Density
Properties of Matter

Measuring density

for 14-16

Density is a derived quantity - we measure two other quantities (mass and volume) and then calculate density. A first time learner, however, deserves first to understand the concept - and that takes careful teaching and a little time. Moreover, different techniques are required for solids, liquids and gases.

Many students imagine that a gas such as air has no mass. Carefully 'weighing' air can help students later to answer some startling questions: What's the mass of air in a classroom? The height of the atmosphere? The average speed of air molecules?

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Measuring and weighing solid blocks

Density
Properties of Matter

Measuring and weighing solid blocks

Practical Activity for 14-16

Class practical

A group of blocks, all the same size but made from different materials, is used to introduce the concept of density.

Apparatus and Materials

For each student group

  • Ruler marked in centimetres (not millimetres)
  • Lever-arm balance
  • Blocks of materials (see Technical notes)

Health & Safety and Technical Notes

Be careful that larger blocks of dense materials do not drop onto feet.

Read our standard health & safety guidance


Materials should be provided in the form of blocks with edge lengths equal to an integer number of centimetres. There should be two types:

  • Blocks of standard size (for example, 5 cm x 4 cm x 3 cm), allowing direct comparison;
  • Blocks of various other sizes.
  • Suitable materials:
  • Soft wood
  • Hard wood
  • Aluminium
  • Iron
  • Lead
  • Brass
  • Polystyrene foam (Styrofoam)
  • Paraffin wax
  • Perspex
  • Slate
  • Glass
  • Marble

You could also supply polystyrene foam sheet (5 cm x 30 cm x 40 cm), marked to show that it is the equivalent of 10 x 10 or 100 of the smaller polystyrene blocks.

Procedure

  1. Compare the masses of two standard-sized blocks of different materials, first by ‘weighing’ them, one in each hand, and then using a balance to find their masses.
  2. Draw up a list, putting the materials in order from lightest to heaviest.
  3. Investigate blocks of other materials which are not of the standard size. Devise a method for including these materials in your list.

Teaching Notes

  • The approach suggested introduces the concept of density. The aim is for students to develop a familiar feeling for density as something we know about a material; a useful qualitative concept rather than a quantitative definition and a scheme of measurement. With an average or slow group of students this might simply take the form of comparing equal-sized blocks of several materials; first by holding them (perhaps blindfold}, then by weighing them with a direct reading balance. Even if you go no further, this will establish general knowledge such as iron weighs three times as much as aluminium, for the same chunk, and many times more than plastic foam.
  • A blindfold guessing game will make a very good start. Ask students to guess which block of material they have been given and maybe even to say which block is the heavier.
  • It will help if the rulers are only marked in centimetres and not also marked in millimetres. If the rulers are only marked in centimetres, students will measure to the nearest centimetre. If there are millimetres as well, some will measure a width as 1.9 cm and thereby ruin the whole experiment in a mass of complicated arithmetic in a misplaced attempt to achieve accuracy.
  • The difficulty of weighing the small foamed polystyrene block (5 cm x 3 cm x 4 cm) should lead the students to the need to weigh a large number of these. The large polystyrene sheet is 5 cm x 30 cm x 40 cm (i.e. 100 small blocks). The sheet should be supplied ruled with markings to show that it is the same as 100 small blocks, which can then be weighed all at once.
  • There should be sufficient balances so that there is one for every four students. These should read up to 1000 g. Chemical balances (or other equal-arm balances) should not be used. For some students it may help considerably if the second scale on the balance is covered over with black masking tape.
  • You could lead the teaching on to a discussion of weighing different sizes and working out the mass of some standard size, such as a unit cube (cm 3) but this may create barriers for some students. Students can calculate volumes, particularly when the measurements are simple, whole numbers. They can divide mass/volume but many do not see the point; they do it but do not see the necessity. Finding the mass of a unit cube seems to have more purpose when it can be used to compare different materials.

This experiment was safety-tested in July 2007

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Measuring the density of regular solid shapes

Density
Properties of Matter

Measuring the density of regular solid shapes

Practical Activity for 14-16

Class practical

Apparatus and Materials

For each student group

  • Ruler, 15 or 30 cm
  • Micrometer
  • Vernier callipers
  • Chemical balance (0-1 kg )
  • Access to cuboidal samples of different metals and non metals, e.g. materials kit
  • Eureka can (OPTIONAL)
  • Aluminium cooking foil, sheet of (OPTIONAL)
  • Paper, sheet of (OPTIONAL)
  • ‘Unknown’ object (a covered or painted regular object made of the same material as one of those used above OPTIONAL)

Health & Safety and Technical Notes

When handling materials such as lead, wash your hands afterwards.

Read our standard health & safety guidance


Procedure

  1. Measure the three dimensions of each regular object, repeating each measurement at two or more places. Depending on the size of the measurement, use the ruler, vernier callipers or a micrometer.
  2. Calculate the volume, V, of the object.
  3. Measure the mass, m, of the object, using the balance.
  4. Calculate the density, D, of each sample, using D = m/V.
  5. Compare your results with the accepted values for each material.

Teaching Notes

  • Students will need to have studied density previously and be familiar with the density equation. Examples may have used cm 3 as the unit of volume and g/cm 3 as the unit of density, or m 3 and kg/m 3 . Either sets of units are generally acceptable, but all length measurements must use the same unit.
  • The displacement method with a eureka can be used as a second method, particularly if the sample shape is not rectangular. If both methods are used for the same sample, students could compare their two values and decide which one is more accurate.
  • How Science Works Extension: In this experiment choose materials whose density is known to quite a high degree of accuracy, so that it is possible for students to perform a numerical comparison between the 'true' and measured values. They could be expected to find this data themselves or be provided with it. A good comparison will express the difference between the measured value and the true value as a percentage of the true value.
  • As an extension activity, you could ask students whether or not they can be confident that two different objects are made of the same material. Ask them to find the density of a sheet of aluminium cooking foil and compare it to that of the solid aluminium block. This same approach could be taken with a block of wood and a piece of paper. You might say "isn't paper made from wood?" as you pass the student the materials. Alternatively, you could give the students a sample of an unknown material and then samples of known materials, perhaps as a forensic test. The unknown object may need to be painted or coated in some way so that its surface does not provide other hints.
  • A critical point to make here is that, whilst they may be able to state with a great deal of confidence that two materials have the same density (or not), this does not mean that they are made from the same material. Further (chemical) testing would be needed to make an absolute statement of that kind.

This experiment was safety-checked in December 2006

Up next

Weighing liquids

Density
Properties of Matter

Weighing liquids

Practical Activity for 14-16

Class practical

Comparing the masses of equal volumes of liquids, to develop a general feeling for density.

Apparatus and Materials

For each student group

  • Perspex containers (rectangular boxes, open at top), 2
  • Balance, lever-arm or top-pan
  • Ruler
  • Marker pen
  • Sand, wheat, rice or dried peas
  • Liquids (e.g. water, paraffin, brine, cooking oil and syrup)

Health & Safety and Technical Notes

Choose liquids that are low risk.

Read our standard health & safety guidance


Procedure

  1. The transparent containers will allow you to measure and weigh standard quantities of liquids and other pourable substances. Mark a line on the outside of each container at a height of, say, 8 cm.
  2. Fill a container to the line with sand. Weigh the container to find its mass.
  3. Repeat with water, and then another liquid. How do the different substances compare?

Teaching Notes

  • The object of this experiment is not to make systematic measurements of liquid densities but rather to extend a general developing feeling for density. It can be used as an extension activity to following ...

    Measuring and weighing solid blocks


    ...The liquids are placed in rectangular containers so that the counting cubes approach to volume measurement is relatively easy
  • Those students who raise the question of the weight of the box itself will weigh it of their own accord, but the teacher should not give a general instruction to do this.
  • Weighing sand might lead to a discussion of an average density for the sand and spaces combined. Start such a discussion by trying wheat or dried peas. Considering water after this may lead to a discussion once again of atoms.
  • When the box is filled with sand or rice then there are spaces between the grains. A discussion about the average density of air and sand is necessary. However when the box is filled with water, what is it they are measuring? What is between the atoms? [Empty space.]

This experiment was safety-tested in July 2007

Up next

Understanding measuring cylinders

Density
Properties of Matter

Understanding measuring cylinders

Practical Activity for 14-16

Demonstration

An introduction to measuring cylinders, using known volumes of water from a rectangular box.

Apparatus and Materials

  • Measuring cylinder, 250 ml
  • Measuring cylinder, 1000 ml
  • Rectangular Perspex box (10 cm x 10 cm x 11 cm)
  • Ruler
  • Marker pen

Health & Safety and Technical Notes

Read our standard health & safety guidance


Procedure

  1. Mark the Perspex box at a level of, say, 4 cm. Measure the width, breadth and depth of this volume, and calculate the volume (in cm 3 ).
  2. Fill the container with water to the marked level. Pour the water into the 250 ml measuring cylinder and note the level.
  3. The process may be repeated, different depths can be tried, and the larger measuring cylinder can be used. Pour water from measuring cylinder to measuring cylinder.

Teaching Notes

  • This is an introduction to the measuring cylinder. It is important the demonstration should not turn into a precise drill and thereby labour the discussion.
  • A volume of water measured in the rectangular box is transferred into the measuring cylinder and the reading on the scale compared with the volume of water poured from the box. This is a way of calibrating a measuring cylinder from first principles by counting the cubes.
  • The graduations on the measuring cylinder could be covered up with a paper scale, which could be filled in as volumes of liquid are poured from the box into the cylinder. Pouring from one cylinder into different sized cylinders will show the same reading providing no water has been lost!
  • Piaget's theory on the conservation of volume of liquids by young students is valuable here. (Remember Piaget’s theories were based on different teaching methods so the ages at which students develop their cognition may vary.)
  • Gases of course are not invariant in volume as they change containers.

This experiment was safety-tested in July 2007

Up next

Measuring the density of liquids

Density
Properties of Matter

Measuring the density of liquids

Practical Activity for 14-16

Class practical

A simple method for comparing the density of liquids.

Apparatus and Materials

For each group

  • Measuring cylinders, 100 ml or 250 ml, clean and dry, 2 or more
  • Chemical balance
  • Access to water and vegetable or olive oil
  • Any other liquids that are safe to handle (OPTIONAL)

Health & Safety and Technical Notes

Take care with any spillages, particularly with the oil, which can create a slip hazard.

Read our standard health & safety guidance


Procedure

  1. Take the measuring cylinder and measure its mass, in grams, as accurately as possible.
  2. Take the measuring cylinder off the balance and add the water carefully, either by careful pouring or with a pipette until the level is as close to the 10 ml mark as possible. Put the measuring cylinder back on the balance. Measure and record the new mass (cylinder plus water), in grams.
  3. Repeat the procedure, adding 10 ml at a time as accurately as possible and recording the volume and total mass, until the measuring cylinder is full. Then, for each volume calculate the mass of the liquid alone.
  4. NOTE: If a 250 ml measuring cylinder is being used you may wish to use 20 ml or 25 ml intervals.
  5. Repeat steps 1 to 3 for the oil (and any other liquids being tested).
  6. Draw a graph of mass of liquid (y-axis) against volume (x-axis). Try to scale the graph so that you can plot all your data sets on a single graph.
  7. For each set of data try and draw a straight ‘best fit’ line passing through the origin. Calculate the density of each liquid from the gradient of its graph line.

Teaching Notes

  • Students will need to have studied density previously and be familiar with the density equation. Examples may have used cm 3 as the unit of volume and g/cm 3 as the unit of density, or m 3 and kg/m 3 . Either sets of units are generally acceptable, but all length measurements must use the same unit. Students may need to be told that with a measuring cylinder 1 ml =1 cm3.
  • The density of water is measured before the oil because water can be easily and quickly rinsed out of the measuring cylinder and oil cannot. When adding the oil to the measuring cylinder, instruct students to try and avoid pouring it down the side otherwise it will form a coating on the sides which will increase the mass without raising the level from which the volume is read, so dry the measuring cylinder before weighing.
  • If there are limitations to the number of balances available then it is still possible to carry this out with students sharing a balance, although care needs to be taken that there are no spillages. If students are not familiar with the meniscus that is formed, show them how to take volume readings correctly.
  • How Science Works extension: If asked to find the density of a liquid, students may take only a single set of readings. The ease with which water and other liquids can be poured allows the refinement of this method to collect multiple results and use a graphical method to minimize the effect of any systematic error in the measurements.
  • Finding densities of liquids and their behaviour is important to food scientists. You could illustrate this by having students measure the density of vinegar, making and measuring the density of a vinaigrette, and then predicting which of these will sit on top when they are poured into a single container.

This experiment was safety-checked in January 2007

Up next

Measuring the average density of a student

Up next

Weighing a sample of air – a rough estimate

Density
Properties of Matter

Weighing a sample of air - a rough estimate

Practical Activity for 14-16

Demonstration

Showing that the mass of air in a bottle is small.

Apparatus and Materials

  • Round-bottomed flask (1 litre) or glass bottle
  • Bung and tube to fit flask or bottle
  • Vacuum pump
  • Balance, lever-arm or top-pan
  • Measuring cylinder, 1 litre
  • Length of pressure tubing, 1 m
  • Screw clip

Health & Safety and Technical Notes

Wear eye protection when evacuating glass containers or handling them. Protect observers by using a safety screen. Use a flask that can withstand the pressure difference. Check it has no specks or marks.

Read our standard health & safety guidance


Instead of the special nature of the Pyrex flask, you may prefer to use a more familiar glass bottle.

For connection to the vacuum pump, it will be necessary to use pressure tubing.

Procedure

  1. The one litre flask must have a well-fitting rubber bung with a glass tube through it, to which is attached a rubber tube with a screw clip. Weigh the flask.
  2. Attach the rubber tubing to the pump and evacuate the flask. Reweigh the flask. (The volume can be found by filling the flask with water and pouring into the measuring cylinder.)

Teaching Notes

  • It is a useful order of magnitude to remember that 1 cm3 of water has a mass of 1 gram and a litre of air has a mass of just over 1 gram (1.2 gram/litre). Students should be able to come up with their own ideas for how to make the measurements.
  • A one litre round-bottomed Pyrex flask weighs approximately 350 g. The change in mass when the air is removed is of the order of 1.2 g . On the single-pan lever-arm balance this amount will scarcely be appreciated. The object of this experiment is to show how small the mass difference is, in fact too small for the balance to measure.
  • The suggestion of weighing a balloon empty, then blowing it up full of air and weighing it again may arise. This is a fallacy quoted by Galileo. A bladder full of air gives the same reading on the balance as it does when squashed flat, because the buoyancy of the surrounding air just compensates for the weight of the air inside in the first case. A rubber balloon does weigh a little more when inflated, because the air inside is at slighter greater pressure, but that is not sufficient for use here. The volume of the container must not change much between our two weighings because it is ignored.
  • The lever-arm balance will probably not register a change in mass when the flask is evacuated but a top pan electronic balance will. This is a useful lesson to learn - that of choosing the best instrument for the job.

This experiment was safety-tested in July 2007

Up next

Evacuating a bottle

Atmospheric Pressure
Properties of Matter

Evacuating a bottle

Practical Activity for 14-16

Demonstration

An approach to the question: Does air have mass?

Apparatus and Materials

  • Ordinary bottle of clear glass with a well-fitting rubber stopper and glass tube
  • Motor-driven vacuum pump
  • Length of pressure tubing, 1 m
  • Hoffman clip
  • Large transparent trough (glass or plastic)

Health & Safety and Technical Notes

Use a bottle strong enough to withstand the pressure difference. Check it has no nicks or scratches.

Wear safety spectacles and use a safety screen to protect observers.

Read our standard health & safety guidance


Reject any old or perished rubber bungs or tubes for this demonstration, as they develop cracks and will not hold the vacuum.

It is advisable to use coloured water in step 3.

Procedure

  1. Connect the rubber tubing to the vacuum pump with the clip open. The bung and glass tube must be tight fitting.
  2. Remove the air by pumping and then close the clip on the rubber tubing.
  3. To show that the air has been removed, immerse the neck of the bottle (including the rubber tubing) under water and remove the clip. Water will rush in to fill the space. If the vacuum is a good one, there should be very little air inside the bottle. If the pump was not very effective or if there was a leak, then the water will not completely fill the bottle and some air will be seen in it. There will always be a small bubble left, however well the bottle is evacuated, due to air that was dissolved in the water.
  4. Repeat the experiment without pumping air out of the bottle before immersing it, in order to show what happens in that case. This should be done second to avoid using the pump with a wet bottle.

Teaching Notes

  • At this point the emphasis is on whether air has mass. How could the method of measuring the mass of a liquid be adapted to measure the mass of air? (By measuring the mass of the beaker plus liquid and then the mass of the beaker and subtracting to find the mass of the liquid.)
  • How do you know that the pump has done its job? (By putting something else, which we can see, into the empty space.) If the pump is a good one, if there are no leaks and the pump has been pumping for long enough (the pumping noise changes) then the water will fill the bottle when it is opened under water. A very small bubble of air will appear at the top, which was the air left in the bottle at evacuation or dissolved in the water.
  • It is essential to show what happens if a bottle full of air is opened under water - the water will not enter the bottle.
  • Be patient discussing the idea of a vacuum. It does not occur naturally to students, and when they have been given the idea they still do not picture it easily. It is an artificial intellectual concept. Remember that they take the air itself for granted as invisible and almost absent, as did our ancestors, including the great Greek philosophers. It was only at a late stage in the development of physical science that scientists realized that we live at the bottom of an ocean of air, which has density and exerts pressure.
  • If students ask what the pump does, the following discussion may help:
  • "The pump acts rather like a lift that is getting people out of the top floor of a tall building. A lift doesn't pull people out. It just offers them the chance to get in the lift, and the lift carries them out."
  • "The lift goes up to the top floor, the lift opens its doors and waits until a few people have wandered in. Then the door slams shut and down the lift goes. Out go the people; walking out if they are human beings, but pushed out by a moving piston in the case of air molecules in the pump. Up goes the lift again; open the doors; more people wander into the lift; out go the people. Up goes the lift... and so on. Think of that happening with a pump taking out air molecules, batch after batch, trip after trip. At that rate you will never get all the molecules out, but a pump does a very good job."

The experiment was safety-checked in July 2007

Up next

Weighing air

Density
Properties of Matter

Weighing air

Practical Activity for 14-16

Demonstration

Weighing air by pumping it into a container and then releasing it into a vessel under water.

Apparatus and Materials

  • Plastic container with tap (at least 30 cm x 30 cm x 30 cm)
  • Foot pump with pressure-gauge
  • Measuring cylinder, 1 litre
  • Lever-arm balance
  • Large transparent trough (glass or plastic)

Health & Safety and Technical Notes

The container should be tested carefully beforehand to find out how much extra pressure it will take. It should take about 0.5 atmospheres (= 51K Ρa = 7.3 psi) extra.

Read our standard health & safety guidance


The container may be a water carrier, or of the type in which beer is sold in bulk. It should be sufficiently rigid to retain its shape when filled with air at atmospheric pressure. An alternative could be a large ball and needle valve.

Two plastic containers are recommended for this experiment, so that there is a spare one available if the first should leak.

Procedure

  1. Weigh the plastic container on the pan of the lever-arm balance.
  2. Attach a 1m length of rubber tubing to the outlet, the air inside being at normal atmospheric pressure.
  3. Pump air into the container using the foot pump. Close the tap. The more air that can be got inside the better.
  4. Weigh the container again. With the containers provided, a difference of about 8 g is possible (corresponding to an excess pressure of about half an atmosphere) and this can be measured on the lever-arm balance.
  5. Fill the measuring cylinder with water and immerse, mouth downwards, in a large trough of water with the open side downwards.
  6. Put the rubber tubing in the water with the end of the tubing well under the inverted cylinder.
  7. Open the tap until the cylinder is filled with air. Close the tap, refill the cylinder with water and repeat until the container is empty.

Teaching Notes

  • It is essential that the volume of the container should not change significantly between the two weighings. Any increase in volume will increase the buoyancy (due to the external air), reducing the measured weight.
  • Students may be amazed by the large volume of air, at atmospheric pressure, which will come out of the pumped-up container.
  • Knowing the mass of the excess air and its volume at atmospheric pressure, the density can be found.
  • As soon as the density is known, you can ask how much mass the air in the room has. For a room 10 m x 10 m x 2 m it is 2 x 10 5 litres x 1.2 g = 240 kg , or about ¼ of a tonne. Not something to dismiss.
  • If students have now measured the density of solids, liquids and gases, they could draw up a table of their results and include the values of other common substances from a data book. The differences could lead to a discussion on why solids and gases have such different densities. Atomic structure and separation of atoms are points to draw out.

This experiment was safety-tested in July 2007

Up next

Introducing the concept of density

Density
Properties of Matter

Introducing the concept of density

Teaching Guidance for 11-14

There is a strong tradition of beginning physics with careful measurements of volumes and masses and calculations of densities, and with considerable care over arithmetic.

Generally, students find the measuring a fairly interesting routine which does not require much thought. The calculation of density appears an unnecessary interruption of interesting experiments.

Many teachers find the calculation simple and so are tempted to rush to arrive at a characteristic physical quantity. Later, when they discover their students are having difficulties, they may revisit the concept of density with greater care. By that time, it's likely that damage has been done to the picture of science which students are forming.

Another approach

Solid blocks of material, which are the same size, aid the comparison of density of different materials. Rectangular shaped containers for liquids and gases help in the measurement and calculation of volume. Using spreadsheets to cope with arithmetic problems enables you to emphasize the concept of density and handle materials of different densities. Once the concept has a secure foundation, the arithmetic skill can be introduced quickly, at a later stage.

We suggest, with an average group, that you get students to compare blocks of the same dimensions. Then ask them how they can bring other blocks into comparison. If they are interested (if you've succeeded in making it an intriguing problem), you can coax the class into a discussion of ways and means. If necessary, offer the suggestion of finding out the mass of one little block of 1 cm x 1 cm x 1 cm.

"Yes. If you had them, you could weigh little blocks like that little cube. But you have not got them. No, we cannot cut them up with a saw. That would take too long and spoil the big blocks. Can you count the cubes in a big block without cutting it up?"

At this point, draw pictures on the blackboard/whiteboard in a progression of problems, or give problems for homework, such as the following:

"Here is a block of wood 2 cm long, 1 cm wide and 1 cm high. Here is a little cube of Plasticene 1 cm by 1 cm by 1 cm. How many little cubes are there in this block?"

"Here is a block 2 cm long, 3 cm wide and 1 cm high. How many little cubes would fit along the 2 cm by 3 cm face? How many layers of cubes from front to back? How many cubes altogether will fit into the block?"

"So it is all a matter of counting cubes by multiplying length x breadth x height."

Of course, students will have learnt this in mathematics but you hope to have produced some practical meaning.

A well-packed box of sugar cubes helps this discussion. There may also be a set of Tillich's bricks, either in the science preparation room or in the mathematics department. These bricks are 1 cm3 and have a density of 1 g/cm3.

Density is just the name that scientists use for the mass of a unit cube.

Up next

Rough and ready measurements

Mass
Properties of Matter

Rough and ready measurements

Teaching Guidance for 14-16

To many students, the image of science is one of exactness and perfection. And yet, good scientists make rough estimates again and again, sometimes without ever making a precise measurement. It is important to teach students that rough measurements are respectable.

Of course, high precision is of the essence in many cases. A modern mass spectrograph must yield measurements of high precision if tiny mass-differences between one atomic nucleus and another are to be interpreted as energy-differences using E = mc 2.

Yet when Chadwick measured the nuclear charges of copper, silver and platinum, by alpha scattering in 1920, relatively rough measurements showed Rutherford's atomic model was correct. Chadwick showed that the nuclear charge (in electron units) is just equal to the atomic number, the number of the element in the periodic table, a series arranged in order of atomic masses. Those answers were suspected from the general pattern of theory and had to be whole numbers since a complete atom (of nucleus plus outside electrons) is neutral. Much more precise measurements were neither needed, nor at the time, possible. Even before that, the first hint of atomic number measurements came in 1906, from Barkla's attempt to measure the number of electrons in a carbon atom by scattering X-rays. His measurements suggested a number of about 6 electrons per atom, in fact somewhere between 5 and 7, yet this rough estimate enabled the founding of atomic theory to proceed.

Galileo made the roughest measurements for his test of constant acceleration down an incline. He knew he was right in his simple summary of natural behaviour. He just wanted to convince some people by quoting an experiment.

Rough estimates are not just a misfortune peculiar to early, clumsy experimenters. They are the right thing in some parts of a growing science. Nuclear physicists and some cosmic ray physicists make very precise measurements. In other cases, they seek only a rough estimate to settle an essential point in the progress of their knowledge.

You cannot give the above examples to students if they do not know the science. In that case, the following may be some help.

"An invading army is about to go into a foreign land and the general wants to know the size of the enemy's forces. He learns that it is 18 000. Does it matter much to his plans if it is 19 000 or 15 000? What he wants to know is that it is about 18 000 and not 30 000. If he waits for his staff to carefully sift through reports and add up the guesses and check them and find that the enemy really has 18 473 men, then the general may set out too late to win the battle."

Other examples include:

  • estimating how many snow ploughs are needed to clear a snowfall in the middle of the night;
  • the Chancellor of the Exchequer makes a clever guess on the number of road vehicle licences which will be paid in the next year;
  • a rough guess that the Sun is 300 000 times as massive as the Earth suffices to tell astronomers that the Earth is not massive enough to affect the orbit of the planet Venus, significantly.
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