The Direction of Light: Exploring Polarisation and Hidden Stress

Most of us think of light in terms of brightness and colour.

A lamp can be bright or dim. Light can be red, green or blue. It can be reflected, refracted, absorbed or scattered.

But light has another property that we rarely notice in everyday life.

Light has direction.

Not simply the direction in which it is travelling, but the direction in which its electromagnetic field is oscillating.

This property is called polarisation, and it provides one of the most elegant demonstrations that light behaves as a transverse wave.

Even better, polarisation is something we can investigate with surprisingly simple equipment. Two polarising filters, a phone screen and a few pieces of transparent plastic can reveal an invisible world of patterns, stresses and colours.

It is one of those areas of science that deserves rather more attention than it normally receives at GCSE and A Level.

What Does It Mean for Light to Be Polarised?

Imagine shaking one end of a rope.

If you move your hand up and down, a wave travels along the rope while the rope itself vibrates vertically.

If instead you move your hand from side to side, the wave still travels along the rope, but the vibration is now horizontal.

The vibration takes place at right angles to the direction in which the wave travels.

That is the defining characteristic of a transverse wave.

Light behaves in a similar way.

The electromagnetic fields making up a light wave oscillate at right angles to the direction in which the light is travelling.

Ordinary light from the Sun, a lamp or many other sources contains waves vibrating in many different orientations.

We describe this light as unpolarised.

A polarising filter selects one preferred direction of vibration.

After passing through the filter, much of the remaining light is polarised.

That simple idea leads to some remarkable experiments.

Experiment 1: Two Polarising Filters

Perhaps the best introduction to polarisation requires nothing more than two polarising filters.

Hold one filter in front of a bright light source.

Some of the light is absorbed, so the view becomes slightly darker.

Now place a second polarising filter behind the first.

Initially, plenty of light may still pass through.

Slowly rotate one filter.

The transmitted light becomes progressively dimmer.

Continue rotating until the two filters are approximately 90 degrees apart.

The view can become almost completely dark.

This is known as using crossed polarisers.

Rotate the filter through another 90 degrees and the light returns.

It is an extraordinarily simple experiment.

Nothing has been switched off.

The lamp is still shining.

The filters are still transparent.

Yet their relative orientation determines whether the light gets through.

Malus's Law

The effect can be described quantitatively by Malus's Law:

I = I0 cos^2(theta)

where:

I = transmitted light intensity

I0 = maximum transmitted intensity

theta = angle between the polarisation directions of the two filters

When theta = 0 degrees:

cos^2(0) = 1

so the transmitted intensity is at its maximum.

When theta = 90 degrees:

cos^2(90) = 0

so ideally no light should pass through.

Real polarising filters are not perfect, so a small amount of light may remain visible.

This makes a good investigation for an A Level student.

A light sensor could be placed behind the filters and the intensity measured every 10 degrees as one filter is rotated.

The resulting graph should follow the cos^2(theta) relationship reasonably closely.

Suddenly an attractive visual demonstration has become a quantitative physics experiment.

Experiment 2: Your Phone Screen Is Already Helping

One of the most convenient sources of polarised light may already be sitting in your pocket.

Many LCD screens produce strongly polarised light.

Display a bright white image on a phone, tablet or computer monitor.

Now look at the screen through a polarising filter.

Rotate the filter.

At some orientations the display will look bright.

At others it may become much darker.

Depending on the construction of the screen, it may become almost black at a particular angle.

This is a wonderful demonstration because there is no obvious reason why rotating a transparent filter should make a glowing electronic screen apparently disappear.

It gives us an opportunity to discuss the physics hidden inside modern technology.

Why LCD Screens Need Polarisers

Liquid crystal displays depend on controlling the polarisation of light.

A simplified LCD contains polarising layers with liquid crystal material between them.

Electrical signals change the orientation of the liquid crystal molecules.

That changes how the polarisation of the light is modified as it passes through the display.

The second polarising layer then determines how much of that light reaches your eye.

Millions of tiny pixels can therefore be controlled independently.

Something as familiar as a laptop screen ultimately depends upon a property of light that many people have never consciously observed.

Experiment 3: Put Plastic Between Crossed Polarisers

Now things become much more colourful.

Set up two crossed polarising filters so that very little light passes through.

Then place a transparent plastic object between them.

Try:

• a transparent ruler;

• plastic cutlery;

• clear packaging;

• a CD case;

• safety glasses;

• transparent plastic sheet;

• a plastic protractor;

• pieces of adhesive tape;

• moulded plastic components.

Instead of remaining dark, the plastic may suddenly produce brilliant bands of colour.

Blues, reds, greens, yellows and purples can appear.

Some objects show beautiful rainbow fringes.

Others reveal bright regions around corners, holes and moulded features.

These colours are not pigments inside the plastic.

They are being produced by interactions between polarised light and the material itself.

Seeing Stress That Is Normally Invisible

This technique is called photoelasticity.

Some transparent materials become optically anisotropic when they are under mechanical stress.

Put more simply, light travelling through stressed plastic can behave differently depending upon its direction of polarisation.

Different parts of the light wave travel through the material at slightly different speeds.

When the components of the light recombine, they interfere.

Because different wavelengths of visible light are affected differently, coloured patterns appear.

What makes this particularly interesting is that the colour pattern can correspond to stresses inside the object.

Areas that look perfectly normal to the naked eye can contain significant internal stress.

Suddenly the invisible becomes visible.

A Simple Engineering Investigation

Take several transparent plastic rulers from different manufacturers.

Place each one between crossed polarisers.

Do they show the same pattern?

Probably not.

Now gently bend one ruler.

Watch how the colours change.

The stress distribution inside the plastic has changed, and the polarised light reveals it.

Release the ruler and much of the pattern may return towards its original state.

This immediately connects classroom physics with engineering.

Engineers need to know where stresses concentrate.

Corners, holes, notches and sudden changes in shape can produce regions where stresses become much larger than expected.

Historically, transparent models viewed through polarised light provided an important way of investigating these stress concentrations.

Modern engineers have sophisticated computer modelling techniques such as finite element analysis, but photoelasticity remains an elegant demonstration of the underlying principles.

Try Adhesive Tape

One of my favourite versions of the experiment requires something even simpler.

Take a clear piece of plastic or glass and place several layers of transparent adhesive tape across it.

Allow some pieces to overlap.

Rotate some strips relative to others.

Place the result between crossed polarisers.

The overlapping layers can produce remarkably strong colours.

Different thicknesses and orientations create different optical effects.

It begins to look almost like stained glass.

Yet the picture has been produced through physics rather than coloured pigments.

This would make an excellent practical activity because students can deliberately design their own polarisation artwork while simultaneously investigating interference and optical anisotropy.

Science and art suddenly become connected.

Polarisation by Reflection

Polarising filters can also reveal something interesting about reflected light.

Look at reflections from:

• water;

• glass;

• polished surfaces;

• wet roads;

• car windscreens.

Now view the reflection through a polarising filter and rotate it.

At certain angles the reflection becomes dramatically weaker.

Reflected light can be partially polarised.

This is particularly noticeable when light reflects from non-metallic surfaces such as water or glass.

The effect explains one of the most familiar applications of polarisation.

Why Polarised Sunglasses Work

Ordinary sunglasses simply reduce the amount of light entering the eye.

Polarised sunglasses do something more useful.

Reflections from roads, water and other horizontal surfaces tend to contain a strong horizontally polarised component.

The polarising material in the sunglasses is arranged to block much of that component.

The result is reduced glare.

This is why polarised sunglasses can be particularly effective for:

• driving;

• sailing;

• fishing;

• skiing;

• photography;

• activities around water.

For someone involved in sailing, the effect is particularly obvious.

Bright sunlight reflected from the surface of the water can produce intense glare. A good pair of polarised glasses can reduce much of that reflection and make it easier to see detail on and sometimes just below the surface.

Here a piece of wave physics becomes immediately useful.

A Quick Test for Polarised Sunglasses

There is a simple experiment you can perform.

Look at an LCD screen while wearing polarised sunglasses.

Tilt your head slowly sideways.

The brightness of the screen may change dramatically.

At approximately 90 degrees it may become very dark.

You are effectively rotating one polarising filter relative to another.

It is the same experiment we started with, except one polariser is inside your sunglasses and the other is part of your screen.

Polarisation in Photography

Photographers make extensive use of polarising filters.

A circular polarising filter fitted to the front of a camera lens can reduce unwanted reflections from:

• water;

• leaves;

• glass;

• painted surfaces;

• wet rocks.

It can also deepen the appearance of a blue sky under suitable conditions and improve colour saturation in landscape photographs.

Unlike many digital photographic effects, this cannot always be reproduced convincingly afterwards in software.

If light reflected from the surface of a lake hides what is beneath the water, the camera sensor never receives the missing information.

Reducing the reflection before taking the photograph can therefore reveal detail that would otherwise be lost.

The effect changes as the filter rotates, so the photographer can adjust it while looking through the camera.

It is another good example of physics becoming a practical creative tool.

Polarisation and Microscopy

Polarised light is also important in microscopy.

Minerals, crystals and biological structures can interact with polarised light in distinctive ways.

A thin mineral section placed between crossed polarisers can produce spectacular colours.

Geologists can use those patterns to help identify minerals and investigate the internal structure of rocks.

Polarised light microscopy is also used to investigate:

• crystals;

• fibres;

• polymers;

• biological tissues;

• industrial materials.

Once again, properties invisible under ordinary illumination become visible by controlling the direction of the light.

Polarisation in Astronomy

Even light that has travelled across enormous astronomical distances can carry polarisation information.

Astronomers can analyse the polarisation of light to investigate magnetic fields, scattering by dust and conditions around distant astronomical objects.

Polarisation measurements can contribute to studies of:

• stars;

• nebulae;

• galaxies;

• interstellar dust;

• black hole environments;

• the cosmic microwave background.

We started with two pieces of plastic held in front of a lamp.

The same underlying physics can help us investigate the Universe.

That is one of the things I find so appealing about experimental physics.

A relatively simple classroom observation can connect directly to some of the most sophisticated scientific measurements being made today.

Polarisation in Communications

Polarisation can also carry information.

Radio waves and microwaves are electromagnetic waves, just like visible light, although they have much longer wavelengths.

Their polarisation therefore matters too.

A transmitting aerial and receiving aerial generally work best when their orientations correspond appropriately.

Turn a receiving aerial through 90 degrees and the signal can decrease dramatically.

Satellite communications frequently make use of different polarisations to help separate signals.

The idea demonstrated with two optical polarising filters therefore extends far beyond visible light.

A Small Investigation for Students

A useful mini-project would be to collect a range of transparent household materials and investigate them systematically.

For each material, record:

1. What does it look like normally?

2. What does it look like between parallel polarisers?

3. What does it look like between crossed polarisers?

4. Does rotating the object change the pattern?

5. Does gently bending or squeezing it change the pattern?

6. Do different thicknesses produce different colours?

7. Can the observed patterns be related to the way the object was manufactured?

Students could photograph their results and create a gallery.

Possible objects might include:

• rulers;

• food packaging;

• spectacle lenses;

• plastic containers;

• protractors;

• disposable cutlery;

• adhesive tape;

• transparent 3D prints;

• plastic clips.

You soon discover that apparently ordinary plastic objects contain an extraordinary amount of hidden optical information.

Going Further: Measuring the Light

For a more advanced experiment, use a light sensor.

Place:

light source -> polariser 1 -> polariser 2 -> light sensor

Keep the first polariser fixed.

Rotate the second polariser through angles from 0 to 180 degrees.

Measure the intensity at perhaps 10-degree intervals.

Plot:

light intensity against angle

Then compare the experimental results with:

I = I0 cos^2(theta)

Students can investigate whether the measured relationship agrees with Malus's Law.

Possible sources of error include:

• background light;

• inaccurate angle measurement;

• imperfect polarising filters;

• sensor alignment;

• variations in the light source.

This turns a visually impressive demonstration into a proper experimental investigation involving measurements, graphs, mathematical modelling and evaluation.

Why Polarisation Matters Educationally

Students are frequently told that light is a transverse wave.

That statement can easily become another fact to memorise for an examination.

Polarisation changes that.

It provides experimental evidence.

Longitudinal waves cannot be polarised in the same way because their oscillations occur along the direction of travel.

The fact that light can be polarised therefore provides powerful evidence for its transverse nature.

There is an important educational distinction here.

Knowing that light is transverse is useful.

Seeing evidence that light is transverse is science.

And being able to design an experiment to investigate that evidence is even better.

Science Beyond the Syllabus

This is precisely why I enjoy exploring scientific ideas beyond the minimum required by an examination specification.

Specifications inevitably have limits.

There is only so much that can be taught in the available time.

But science itself does not stop at the edge of the syllabus.

Polarisation connects wave theory with engineering, photography, computing, materials science, astronomy, communications and everyday technology.

It can begin with equipment costing only a few pounds.

Yet it leads to some remarkably deep ideas.

That is exactly the sort of science I want students to experience.

Not simply:

"What do I need to remember for the examination?"

but:

"Why does that happen?"

"How could we test it?"

"What else could we discover?"

Conclusion: Light Has More to Tell Us

Light is far more complicated than simply something that allows us to see.

It has wavelength.

It has frequency.

It carries energy and momentum.

It reflects, refracts, diffracts and interferes.

And it can be polarised.

Two simple filters reveal that light has an orientation.

Add a piece of transparent plastic and suddenly invisible mechanical stresses can appear as brilliant bands of colour.

Look at reflected sunlight and we discover why polarised sunglasses work.

Turn to photography and we can control reflections before they reach the camera.

Look inside an LCD screen and polarisation becomes part of the technology we use every day.

Turn towards astronomy and the polarisation of light arriving from space can reveal information about objects millions or billions of kilometres away.

That is quite a journey from two small pieces of polarising plastic.

And it is an excellent reminder that some of the most interesting science begins when we look at something familiar and ask a slightly different question:

Light travels in a direction — but in which direction does it vibrate?

 

Why Geometrical Proof Still Matters in GCSE and A Level Maths

When I went to school, geometrical proofs seemed to be everywhere.

We did not simply learn that the angles in a triangle add up to 180 degrees. We were expected to understand why.

We did not simply memorise facts about parallel lines, similar triangles or circles. We drew diagrams, labelled angles, constructed arguments and proved results from facts we already knew.

Modern GCSE and A Level Mathematics certainly still contains proof. Students may meet algebraic proof, vector proof, proof by contradiction and some geometrical reasoning. However, my impression from teaching students today is that geometrical proof occupies a much smaller part of their mathematical experience than it once did.

I think something valuable has been lost.

The important point is not that students should memorise dozens of complicated classical proofs.

It is that proving something teaches a fundamentally different mathematical skill from simply using it.

There is an enormous difference between saying:

"The angles in a triangle add up to 180 degrees."

and being able to answer:

"Why?"

That single question takes us towards the real heart of mathematics.

---

Mathematics Is More Than a Collection of Rules

A student can sometimes get surprisingly far in GCSE Mathematics by learning procedures.

They learn:

Area of a triangle = 1/2 x base x height

They learn:

Pythagoras:

a² + b² = c²

They learn:

Circumference = pi x diameter

They learn various angle rules.

With sufficient practice, they can recognise the type of question, select the appropriate rule and calculate the answer.

That is useful.

But it is not quite the same as understanding mathematics.

A formula tells you what to do.

A proof tells you why you are allowed to do it.

That distinction becomes increasingly important as mathematics becomes more advanced.

---

A Simple Example: Why Do the Angles in a Triangle Add to 180 Degrees?

Most students can tell me:

Angles in a triangle = 180 degrees.

But suppose I ask them to prove it.

Draw any triangle ABC.

Now draw a straight line through the top vertex A which is parallel to the base BC.

Because the new line is parallel to BC, the angles created at A correspond to the two base angles of the triangle.

Along the straight line through A, the three angles together equal 180 degrees.

Therefore:

angle A + angle B + angle C = 180 degrees

That is a very simple proof.

But look at how much mathematics is contained inside it.

The student has used:

• the properties of parallel lines;

• alternate or corresponding angles;

• angles on a straight line;

• logical deduction;

• mathematical notation;

• a diagram;

• and previously established facts.

One fact has been built from other facts.

That is mathematics.

---

Proof Turns Mathematics Into a Connected Subject

Without proof, mathematics can sometimes appear to students as hundreds of unrelated rules.

Triangles have one set of rules.

Circles have another.

Parallel lines have another.

Algebra has another.

Trigonometry brings another collection of formulae.

Proof begins to reveal that these ideas are connected.

One result follows from another.

A theorem is not an arbitrary instruction invented by somebody writing an examination paper.

It is a consequence of ideas that came before it.

This is one of the reasons I like geometrical proof so much as a teaching tool.

You can often see the logic.

---

Why Are Vertically Opposite Angles Equal?

Here is another result students often learn as a rule:

Vertically opposite angles are equal.

But again we can ask:

Why?

Imagine two straight lines crossing.

Call two neighbouring angles A and B.

Because they form a straight line:

A + B = 180 degrees

Now take angle C on the other side.

B + C = 180 degrees

Therefore:

A + B = B + C

Subtract B from both sides:

A = C

So the vertically opposite angles must be equal.

A familiar angle rule has suddenly become a small piece of algebraic reasoning as well as geometry.

That connection is valuable.

---

Proof Teaches Students How to Think in Steps

One of the biggest difficulties I see students encounter in mathematics is not arithmetic.

It is reasoning.

They may be perfectly capable of performing individual calculations but struggle with a problem requiring several linked steps.

Proof develops exactly this ability.

A geometrical proof might require the student to think:

I know these lines are parallel.

Therefore these two angles are equal.

If those angles are equal, these two triangles might be similar.

If the triangles are similar, their corresponding sides are proportional.

Therefore I can establish the required result.

Each statement depends on the previous statement.

Students cannot simply jump to the answer.

They have to construct a chain of reasoning.

That is enormously useful beyond geometry.

---

The Difference Between Evidence and Proof

Geometry also provides an excellent opportunity to teach an important distinction.

Something looking true is not the same as proving that it is true.

Suppose I draw ten triangles and measure their angles.

Every time I obtain approximately:

180 degrees.

Have I proved that the angles in every possible triangle add to 180 degrees?

No.

I have gathered evidence.

Perhaps my measurements are inaccurate.

Perhaps there is some unusual triangle I have not considered.

Measurement can suggest a mathematical relationship.

Proof establishes it.

This distinction becomes particularly important in A Level Mathematics.

Trying examples can help us discover a pattern.

But a mathematical proof needs to establish that the result works in every case covered by the claim.

---

Dynamic Geometry Can Actually Make Proof More Interesting

Modern technology could make geometrical proof more engaging rather than less important.

Programs such as GeoGebra allow students to construct a triangle and then drag its vertices around the screen.

The triangle can become:

acute;

obtuse;

right-angled;

almost flat;

isosceles;

scalene.

The angles continue to add to 180 degrees.

This creates a wonderful teaching sequence.

First ask:

"What do you notice?"

Then:

"Do you think this will always happen?"

Then:

"How could we prove it?"

Technology discovers the pattern.

Proof explains the pattern.

The two approaches complement each other.

---

A Practical Investigation: Exterior Angles

Another useful classroom investigation involves the exterior angle of a triangle.

Draw a triangle and extend one side.

Measure the exterior angle.

Then measure the two opposite interior angles.

Students should discover that:

exterior angle = sum of the two opposite interior angles

Rather than giving them the rule immediately, let them find it.

Then prove it.

Suppose the triangle angles are A, B and C, with exterior angle D adjacent to C.

We know:

A + B + C = 180 degrees

We also know:

C + D = 180 degrees

Therefore:

A + B + C = C + D

Subtract C:

A + B = D

The exterior angle theorem has now emerged from two much simpler angle facts.

That is far more satisfying than simply memorising another rule.

---

Why Does Pythagoras' Theorem Work?

Pythagoras' theorem is perhaps the perfect example of something millions of students can use without really understanding.

For a right-angled triangle:

a² + b² = c²

Students substitute numbers into it.

They calculate missing sides.

They rearrange it.

But why should the squares of the two shorter sides add to the square of the hypotenuse?

There are hundreds of known proofs of Pythagoras' theorem.

Some use similar triangles.

Others rearrange squares and triangles.

One particularly visual approach places four identical right-angled triangles inside a large square.

Depending on how those triangles are arranged, the remaining area can be represented in two different ways.

Comparing those areas leads directly to:

a² + b² = c²

Suddenly the theorem is no longer a mysterious formula.

It has an explanation.

For a student who has only ever used Pythagoras as a calculator procedure, seeing a proof can completely change their view of it.

---

Similar Triangles Are a Gateway to Powerful Mathematics

Similarity is another area where proof has enormous value.

If two triangles have the same angles, their corresponding sides are in the same ratio.

That simple geometrical idea eventually feeds into:

trigonometry;

scale drawings;

surveying;

optics;

astronomy;

coordinate geometry;

and many areas of physics.

It was historically possible to estimate the height of a building or tree using shadows because similar triangles preserve ratios.

Imagine a 1 metre vertical stick casting a 1.5 metre shadow.

At the same moment a tree casts a 12 metre shadow.

Using similarity:

tree height / 12 = 1 / 1.5

Therefore:

tree height = 8 metres

The calculation is easy.

The more interesting mathematical question is:

Why are we entitled to compare those ratios?

Because the Sun's rays are effectively parallel over these distances, producing triangles with matching angles.

The geometry justifies the calculation.

---

Circle Theorems Are Much Easier When They Are Connected

Circle theorems are an area where students are often tempted simply to memorise a collection of diagrams.

Angle at centre = twice angle at circumference.

Angle in a semicircle = 90 degrees.

Opposite angles in a cyclic quadrilateral add to 180 degrees.

Angles in the same segment are equal.

Tangent is perpendicular to radius.

Alternate segment theorem.

For some students this becomes a collection of unrelated pictures.

But many of the results can be connected and derived from one another.

Once students see those relationships, there is less to memorise.

They begin to understand the structure behind the rules.

That is one of the hidden advantages of proof:

understanding can reduce the burden on memory.

---

Proof Is Not Just About Geometry

The skills developed through geometrical proof continue throughout mathematics.

At GCSE, students may be asked to prove algebraic statements.

For example:

Prove that the sum of two consecutive odd numbers is divisible by 4.

Let the first odd number be:

2n + 1

The next odd number is:

2n + 3

Their sum is:

2n + 1 + 2n + 3

= 4n + 4

= 4(n + 1)

Because the answer is 4 multiplied by an integer, it must be divisible by 4.

The same logical habits appear again:

define;

reason;

deduce;

conclude.

Geometry is simply one of the most accessible places to learn them.

---

At A Level, Proof Becomes Even More Important

A Level Mathematics demands increasingly sophisticated reasoning.

Students encounter ideas such as:

• proof by deduction;

• proof by exhaustion;

• proof by contradiction;

• vector proofs;

• trigonometric identities;

• algebraic proof.

Further Mathematics takes this considerably further.

The student who has spent years asking "why?" often finds this transition easier than the student who has mainly learnt mathematics as a collection of procedures.

Consider a trigonometric identity.

A student may be asked to show that one expression is equivalent to another.

They cannot simply put numbers into a calculator.

They have to transform one expression logically until it becomes the other.

That is proof.

The same thinking appears when manipulating vectors, deriving results and solving unfamiliar problems.

---

Geometry Also Teaches Students How to Communicate Mathematics

A good proof has to be understandable to somebody else.

That means students must learn to write things such as:

AB is parallel to CD.

Therefore angle ABC = angle BCD because alternate angles are equal.

Triangle ABC is congruent to triangle DEF by SAS.

Therefore corresponding sides are equal.

This is mathematics being used as a language.

A student may know the answer intuitively but still need to explain how they know.

That ability is valuable.

In mathematics, science and many other subjects, explaining your reasoning is often as important as reaching the correct conclusion.

---

Proof Helps With Unfamiliar Examination Questions

This may be the most practical reason for students to take proof seriously.

Examination questions are not always exact copies of examples they have practised.

A student who relies entirely on pattern recognition can become stuck when a problem looks unfamiliar.

They think:

"I haven't been shown one like this."

A student with stronger reasoning habits is more likely to ask:

"What do I know?"

"What can I work out?"

"What follows from that?"

"Which facts connect these pieces of information?"

That is essentially the thought process behind proof.

It is also the thought process behind good mathematical problem-solving.

---

There Is Value in Struggling With a Proof

Proofs should not always be demonstrated immediately by the teacher.

Sometimes the best lesson comes from allowing students to wrestle with the problem.

Give them a diagram.

Give them a few facts.

Then ask:

"Can you prove this?"

At first there may be silence.

Students draw extra lines.

They calculate angles.

They make suggestions that do not work.

They start again.

Eventually someone notices the key connection.

That moment matters.

The student has not simply received mathematics.

They have created an argument.

I think we sometimes underestimate the educational value of this kind of productive struggle.

---

A Good Home Challenge: Prove Something You Already Know

Students can try this without any specialist equipment.

Choose a geometrical fact that you think you already know.

For example:

• angles in a triangle add to 180 degrees;

• vertically opposite angles are equal;

• the exterior angle of a triangle equals the two opposite interior angles;

• the base angles of an isosceles triangle are equal;

• the angle in a semicircle is 90 degrees;

• opposite angles in a cyclic quadrilateral total 180 degrees.

Do not look up the proof immediately.

Instead ask:

1. What facts am I allowed to use?

2. Can I add a construction line?

3. Are there parallel lines?

4. Are there equal lengths?

5. Can I find congruent or similar triangles?

6. What must follow from what I already know?

Even if you eventually need to look at a hint, the attempt itself develops mathematical reasoning.

---

Could Students Rediscover a Theorem?

One of my favourite approaches is to turn the traditional order around.

Instead of:

Here is the theorem.
Here is the formula.
Now answer twenty questions.

Try:

Here is a mathematical situation.
Investigate it.
Look for a pattern.
Make a conjecture.
Test your conjecture.
Now prove it.

That sequence is much closer to how mathematics is actually developed.

Observation leads to conjecture.

Conjecture leads to proof.

Proof leads to a theorem.

The student sees mathematics as something that can be discovered rather than merely something printed in a textbook.

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The Wider Value of Learning to Prove Things

There is also a benefit that extends beyond mathematics.

Proof teaches students not to accept a claim merely because somebody says it is true.

Instead they learn to ask:

What is the evidence?

What assumptions are being made?

Does the conclusion actually follow?

Could there be another explanation?

Have all possibilities been considered?

Those are valuable habits in science, computing, economics and everyday life.

They are increasingly valuable in a world filled with statistics, graphs, algorithms, social media claims and AI-generated information.

Mathematical proof is one of the purest forms of disciplined reasoning we can teach.

---

Perhaps We Should Bring More Proof Back Into Mathematics

I am not suggesting that GCSE students need to spend months studying the classical geometry of Euclid.

Nor do we need to return uncritically to the way mathematics was taught decades ago.

Modern mathematics education has many advantages.

We have dynamic geometry software, graphical calculators, computer algebra, interactive demonstrations and enormous collections of mathematical resources.

But perhaps we should combine those modern tools with one very old mathematical question:

Why is this true?

When a student gives an answer, ask why.

When they quote a theorem, occasionally ask them where it comes from.

When they spot a pattern, ask whether it will always work.

When they measure something, ask whether measurement is enough to establish it.

And sometimes, instead of giving them another calculation, simply give them something to prove.

---

Conclusion: Knowing the Answer Is Not the Same as Understanding It

There is an important difference between knowing a mathematical fact and understanding why that fact must be true.

A student can memorise:

a² + b² = c²

They can memorise:

angles in a triangle = 180 degrees

They can memorise dozens of circle theorems and algebraic rules.

But proof takes them one level deeper.

It turns:

"I know the rule"

into:

"I understand why the rule works."

That is one of the transitions from doing mathematics mechanically to thinking mathematically.

Perhaps geometrical proof deserves a little more space in GCSE and A Level Mathematics, not because we should recreate mathematics lessons from the past, but because the ability to construct a logical argument is as important today as it has ever been.

The calculator can give a numerical answer.

Software can draw the diagram.

AI can even suggest a solution.

But the most interesting mathematical question remains one that is thousands of years old:

Can you prove it?

 

Galileo’s Inclined Plane: Recreating the Experiment That Helped Create Modern Mechanics

There are some physics experiments that are interesting because they demonstrate a particular equation.

Others are more important because they show us how physics itself developed.

Galileo’s inclined-plane experiment belongs firmly in the second category.

At first sight, it could hardly be simpler: put a ball at the top of a gently sloping track, release it and measure how it moves.

Yet hidden inside that simple experiment is an extraordinarily important idea:

motion can be measured, represented mathematically and used to discover laws of nature.

Today, we can repeat the experiment with a smartphone camera, video-analysis software, light gates or PASCO sensors and generate a graph within seconds.

Galileo had none of those things.

He did not even have a modern stopwatch.

That makes recreating his experiment particularly valuable for students. We can perform it twice: first as twenty-first-century physicists and then try to solve the problem with the technology available more than four centuries ago.

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The Problem Galileo Was Trying to Solve

Drop a ball vertically and it falls very quickly.

That creates a serious experimental problem.

Suppose you want to investigate whether a falling object moves at constant speed or accelerates.

You need to measure its position at different times.

But if the entire fall lasts only a fraction of a second, that is extremely difficult without electronic timing.

Galileo's inspired solution was effectively to slow gravity down.

Instead of allowing the ball to fall vertically, let it roll down a shallow slope.

The gravitational effect pulling it along the slope is smaller, so the motion takes considerably longer and becomes measurable.

Galileo described a long wooden channel, carefully smoothed, down which a rounded bronze ball could roll. His published account appeared in Two New Sciences in 1638.

It is wonderfully recognisable to a modern physics student.

Ramp.

Ball.

Distance measurements.

Repeated trials.

Timing.

Data.

The equipment has changed enormously.

The experimental thinking has not.

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Part One: Do the Experiment the Modern Way

I would begin by giving students the simplest possible arrangement.

Equipment

You could use:

• a long wooden track, guttering or dynamics track;
• a steel or glass ball;
• metre rule or tape measure;
• clamps or blocks to raise one end;
• smartphone capable of recording video;
• a contrasting background or distance markers;
• video-analysis software if available.

In my own laboratory I would also be tempted to repeat the experiment with PASCO equipment. A motion sensor, photogates or suitable position-measuring equipment makes it possible to collect a large amount of high-quality data very quickly.

But I would not start with the technology.

I would start with the ball.

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Make the Slope Gentle

Raise one end of the track by only a relatively small amount.

The ball should accelerate clearly but take long enough to travel along the track that its motion is easily observed.

Mark perhaps:

0.10 m

0.20 m

0.30 m

0.40 m

0.50 m

and so on.

Release the ball from rest.

Do not push it.

That apparently trivial instruction matters enormously.

A push gives the ball an initial velocity and changes the experiment.

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What Should the Students Notice?

Many students initially expect one of two things.

They may expect the ball to travel approximately equal distances during equal time intervals.

That would mean constant velocity.

Or they may simply say:

"It gets faster."

That observation is correct, but physics requires us to go further.

How does it get faster?

That is where measurement begins.

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Position Against Time

Suppose video analysis gives results something like this:

Time, t (s)	Distance, s (m)
0.0	0.000
0.2	0.012
0.4	0.048
0.6	0.108
0.8	0.192
1.0	0.300

The precise values will depend on the ramp, ball and angle.

What matters is the pattern.

Doubling the time does not double the distance.

For motion starting from rest under constant acceleration:

s=½at²

So:

s is proportional to t²

This is the central discovery.

Galileo's published description reports comparing different fractions of the ramp and finding that the distances travelled followed the squares of the corresponding times.

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A Better Graph

Plotting distance against time gives a curve.

That is useful, but we can do something even better.

Calculate t² and plot:

s against t²

For uniformly accelerated motion from rest, we expect:

s=½at²

So a graph of s against t² should be approximately a straight line.

Its gradient is:

gradient = ½ a

Therefore:

a = 2 x gradient

Suddenly, a rolling ball has given us a measurable acceleration.

This is an excellent opportunity to show students why physicists sometimes transform data before plotting it.

We are not simply producing a pretty graph.

We are asking:

What graph should be straight if our proposed physical model is correct?

That is a much more scientific question.

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The Really Interesting Question: How Did Galileo Measure Time?

This is where I think the experiment becomes much more memorable.

We can collect our data electronically and obtain times to perhaps thousandths of a second.

Galileo couldn't.

There were no electronic sensors.

There was no smartphone.

There was no stopwatch as we understand it.

So ask the students:

How would you measure a short period of time in the early 1600s?

It is worth letting them think.

Someone may suggest counting.

Someone may suggest a pendulum.

Someone may suggest the human pulse.

Someone may eventually suggest water.

And that takes us remarkably close to Galileo's own published method.

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Timing Motion With Water

Galileo described placing a large vessel of water above the apparatus with a narrow outlet producing a thin stream.

During the ball's descent, water was collected in another vessel.

The collected water was then weighed.

More water meant more elapsed time.

Because the flow was approximately steady, the mass of water provided a measure of the duration of the experiment.

That is a beautifully ingenious piece of experimental physics.

The students do not actually need to know the time in seconds.

They simply need something proportional to time.

If water flows at a constant mass flow rate:

m proportional to t

Therefore:

t proportional to m

And because:

s proportional to t²

we should also expect:

s proportional to m²

That gives us the opportunity to repeat Galileo's reasoning without ever using a clock.

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Building a Galileo-Inspired Water Timer

For a modern reconstruction I would use something slightly easier to control than an ordinary household tap.

A large reservoir with a narrow outlet works better because we want the flow rate to remain as steady as possible.

You could use:

• a large container of water;
• a narrow tube or outlet;
• a collecting beaker;
• an electronic balance;
• the inclined track;
• the rolling ball.

One person releases the ball.

At the same instant another begins collecting the water.

When the ball reaches the end, collection stops.

Measure the mass of water collected.

Repeat several times.

Then change the distance travelled.

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Why Mass Is Better Than Simply Looking at Water Height

You could collect the water in identical narrow tubes and compare its height.

That can work as a classroom visualisation provided the tubes have a uniform cross-sectional area.

Then:

volume proportional to height

and therefore approximately:

time proportional to height

But weighing the collected water is closer to Galileo's published description and gives more useful quantitative data.

It also introduces another important scientific principle:

Sometimes we measure one quantity indirectly by measuring another quantity that is proportional to it.

Modern physics is full of this.

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Calibrating the Water Clock

There is another experiment hidden inside the experiment.

Before trusting the water timer, test it.

Collect water for:

5 seconds

10 seconds

15 seconds

20 seconds

using a modern stopwatch.

Measure the mass each time.

Plot:

mass of water against time

If the water flows at a reasonably constant rate, the graph should be close to a straight line.

For example:

flow rate = mass / time

If 100 g of water is collected in 10 seconds:

flow rate = 100 / 10

flow rate = 10 g/s

Now if another experiment collects 36 g:

time = mass / flow rate

time = 36 / 10

time = 3.6 s

We have effectively built a primitive clock.

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The Human Difficulty Is Part of the Experiment

This is also where students discover something important about experimental science.

Starting the water flow at exactly the same moment that the ball begins moving is difficult.

Stopping it at exactly the right moment is difficult too.

Modern reconstructions of Galileo's apparatus have found precisely this problem: synchronising the ball and the water timing system can become an important source of uncertainty.

That makes the experiment even better educationally.

Instead of hiding experimental error, we can investigate it.

Ask:

• Does the same person release the ball and control the water?
• Would two people be better?
• What cue should signal the end of the run?
• Could the ball strike something and produce a sound?
• How many repetitions should we perform?
• Should we calculate a mean?
• How much variation occurs between trials?

Now we are doing much more than mechanics.

We are learning experimental design.

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Repeat It Again and Again

Galileo emphasised repeated measurements in his account.

That is another important lesson.

One successful run proves very little.

Suppose five measurements give collected water masses of:

42.1 g

40.8 g

41.7 g

42.5 g

41.4 g

Instead of selecting the result we like best, calculate the mean.

mean = total / number of readings

Repeated measurements help reveal random uncertainty.

Students can then compare the spread of their seventeenth-century measurements with those obtained using electronic sensors.

I suspect many will suddenly develop a greater appreciation for their motion sensors.

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Galileo Versus PASCO

This would make an excellent two-part practical.

Experiment A — Galileo's technology

Measure time using collected water.

Record:

• distance travelled;
• mass of water collected.

Look for the relationship:

s proportional to m²

Experiment B — Modern technology

Use video analysis, photogates or PASCO sensors.

Measure:

• position;
• time;
• velocity;
• perhaps acceleration.

Look for:

s proportional to t²

Then compare the two sets of results.

The physics should agree.

The precision probably will not.

And that is precisely the point.

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Now Increase the Gradient

Once students have established the basic behaviour, increase the angle of the track.

Repeat the experiment.

The ball accelerates more rapidly.

Increase the angle again.

Again the acceleration increases.

Why?

Gravity acts vertically downwards, but part of the gravitational effect acts along the slope.

For an ideal object sliding without friction:

a = g sin(theta)

where:

a = acceleration along the slope

g = gravitational field strength

theta = angle of the slope

As theta increases, sin(theta) increases.

At:

theta = 0 degrees

sin(theta) = 0

so there is no gravitational acceleration along a horizontal surface.

As the slope becomes steeper, the acceleration along it increases.

This provides the conceptual bridge towards free fall.

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But There Is a Beautiful A-Level Complication

If we are using a rolling ball, there is a subtlety worth discussing.

The ball is not merely moving down the track.

It is also rotating.

Some of the gravitational potential energy therefore becomes rotational kinetic energy.

For an ideal solid sphere rolling without slipping:

a = (5/7)g sin(theta)

rather than simply:

a = g sin(theta)

That is not a reason to avoid the ball.

Quite the opposite.

It creates a superb extension question:

Why is the measured acceleration smaller than g sin(theta)?

Students can then distinguish between:

• a sliding particle;
• a dynamics trolley;
• a rolling sphere.

That turns a classic GCSE-style demonstration into an excellent A-Level mechanics investigation.

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Can We Really Turn the Ramp Vertical?

Conceptually, the inclined plane helps us understand free fall because increasing the slope increases the component of gravity acting along the direction of motion.

But I would be careful about saying that we simply keep tilting a rolling-ball track until it reaches 90 degrees.

At that point the physical situation has changed.

A ball constrained to roll along a track is not quite the same system as an object falling freely.

A better advanced investigation would therefore be to measure acceleration for several angles and investigate the relationship between:

a and sin(theta)

Then discuss what the model predicts as:

sin(theta) approaches 1

For a sliding object, the prediction approaches:

a = g

That provides the mathematical connection with free fall.

The experiment has therefore taken us from a slow ball moving down a gentle ramp to one of the fundamental constants of mechanics.

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Measuring g From the Experiment

Students could go further.

Measure the inclination angle.

Find the acceleration from the gradient of the:

s against t^2

graph.

For an appropriate sliding or low-friction system:

a = g sin(theta)

Therefore:

g = a / sin(theta)

Repeat at several different angles.

Or plot:

a against sin(theta)

The gradient should give an estimate related to g.

With the rolling sphere, the gradient will instead reflect the rotational factor as well.

This is exactly the sort of result I like in practical physics because an apparent "failure" to obtain the expected value can lead to more physics rather than less.

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A Historical Investigation Rather Than Just a Demonstration

There is another fascinating dimension to this experiment.

Historians have discussed exactly how Galileo achieved the precision claimed in accounts of his experiments. His published description includes the water method, while historical work has also considered timing by pulse and possible use of musical rhythm. Reconstructions show that these questions about technique, accuracy and experimental skill are genuinely interesting rather than merely historical trivia.

That gives students three different questions to investigate.

The physics question

What mathematical relationship describes accelerated motion?

The experimental question

How accurately can we measure it?

The historical question

How could someone establish the relationship without modern instrumentation?

That combination is what makes this experiment special.

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What I Would Ask Students Before Giving Them the Equation

I would resist the temptation to begin by writing:

s = 1/2 at^2

on the board.

Instead I would give them the data.

Then ask:

What happens if you double the time?

What happens if you triple it?

Does:

s / t

remain constant?

What about:

s=½at²

Can you produce a straight-line graph?

What does the gradient mean?

Only after that would I introduce the familiar equation.

The student has then partly discovered the equation rather than merely being told it.

That is much closer to the intellectual spirit of the experiment.

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Some Excellent Extension Investigations

Once the basic apparatus exists, there are many experiments available.

Change the angle.
How does acceleration depend on inclination?

Change the ball.
Compare steel, glass, wood and different diameters.

Compare rolling and sliding.
Does the same theory describe both?

Investigate surface roughness.
Does the ball roll without slipping?

Compare timing methods.
Water clock versus video versus electronic sensor.

Investigate uncertainty.
Which method produces the smallest percentage uncertainty?

Try Galileo's distance ratios.
If one distance takes time t, what distance should the ball cover in 2t?

Since:

s proportional to t²

then:

2t gives 4s

and:

3t gives 9s.

This produces the famous sequence:

1, 4, 9, 16, 25...

for distances travelled from rest at equal elapsed times under constant acceleration.

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From a Wooden Ramp to Modern Mechanics

There is something rather satisfying about putting a ball at the top of a piece of wood and realising how much physics can emerge from it.

Acceleration.

Graphs.

Mathematical modelling.

Gravity.

Energy.

Rotation.

Uncertainty.

Experimental design.

Data analysis.

History of science.

And perhaps most importantly, the idea that nature's behaviour can be described mathematically.

Galileo did not have a PASCO sensor capable of sending hundreds of readings per second to a computer.

He had a ball, a carefully constructed inclined plane, water, balances, measurement and an exceptionally important question.

Modern equipment allows us to see his result with extraordinary clarity.

But recreating the experiment using water reminds students that the crucial piece of scientific apparatus was not the clock.

It was the reasoning.

That is why Galileo's inclined plane deserves to remain in the physics laboratory more than four hundred years later.

It is not simply an old experiment.

It is one of the experiments that shows us how experimental physics became physics.