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7500 /www/nrich/html/content/id/7500/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/7418">Warm-up</a></li> <li><a href="http://nrich.maths.org/6145">Try this next</a></li> <li><a href="http://nrich.maths.org/7275">Think higher</a></li> <li><a href="http://nrich.maths.org/7605">Read: mathematics</a></li> <li><a href="http://www.zmescience.com/science/biology/endangered-species-number-24082011/">Read: science</a></li> <li><a href="http://plus.maths.org/content/cat-count">Explore further</a></li> </ul> <div> </div> Have a look at the sets of four quantities below. Can you rank them in order from smallest to largest? To help you to decide on your rankings, you may need to find extra information or carry out some experiments. Can you provide convincing evidence of your final rankings? <table style="border-spacing:10px;" border="1"> <tbody> <tr> <td><mdo:image height="100" src="clock.gif" width="100"></mdo:image></td> <td>Time For light to reach us from the nearest star other than the sun That a pet rat lives Between two football world cups For all of the red blood cells in your body to be replaced....10 times</td> </tr> <tr> <td><mdo:image height="100" src="ruler.jpg" width="100"></mdo:image></td> <td>Distance You can hop in 5 seconds A car can drive on $1$cm$^3$ of fuel From the top of a 6 storey building to the ground The distance from the penalty spot to the goal on a football field</td> </tr> <tr> <td><mdo:image height="100" src="speed.gif" width="100"></mdo:image></td> <td>Speed Of a racehorse running at maximum speed Of the fastest sprinter in the world Of the fastest cyclist in an Olympic cycle sprint race Of a bus going past the school</td> </tr> <tr> <td><mdo:image height="100" src="scales.jpg" width="100"></mdo:image></td> <td>Mass Of a standard family car Of a million cubic centimetres of water Of a team of international male rugby players (15 players) Of enough potatoes to make chips to feed everyone in the school for a week.</td> </tr> <tr> <td><mdo:image height="100" src="volume.gif" width="100"></mdo:image></td> <td>Volume of water In a half-filled bath Used in a 10 minute shower Used in 5 dishwasher cycles Used to flush the toilet 20 times </td> </tr> <tr> <td><mdo:image height="100" src="number.jpg" width="100"></mdo:image></td> <td>Number of Mars bars with the same total weight as your whole class Steps when walking at a leisurely pace for 1 hour Stitches in a handknitted jumper People at a cup final in a large stadium </td> </tr> <tr> <td><mdo:image height="100" src="area.jpg" width="100"></mdo:image></td> <td>Area Of the sheets of paper in a ream (500 sheets) of A4. Of the floor of the classroom Covered by 5 litres of paint Of fabric needed to make school uniforms for the whole class.</td> </tr> </tbody> </table> Perhaps you might like to make 4 ranked quantities of your own to challenge a friend. For more advanced science, see <a href="http://nrich.maths.org/6505&part=">Approximately Certain</a>. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We received many solutions to this problem, including carefully considered solutions from Patrick from Otterbourne School, the Math Club from DMCS in Canada, Siannah from Ricards Lodge School, members of the Pythagorean Club at All Saints Dagenham School, and from Kartik and Mustafa from Wilson's School. Many of you pointed out that the rankings depended on estimates and that different contexts and assumptions might lead to different conclusions (e.g. the size of the bath, the flow of water from a shower, how well the pet rat is looked after, the size of the classroom, the pace at which a leisurely walker travels...). Bearing this in mind, here is a suggested ranking with the evidence that was provided: Time That a pet rat lives 2-3 years For all of the red blood cells in your body to be replaced....10 times 3 years Between two football world cups 4 years For light to reach us from the nearest star other than the sun 4.2 years Distance You can hop in 5 seconds 8 metres The distance from the penalty spot to the goal on a football field 11 metres From the top of a 6 storey building to the ground 18 metres A car can drive on 1$cm^{3}$ of fuel 20 metres Speed Of the fastest sprinter in the world 10m/s Of a bus going past the school (30mph = 13.4m/s) Of a racehorse running at maximum speed 15m/s Of the fastest cyclist in an Olympic cycle sprint race 19m/s Mass Of enough potatoes to make chips to feed everyone in the school for a week. (1000kg for 1000 students) Of a million cubic centimetres of water 1000kg Of a standard family car 1500kg Of a team of international male rugby players (15 players) 1500kg Volume of water used in a 10 minute shower 90 litres used in 5 dishwasher cycles 100 litres in a half-filled bath 100 litres used to flush the toilet 20 times 120 litres (modern toilet) Number of Steps when walking at a leisurely pace for 1 hour 8000 Mars bars with the same total weight as your whole class 20000 (for a class of 30 students weighing about 40kg each) Stitches in a handknitted jumper 75000 People at a cup final in a large stadium 80000 Area Of the sheets of paper in a ream (500 sheets) of A4. 30m$^2$ Covered by 5 litres of paint 50m$^2$ Of the floor of the classroom 60m$^2$ Of fabric needed to make school uniforms for the whole class will depend on the size of the class, the width of the fabric and the design of the uniform! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/7500">This activity</a> offers an opportunity to combine skills from mathematics and science. Students are required to make estimates, understand and calculate with units, think about orders of magnitude, and find information to draw conclusions.</div> <div> </div> <div>This problem highlights the fact that in science it is rather hard to calculate anything without making some sort of assumptions. Good science will clearly state and be aware of these assumptions; bad science will ignore them. </div> <h3>Possible approach</h3> <div>Hand out <a href="/content/id/7500/Place%20your%20orders.pdf">this worksheet</a> with all sets of quantities, and give students a few minutes on their own or in pairs to rank them in order of magnitude. Reassure students that at this stage, they do not need to perform any written calculations and it doesn't matter if they are unsure of their rankings. Alternatively, hand out these different sets of quantities to different groups: <a href="/content/id/7500/Place%20your%20orders%20-%20Area.doc">Area</a>, <a href="/content/id/7500/Place%20your%20orders%20-%20Distance.doc">Distance</a>, <a href="/content/id/7500/Place%20your%20orders%20-%20Mass.doc">Mass</a>, <a href="/content/id/7500/Place%20your%20orders%20-%20Number%20of.doc">Number</a>, <a href="/content/id/7500/Place%20your%20orders%20-%20Speed.doc">Speed</a>, <a href="/content/id/7500/Place%20your%20orders%20-%20Time.doc">Time</a>, <a href="/content/id/7500/Place%20your%20orders%20-%20Volume%20of%20Water.doc">Volume</a>.</div> <div> </div> <div>Next, give each group two sets of quantities to work on in more detail, so that each set is looked at by at least two groups.</div> <div>"Your task is to come up with a rank order for the quantities you have been given, together with a convincing presentation of evidence to justify your order."</div> <div>Allow students access to reference materials, measuring equipment, and anything else that might be useful, and give them plenty of time for research and experiment. This could be part of a homework task.</div> Once they have finished, take each set of quantities and invite the different groups to present their rankings and reasoning. Ask the rest of the class to judge the different presentations on the strength of the evidence they have offered for their rankings. <h3>Key questions</h3> <div>What is precisely stated and what is not precisely stated?</div> <div>Can you give quick, sensible lower and upper bounds on the quantities before calculating?</div> <div>Is a detailed calculation necessary for all of the parts of the problem?</div> What can be resolved by experiment? By measurement? By looking something up? <h3>Possible extension</h3> <div>Ask students to produce definite upper and lower bounds for the quantities. The ordering is only guaranteed when the intervals do not overlap.</div> <div> </div> <div><a href="http://nrich.maths.org/6505&part=">Approximately Certain</a> uses a similar idea with more advanced scientific content.</div> <h3>Possible support</h3> Reassure students that there are often no exact values (so no "wrong" answers), and their task is simply to find some evidence to convince others of the rankings. Encourage them to experiment, and offer some guidance on how to search for suitable data. <a href="http://nrich.maths.org/7341&part=">In Order</a> has quantities to rank using simpler scientific contexts. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">How could you estimate each of the quantities? What extra information might you need to find out? For some questions, you may need to be more accurate in your estimations than others to arrange the list in order of magnitude. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Time For light to reach us from the nearest star other than the sun 4.2 years That a pet rat lives 2 years Between two football world cups 4 years For all of the red blood cells in your body to be replaced....10 times 3 years Distance You can hop in 5 seconds 8 metres A car can drive on 1$cm^{3}$ of fuel 20 metres From the top of a 6 storey building to the ground 18 metres The distance from the penalty spot to the goal on a football field 11 metres Speed Of a racehorse running at maximum speed 13m/s Of the fastest sprinter in the world 10m/s Of the fastest cyclist in an Olympic cycle sprint race 16m/s Of a bus going past the school (30mph = 13.4m/s) Mass Of a standard family car 1500kg Of a million cubic centimetres of water 1000kg Of a team of international male rugby players (15 players) 1500kg Of enough potatoes to make chips to feed everyone in the school for a week. (~1000kg for 1000 students) Volume of water in a half-filled bath 150 litres used in a 10 minute shower 90 litres used in 5 dishwasher cycles 75 litres used to flush the toilet 20 times 120 litres (modern toilet) Number of Mars bars with the same total weight as your whole class 25000 Steps when walking at a leisurely pace for 1 hour 10000 Stitches in a handknitted jumper 75000 People at a cup final in a large stadium 90000 Area Of the sheets of paper in a ream (500 sheets) of A4. 30m^2 Of the floor of the classroom 70m^2 Covered by 5 litres of paint 60m^2 Of fabric needed to make school uniforms for the whole class. </mdoxml> 2 3 0 0 1 0 0 Place your orders Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings? ajk44 resources Zoe found Measures and Mensuration Time Measures and Mensuration Mass and weight Measures and Mensuration Length/distance Measures and Mensuration Volume and capacity Measures and Mensuration Area Measures and Mensuration Speed Using, Applying and Reasoning about Mathematics Selecting and using information Applications STEM - General Applications Maths Supporting SET Secondary Mapping Document MD Number operations and calculation methods Secondary Mapping Document MD Units of measurement Secondary Mapping Document DisplayCabinet STEM mapping STEM - maths supporting SET STEM mapping STEM - Length-Area-Volume STEM mapping STEM - Distance-Speed-Time STEM mapping STEM - Estimation