Stimulus question 44
Rhayader has Moved!
On the A44 travelling west between Llanfihangel-nant-Melan and Penybont in Powys there are two road signs that indicate the mileage to the towns of Llandrindod Wells and Rhayader. From Llanfihangel-nant-Melan there is only one direct route using major roads to reach these towns.
The first, just under half a mile after the Fforest Inn junction with the A481 shows:
A44 Llandrindod Wells 12 Rhayader 15
The second sign is situated 5.3 miles further along the same road, just after Penybont. It shows:
A44
Llandrindod Wells 5 Rhayader 10
Question Is it possible that all the mileages are correct or is there a mistake? Show your working clearly.
Extension Question If two signposts are accurately marked as showing:
Llandrindod Wells 15 Llandrindod Wells 8 Rhayader 20 Rhayader 12
What could be the distance between the two signposts? What are the maximum and minimum possible distances between the two signposts?
1 ‘Rhayader has Moved’ Question – scoring Full credit Shows the minimum and maximum possible mileages for the two towns for both signs: Llandrindod Wells: 11.5 miles to 12.5 miles, and 4.5 miles to 5.5 miles. Rhayader: 14.5 miles to 15.5 miles, and 9.5 miles to 10.5 miles.
Concludes that although the mileages for Rhayader may be correct, they cannot be correct in both cases for Llandrindod Wells, if the distance between the signs is 5.3 miles. Even if the actual distance at the first sign was 11.5 miles, to then travel 5.3 miles means that the second sign is still 6.2 miles from Llandrindod Wells and so should read 6 miles at least. OR Suggests that the first sign is incorrect, and should read 10 or 11 miles, showing working.
Partial credit States that the mileages for Rhayader can be correct, but both mileages for Llandrindod Wells cannot, as the difference is 7. Makes no reference to rounding, or that the mileages could be out by up to half a mile each way. OR Shows the minimum and maximum possible mileages for the two towns for both signs, but goes no further.
No credit Any other response
Support Provide a copy of the map of this part of Powys. Using the scale, find the actual distances. Advise learners to draw a diagram of the road, and show the upper and lower bounds of the distances at each signpost. E.g. a starting point could be the diagram on the PowerPoint slide: A44
Rhayader Penybont Llanfihangel -nant-Melan
Llandrindod Wells
Solutions There is a mistake on the signs. The actual distances (taken from a map) to each town along the A44 and A438 at each sign are:
Llandrindod Wells 10.4 Llandrindod Wells 5.1
Rhayader 15.3 Rhayader 10.0
2 It is likely that learners will choose any of several methods that could solve the problem. Some may be more efficient than others, using alternative mathematical techniques. A sheet with two examples of possible working is enclosed. Teachers may wish to share these with learners for them to discuss and evaluate, comparing these with the learners’ own solutions. Learners should therefore be encouraged to look for efficiency in the different solutions, and discuss which may be an optimum strategy.
Developing numerical reasoning Identify processes and connections • Identify what further information might be required and select what information is most appropriate; • Select appropriate mathematics and techniques to use.
Represent and communicate • Explain results and procedures precisely using appropriate mathematical language.
Review • Draw conclusions from data and recognise that some conclusions may be misleading or uncertain.
GCSE Subject Content Foundation Intermediate Higher
Recognising that measurement is approximate and that a measurement
expressed to a given unit is in possible error of half a unit.
‘Rhayader has Moved’ Extension Question – solution Any value between 7 and 8 miles.
Full credit Shows the minimum and maximum possible mileages for the two towns for both signs: Llandrindod Wells: 14.5 miles to 15.5 miles, and 7.5 miles to 8.5 miles. Rhayader: 19.5 miles to 20.5 miles, and 11.5 miles to 12.5 miles.
Shows that for Llandrindod mileage to be correct, the signposts could be between 6 and 8 miles apart, but for Rhayader to be correct the signposts could be between 7 and 9 miles apart. For both to be correct, the solution is between 7 and 8 miles.
Partial credit Shows the minimum and maximum possible mileages for the two towns for both signs, but goes no further. Shows one specific solution, rather than a range.
No credit Any other response
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