Read a line graph with correctly spaced time inputs
Read paired measurements at their recorded times · Use unequal horizontal spacing for unequal time gaps · Distinguish observations from interpolation between them
Section Paper 3 · Historical Core · 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
- Cartesian coordinates
- Rates of change
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
- Read paired measurements at their recorded times
- Use unequal horizontal spacing for unequal time gaps
- Distinguish observations from interpolation between them
Use the time scale
The illustrative temperature readings at 0,1,3,6 hours are 2,6,10,7°C. Horizontal gaps are one, two and three hours, not three equal steps. A line joins successive recorded values to display a trend; an intermediate value on that line is an interpolation model, not an additional observed measurement.
Change per unit time
Between 0 and 1 hours temperature rises 4°C, averaging 4°C/h. Between 1 and 3 it rises 4°C over two hours, averaging 2°C/h. Between 3 and 6 it falls 3°C over three hours, averaging −1°C/h. Equal vertical changes can have different rates. A straight join does not prove the actual temperature changed steadily throughout the interval.
Sources for this lesson
Pause and explain
Should times 0,1,3,6 be plotted with equal horizontal gaps?
Worked example
Find the average temperature change from hour 1 to hour 3.
Show the worked solution
- Change=10−6=4°C.
- Time interval=2 hours.
Answer 2°C/h.
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.
Read the temperature at hour 3 and the change from hour 3 to hour 6.
Give me a hint
Use the two plotted observations.
Compare my reasoning
- At hour 3 it is 10°C; at hour 6 it is 7°C.
- Change is 7−10=−3°C.
10°C; a fall of 3°C.
Look for these in your work
- Read the axis units
- Retain the direction of change
Using straight-line interpolation on the illustrated join, estimate the temperature at hour 2. Is it a recorded observation?
Give me a hint
Hour 2 is halfway between hours 1 and 3.
Compare my reasoning
- The interpolated value is halfway between 6 and 10°C: 8°C.
- No reading at hour 2 was supplied.
Estimated 8°C under the straight-join model; not a recorded observation.
Look for these in your work
- State the interpolation assumption
- Do not invent an observation
Compare the average changes over hours 0–1 and 1–3. Why do equal rises not mean equal rates?
Give me a hint
Divide each rise by its own elapsed time.
Compare my reasoning
- Both rises are 4°C.
- Their durations are 1 and 2 hours, giving 4 and 2°C/h.
4°C/h then 2°C/h; the second rise took twice as long.
Look for these in your work
- Use the actual time intervals
- Separate change from rate
Common mistakes
- Use actual time gaps, not point positions in a list.
- A connected segment is not evidence of constant change between observations.
What can you explain now?
Should times 0,1,3,6 be plotted with equal horizontal gaps?
Compare with the explanation
No; use gaps proportional to 1,2,3 hours.
Horizontal distance must follow the numerical time scale.
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Make it stick
Should times 0,1,3,6 be plotted with equal horizontal gaps?
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.