James Osborne Director of Mathematics Bradfield School, Sheffield josborne@bradfield.sheffield.sch.uk

http://www.nanozine.net/enrico-fermi.html ‘Fermi’ questions are those which make use of estimates of quantities which are difficult or perhaps impossible to measure directly. True ‘Fermi’ questions usually involve complicated Physics but can often be readily simplified to an appropriate level for students of all ages and abilities. The hyperlinks on the slide above lead to more detailed descriptions and examples of ‘Fermi’ questions. ‘Fermi’ questions http://www.physics.uwo.ca/science_olympics/events/puzzles/fermi_questions.html http://www.fermiquestions.com/tutorial

‘Fermi’ questions How much of a person’s lifetime is spent watching television? How many years would a teacher’s career have to last to earn the same total amount of money as a Premiership footballer’s career? What weight of sugar would fill this room? How many times would you need to fold a piece of paper until its thickness was the distance from the earth to the sun? Examples of ‘Fermi’ questions that I have used with students of all ages and abilities for group work and whole-class discussions. These problems require a mixture of common sense, general knowledge, sensible estimation, specific knowledge, consideration of ‘representative’ values, ..... http://becuo.com/confused-child

‘Fermi’ questions What weight of sugar would fill this room? Room volume  7 x 11 x 3  240 m3 Standard bag of sugar weighs 1kg Volume of sugar  0.2 x 0.1 x 0.05 = 0.001 m3 per bag 240 ÷ 0.001 = 240 000 bags of sugar 240 000 kg or 240 tonnes A possible solution to the sugar problem (based on the room used at SHU for the session). ‘Fermi’ questions have solutions which are the power of 10 associated with their order of magnitude in standard form. In this case 150 000 = 1.5 x 10^5 so the ‘Fermi’ solution would be 5. The slightly pedantic units of ‘m3 per bag’ are useful to help students consider what the answer to a particular calculation represents. Dimensional analysis of the 150  0.001 calculation gives “m3”  “m3 per bag” leading to bags (of sugar) being the units of the answer. http://becuo.com/confused-child

‘Fermi’ questions in our developing classification 4. Explaining and reasoning   Pupils should be taught to: Use calculations to support explanation and argument Look for a counter example to define the conditions and limits of a rule Use a relationship or pattern to justify or confirm others. Use properties and relationships to reason and deduce. Use a diagram to support an explanation How ‘Fermi’ questions might fit into our developing classification of problem solving skills that students need to be taught.

Q4, Edexcel Sample Assessment Material GCSE (9-1) Higher Paper 3 An example of a question from the new GCSE (9-1) SAMs released by Edexcel where the dimensional analysis skills developed through tackling ‘Fermi’ problems would be useful: e.g. 140 (chickens) x 0.1 (kg of food per chicken per day) = 14 kg of food per day e.g. 6.75 (£ per bag)  25 (kg of food per bag) = 0.27 (£ per kg of food) e.g. 6 (eggs per chicken per week) x 2 weeks per fortnight = 12 eggs per chicken per fortnight Q4, Edexcel Sample Assessment Material GCSE (9-1) Higher Paper 3

Real life representation Length Real life representation 1 000 000 metres 100 000 metres 10 000 m 1000 m (1km) 100 m 10 m 1 m Length of a school desk 0.1 m 0.01 m (1cm) 0.001 m (1mm) 0.000 1 m 0.000 01 m 0.000 001 m (1m) Students find the process of estimating quantities difficult because there is often a disconnect between their perception of the world and the more formal measurement of quantities in standard units. This activity, adapted from the Standards Unit N4: Estimating Length Using Standard Form, is a useful pre-cursor to several ‘content’ topics but also supports the teaching of some of the problem solving skills required to tackle ‘Fermi’-esque problems. Ask students to consider real-life things that are approximately the same size as the given lengths. With some students it may be necessary to spend time revisiting units of measurement and the x10 and 10 relationships between them. Encourage students to start from the middle and work outwards getting either larger or smaller. Adapted from Standards Unit N4 Estimating Length Using Standard Form

Real life representation Length Real life representation 1 000 000 metres Length of Great Britain 100 000 metres 10 000 m 1000 m (1km) 100 m 10 m 1 m Length of a school desk 0.1 m 0.01 m (1cm) 0.001 m (1mm) 0.000 1 m 0.000 01 m 0.000 001 m (1m) Length of a bacteria If students are struggling then it can be useful to give them the ‘extreme’ values in this particular set. These limits represent the extent of what the majority of students might reasonably be expected to be aware of but could be extended or reduced as appropriate. Adapted from Standards Unit N4 Estimating Length Using Standard Form

Real life representation Length Real life representation 1 000 000 metres Length of Great Britain 100 000 metres Distance from Sheffield to Leicester 10 000 m Distance from Sheffield to Rotherham 1000 m (1km) Length of Granville Road (S2 2RJ) 100 m Length of a football pitch 10 m Length of a classroom 1 m Length of a school desk 0.1 m Length of an finger 0.01 m (1cm) Width of a finger 0.001 m (1mm) Thickness of a pencil lead 0.000 1 m Thickness of human hair 0.000 01 m Mist water droplet 0.000 001 m (1m) Length of a bacteria A possible set of representations is shown. Insist on linear representations (e.g. A representation of 1000 m as being 2.5 laps of an athletics track is not helpful as it cannot easily be visualised in full) Insist on single items (e.g. A representation of 10 m as being “10 school desks” is not allowed, though it does provide a useful link to unit conversion) however, encourage them to use this approach as a starting point if they are unable to progress to the next order of magnitude (e.g. imagine what 10 desks might look like end-to-end - what individual item might be the same length as this?) Adapted from Standards Unit N4 Estimating Length Using Standard Form

Real life representation Length Real life representation 1 000 000 metres Length of Great Britain 100 000 metres Distance from Sheffield to Leicester 10 000 m Distance from Sheffield to Rotherham 1000 m (1km) Length of Granville Road (S2 2RJ) 100 m Length of a football pitch 10 m Length of a classroom 1 m Length of a school desk 0.1 m Length of a finger 0.01 m (1cm) Width of a finger 0.001 m (1mm) Thickness of a pencil lead 0.000 1 m Thickness of human hair 0.000 01 m Mist water droplet 0.000 001 m (1m) Length of a bacteria Students of all ages and abilities (without specific knowledge) will find it difficult to give reasonable representations beyond the green section. This is due to the limits of perception of the human eye and the curvature of the Earth (see later slides). Adapted from Standards Unit N4 Estimating Length Using Standard Form

‘Content’ lessons that this task might link to Consider which ‘content’ this activity might link to.

‘Content’ lessons that this task might link to Standard Form Place value Multiplying and dividing by powers of 10 Rounding/accuracy of measurement Estimation Units of measurement Circle parts/properties Pythagoras & Trigonometry Direct proportion Cross-curricular links to science, geography, philosophy, social sciences Some possible links.

Heights of students (cm) “Why do we have to round our answers?” Heights of students (cm) 1.47 1.54 1.56 1.62 1.65 1.66 1.67 1.7 1.75 1.82 1.83 Mean height = 1.66357142857143 1 desk + 6 finger lengths + 6 finger widths + 3 pencil leads + 5 hair thicknesses + ..... An example of its link to sensible rounding. ‘Sensible’ rounding should become quickly apparent in this context.

Names of units of measure – students love these. http://en.wikipedia.org/wiki/Metric_prefix

Under normal light conditions with the object at a comfortable reading distance: 0.1 mm [Adapted from Wikipedia] Limits of perception [possible links to Science (microscopes), Philosophy (if we can’t see something, how do we know it is there? Social Sciences (our perceptions impact upon our behaviour)] http://cache2.asset-cache.net/gc/184873635-blue-eye-macro-gettyimages.jpg?v=1&c=IWSAsset&k=2&d=A6%2FBucwOpXozyUoS75rJT9%2FencgysWspzBvB9%2B03bFE%3D

Distance to horizon, d kilometres viewed from a height, h metres Limits of perception (further links to Science (how high would we have to be to see half of the Earth’s surface?), Geography (what’s the furthest that could be seen from Britain?) The 1344 metres is the height of Ben Nevis, the tallest mountain in Britain. The formula could be linked to content: proportion “The distance to the horizon in km is directly proportional to the square root of height of the observer’s eyes above sea level in metres. If a height of 1.7 m gives a horizon distance of 4.7 km, what height is required to be able to see 25 km?” [Adapted from http://science.howstuffworks.com/question198.htm]

h d Earth’s radius [Adapted from http://en.wikipedia.org/wiki/Horizon] The observed distance of the horizon can also be approximated by Pythagoras Theorem. The method assumes that the Earth is a perfect sphere (it isn’t) and that the horizon is at sea-level (not in Sheffield!) There is more detail on this at http://en.wikipedia.org/wiki/Horizon

In conclusion: To answer exam questions requiring multi-step calculations in increasingly complex real-life contexts, students must be able to make appropriate links between the given pieces of information. ‘Fermi’ questions encourage students to make these links between quantities by using sensible approximations of everyday quantities so that the focus can be on the choice of operation rather than carrying out the operation. The use of appropriate units is critical. Students find it hard to make sensible approximations and need to be taught explicitly to make sense of the world around them. Students find it hard to link quantities together and need to be taught explicitly to make use of units of measurement to understand what an answer to a particular calculation represents. http://www.quotessays.com/enrico-fermi.html