This article explains circle theorems, including tangents, sectors, angles and proofs (with thanks to Revision Maths).
Two Radii and a chord make an isosceles triangle.
Perpendicular Chord Bisection
The perpendicular from the centre of a circle to a chord will always bisect the chord (split it into two equal lengths).
Angles Subtended on the Same Arc
Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
Angle in a Semi-Circle
Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.
We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.
We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.
But all of these angles together must add up to 180°, since they are the angles of the original big triangle.
Therefore x + y + x + y = 180, in other words 2(x + y) = 180. and so x + y = 90. But x + y is the size of the angle we wanted to find.
A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).
A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.
Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.
Angle at the Centre
The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.
You might have to be able to prove this fact:
OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a Similarly, ∠OXB = b
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a Similarly, ∠BOX = 180 – 2b Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX = 360 – (180 – 2a) – (180 – 2b) = 2a + 2b = 2(a + b) = 2 ∠AXB
Alternate Segment Theorem
This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.
You may have to be able to prove the alternate segment theorem:
We use facts about related angles
A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90. The angle in a semi-circle is 90, so ∠BCA = 90. The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180 Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90 But OAC + x = 90, so ∠OAC + x = ∠OAC + y Hence x = y
A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.
Area of Sector and Arc Length
If the radius of the circle is r, Area of sector = πr2 × A/360 Arc length = 2πr × A/360
In other words, area of sector = area of circle × A/360 arc length = circumference of circle × A/360
And, no, this is nothing to do with Harry Potter’s invisibility cloak…
Times tables can be tricky, and there’s no substitute for learning them by heart. However, the divisibility rules can at least tell you whether an answer is definitely wrong. I’m a great believer in ‘sanity checking’ your work. Just ask yourself, “Is this crazy?” If it is, you’ll have to do the question again!
The divisibility rules are quite simple (except for the ones for 7 and 8). They tell you whether a number can be divided by any number from 1 to 10. They’re most useful when simplifying fractions…or when you’re struggling to remember your times tables!
Must be a whole number, eg 2, but not 2.5.
Must be an even number ending in 0, 2, 4, 6 or 8, eg 22, but not 23.
The sum of the digits must be divisible by 3, eg 66, but not 67.
The number formed by the last two digits must be ’00’ or divisible by 4, eg 500 or 504, but not 503.
Must end in 5 or 0, eg 60, but not 61.
Must be divisible by both 2 and 3, eg 18, but not 23.
If you double the last digit and take this away from the number formed by the rest of the digits, the result must be 0 or divisible by 7, eg 672 (2 x 2 = 4, and 67 – 4 = 63, which is divisible by 7), but not 674.
The number formed by the last three digits must be ’00’ or divisible by 8, eg 5,000 or 5,008, but not 5,003.
The sum of the digits must be divisible by 9, eg 666, but not 667.
“If I’d known I’d have to go back to school, I’d never have become a teacher!”
The QTS numeracy and literacy tests are not very popular, but trainee teachers still have to pass them before they can start teaching in the state sector, so I thought I’d try and help out. There is always more than one way of doing a Maths question, but I hope I’ll demonstrate a few useful short cuts and describe when and how they should be used.
The point of short cuts is that, even though you may have to do more sums, they’ll be easier sums that can be done faster and more accurately. The numeracy test consists of two sections – mental Maths and interpreting charts – and I’m going to focus on the first of these.
Fractions to Percentages – Type 1
There are a number of typical types of questions in the numeracy test, and a lot of them involve multiplication – so knowing your times tables is an absolute must! One of the most common kinds of question involves converting fractions to percentages.
These are just two ways of showing the same thing, but to answer these questions you’ll need to try different approaches. First of all, have a look to see if the denominator (or the number on the bottom of the fraction) is a factor or a multiple of 100.
If it is, you can simply multiply or divide the numerator (the number on the top) and the denominator by whatever it takes to leave 100 on the bottom. Any fraction over 100 is just a percentage in disguise, so you just need to put the percentage sign after the numerator, eg what is the percentage mark if:
a pupil scores 7 out of a possible 20? Answer: 20 x 5 = 100, so 7 x 5 = 35%.
a pupil scores 18 out of a possible 25?
a pupil scores 7 out of a possible 10?
a pupil scores 9 out of a possible 20?
a pupil scores 130 out of a possible 200?
Fractions to Percentages – Type 2
If the denominator is not a factor of 100, check if it’s a multiple of 10. If it is, you can convert the fraction into tenths and then multiply the top and bottom by 10 to get a fraction over 100, which, again, is just a percentage in disguise, eg what is the percentage mark if:
A pupil scores 24 marks out of a possible 40? Answer: 40 ÷ 4 = 10, so 24 ÷ 4 = 6 and 6 x 10 = 60%.
A pupil scores 12 marks out of a possible 30?
A pupil scores 32 marks out of a possible 80?
A pupil scores 49 marks out of a possible 70?
A pupil scores 24 marks out of a possible 60?
Fractions to Percentages – Type 3
If neither of the first two methods works, that means you have to simplify the fraction. Once you’ve done that, you should be able to convert any common fraction into a percentage in your head. The most commonly used fractions are halves, quarters, fifths and eighths, so it’s worth learning the decimal and percentage equivalents off-by-heart, ie
½ = 0.5 = 50%
¼ = 0.25 = 25%
¾ = 0.75 = 75%
⅕ = 0.2 = 20%
⅖ = 0.4 = 40%
⅗ = 0.6 = 60%
⅘ = 0.8 = 80%
⅛ = 0.125 = 12.5%
⅜ = 0.375 = 37.5%
⅝ = 0.625 = 62.5%
⅞ = 0.875 = 87.5%
To simplify the fractions, check first to see if the numerator goes into the denominator. If it does, you can simply divide both numbers by the numerator to get what’s called a ‘unit fraction’, in other words, a fraction with a one on top, eg 4/8 divided by four top and bottom is ½.
By definition, a unit fraction can’t be simplified, so then you just have to convert it into a percentage.
If the numerator doesn’t go exactly, divide it by the first prime number (two) and then try to divide the denominator by the result, eg 6 ÷ 2 = 3, so 6/9 divided by three top and bottom is 2/3.
If that doesn’t work, try dividing the numerator by the next prime number (three) and so on and so on…
This will guarantee that the fraction ends up in the lowest possible terms, at which point it should be in the list above, which means you can easily convert it into the correct percentage, eg what is the percentage mark if:
a pupil scores 7 out of a possible 28? Answer: 7 goes into 28 four times, so the fraction is 1/4, which is 25%.
a pupil scores 27 out of a possible 36? Answer: 27 doesn’t go into 36, but 27 ÷ 3 = 9, so 27/36 divided by 9 top and bottom makes 3/4, which is 75%.
a pupil scores 24 out of a possible 48?
a pupil scores 8 out of possible 32?
a pupil scores 9 out of a possible 24?
Multiplying Three Numbers Involving Money
There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it, eg 1 x 2 x 3 is the same as 3 x 2 x 1.
The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning and then multiply the remaining three numbers together, eg a number of pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get a certain number of correct spellings, and the average amount of sponsorship is shown for each.
How many pounds would the class expect to raise for charity if the basic sum is:
20 x 30 x 5p? Answer: 2 x 3 x 5 = 6 x 5 = £30.
40 x 500 x 7p?
30 x 400 x 6p?
50 x 40 x 8p?
60 x 20 x 9p?
Division by Single-digit Numbers
This is what I call the ‘wedding planner problem’. There are three ways of doing this type of question:
Method A: Use the ‘bus stop’ method to divide the total number of guests by the number of seats per table – remembering to add one if there is a remainder.
Method B: Go straight to the end of your times tables by multiplying the number of seats by 12, then calculating the remainder and dividing by the number of seats per table, again remembering to add one if there is another remainder.
Method C: Use trial and error by estimating the number of tables needed using a nice, round number such as 5, 10 or 20 and working out the remainder as before.
Dining tables seat 7 children. How many tables are needed to seat 100 children? Answer: Method A) 100 ÷ 7 = 14 r 2, so 14 + 1 = 15 tables are needed. Method B) 7 x 12 = 84, 100 – 84 = 16, 16 ÷ 7 = 2 remainder 2, 12 + 2 + 1 = 15 tables. Method C) 10 x 7 = 70, which is too small, 20 x 7 = 140, which is too big, 15 x 7 = 70 + 35 = 105, which is just right as there are only 5 seats to spare.
Dining tables seat 6 children. How many tables are needed to seat 92 children?
Dining tables seat 5 children. How many tables are need to seat 78 children?
Dining tables seat 9 children. How many tables are needed to seat 120 children?
Dining tables seat 6 children. How many tables are needed to seat 75 children?
Division by Two-digit Numbers
If the number of seats is outside your times tables, the best option is just to use trial and error, starting with 5, 10 or 20, eg
It is possible to seat 40 people in a row across the hall. How many rows are needed to seat 432 people? Answer: 40 x 10 = 400, 432 – 400 = 32, so one more row is needed, making a total of 10 + 1 = 11 rows.
It is possible to seat 32 people in a row across the hall. How many rows are needed to seat 340 people?
It is possible to seat 64 people in a row across the hall. How many rows are needed to 663 people?
It is possible to seat 28 people in a row across the hall. How many rows are needed to seat 438 people?
It is possible to seat 42 people in a row across the hall. How many rows are needed to seat 379 people?
Percentages to Fractions
This is a type of question that looks hard at first but becomes dead easy with the right short cut. All you need to do is to work out 10% first and then multiply by the number of tens in the percentage.
Another way of saying that is just to knock one zero off each number and multiply them together, eg a test has a certain number of questions, each worth one mark. For the stated pass mark, how many questions had to be answered correctly to pass the test?
?/30 = 40% Answer: 3 x 4 = 12 questions (ie 10% of 30 is 3 questions, but we need 40%, which is 4 x 10%, so we need four lots of three, which is the same as 3 x 4).
?/40 = 70%
?/50 = 90%
?/80 = 70%
?/300 = 60%
Ratio – Distance
There are two ways of converting between different units of distance from the metric and imperial systems:
Method A: Make the ratio into a fraction and multiply the distance you need to find out by that same fraction, ie multiply it by the numerator and divide it by the denominator. (Start with multiplication if doing the division first wouldn’t give you a whole number.)
Method B: Draw the numbers in a little 2 x 2 table, with the figures in the ratio in the top row and the distance you need to find out in the column with the appropriate units, then find out what you need to multiply by to get from the top row to the bottom row and multiply the distance you have to find out by that number to fill in the final box.
8km is about 5 miles. How many kilometres is 40 miles? Answer: Method A) 8:5 becomes 8/5, and 40 x 8/5 = 40 ÷ 5 x 8 = 8 x 8 = 64km. Method B) Miles km 5 8 x 8 40 8 x 8 = 64km
6km is about 4 miles. How many kilometres is 36 miles?
4km is about 3 miles. How many kilometres is 27 miles?
9km is about 7 miles. How many miles is 63 kilometres?
7km is about 4 miles. How many kilometres is 32 miles?
Ratio – Money
You can use the same methods when converting money, except that the exchange rate is now a decimal rather than a fraction. Just remember that the pound is stronger than any other major currency, so there will always be fewer of them. It’s easy to get things the wrong way round, so it’s worth spending a couple of seconds checking, eg
£1 = €1.70. How much is £100 in euros? Method A) 100 x 1.70 = €170. Method B) £ € 1.00 1.70 x 100 100 1.70 x 100 = €170
£1 = €1.60. How much is £200 in euros?
£1 = €1.50. How much is €150 in pounds?
£1 = €1.80. How much is €90 in pounds?
£2 = €3.20. How much is £400 in euros?
Time – Find the End time
The most useful trick to use here is rounding. If the length of a lesson is 45 minutes or more, then just round up to the full hour and take the extra minutes off at the end. This avoids having to add or subtract ‘through the hour’, which is more difficult.
If the lessons are less than 45 minutes long, just work out the total number of minutes, then convert into hours and minutes and add to the start time, eg
A class starts at 9:35. The class lasts 45 minutes. What time does the class finish? Answer: 9:35 + 1 hour – 15 minutes = 10:35 – 15 minutes = 10:20.
A class starts at 11:45. There are three consecutive classes each lasting 25 minutes and then half an hour for lunch. What time does lunch finish? Answer: 11:45 + 3 x 25 + 30 = 11:45 + 75 + 30 = 11:45 + 1 hour and 15 minutes + 30 minutes = 13:30.
Lessons start at 11:15. There are two classes each lasting 40 minutes and then lunch. What time does lunch start?
Lessons start at 2:00 in the afternoon. There are four 50-minute classes with a 15-minute break in the middle. What time does the day finish?
Lessons start at 9:40. There are two classes of 50 minutes each with a break of 15 minutes in between. What time do the classes finish?
Time – Find the Start Time
It’s even more important to use rounding when working backwards from the end of an event, as subtraction is that bit more difficult, eg
A school day finishes at 3:15. There are two classes of 50 minutes each after lunch with a break of 15 minutes in the middle. What time does lunch end? Answer: 3:15 – 2 hours + 2 x 10 minutes – 15 minutes = 1:15 + 20 minutes -15 minutes = 1:20.
A school day finishes at 4:30. There are two classes of 40 minutes each after lunch. What time does lunch finish? Answer: 4:30 – 2 x 40 = 4:30 – 80 minutes = 4:30 – 1 hour and 20 minutes = 3:10.
Lunch starts at 1:05. There are two classes before lunch of 55 minutes each. What time do the classes start?
Lunch starts at 1:15. There are three classes before lunch of 45 minutes each. What time do the classes start?
A school bus arrives at school at 8:45. It picks up 20 children, and it takes an average of four minutes to pick up each child. What time is the first child picked up?
Percentage to Decimal
A decimal is a fraction of one unit, but a percentage is a fraction of 100 units, so, to convert from a percentage to a decimal, you just need to divide by 100, eg
What is 20% as a decimal? Answer: 20 ÷ 100 = 0.2.
What is 30% as a decimal?
What is 17% as a decimal?
What is 6% as a decimal?
What is 48% as a decimal?
Decimal points can be confusing, so the best way to do these sums is to take out the decimal point and put it back at the end. You just need to remember to make sure there are the same number of decimal places in the answer as in both numbers in the question, eg
1.5 x 1.5 Answer: 15 x 15 = 10 x 15 + 5 x 15 = 150 + 75 = 225, but there are two decimal places in the numbers you’re multiplying together, so the answer must be 2.25.
3 x 4.5
4.7 x 8
7.5 x 7.5
2.5 x 6.5
Multiplying Decimals by a Power of 10
Because we have 10 fingers, we’ve ended up with a ‘decimal’ number system based on the number 10.
That makes it really easy to multiply by powers of 10, because all you have to do is to move the decimal point to the right by a suitable number of places, eg one place when multiplying by 10, two when multiplying by 100 etc. (You can also think of it as moving the digits in the opposite direction.)
This type of question is therefore one of the easiest, eg
4.5 x 10 Answer: 45.
3.8 x 100
7.6 x 1000
4.6 x 100
3.5 x 10
Percentage of Quantity
Finding a percentage is easy if it ends with a zero, as you can start by finding 10% (Method A). If you happen to know what the fraction is, you can also divide by the numerator of that fraction (Method B), so 20% is 1/5, so you just need to divide by five, eg
Find 20% of 360 Answer: Method A) 360/10 x 2 = 36 x 2 = 72. Method B) 360 ÷ 5 = 72 (or 360 x 2 ÷ 10 = 720 ÷ 10 = 72).
Find 20% of 45
Find 30% of 320
Find 60% of 60
Find 80% of 120
Just because this is the ‘mental Maths’ section of the test doesn’t mean that you can’t work things out on paper, and these simple multiplication sums can be done like that.
Alternatively, you can use ‘chunking’, which means multiplying the tens and units separately and adding the results together, and the short cut for multiplying by five is to multiply by 10 and then divide by two, eg
23 x 7 Answer: 20 x 7 + 3 x 7 = 140 + 21 = 161.
42 x 5 Answer: 42 x 10 ÷ 2 = 420 ÷ 2 = 210
34 x 6
56 x 8
34 x 8
Again, working these sums out on paper is probably quicker (and more reliable), although the easiest way to divide by four is probably to halve the number twice, and the short cut for dividing by five is to multiply by two and then divide by 10.