Using mathematics skills in science is challenging for some students – some may generally struggle with mathematics and some may compartmentalise their mathematics knowledge. Language skills can also become a barrier to students’ learning, especially with the added challenge of technical vocabulary.
In our ‘How to teach Cambridge IGCSE™ Physics’ webinar you asked lots of interesting questions about mathematics and language skills. Here, webinar host and author of our Cambridge IGCSE™ Physics Digital Teacher’s Resource, Michael Smyth, offers his answers to your questions.
Mathematics skills for Cambridge IGCSE™ Physics
General tips
Students should be encouraged to view maths as a language. We use language to communicate ideas, but we need to gain sufficient ability for our needs. We use this language every day: telling the time, use of dates, working out costs of items etc.
Typically, students find it challenging to:
- Rearrange equations that already have fractions (such as the transformer equation)
- Use standard form correctly on the calculator
- Understand the difference between significant figures and decimal places
Maths lesson methods
In physics, we need to be aware of the methods that have already been taught in maths lessons. For some learners, it can be useful to reinforce these same methods, perhaps explaining them slightly differently. For others finding difficulty with these methods, it can be useful to show alternative methods. For example, in the transformer equation:
There are two methods to make NS the subject.
The method probably shown in maths lessons would start by saying ‘multiply both sides by NS‘. This will remove NS from the denominator on the right, but it will still leave anther fraction on the left, needing another ‘multiply both sides by…’ step.
An alternative method, which is easier for some learners, is called cross multiplication. It involves multiplying across diagonals that pass through the ‘=’ sign.
Cross multiplication gives the result
Vp Ns = Vs Np
which is then much simpler for students to rearrange.
How can I support students with their maths skills in relation to topics like radioactivity and simple harmonic motion?
First, remember not to go beyond the syllabus with the maths in these topics.
Some learners will understand what they are doing and be able to apply their maths skills to new situations. However, for learners who need more support, providing a method that they can learn is the most useful.
For example, in radioactivity learners need to be able to calculate the activity of a sample after a certain time, given the half-life:
e.g. a sample has an activity of 4000 counts per second and a half-life of 20 minutes. What is the activity after 2 hours?
Method:
Step 1 – work out the number of half-lives: in this case there are six 20 minute periods in 2 hours (60 minutes x 2 = 120 minutes; 120 divided by 20 = 6)
Step 2 – divide the initial activity by 2, and do this the same number of times as there are half-lives: so we divide 4000 by 2, divide the answer again by 2 and so on, until we have used the ‘divide by 2’ function 6 times.
Learners who need further challenge can see this as dividing 4000 by (2 x 2 x 2 x 2 x 2 x 2) or 26, so they can learn that we divide the initial activity by ‘2 to the power of the number of half-lives’.
Demonstrating half-life
Demonstrate half-life, and the random nature of radioactive decay, with a large group of learners by giving each learner a coin or dice and telling them all to stand.
Each time you tell them to toss the coin or roll the dice, one half-life has elapsed. If they toss a coin, then any learner getting heads must sit. If rolling a dice, then any learner getting an odd number must sit. Continue this a few times and record the numbers still standing each time. What is the trend? Does this number drop by exactly half each time?
This is a good example of differentiation in action. Different methods will work better for different learners.
Wave properties
Simple harmonic motion does not come into the Cambridge IGCSE™ syllabus, but wave properties do. Here, learners need to know how to interconvert frequency and time period. Some may find the concept of reciprocals challenging, so show them an example, such as the frequency of a wave is ‘1 divided by the time period’ and vice versa.
When measuring the time period of a pendulum, check that learners are not measuring the time between successive passes through the centre (in each direction), as this will result in a time period that is too short by half.
You could ask them to plot T against l, but this will not be a straight line. Alternatively, they could plot T2 against l but you don’t need to discuss the reason. You can let them know that this will be covered at Cambridge International AS & A Level.
Average speed
We can also remove any mystery from equations that appear complex. An example of this is the equation for average orbital speed:
Ask learners to recall the equation for average speed that they learnt toward the beginning of the course (distance divided by time). Then ask ‘what is the distance around a circle? (the circumference). ‘What is the equation for the circumference of a circle?
‘So, if the time to go around that circumference is T, then what is the average speed?
What are the number of significant figures or decimal places at Cambridge IGCSE?
First, you need to establish that learners understand how to determine the number of significant figures (s.f.). For example, that 1, 1.0, 1.00 and 1.000 do not have the same meaning. In this example, a distance of 1.0 m means that distance is known to the nearest 10 cm, whereas 1.000 m means the distance is known to the nearest 1 mm.
There are several amusing stories that can help illustrate this. One being that of a museum tour guide. She tells her tour group that ‘this fossil is 450,000,006 years old.’
One of the group says ‘that is extremely accurate!’ She replies ‘oh yes. I was told it was 450,000,000 years old when I started working here, and that was 6 years ago!’.
The difference here is the number of significant figures. The age she was told was 2 s.f., so the age was known to the nearest 10 million years. The age she quoted was 9 s.f., so the age is implied to be known to the nearest year. In this example, it is probably sufficient to know the age to only the nearest 10 million years.
Significant figures only apply to measurements or calculated values and not to counting numbers, or to values that can only be whole numbers. For example, if there are 35 chairs in the room, then significant figures have no meaning in the value 35.
The rule that learners should use in calculations is to look at the number of s.f. in the question. For example, if the quantities are 2.6 (2 s.f.) and 345 (3 s.f.) then we should not exceed the larger one of these in our answer. In this
example,
shows on the calculator as 132.6923… The answer should be rounded to 3 s.f., so given as 133.
Making measurements
When making measurements, learners should use the number of s.f. appropriate to the equipment they are using. For example, a metre rule calibrated in mm is used to measure the length of an inextensible string.
A length could be quoted as 0.983 m because it can be known to the nearest mm, or 0.001 m. However, measuring the distance moved by a toy car down a ramp in a certain time with this same metre rule should not be given to the nearest 1 mm as we cannot judge this by eye.
In this case, to the nearest 10 mm would be more appropriate. Similarly, many stopwatches give times to the nearest 10 ms, or 0.01 s. Human reaction time is typically close to 0.3 s, so manual timings should be given to the nearest 0.1 s. However, if light gates are used, then timings to the resolution of the stopwatch are appropriate.
How do you teach students energy mass equation at this level?
The syllabus requires only qualitative treatment of mass-energy changes, so no equations are needed. However, the famous equation E = mc2 is said to be the one most widely known to the general public, so learners may ask about this. It is sufficient to explain that mass is a store of energy, in a similar way to sugar being a store of chemical energy.
However, when a tiny quantity of mass is changed, a vast quantity of energy is changed. This can be seen by the term c2 in the equation, which is the speed of light (already a very large number) squared. This can be illustrated by stating that the Sun loses 5.5 million tonnes of mass every second, and most of this is transferred to space as radiation energy (the remaining quantity is lost as particles that are sent out into space).
When determining impulse, if you are given a question in which an object with a mass of 3kg slows down from say 10m/s to 6m/s, why does Cambridge IGCSE typically prefer to solve this as Ft= 3(10-6) instead of 3(6-10)?
While it is more mathematically correct to use the latter (final velocity minus initial velocity), this will yield a negative answer. Impulse is indeed a vector quantity (force x time) but learners at this level do not need to be concerned about the direction. Hence, the magnitude only is given.
There is nothing to prevent teachers encouraging those learners who need more challenge to use the latter method, especially, if these learners may progress to the Cambridge International AS & A Level 9702 course.
How do you determine if the line of a graph should be a curve or straight line when it looks like it could go either way?
This depends on the pattern of the points. Learners should always draw the line according to their results and not according to knowledge of theory. The reason for this is the possibility of experimental or systematic errors, which can affect the results. For example, a graph of first bounce height (on the y-axis) against drop height for a ball may be expected to be a straight line passing through the origin.
However, if the height is measured from the top of the ball, then the graph will not pass through (0, 0). Learners should not force their line to pass through the origin for this reason. Also, in this case if a table tennis ball is used, and the drop height extends up to 1.5 m, then the effects of air resistance may be seen which will cause the line to curve slightly in the upper portion.
Given the same set of data, it may be possible for one learner to interpret the graph as a curve, while another learner decides that one of two points are anomalous, and draws a straight line of best fit.
Learners should draw a line or curve according to how their points appear.
Language Skills for Cambridge IGCSE™ Physics
General tips
Remember that scientific English is not the same as conversational English, and must be learned as an additional skill. You can find an excellent resource on language awareness on the Cambridge International site.
How do you help learners improve their scientific language skills?
First, ensure you set a good example yourself. Use terms like weight and mass consistently and correctly. ‘You will add one 100 g mass each time.’ or ‘You will increase the weight by 1 N each time.’
Encourage learners to construct sentences or phrases using key words rather than just learning the key words alone. Give examples of these.
Take great care with similar words such as magnetic and magnetised. For example, ‘the soft iron core in the electromagnet is magnetic, but when current flows in the coil, the core becomes magnetised.’ Many learners make the mistake of saying it becomes magnetic when current flows.
Give learners quizzes where they have to write the scientific meaning and the everyday meaning for the same word: moment, pressure, cell, resistance, current, power etc., all have meanings outside of physics.
What is the difference between momentum and impulse?
Momentum is the product of the mass and velocity of a moving object, mv, and is a vector quantity with unit Ns. Expressed as a base SI unit, this is kg m/s.
Impulse is the product of a resultant force acting to cause movement and the time it acts, Ft. It is also a vector quantity and also has the unit Ns, hence the possible source of the confusion here.
Impulse is the change in momentum of an object, but not the momentum itself. Impulse is a consequence of Newton’s second law, which you can summarise as F = ma. Acceleration is change in velocity,
so
Hence
The last equation also shows why the units of impulse and change in momentum, or just momentum, must be the same.
For more help with maths and language skills in science check out our new workbooks, now available for pre-order:
Maths skills workbook for Cambridge IGCSE Biology, Chemistry and Physics
English language skills workbook for Cambridge IGCSE Biology, Chemistry and Physics
