for the official examinable syllabus of each school year, always confirm with the school, tutor, and current Ministry/IEP announcements.