Transform data and spatial representations
Match equivalent charts, interpret gradients and track features under a stated rotation.
Original StudyForA scenarios and formative model answers; no Cambridge marks, predicted grades or certified mastery. Alternative reasoned answers may be valid. Full-length unseen papers, a wider mixed-problem bank and calibrated specialist teacher marking remain outstanding.
Before you begin
- Read tables and instructions in English
- Use arithmetic, ratios and whole-number constraints
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before calculating, what should you identify?
What you will learn
- Recognise equivalent representations of data
- Track spatial features and interpret a graph in context
Keep the same quantities across representations
A count table, proportional bar chart and pie chart can represent the same distribution. Check the total and each category’s proportion, rather than whether the pictures look similar. A pie angle is category count divided by total, multiplied by 360°.
Counts A=12, B=18 and C=30 total 60. They give proportions 20%,30%,50% and pie angles 72°,108°,180°. A chart with angles 120° each would describe equal counts, so it is not equivalent.
Track orientation and axis meaning
For a square tile, label corners before rotating. A 90° clockwise rotation sends top-left to top-right, top-right to bottom-right, bottom-right to bottom-left and bottom-left to top-left. Reflection uses a different rule and can reverse handedness.
A graph’s gradient is vertical change divided by horizontal change, with contextual units. If cost rises from 8 to 14 as distance rises from 2 to 5 km, the gradient is 2 credits/km. A vertical intercept may represent a fixed charge, but only if the stated model and domain support that interpretation.
Sources for this lesson
Pause and explain
Where does a top-left mark go after a 90° clockwise tile rotation?
Worked example
Convert the A,B,C counts to pie angles and verify the total.
Show the worked solution
- Use total 60.
- Calculate 12/60×360=72, 18/60×360=108, 30/60×360=180.
- Add the angles to 360.
Answer The angles are 72°,108° and 180° and sum to 360°. The proportions match the original counts exactly.
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.
Guided: What pie angle represents B=18 out of 60?
Give me a hint
Convert its fraction of the total.
Compare my reasoning
- 18/60=0.3.
- 0.3×360=108.
- Label the angle in degrees.
B occupies 108°, representing 30% of the observations. Using 18° would mistake a count for an angle.
Look for these in your work
- Uses the total
- Converts to angle
- Labels the result
Independent: A tile has a dot top-left and a stripe bottom-right. Locate both after 180° rotation.
Give me a hint
Each corner moves to its opposite.
Compare my reasoning
- Top-left moves to bottom-right.
- Bottom-right moves to top-left.
- Keep the two features distinct.
The dot moves to bottom-right and the stripe to top-left. A half-turn swaps opposite corners without reflecting the tile.
Look for these in your work
- Tracks the dot
- Tracks the stripe
- Uses rotation rather than reflection
Application: A linear hire graph passes through (2 km,8 credits) and (5 km,14 credits). Find the per-kilometre rate and fixed charge in cost=a+bd.
Give me a hint
Use the gradient, then substitute a point.
Compare my reasoning
- b=(14−8)/(5−2)=2 credits/km.
- 8=a+2×2 gives a=4.
- Check 4+2×5=14.
The model is cost=4+2 d, with a 4-credit fixed charge and 2 credits per kilometre. The second point verifies the model within the stated linear rule.
Look for these in your work
- Finds the gradient
- Finds the fixed charge
- Checks another point
Common mistakes
- Matching charts by appearance only
- Confusing rotation with reflection
- Giving a gradient without its contextual units
What can you explain now?
Where does a top-left mark go after a 90° clockwise tile rotation?
Compare with the explanation
Top-right
Track the corner around the centre in the specified direction.
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Make it stick
Where does a top-left mark go after a 90° clockwise tile rotation?
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.