Contextualised task 34 The Snowdon Horseshoe
Teaching Notes
This task is based on the mathematics involved in creating a route card for a journey in the mountains. In particular students will investigate methods of estimating timings. This will involve using worded formulae and, ultimately, trigonometry. While not essential, the whole task could be completed using a spreadsheet. If this approach is to be used, all the data is provided in an accompanying file.
Task A: Naismith’s Rule
Outline Students are presented with information about the route known as the Snowdon Horseshoe. They use Naismith’s Rule to calculate an estimate of the time taken on sections of the route. They have the option of creating a route profile.
You will need: Teachers’ script PowerPoint Question sheet Spreadsheet (optional) Mark scheme
Task B: Langmuir’s Refinements
Outline Students consider a refinement to Naismith’s Rule which provides an improved model for estimating timings. They calculate angles of elevation that are needed to enable ‘Langmuir’s Refinement’ to be used.
You will need: Question sheet Spreadsheet (optional) Mark scheme
Task A: Teachers’ script for PowerPoint presentation
The text in the right-hand boxes provides a possible script to be read to students. However, it is probably preferable to use your own words and elaboration. When questions are asked, time for discussion in pairs / groups should be provided. Ensure that students are given opportunity to explain their reasoning in response to these questions. All students need to understand the concepts in order to make progress with the task.
Slide Keep this slide on the screen until you are ready 1 to start the presentation The Snowdon Horseshoe
Slide Has anyone ever climbed Snowdon? Do they 2 know which way they climbed it?
There are a number of different ways to reach the top of Snowdon – the highest point in Wales.
The Snowdon Horseshoe is widely regarded as the best mountaineering route in the whole of England and Wales. As this photo shows, it is a challenging walk and not really suitable for inexperienced walkers or people who are nervous of heights!
Slide Are any students taking part in the Duke of 3 Edinburgh award? Or are any involved in other activities that involve planning an expedition?
When planning a walking or mountaineering route it is often useful to produce a route card.
A route card will divide the walk into ‘legs’. Each leg starts and finishes at a clear point of reference on the map. This Snowdon Horseshoe route starts at Pen-y-pass Youth Hostel.
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The six-figure grid reference of each point is noted.
It may be appropriate to have a discussion about how six-figure grid references work.
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The height above sea level, in metres, is the next important piece of information to go on the route card. Why do you think this is?
e.g. it aids navigation and time-planning
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The bearing is also worked out. This will be the bearing on which you would start walking for the leg. (Note that the bearing would almost certainly not stay the same throughout the leg.)
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The distance for the leg is also worked out. Here it is stated in metres. You need to understand how to use a map scale to work out these distances.
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Other information is then worked out. The ascent is the total height gain in that leg. What is the ascent on the leg from Bwlch Glas to Snowdon summit?
92 metres
Find another ascent figure that should be added to the route card. And another.
In this task you will be exploring ways of estimating time for each leg in a journey.
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The route card can also be sued to work out a ‘route profile’. On the route profile, which peak is Y Lliwedd West Peak? How do you know?
It is the fourth peak from left to right
Slide William Naismith was a Scottish mountaineer 4 born in 1856. He created a way to estimate Naismith’s Rule journey times when walking in the hills and mountains. This rule is now named after him. For a fit hillwalker:
• Allow one hour for every 4 kilometres travelled
• Add 10 minutes for every 100 metres of ascent Naismith thought that a fit walker should walk at an average speed of 4 kilometres per hour. The Snowdon Horseshoe But he also knew that people slow down when 11 kilometres with 1020 metres of ascent walking uphill. He made an adjustment: 10 minutes should be added to the journey time for every 100 metres of ascent.
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The Snowdon Horseshoe walk is 11 km long. There are 1020 metres of ascent throughout the walk. Use Naismith’s Rule to estimate the length of time that the walk should take. (Note that students will have to be careful with units of time here)
4 hours and 27 minutes
Slide In 1984, Eric Langmuir suggested some 5 refinements to Naismith’s rule. These Langmuir’s refinements refinements take account of the fact that travelling downhill also changes the walking For a fit hillwalker:
• Allow one hour for every 4 kilometres travelled pace. A steady downhill makes walking quicker,
• Add 10 minutes for every 100 metres of ascent but a steep downhill slows the pace.
• Subtract 10 minutes for every 300 metres of descent if the slope is between 5° and 12° The angle of slope needs to be known for • Add 10 minutes for every 300 metres of descent if the slope is greater than 12° Langmuir’s refinements to be calculated.
Task 1: Question sheet
Here is the route card for the Snowdon Horseshoe.
a) Complete the ascent column. State any descents using negative numbers. b) Use Naismith’s Rule to estimate the time for each leg
Optional: if you use a spreadsheet to help with this task, create a route profile diagram.
Leg To Grid Ref Height (m) Bearing (°) Distance (m) Ascent (m) Time START Pen-y-pass 647557 359 1 Bwlch y Moch 633553 550 256 1600 2 Crib Goch (921 m) 625553 921 287 825 3 Bwlch Coch 622552 858 239 425 4 Crib y Ddysgl 611552 1065 280 1000 5 Bwlch Glas 607547 993 240 400 6 Snowdon summit 610544 1085 152 450 7 Junction 608543 1020 217 200 8 Bwlch y Saethau 615542 780 90 825 9 Bwlch Ciliau 619536 744 122 575 10 Y Lliwedd West Peak 622533 898 131 500 11 Lliwedd Bach 627532 818 132 525 12 Left turn off ridge 631535 700 49 400 13 Llyn Llydaw (ford) 633544 435 0 850 14 Jn with Miner's Track 635545 440 33 275 15 Junction 647544 380 95 1900 16 Pen-y-pass 647557 359 354 250
Naismith’s Rule Allow one hour for every 4 kilometres travelled Add 10 minutes for every 100 metres of ascent
Task A: Mark scheme
The information below is intended as a guide only
Full credit
Finds all ascents
AND
Uses Naismith’s Rule to find the estimates as stated below (to one decimal place):
Leg To Grid Ref Height (m) Bearing (°) Distance (m) Ascent (m) Time START Pen-y-pass 647557 359 1 Bwlch y Moch 633553 550 256 1600 191 43.1 2 Crib Goch (921 m) 625553 921 287 825 371 49.5 3 Bwlch Coch 622552 858 239 425 -63 6.4 4 Crib y Ddysgl 611552 1065 280 1000 207 35.7 5 Bwlch Glas 607547 993 240 400 -72 6 6 Snowdon summit 610544 1085 152 450 92 16 7 Junction 608543 1020 217 200 -65 3 8 Bwlch y Saethau 615542 780 90 825 -240 12.4 9 Bwlch Ciliau 619536 744 122 575 -36 8.6 10 Y Lliwedd West Peak 622533 898 131 500 154 22.9 11 Lliwedd Bach 627532 818 132 525 -80 7.9 12 Left turn off ridge 631535 700 49 400 -118 6 13 Llyn Llydaw (ford) 633544 435 0 850 -265 12.8 14 Jn with Miner's Track 635545 440 33 275 5 4.6 15 Junction 647544 380 95 1900 -60 28.5 16 Pen-y-pass 647557 359 354 250 -21 3.8
If students try to plot a route profile diagram on a spreadsheet they will need to work out, additionally, total distance. This should be plotted against ascent using the option of ‘scatter diagram with line’.
Partial credit
Finds all ascents
AND
Uses Naismith’s Rule to find at least eleven correct estimates (which will ensure that they have dealt with at least one ascent correctly)
Limited credit
Finds all ascents
AND
Shows understanding of how to convert appropriately between minutes and hours, and metres and kilometres, when using Naismith’s Rule; e.g. for the first leg:
Time = distance ÷ speed Time = 1600 ÷ 4000 (or 1.6 ÷ 4) = 0.4 0.4 (hours) × 60 = 24 (minutes)
10 × 191/100 = 19.1 (minutes)
Total time for leg: 24 + 19.1 = 43.1
No credit
Any other response Task B: Question sheet
Today, Naismith’s Rule is typically seen as useful for estimating the minimum time needed for a walk.
Several modifications and refinements to it have been suggested in recent years.
Langmuir’s refinements take account of the effect of descent on walking pace.
Langmuir’s refinements • Allow one hour for every 4 kilometres travelled • Add 10 minutes for every 100 metres of ascent • Subtract 10 minutes for every 300 metres of descent if the slope is between 5° and 12° • Add 10 minutes for every 300 metres of descent if the slope is greater than 12°
Use Langmuir’s approach to recalculate the estimates of the time for each leg.
You will need to consider how to estimate each angle of slope.
Task B: Mark scheme
The information below is intended as a guide only
Full credit
Calculates an estimate of all angles of elevation using tan-1(ascent ÷ distance) as stated below
AND
Uses Langmuir’s refinements to find estimates of times as stated below (to one decimal place)
Angle of Time Leg To Distance (m) Ascent (m) elevation (°) (minutes) START Pen-y-pass 1 Bwlch y Moch 1600 191 6.8 43.1 2 Crib Goch (921 m) 825 371 24.2 49.5 3 Bwlch Coch 425 -63 -8.4 4.3 4 Crib y Ddysgl 1000 207 11.7 35.7 5 Bwlch Glas 400 -72 -10.2 3.6 6 Snowdon summit 450 92 11.6 16 7 Junction 200 -65 -18 5.2 8 Bwlch y Saethau 825 -240 -16.2 20.4 9 Bwlch Ciliau 575 -36 -3.6 8.6 10 Y Lliwedd West Peak 500 154 17.1 22.9 11 Lliwedd Bach 525 -80 -8.7 5.2 12 Left turn off ridge 400 -118 -16.4 9.9 13 Llyn Llydaw (ford) 850 -265 -17.3 21.6 14 Jn with Miner's Track 275 5 1 4.6 15 Junction 1900 -60 -1.8 28.5 16 Pen-y-pass 250 -21 -4.8 3.8
Note that, prompted by the angles of elevation, some students may recognise that distances are taken from the map and are therefore horizontal distances. However, the distance travelled can be thought of as the hypotenuse of a right-angled triangle and is therefore further. It might help to remind them that the distances are estimates, and they could be encouraged to explore the percentage error in any results. All errors are less than 5%.
Partial credit
Finds all angles of elevation
AND
Uses Langmuir’s refinements to find all correct estimates for at least two of the four categories stated below (and colour-coded above):
ascent descent with no adjustment descent between 5 and 12 degrees descent greater than 12 degrees.
OR
Uses Langmuir’s refinements to find all correct estimates based on incorrect angles of elevation.
Limited credit
Finds all angles of elevation
OR
Shows understanding of how to apply Langmuir’s refinements; e.g. in the third leg:
Time = distance ÷ speed Time = 425 ÷ 4000 (or 0.425 ÷ 4) = 0.10625 0.10625 (hours) × 60 = 6.375 (minutes)
Angles implies a need to subtract 10 minutes for every 300 metres of descent 10 × 63/300 = 2.1 (minutes)
Total time for leg: 6.375 – 2.1 = 4.275 minutes
No credit
Any other response.
Progression in reasoning Identify processes and connections use a scientific calculator Consider which of mental, Calculate with increasing Calculate with complete to carry out calculations written calculator or efficiency and accuracy, and on efficiency and accuracy, and on effectively and efficiently spreadsheet methods should be a scientific calculator use some a scientific calculator use the using the available range used. Conduct each accurately. function keys appropriately. full range of function keys of function keys e.g. selects to use a spreadsheet e.g. creates and copies a formula appropriately. to calculate ascent e.g. creates and copies formulae to calculate timings in both parts of the task Represent and communicate explain results and procedures precisely Explanations are clear – both A wider range of appropriate Orally and in writing: use using appropriate orally and in writing, using mathematical vocabulary is mathematical vocabulary mathematical language some mathematical vocabulary. used in explanations. precisely. e.g. presents all solutions in an Arguments are supported with e.g. provides a thorough organised way that allows easy evidence. justification for their method of comparisons to be made e.g. provides evidence to explain calculating angle of slope between legs of the route why a spreadsheet (or calculator) method has been chosen Review interpret answers within e.g. Considers every answer e.g. Systematically checks results e.g. Recognises that, due to the context of the problem found and investigates any given by every variant of a ascent and descent, the distance and consider whether anomalies that arise formula in a spreadsheet travelled is greater than the answers, including actual distance found from a calculator, analogue and map. Investigates the size of digital displays, are errors that could occur. sensible
GCSE Content GCSE Mathematics – Numeracy and GCSE Mathematics GCSE Mathematics only Understanding number and place value Rounding whole numbers to the nearest 10, 100, 1000, etc. Rounding decimals to the nearest whole number or a given number of decimal places. Understanding and using directed numbers, including ordering directed numbers. Understanding number relationships and methods of calculation Using the facilities of a calculator, including the constant function, memory and brackets, to plan a calculation and evaluate expressions. Using calculators effectively and efficiently. Using appropriate trigonometric functions on a calculator. Solving numerical problems Interpretation and use of mathematical information presented in written or visual form when solving problems. Giving solutions in the context of a problem, selecting an appropriate degree of accuracy, interpreting the display on a calculator, and recognising limitations on the accuracy of data and measurements. Rounding an answer to a reasonable degree of accuracy in the light of the context. Understanding and using equations and formulae Substitution of positive and negative whole numbers, fractions and decimals into simple formulae expressed in words or in symbols. Understanding and using properties of shape Using trigonometric relationships in right-angled triangles to solve problems, including those involving bearings and angles of elevation and depression. Calculating a side or an angle of a right-angled triangle in 2-D. Understanding and using properties of position, movement and transformation Using and interpreting maps. Understanding and using measures Using compound measures including speed Using compound measures such as m/s, km/h
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