KS3 Physics: 5. Mathematical & Experimental Methods
Our science lessons so far at Bobby Moore Academy have addressed the key concepts that explain much of how the world works. You may have noticed that we covered three distinct areas of science:
- Biology, the study of living things. This included our work on cells, the human body, plants and ecosystems, and genes and variation.
- Chemistry, the study of how atoms interact with each other and form new substances. This included our study of atomic structure, the periodic table, and chemical reactions.
- Physics, the study of matter and motion. This included our work on heating and cooling, energy, sound and light, forces and space, and electricity and magnetism.
(We also studied the Earth’s structure and rocks — this is part of the science of geology, which is only classed as a separate science when studied at university level.)
The lessons mentioned above covered different topics, but there was not much difference in how we approached each one. Mostly, we tried to think qualitatively about the phenomena we were studying. (Phenomena are simply things that happen, and to think about them qualitatively means to explain them using words or qualities.) This means most lessons ended with questions to answer in words, often in full sentences or a paragraph. The qualitative study of science is very important, and allows us to build a wide vocabulary of keywords to be used later on, but it does not capture what makes science unique. We will move on to science’s distinguishing feature in this unit and for the rest of your time at BMA.
Quantitative Science
Science is the quantitative study of the world. To think about phenomena quantitatively means to explain them using numbers, or quantities. Instead of trying to explain things in terms of their qualities, scientists assign numbers to them and try to work out how these numbers affect each other. Assigning numbers generally means making measurements, and this is often the most difficult part. Let’s start with a simple example. We may decide at some point that our diet is not very good and we are getting fat as a consequence. We might say: ‘if you eat junk food, then you will get fat’. This is a qualitative way of thinking about the problem: we are saying that the foods we eat are bad for us (they are ‘junk’ food) and that our body’s shape is changing (we are becoming fat). The sentence describes our food and ourselves in terms of words or qualities. If we were to think about this problem quantitatively, on the other hand, we would need to make measurements.
In Year 7, we saw that scientific knowledge is created in a three step process. We showed this using the mystery box activity, in which we had to try to work out what was inside a box without being able to look inside. First, we guessed what might be inside using an as if sentence, e.g. the box sounds as if there is a ping pong ball inside. Next, we deduced (worked out) what we would expect to happen if our guess was correct. We used if … then … sentences for this stage: if the box contained a ping pong ball, then we would expect to hear it roll if we tilted the box. Finally, we checked whether our deduction was correct: if we tilt the box, does the object inside make a rolling sound? Checking means doing an experiment. If the object does make a rolling sound in our experiment, then our guess could still be true. If it does not, then our guess is wrong: we need to make another one. This is roughly how our scientific understanding of the world develops.
We describe science as a quantitative way of thinking about the world because the last stage of the process — checking by experiment — means we must find some way to measure the thing we are thinking about. It is not scientific to say ‘if you eat junk food, then you will get fat’ because we cannot check this statement by experiment. Instead, we would need to create a sentence involving measurements, e.g. ‘if you consume more sugar, then your mass is likely to increase, (because the body converts any sugar you do not use to fat).’ This statement is called a hypothesis: it states what we would expect to happen, based on our previous knowledge. A hypothesis contains a cause and an effect. The cause is the thing that changes first, which is the first part of the sentence. The effect is the thing that changes as a consequence of the first change, which is the second part of the sentence. In our simple example, consuming more sugar is the cause, increasing in mass is the effect. It would not make sense to say ‘if your mass increases, you are likely to eat more sugar’. A person whose mass increases does not automatically consume more sugar; it happens the other way around.
Most scientific work that takes place in universities and other places is thus focused on making and improving measurements of things: this can be called normal science or experimental science. Their checks must be accurate and reliable, words we will learn about in more detail later. A very small number of scientists focus only on guesses and deductions. They make guesses about the big ideas we have been learning about so far in science: that matter behaves as if it is made up of tiny particles; that the Earth behaves as if its surface is made up of tectonic plates that move around; that light behaves as if it is made up of waves. The most famous physicists, like Newton and Einstein, made new guesses that explained lots of phenomena; these wide-reaching new guesses are sometimes called scientific revolutions. Guesses and deductions do not involve experiments, so this part of the subject is called theoretical science.
Scientific Experiments
Before carrying out an experiment, we must write a method to explain to other scientists how we have made measurements based on our hypothesis. This is a set of instructions, a bit like a recipe, written in the passive voice (i.e. no personal pronouns like I, we, you, and as few descriptive words as possible). In experiments, we call the cause in our hypothesis the independent variable, and the effect the dependent variable. They are both called variables because they are things that can vary or change. The effect is described as dependent because it depends on the cause. The cause is described as independent because it does not depend on the effect (as we saw above, increasing in mass does not necessarily mean eating more sugar).
We must specify the variables in our method. In our example, the mass of sugar we consume is the independent variable. Our mass is the dependent variable. To check whether the statement ‘if you consume more sugar, then your mass is likely to increase’ is true, we would need to change the mass of sugar we consume and measure how that change affects our mass. To ensure that only the change in sugar affected our mass, we would also need to make sure that all other possible causes were kept the same (in primary school, we call this a fair test). These other possible causes are called control variables, and we specify these in our method too. In this example, we would have to keep the volume of water consumed, the time taken exercising, and the time period over which measurements were taken the same. (This example would actually be very difficult to check by experiment, because in real life it is almost impossible to keep all other possible causes of weight gain the same).
When preparing for an experiment, we must ensure we will stay safe while making our measurements. We must also ensure the safety of other people using the laboratory. In school, we use hazardous equipment like bunsen burners, highly reactive chemicals (e.g. potassium or sulphuric acid), and sharp scalpels. An object is a hazard if it has the potential to cause harm. The harm it causes is called a risk. For example, a bunsen burner is a hazard; the risk is that it burns your skin. To reduce the risks to ourselves and others during experiments, we must add control measures to our methods. These are steps taken to ensure nobody is harmed while measurements are being made. Control measures for a bunsen burner might include: not putting hands in the flame, not touching the hottest part of the burner, tying hair back to avoid it catching fire, leaving the bunsen burner to cool for five minutes after it is switched off. Every time we do an experiment, we must identify any major hazards and risks and suggest control measures to reduce the chance of them causing harm.
As we carry out our experiment, we should be recording the measurements we make. To record something means to measure or observe it and store it for future reference, like when we record a video of an event on our phones. We call each individual measurement we make a data point, and these should be recorded in a results table, usually written down on paper or typed into a computer. In most experiments, we put measurements of the independent variable in the left-hand column of the results table, and measurements of the dependent variable in the right-hand column. (Sometimes we include extra columns for repeats or further calculations in our results table — more on this later). All measurements must have units: these tell us exactly what we are measuring and allow us to compare our measurements to those made by other scientists. For example, we would probably measure the mass of sugar in our experiment in grams, and we would measure our mass in kilograms. (As we will see later, we often need to convert units so they have the same prefix, in this case kilo). In our results table, we write the units in the column headings, so that we can just write numbers in the table itself.
Data Analysis
Once we have recorded our measurements, we need to analyse our data. This involves looking for a pattern or trend. We can state patterns or trends in scientific sentences: usually these are written in the form ‘The higher the …, the higher / lower the …’, or ‘As … increases, … increases / decreases’. These are different to a hypothesis, which uses if … then … sentences. A hypothesis is a statement made about something we would expect to happen in the future, whereas a scientific sentence refers to measurements made in the past. The two are similar, however, in that they both contain a cause and an effect. We can say our results show a trend if we have made at least five measurements of both independent and dependent variables, and all or most of them follow the same pattern. For example, if we made five measurements of a) the mass of sugar consumed by a person, and b) the mass of the person, and all of them fit the pattern ‘The more sugar consumed, the greater the increase in mass of the person’, then we could state this as a trend we have observed (provided all control variables were satisfactorily kept the same).
Another way we can analyse our data is to plot a graph. This allows us to see the pattern or trend more clearly. There are three common types of graph we use in science. The one we choose depends on the data points we have recorded.
- Categorical data is produced by variables that can only have certain values. Imagine doing a simple experiment to find out how many cars of different colours passed Bobby Moore Academy each day. There is no scale containing all the colours, at least in the sense that we usually think about them: we would instead say the cars could be red, blue, green, white, black, and so on. If one of the variables in an experiment produces categorical data, we plot a bar graph or pie chart to present our results.
- If measurements of a variable can have any value, we say it produces continuous data. In our experiment, both variables gave us continuous data, because the mass of sugar consumed and our mass could both have any value (our mass could be 60kg, 73.25kg, or 49.485800423423kg — it just depends on how accurate our scales are). If both variables give us continuous data, we plot a line graph or scatter graph to present our results. We will look in more detail at these graphs later on.
When we plot line graphs, the general rule is that the independent variable is plotted on the horizontal (x) axis and the dependent variable is plotted on the vertical (y) axis. An easy way to remember this rule is that the independent variable is the cause, and therefore comes first in our scientific sentence. X comes before Y in the alphabet and, likewise, the independent variable goes on the x axis. (If our independent variable is time, we plot a line graph with time on the horizontal axis. This is called a time series graph.) When we plot bar graphs, it doesn’t particularly matter which way round we plot the axes, but it is more common in science to have the bars going vertically upwards than across.
Accuracy & Errors
We might think that making measurements sounds simple. We all know how to measure the length of a line using a ruler, for example. As we carry out increasingly complicated experiments, however, making measurements can become more difficult. The word we use to describe how well we have measured something is accuracy. Imagine you want to measure your height. One person measures your height using the ruler from their pencil case. It’s only 15cm long, so they have to make multiple measurements, moving the ruler each time. Another uses a tape measure, but they’re not sure they’ve got it straight or fully stretched out, or whether they’ve measured exactly from the top of your head. A final person uses a stadiometer (see picture below), borrowed from a GP’s surgery. Who do you think made the most accurate measurement? It was almost certainly the final person, because their measurement would be the best representation of the your actual height. We might say: an accurate measurement is one that is close to the actual value.
There are three ways we can improve the accuracy of our measurements, which we can remember using the acronym LET. The first (L) is to draw a line of best fit. When we have two variables that produce continuous data, we plot them onto a scatter graph on which we do not join the dots. Instead, we draw a straight line going through or near all the points. (Sometimes we draw a curve of best fit, if the points make a smooth curve). If properly drawn, there should be roughly the same number of points above the line as below it. A line of best fit shows the general trend of the data, ignoring the effect of errors or uncertainties in measurements. By drawing a line of best fit, we therefore reduce the effect of random errors. These are errors in measurement that can’t be predicted in advance. Random errors happen in every experiment; as long as they are small relative to the measurements being made, they are nothing to worry about. If the random error in a reading is large, so that the data point does not fit the pattern (i.e. it is a long way from the line of best fit), then we call it an anomalous result or an outlier. We remove this result from our results table and graph and do not include it in any further calculations.
The second way to improve accuracy (E) is to improve the equipment used. The case above, where the person used a specialist piece of equipment to measure height (the stadiometer), is a good example of this. Other examples might include: using a digital thermometer instead of the ones we normally use in the lab; using video technology to record the position of a moving object; or using special pieces of equipment to measure lengths too small to be measured using an ordinary ruler (e.g. we would use a micrometer to measure the width of a piece of wire or a human hair). When considering which equipment to use, we must look at its resolution, which is the smallest measurement we can make using the equipment. The resolution should be much less than the measurement being made. This is why we could not measure the width of a human hair with an ordinary ruler: the resolution of the ruler is 1mm, and a human hair is much narrower than this. We must also look at the range of the measuring instrument: i.e. what is the highest and lowest reading it can measure. This is why the person trying to measure your height with a 15cm ruler would be unlikely to give you an accurate measurement: the range of their ruler is not large enough to measure the height of a person.
Even if the equipment we are using has a suitable resolution and range, sometimes it may give us inaccurate results because it has not been set up properly. This type of error is typically not random: the difference between the measured value and the actual value is the same every time. We call this type of error a systematic error. The most common type of systematic error is a zero error. Imagine you want to measure your mass using a set of scales. Before you step on the scales, they should read 0.0 kg. What if they read 0.2 kg? Let’s imagine you stepped onto these scales, and measured your mass as 53.2 kg. The reading has gone up by 53.0 kg, meaning this is your actual mass. One way of accounting for a zero error is thus to subtract the zero error from your reading. (If your initial reading is negative, you are subtracting a negative number, meaning you are effectively adding it to your reading). The other way of accounting for a zero error, of course, is to reset the measuring device to zero before making a measurement.
The third and final way to improve accuracy (T) is to use an appropriate technique. By far the most common technique used to improve accuracy is to take repeats and calculate a mean. Let’s go back to our measurements of height and imagine each of the three people measuring did repeats.
To the nearest whole number, the data suggests that the person whose height is being measured is 170cm. This was the mean reading of the most accurate method, method 3, which used the stadiometer. Method 3 was also the most precise method. Precision is a measure of how close together repeated readings are. The closer the readings are together, the smaller the random errors in the measurements.
We find precision by calculating the range, i.e. the difference between largest and smallest readings. The first method was much less precise: there was a much larger range in these readings, which suggests each one contains significant random errors. When the mean is calculated, however, it is relatively close to the readings of the other methods. Doing repeat readings and calculating a mean has therefore reduced the effect of random errors. It has given us a reasonably accurate mean reading, in spite of there being large random errors in each one. One final point about calculating means: if any of the readings in the table above was a long way from the others, we would consider it anomalous and it would not be included in the mean calculation.
Reliability & Validity
A related but distinct concept to accuracy is reliability. Reliability tells us whether we would get the same reading if we were to make a measurement more than once. Simple measurements, like measuring length with a ruler, temperature with a thermometer, or mass using a balance, tend to be fairly reliable. In fact, we can say they are both repeatable and reproducible. A repeatable measurement is one that gives the same reading if the same person does it again. Reproducible means that whoever makes the measurements, and whenever and wherever they make them, they get the same reading. Other measurements are harder to make reliably. When you do an exam, your teachers are effectively trying to measure how much you know. Imagine you write an essay for an exam. Three different teachers mark it, and they all give you different grades. This suggests the exam was not a reliable measure of how much you knew. The grade was not a reproducible result, because when different people carried out the measurement (by marking the exam), they did not get the same results. (The reason exams are often unreliable is that knowledge is very hard to define. This is a problem in many psychological experiments. Experimental psychology is the scientific study of the mind; psychological experiments try to measure feelings, emotions and states of mind. These are also very difficult to define; for example, how can we measure guilt, love or shame? Often, the measurements that are made are difficult to reproduce when the experiment is repeated.)
Even if we make accurate measurements that can reliably be made again, the conclusions we draw from them may not be valid. The most common reason for not coming to valid conclusions is that we did not satisfactorily keep our control variables the same. For example, imagine I am testing the reactivity of metals. I use a cube of aluminium, a spatula of magnesium powder, a thin ribbon of lithium and a tiny speck of potassium. I cannot come to a valid conclusion about which is most reactive because I have not kept the mass and surface area of the metals the same in each test. Validity is also a serious problem for psychologists or other scientists making measurements on humans, because there are always a huge number of variables that are very difficult to control. Imagine a doctor wishes to know what effect eating broccoli has on the human body. To come to a truly valid conclusion, she would have to find a group of people with exactly the same attributes (mass, body fat, fitness, existing health conditions, etc); make them eat only broccoli (so she knew that only the broccoli was causing any changes happening in their bodies); and ensure they all did exactly the same activities during the test (walking, sleeping, drinking water, etc). This would be impossible, as well as unethical (considered morally wrong).
The final common reason for invalid conclusions to be made is confusing correlation with causation. Causation is what we discussed at the beginning of this unit: how a change to one variable causes a change to another. When the temperature of a cup of tea increases, this causes the time taken for sugar to dissolve to decrease. We can explain this in terms of scientific theories (particles have more energy, etc). Correlation is a way of describing two things happening together, but it does not necessarily imply that one causes the other. Take an example we saw earlier in this unit: the graph showing students’ French marks vs German marks (see below). Students who get high French marks also tend to get high German marks, and vice versa. This is called a positive correlation between the two variables, i.e as one goes up, so does the other. A positive correlation does not necessarily mean, however, that their high scores in French caused their high scores in German. They probably got high scores in both because they did lots of revision for both, or always paid attention in class, or kept on top of their homework. We cannot explain the correlation adequately in terms of theories that link learning the two languages together. This means that ‘the higher the French score, the higher the German score, because learning French helps you learn German’, is not a valid conclusion.
Units & Prefixes
Earlier, we noted that all measurements must have some units specified, otherwise we would not be able to compare them to those made by other people. Imagine someone tells you that the width of a room is eight paces (a pace is a big step — we could also call it a stride). Do you know how wide the room is? You have an idea, but you don’t have an accurate measurement, because the length of one pace depends on the person who is walking. The length of one pace made by a teacher is undoubtedly much longer than the length of a pace made by a reception student! To ensure everyone in the scientific community can make easy comparisons of measurements, the following Système International (SI) units are used:
The shaded units in bold text are called the base units. All the other units can be expressed in terms of these. For example, we can measure the amount of charge transferred through a wire by multiplying the current (how much charge is transferred every second) by the time over which charge has been transferred. The units of charge are named after Augustin Coulomb (who we came across in Year 8), but Coulombs can also be expressed in base units as Ampѐre-seconds, or As, i.e. the units of current multiplied by the units of time. Some SI units are different to the ones we commonly use in everyday life. For example, in physics we tend to measure temperature in Kelvin, but in everyday life we almost always measure in oC. We must therefore know how to convert between units. In the example of temperature, as we saw in Year 7, we must add 273 onto a temperature reading in oC to find the temperature as measured in Kelvin.
Often, the numbers involved when using these units are very large or very tiny. For example, the width of an atom is approximately 0.0000000001 metres, and the frequency of an X ray wave is approximately 30000000000000000Hz. It is not very practical to use numbers with so many zeroes, so we use a mathematical method called standard form instead. You have learned about this in maths already, but it is very important in science too. We assign prefixes to certain scale factors in standard form. (A prefix is a word that goes before another word.) The most common prefixes used in science are as follows:
It is very important to be able to convert between these prefixes too. For example, to know that there are 1,000 millimetres in 1 metre, and 1,000 metres in 1 kilometre, which means there are 1,000,000 millimetres in 1 kilometre.
Formulae & Graphs
In science experiments at school, we often aim to use our data to generate scientific laws. A law in science is something that is true in all future cases, and is typically expressed as a mathematical formula or equation. It is important to understand that a scientific sentence, a line graph and a mathematical formula are all ways of representing the same trend. To use an example from Year 8, we could do an experiment to test the strength of electromagnets. Our data might give us a scientific sentence: ‘The greater the number of turns on the coil, the greater the number of paperclips picked up.’ We could plot this on a graph, with the number of turns on the x axis and the number of paperclips picked up on the y axis. Or we could try to come up with a mathematical formula. Imagine every extra turn on the coil caused one more paperclip to be picked up, so one turn picked up one paperclip, five turns picked up five, twenty turns picked up twenty, and so on. From these data, we could work out the equation:
Number of paperclips picked up = number of turns
p = t
Now imagine another electromagnet, in which every extra turn on the coil caused two more paperclips to be picked up. One turn would pick up two paperclips, five turns would pick up ten, twenty turns would pick up forty, and so on. From these data, we could work out the equation:
Number of paperclips picked up = 2 x number of turns
p = 2t
In both cases, the scientific sentence would be the same as above, but the equations are different. The graphs would also look different. The second graph would be twice as steep as the first. To use a mathematical word, the gradient of the second graph would be twice the gradient of the first. Gradient is an incredibly important concept in science, as it tells us the rate at which one variable increases relative to another, i.e. how much the dependent variable changes for each change to the independent variable. We will see many examples in which we have to work out what is happening in the world by interpreting changes in gradient on a graph. Other parts of a graph which can be useful to find out are the area under the line and the y intercept; it is also important that we are able to identify these.
In the two examples above, the variables are related by a constant, i.e. a whole number that doesn’t change. These are both therefore examples of directly proportional relationships, where the two quantities are in a constant ratio. The graph representing a directly proportional relationship is always a straight line passing through the origin (where x = 0 and y = 0), like in the example below. It is a straight line because the gradient is constant: the value of the gradient is equal to the constant ratio between the two variables. It passes through the origin because if there were no turns on the coil, then no paperclips would be picked up.
Numbers like the constant ‘2’ in the second equation above are often caused by other variables, ones which were control variables and therefore kept the same in the experiment. When a scientist discovers a proportional relationship between two or more variables, they are said to have discovered a law. For example, Ohm’s law is named after the German scientist George Ohm. His law links current, potential difference and resistance in a wire according to the following formula:
Potential difference =Current x Resistance
V =I R
At a fixed temperature, the resistance of the wire stays constant. For example, we could imagine the resistance of a length of wire to be 10Ω. This would mean that whatever the current in the wire, the potential difference supplied by the cell would be 10x greater. Because the ratio between potential difference and current is always the same, we can say that potential difference is directly proportional to current. If we were to plot a graph of potential difference vs current for a wire at a fixed temperature, it would be a straight line that went through the origin, as in the example below.
Another way to write Ohm’s law is as follows:
Current=Potential difference / Resistance
I=V/R
Now imagine we have a circuit in which the potential difference across the cell remains fixed. If the resistance of the circuit increases, what would you expect to happen to the current? You probably remember from Year 8 that if resistance increases, current decreases. We can see this from the formula above. If potential difference is fixed, e.g at 5V, and the resistance doubles from 10Ω to 20Ω, then the current goes from 5/10 (0.5A) to 5/20 (0.25A). In other words, the current decreases. Not only that, but the current halves. This is an example of an inversely proportional relationship, where the product of the two quantities is constant. If one is doubled, the other halves, as in the example we have just seen. The graph representing an inversely proportional relationship is always a curve that goes close to both the x and y axes but never quite touches either, like in the example below.
Other laws named after famous scientists are Newton’s laws, Kepler’s laws and Boyle’s law. Other famous formulae include a special case of Einstein’s theory of special relativity: Energy = mass x speed of light^2, which is better known as E = mc^2. Most laws like this are to be found in physics, but chemistry and biology also contain a few mathematical formulae. To work with formulae, we must be comfortable substituting numbers (i.e. replacing one of the words or letters with a number), and rearranging (i.e. making one of the words or letters the subject of the formula).
These mathematical methods are fundamental to the deeper study of science. The knowledge you have acquired so far in biology, chemistry and physics is only thought to be true because the guesses of scientists have been confirmed by measurement and observation. Without this checking process, science would have no basis in reality. There would be nothing to stop us suggesting that the Earth was made by magic and the Moon is made of cheese. To ensure our measurements are accurate and reliable, and that we can discover the trends and laws that arise from our observations, we must use the experimental and mathematical methods you have learned about in this unit. As you will discover during the rest of your time studying science at Bobby Moore Academy, scientists think about everything in terms of data points, graphs, formulae and gradients. It is for this reason that science can be thought of as a truly quantitative study of the world.
