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QMHS Y10 Revision Topics
Transcript of QMHS Y10 Revision Topics
and sampling Shape & Space Revision of basic Trigonometry 3D coordinates Pythagoras 2D representations of 3D solids Algebra Solving quadratics Circle Theorems Number Percentages Decimals Factors, multiples and primes Inequalities Frequency polygons Stem and leaf diagrams Cumulative Frequency Box and whisker diagrams (a.k.a. boxplots) Linear equations Circles Trial and improvement Simultaneous equations Standard form and indices Line graphs Parallel and perpendicular Graphical inequalities Properties of shapes Volume and surface area Metric and Imperial Probability Proportion Direct proportion Rearranging
formulae Averages Congruent triangles Similar figures Graphs Exponential graphs Solving equations graphically Constructions and loci Scatter Diagrams Histograms Mean Median Mode To calculate the mean of a set of numbers then add them together and divide by the number of values you have... The median of a set of data is the middle value. To find the median of a list of numbers it is usually quickest to re-write them in order. The mode (or modal value) of a set of numbers is the value which occurs most often. Moving Averages A moving average allows short term fluctuations to be 'ironed out'. It's often used with data which gets added-to over time and is the average of the last few values. An example of how to calculate a simple four-point moving average is shown on the BBC bytesize website at the link below:
http://bbc.in/kiYxTG If you know the averages of multiple sets of data (and also the sizes of the sets) then the average of ALL of the data can also be found. To do this then you have to 'weight' each average as you combine them. Combined Averages Frequency tables A frequency table is a way of presenting lists of data in a convenient and compact form. The simplest frequency tables are called 'ungrouped' and it is a relatively straightforward to find the mean, median and mode from them (though it is often necessary to add some extra columns). Ungrouped Grouped A grouped frequency table is used when the data being analysed is sorted into intervals. (An estimate of) the mean is then calculated in almost the same way as for an ungrouped table but using the mid-points of each interval. The mode and median cannot be calculated beyond finding which interval they lie within. WARNING!
The following video gives a very nice explanation of how to calculate the mean, median and mode from both ungrouped and grouped frequency tables. It does not mention the following points though... When using a grouped frequency table the mean which you calculate is only an estimate. This is because we do not know how the data is distributed within each interval group.
The method used to calculate the median value is slight variation on the way it is calculated in the example shown previously. Either way works but don't mix them together! Completing the square Factorising is probably the quickest method to solve a quadratic but unfortunately not every equation can be factorised... The more you do it the quicker you'll get at spotting whether an expression can be factorised - so practise, practise, practise! The method itself is the same every time - start by looking for factor pairs and be careful not to miss any minus signs! Factorising A special case of quadratic equation is the 'difference of two squares'. With a bit of practice these are a type of quadratic you should be able to spot pretty easily - this is useful as they're nice and easy to factorise! (MyMaths is a great resource here!) Difference of two squares If an expression can't be factorised then a powerful alternative is to 'complete the square'. Much like factorising, this a process which you'll get much faster at the more you practice! A step-by-step guide of how to complete the square on an expression and also how to then use this to solve a quadratic equation are shown below. The final (algebraic) method for solving quadratics is to use the quadratic formula. The quadratic formula works even when the quadratic can't be factorised (i.e. it doesn't have whole-number solutions) and so it's a pretty fool-proof method. The downsides are that (after you've practised with all three methods) it's probably the slowest, and it's pretty easy to make a mistake typing the numbers into your calculator. For this reason then always spend a few seconds looking at your answers if you've used the formula and make sure they look sensible! The quadratic formula Highest Common Factor (H.C.F.) (This is mostly revision of pre-GCSE work - so be sure to check your old red books!) The Highest Common Factor of two numbers, a and b, is the biggest factor of both a and b (i.e. the biggest number that goes into both a and b). To find it then separate a and b into their prime factors and multiply together all those which are common to both.
The video on the right discusses how to calculate both the H.C.F. and L.C.M. The Lowest Common Multiple of two numbers, a and b, is the smallest number which is a multiple of both a and b (it is the first number to appear in the times tables of both a and b). It is found by comparing the prime factors of both a and b and multiplying together the highest power of each.
The video on the right discusses how to calculate both the H.C.F. and L.C.M. Lowest Common Multiple (L.C.M.) Other things worth knowing... The first few prime numbers (those between 1 and 50 are definitely worth committing to memory) - they're helpful all over the place in your GCSE!
Practice spotting factor pairs of numbers - as well as being useful for topics like H.C.F. and L.C.M., it will help in your factorising of quadratics too. Some other things you should probably know from previous years: Using factor trees to find prime factors You need to be able to use a factor tree to break apart a number into its prime factors. Don't forget that you're not finished after drawing the factor tree - you also need to write out the numbers multiplied together. More complicated problems: More complicated problems will often involve applying several percentage increases/decreases in a row, typical problems might be things like:
Complex problems e.g. How much will £500, placed in a 2% savings account, be worth after 2 years? What about after 25 years?
How high will a ball bounce on its fifth bounce if it is dropped from 8m and the height of each bounce is 80% of the previous one?
(Some examples with solutions are included to the left...) Some revision of topics from previous years: [You need to be familiar with both calculator and non-calculator methods here...] You should be comfortable with questions involving finding a number as a percentage of another, percentage increases and decreases, and also be able to calculate the reverse of these processes:
Find one number as a % of another, e.g. 17 children in a class of 31 are girls, what % are boys?
Percentage of a quantity and increase/decrease e.g. find the price of a £40 meal when VAT is added at 20%.
Percentage change e.g. a house rises in value from £56,000 to £67,000 - what is this as a percentage increase?
'Reverse percentage' problems e.g. a car is sold for £5,900, making a loss of 30%. What was the car’s original price?
MAKE SURE YOU KNOW WHAT THE TERM 'DECIMAL MULTIPLIER' MEANS AND WHY IT'S SO USEFUL! Trigonometry Sine and Cosine Rules The Sine Rule The Sine rule is used when you know an angle and its opposite side and you then want to find one of:
Another angle, having been given its opposite side.
Another side, having been given its opposite angle. [Remember that these are for non-right-angled triangles!] The cosine rule is mainly used in the following two situations:
To find any of the internal angles, having been given all three side lengths.
To find a side length, having been given its opposite angle and the two remaining side lengths. The Cosine Rule From the cosine rule it is possible to prove a formula for the area of a triangle (given two sides and the angle between them). You should know this formula and know how to apply it.
A = (1/2)bcCos(A) The area of a triangle There are a number of things you should be able to do with line graphs... Cubic graphs Given an inequality then plot the line which you would get by changing the inequality into an equals sign. This line should be dotted if the original inequality was 'greater than' or 'less than' and should be solid if the original inequality was 'greater than or equal to' or 'less than or equal to'.
Having drawn your line, you need to shade the area containing unwanted points (those for which the inequality does not hold true). The best way to do this is to select a point to check (this must not be a point on the line itself). Substitute the x- and y-coordinates from the point into the inequality and see if the result makes sense: if it does then shade the OTHER side of the line to that on which the point lies, if it doesn't then shade the SAME side of the line to that on which the point lies.
If you're dealing with multiple inequalities simply treat each one separately and shade all the inequalities one-by-one. Any area left unshaded after all inequalities have been drawn contains the points which are true for all the inequalities. Be able to draw a line given its equation... Find the equation of a line Understand line segments Know and understand 'y = mx + c' ...or in implicit form
(e.g. 3x + 4y = 13) ...in explicit form (e.g. y = 2x - 4) Given its gradient and a point on the line Given two points on the line You should be able to find both the midpoint and gradient of a line segment Know that in the form 'y = mx + c', m is the gradient (steepness) and c is the y-intercept. You should know what large/small/positive/negative gradients look like. Some revision of the important properties can be found on MyMaths and through the links below. A lot of this is stuff you'll have done in years 8 and 9 so check your old red note books for more notes!
BBC bytesize (http://bbc.in/mKRka6) - Succinct discussion of the important points.
http://bit.ly/iQtdUP - slightly more detailed discussion of what m and c are.
http://bit.ly/9sdL8o - another more detailed explanation of what all the terms are and how to calculate the gradient in particular. Know the conditions of two lines being parallel or perpendicular Given a line, you need to be able to find the equations of lines which are parallel/perpendicular You also need to be able to construct the perpendicular bisector of a line segment. By combining several 2D inequalities you should be able to solve problems which involve graphing regions to solve a problem. You should understand 2-dimensional inequalities and be able to represent them on a graph [You also need to understand graphical inequalities. This is discussed in the Graphs part of this diagram.] You should know that inequalities can be plotted on axes and how to do this Volume Revision of basic ideas More complicated problems Surface Area Dimensions You need to be confident in finding perimeters and areas of all the 2D shapes you've met in the past. This includes shapes such as rectangles, squares, circles, triangles, parallelograms, trapezia and compound shapes (i.e. shapes which are combinations of these other 'simple' shapes). You should be able to convert between different length scales. You need to know how to calculate the volume of:
Prisms (including cuboids and cylinders).
Pyramids (including cones).
Be prepared for inverse problems (e.g. given the volume of a cone, can you find its height?). As with volume, you need to know how to calculate the surface area of:
Prisms (including cuboids and cylinders).
Again, be prepared for inverse problems (e.g. given the surface area of a cone, can you find its height?). You'll also encounter more complicated problems involving both volume and surface area. A common example involves frustra (a frustrum is just a cone with the top cut off). Solving problems like this aren't too tricky as long as you break them apart into smaller problems...
An example of a GCSE question is shown below, and a worked solution is included alongside. Try and solve the problem yourself before looking at the solution. Question Solution Dimensions tell you a lot about a number. You should remember that length, area and volume involve 1, 2 and 3 lengths multiplied together respectively. You can tell whether a formula will give you a length, area or volume by counting the number of lengths which are multiplied together in it! You should know the key metric to imperial conversions, such as pounds, feet, miles, pints and gallons. Most of the key ones are listed below. Theoretical and experimental You need to be aware of the difference between theoretical and experimental probabilities.
Theoretical probability is what should happen after something is tested.
Experimental probability is what does happen after something is tested.
E.g. If a fair coin is thrown, the theoretical probability of getting a heads is 0.5. If after throwing the coin ten times you get six heads then the experimental probability would be 6/10, or 0.6.
It is IMPORTANT TO REMEMBER that the more events you test, the closer the two probabilities get to one another. You should be confident in topics covered in Years 7-9 such as:
Knowing that probabilities must always add up to 1.
Sample Space Diagrams - what they are and how to use them. Revision of basic ideas You should know how to work out probabilities of an event NOT happening, combining the probabilities to find the probability of them both happening (AND), and combining two probabilities to find the probability of only one of them happening (OR). Knowing the rules below will save you a lot of time in your exams!
Prob(not A) = 1 - Prob(A)
Prob(A and B) = Prob(A) x Prob(B)
Prob(A or B) = Prob(A) + Prob(B) The NOT, AND & OR rules Tree diagrams are really useful for combining the probabilities of two (or more!) events. You should be able to draw tree diagrams and also be able to spot where they're applicable (sometimes a question won't tell you to draw one even if they're the best method to use!). Some key points to remember:
Outcomes should be at the ends of branches (e.g. six, not a six)
Probabilities are written on the branches themselves. These can be fractional or decimal.
When combining probabilities along a branch then you use the AND rule.
When combining different paths through the diagram you use the OR rule. Tree Diagrams Two quantities are directly proportional if, when one is increased, the other increases by the same percentage.
A good explanation of what direct proportionality is, and how to use it, can be found at:
If you're slightly confused by this method then there's an alternative way shown in the sheet below. Two quantities are inversely proportional if one increases as the other decreases (and vice versa). A good explanation can be found at:
As with direct proportion, there's an alternative method shown in the sheet below. Inverse proportion Direct proportion can also involve higher powers such as squares or cubes. More complicated direct proportion More complicated inverse proportion Inverse proportion can also involve higher powers such as squares or cubes. You also need to know about what proportionality relationships look like if they're plotted on a graph. There are some good examples on the BBC bytesize website at the link below:
Another good place to have a look if you're still unsure is MyMaths - you can find the lesson under Number and then Proportion. Graphs and proportion You should be comfortable with re-arranging formulae. There are various things which can catch you out here. The best way to get good at this is practice. You'll almost certainly make some mistakes the first few times you do this on your own so make sure you get these out the way during your revision and not in your exam! Linear expressions with numerical fractions With practice these questions shouldn't be too troublesome. Just make sure that when you multiply away a denominator you have to multiply every term by that denominator. An example of a simpler problem and also a past-paper question are given on the right. The key to getting this type of question right is to take lots of care when you square or square-root! Have a look at the examples on the right if you're not too sure. Expressions involving squares or square roots Expressions when the unknown appears more than once Expressions when the unknown appears more than once are some of the toughest equations to rearrange. As a general rule try to follow the steps below:
Multiply away any fractions and brackets.
Gather all the terms which involve the unknown you're interested in on one side of the equals sign (i.e. the one you're trying to make the subject of the equation). Move all terms which don't involve the unknown onto the other side.
Factorise using the unknown and divide by the term in the brackets.
The video shown on the right shows how to follow these steps to solve a past paper question. See if you can spot each of the steps being done as the question is solved. You need to be able to find averages from both grouped and ungrouped frequency tables. Remember that the mean you find for a grouped frequency table is only an estimate! You need to know the four rules for determining whether two triangles are congruent (i.e. ASA, SAS, SSS and RHS). It's worth having a go at a few past paper questions on this topic as they can sometimes be quite tricky. You also need to take care when laying out your answers to these questions. Question Solution You should understand what it means if two figures are similar. This means you should understand:
How to prove whether two triangles are similar.
How to calculate the length scale factor between two similar shapes and how to use this to calculate missing lengths.
Why two squares or two circles are always similar but two rectangles aren't always similar.
The effect an enlargement has on both the area and volume of a shape/solid by considering scale factors. There are several ways to draw a graph given its equation in explicit form. You should be able to draw a line by:
Drawing a table to find two (or preferably three) points which lie on the line, and then drawing a line through these. (A discussion of this can be found on the BBC bytesize website at
- you don't have to plot quite as many points
as they do though.)
Start from the y-intercept and use the gradient to extend the line in both directions. Given a line in implicit form, the best way to go about drawing the graph is to use a table of coordinates. It's important to remember that this method is much quicker if you pick two points when y = 0 (and then find x) and when x = 0 (and then find y). Given two points on a line, you should be able to find the equation of the line. An example of how to do this is shown below. Given a point on a line, as well as the gradient of the line, you should be able to find the equation of the line in the form y = mx + c. An example of how to do this is shown below. This is really a revision of the things you've done in earlier years. You need to be able to recognise, draw and interpret exponential graphs (you'll see these in science and geography to show population change). A quick reminder of the key points is shown on the right.
Exponential graphs can be a pretty tough topic in GCSE papers and is a common area for A* questions. Below is a video going through a question (the working's a bit messy but it's sometimes useful being talked through a question). You need to be able to know the key characteristics of a cubic graph (i.e. recognise one when you see it).
You should also know how to find the solutions to a cubic equation by checking where it crosses the axes. Remember that wherever a graph crosses the y-axis then x = 0 and wherever a graph crosses the x-axis then y = 0. Revision of basic Pythagoras Perpendicular Bisector When given a straight line, you need to be able to find its mid point and construct its perpendicular bisector. (The quickest way to find the midpoint of a line is to construct the perpendicular bisector!)
The video below (excusing the cheesy music) shows clearly and concisely how to go about doing this. Angle Bisector You need to know how to construct an angle bisector (a line which evenly splits an angle into two). A good explanation as to how to go about constructing an angle bisector can be found on the BBC bytesize website (further down the page after they discuss perpendicular bisectors) at
http://bbc.in/iPVeXJ Alternatively, a good explanation can be found on the BBC bytesize website at
http://bbc.in/iPVeXJ Alternatively, there's a pretty good demonstration of how to construct an angle bisector in the video on the right. (Once again, it's after you're shown how to construct a perpendicular bisector.) You need to be comfortable using the SOH-CAH-TOA relationships and most importantly know WHEN to use them.
JUST LIKE PYTHAGORAS, THEY CAN ONLY BE USED FOR RIGHT-ANGLED TRIANGLES!!
There's plenty of places you can find revision notes on SOH-CAH-TOA. Two are listed below:
MyMaths (see the "Trig Missing Angles" and "Trig Missing Sides" lessons in particular).
http://bbc.in/e8a1iW(the full discussion is over 4 pages on the BBC bitesize site - this link is to the first page)
There are also plenty of videos covering basic trigonometry, though these generally range from bad to terrible in standard. You should be comfortable using Pythagoras' theorem in 2D and know when it can be applied. It's also important to commit some of the simpler Pythagorean triples to memory (e.g. 3-4-5, 5-12-13) and knowing that multiples of these triples also work (e.g. 6-8-10, 9-12-15).
PYTHAGORAS' THEOREM CAN ONLY BE APPLIED TO RIGHT-ANGLED TRIANGLES!!
There's plenty of places you can find revision notes on Pythagoras. Two are listed below, along with a video with a brief explanation and a few examples (note that the MyMaths and BBC bitesize links are probably better explanations - so have a look at those first!)
MyMaths (see the "Pythagoras' Theorem" lesson).
http://bbc.in/e8a1iW(the full discussion is over 4 pages on the BBC bitesize site - this link is to the first page) Which to use? The video below is quite useful if you're getting confused about when to use each of the rules. This is another one of those areas where practice makes perfect... You need to be able to apply Pythagoras' theorem in 3D (a common question involves finding the longest diagonal of a cuboid). If you're not too confident with doing this then you can break the problem up into two steps, but once you get more comfortable with this then there is a shortcut.
The MyMaths lesson on three-dimensional Pythagoras is very good and also includes some interactive exercises - find it under 'Shape -> Pythagoras -> Pythagoras 3D'.
Below you'll also find a basic worked example on 3D Pythagoras, as well as a past paper question + solution. You'll Notice that the exam question also includes some trigonometry - as both topics are applicable to right-angled triangles then mixing them within a single question is quite a natural thing to do - be prepared for it! Pythagoras in 3D You need to understand how to both plot and also interpret a scatter diagram. Some questions to consider when you're drawing one might be:
Have you got enough points to draw conclusions from your plot?
Are your axes the right way around?
You should be able to draw a line of best fit through your points (when a correlation is present!), and then be able to interpret this line. What do the gradient and y-intercept tell you about your data?
A good refresher on scatter diagrams can be found on the BBC bytesize website at:
As usual, MyMaths also has a lesson + exercises on scatter diagrams if you want to find more revision notes and activities (they're call scatter graphs on MyMaths).
[N.B. SCATTER DIAGRAMS AREN'T COVERED UNTIL THE VERY END OF THE SYLLABUS IN YEAR 10 AND SO YOU MAY NEED TO REFER BACK TO YOUR NOTES FROM YEARS 8 & 9.] You should appreciate that histograms are quite different to bar charts and need to know what the differences are. You should be able to construct a histogram, understanding the importance of using frequency density when the data groups are unequal. Typical questions will involve finishing off an imcomplete histogram and then using it to answer further questions.
A good explanation of what Histograms are can be found on MyMaths. A slightly more concise explanation of how to draw one is given on the BBC bytesize website at:
(The page before this explains how to group data if you need reminding.)
An example exam question is shown in the video on the right.
[N.B. HISTOGRAMS AREN'T COVERED UNTIL THE VERY END OF THE SYLLABUS IN YEAR 10.] You should be able to construct and interpret box and whisker diagrams. The following explanation on the BBC bytesize website is a good guide:
It's important to realise that when drawing a box and whisker diagram, the five pieces of information you need are the lowest value, lower quartile, median, upper quartile and highest value. Sometimes an exam question might give you slightly different information and expect you to use this to work out the values you need (e.g. you may be given the lower quartile and inter-quartile range, in which case you would need to calculate the upper quartile). Don't get caught out by not reading the question carefully!
MyMaths has a lesson on how to draw and interpret box and whisker diagrams and is definitely worth running through if you're at all unsure about them. BEING ABLE TO INTERPRET A BOX AND WHISKER DIAGRAM IS EVERY BIT AS IMPORTANT AS BEING ABLE TO CONSTRUCT ONE - DON'T OVERLOOK THIS...
On the right is a short video explaining how to construct a box and whisker diagram (it does not run through how to find the quartiles or the median though). In the video it mentions that the lines representing the lowest/highest values should be shorter than the other vertical lines but you wouldn't lose marks if you didn't do this in your exam (the diagrams in the BBC bytesize explanation have no lines at the end and are just as good as the one shown in the video).
[If you're not too sure about how to find the interquartile range then have a look at http://bbc.in/cZXqX2] Cumulative frequency diagrams show the running total of a data set, adding the frequencies as they go along. They are useful plots as it is very easy to find the median and upper/lower quartiles of a data set. As with the other types of data-plots you have studied, it is important that you can both construct AND interpret cumulative frequency diagrams. They usually have a characteristic 'S' shape. A good explanation of how to construct one is given on the bytesize website:
http://bbc.in/lnLz47 (note that the points are plotted at the upper class boundaries!)
Interpreting cumulative frequency diagrams isn't too bad once you get the hang of it but is something that hasn't always been done particularly well in the past. Don't fall into the trap of not revising this!
[If you're not too sure about how to find the interquartile range then have a look at http://bbc.in/cZXqX2] Stem and leaf diagrams are something you first met back in year 7 and they don't get too much more complicated than the way you learnt about them back then. One of the most common mistakes when drawing a stem and leaf diagram is to forget to include a key!
A good reminder of how to construct them can be found on the BBC bytesize website:
One things that the bytesize example doesn't mention is that you must make sure you write your leaves in neat columns (it does do this even though it hasn't said so!).
Don't forget that it's also important to be able to interpret a stem and leaf diagram, not just to construct one. The perpendicular bisector of a chord passes through the centre of a circle and a perpendicular from a centre to chord bisects the chord.
The angle between a tangent and a radius is 90.
Tangents drawn from an external point are equal in length.
The angle in a semicircle is a right angle.
The angle subtended at the centre is twice the angle at the circumference.
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral are supplementary.
The angle between a chord and a tangent is equal to the angle in the alternate segment. You should be able to use the following properties: [N.B. All these topics will be revisited in year 11 when the proofs of required theorems will be met!] You need to be able to multiply and divide with decimals up to 2 d.p.
You also need to be able find further solutions from others which are given; e.g. if 87 x 132 = 11,484 then what is 87 x 132000? Multiplying and dividing with big and small numbers You should understand that in order to solve 'real-world' problems we will need to work in three, not two, dimensions. To do this we include a third dimension, the z-direction.
When using three dimensions then the z-direction is PERPENDICULAR to both the x-direction and the y-direction (i.e. at right-angles to both the x- and y-directions).
MyMaths gives a clear explanation of three dimensional coordinates and as usual has some good interactive exercises to check your understanding (see 'Algebra -> Coordinates -> 3D Coordinates').
Youtube has several videos which deal with 3D coordinates, however several only cover the basic GCSE topics involved or stray beyond what you need to know and cover some ideas you won't meet until A-Level. If you're going to use Youtube for this topic then be careful and make sure you're confident in the videos you use!