Calculate slope and distinguish zero from undefined
Calculate vertical change divided by horizontal change · Handle horizontal and vertical segments separately · Recognise that one repeated point does not determine a unique line
Section Paper 1 · Historical Core · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Signed coordinate differences
- Division
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Calculate vertical change divided by horizontal change
- Handle horizontal and vertical segments separately
- Recognise that one repeated point does not determine a unique line
Use the same endpoint order
For distinct points with different x coordinates, slope m=(y₂−y₁)/(x₂−x₁). Reverse both differences together and the same slope results. Reversing just one changes the sign incorrectly. Slope is a ratio, not the line length or its angle in degrees.
Check the denominator
A horizontal line has zero vertical change and non-zero horizontal change, so slope 0. A vertical line has zero horizontal change and non-zero vertical change; it has no finite slope, not slope 0. Coincident endpoints give 0/0 and do not define a unique segment direction or line. The diagram separates these cases.
Sources for this lesson
Pause and explain
What is the slope of a vertical line?
Worked example
Find the slope from A(−2,1) to B(2,3).
Show the worked solution
- Vertical change=2; horizontal change=4.
- Slope=2/4=1/2; reversing both changes gives (−2)/(−4)=1/2.
Answer 1/2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find slopes A→B and B→A for A(−2,1), B(2,3).
Give me a hint
Reverse both differences together.
Compare my reasoning
- A→B gives 2/4=1/2.
- B→A gives (−2)/(−4)=1/2.
Both slopes are 1/2.
Look for these in your work
- Match subtraction orders
- Check reversal leaves slope unchanged
Find the slopes of C(3,−2) to D(3,4), and E(−2,−1) to F(2,−1).
Give me a hint
Check which coordinate changes.
Compare my reasoning
- CD is vertical: horizontal change 0, so no finite slope.
- EF is horizontal: slope 0/4=0.
CD: no finite slope; EF: slope 0.
Look for these in your work
- Separate zero and undefined
- Use the denominator condition
Two listed endpoints are both P(0,4). Can these data determine a unique slope or line?
Give me a hint
Both coordinate changes are zero.
Compare my reasoning
- The proposed slope is 0/0, which is undefined.
- Infinitely many lines pass through P, so the repeated point is insufficient.
No; one repeated point does not determine a unique line or slope.
Look for these in your work
- Recognise degenerate data
- State what information is missing
Common mistakes
- Use one consistent subtraction order.
- Do not turn division by zero into slope zero.
What can you explain now?
What is the slope of a vertical line?
Compare with the explanation
No finite slope; its horizontal change is zero.
A non-zero vertical change cannot be divided by a zero horizontal change.
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Make it stick
What is the slope of a vertical line?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.