Today we shift our focus to the Statistics section and focus on the topic of Probability. Here we take an introductory look at probability and using a sample space diagram to identify the number of outcomes. You should know that all probabilities from an event add up to make 1. Take care when calculating probabilities as your answer can be in the form of fractions (simplify them), decimals, and percentages so you may need to convert between them. We cover the 'or' rule for mutually exclusive events and the 'and' rule for independent events.
In today's blog post we cast our eye on Simultaneous equations. A simultaneous equation is where we have two equations containing different unknowns (normally x's and y's) and our job is to find the value of both of these letters. There are two ways to solve simultaneous equations; the first method is the elimination method. In this method the aim is to get one of your variables to have matching coefficients, we achieve this by multiplying one or both of the equations. When the are the same we simply add or subtract the equations to eliminate a variable. Solve the resultant equation to find the value of one variable. Remember to substitute this value into an equation to find the value of the second variable to get full marks.
The second method is the substitution method. In this method, the aim is to rearrange one of the equations so that a variable is equal to something. This something is then substituted into the other equation (this leaves one equation with one variable in it). Solve the resultant equation to find the value of one variable. Remember to substitute this value into an equation to find the value of the second variable to get full marks.
If there are quadratics present, be sure to use the substitution method.
In today's snapshot, we take a look at the topic of Inequalities. Many of the processes involving inequalities you should recognise from equations. In the case of inequalities, the = sign is replaced with four different symbols (<, >, ≤, ≥). You need to remember that inequalities show a range of values that your unknown might be (x) not just one specific value. You need to be able to represent inequalities on a number line deciding whether or not to fill the circle in. You solve and graph inequalities in the same way as an equation.
In today's blog post we are looking at the more complex topic of Completing the square and then using this information to solve quadratics. To complete the square you need to think about halving the coefficient of b, this will form the main part of your completed square. There will be another amount (half the coefficient of b then squared). This will need to be combined with your c value to create a number. From there you have a relatively straightforward equation to solve (remember you are looking for two answers).
Completing the square will also reveal the turning point of the quadratic graph if you were to plot it.
CLICK ON THE APPROPRIATE LINK FOR FAST ACCESS TO 88 SHORT RECAP VIDEOS COVERING THE GCSE MATHS COURSE.
- Place value and number lines
- Addition and subtraction
- Multiplication and division
- Prime numbers, factors & multiples
- Prime factor decomposition, HCF & LCM
- Powers and roots
- Fractions – simplifying, improper, mixed
- Adding and subtracting fractions
- Multiplying and dividing fractions
- Converting decimals to fractions
- Converting fractions to decimals
- Converting recurring decimals to fractions
- Approximations and error intervals - bounds
- Standard form
- Surds and denominator rationalizing
- Units - mass, length, area and volume
- Units – time and money
- Notation and collecting terms
- Laws of indices
- Simultaneous equations
- Expanding brackets
- Linear equations
- Factorising and the quadratic formula
- Completing the square and solving quadratics
- Algebraic fractions
STATISTICS AND PROBABILITY
- Mean, median, mode and range
- Frequency table averages
- Grouped frequency table averages
- Representing data
- Frequency and two-way table
- Cumulative frequency tables and graphs
- Quartiles and box plots
- Scatter graphs
- Types of data and sampling
- Venn diagrams and probability trees
- Frequency and two-way tables
RATIO, PROPORTION AND RATES OF CHANGE
- Quartiles as fractions and % of each other
- Percentages and percentage change
- Simple and compound interest
- Rates of change
- 3D shapes
- Angle facts
- Angles in triangles and polygons
- Angles and parallel lines
- Pythagoras theorem
- Trigonometry functions
- Perimeter and area
- Advanced areas
- Circle definitions
- Area and circumference
- Similarity and congruence
- Congruence criteria for triangles
- Constructing bisectors and loci
- Sine and cosine rules
- Circle theorems
- Equation of a straight line
- Midpoints, parallel lines and perpendicular lines
- Contextual graphs
- Quadratic and cubic graphs
- Reciprocal and exponential graphs
- Equation of a circle
- Translations and reflections
- Using graphs to find solutions
- Estimating areas and gradients under curves
- Trigonometric graphs
- Frequency and two-way tables
In today's blog, we take a look at solving quadratics using two methods. The first method to explore is solving by factorising. This means that we need to put the quadratic into double brackets and then solve to find values for x. If you end up with a quadratic that cannot be factorised then you will need to use the Quadratic formula method.
It is important to establish the values of a, b and c that are needed to substitute into the formula. You will need to memorise the formula as you will not be given this in the exam. Remember that when using the quadratic formula, you will need to do two calculations one where you ADD the square root and one where you SUBTRACT the square root.
Get those Compasses ready as we look at Bisectors and Loci today. You will sometimes be asked to construct perpendicular bisectors; either through a line segment, from a particular point or through a particular point. In each case, you will need to have a line segment to bisect. Remember you need to keep all of your construction markings in. We also take a look at bisecting an angle where a set of intersecting arcs need to be created inside the angle.
Loci are a series of points that satisfy a particular condition (a set distance from a point/line). In some questions, you will be asked to shade in regions based on the loci you have constructed.
On day 61, we focus our attention on the Congruence criteria for triangles. There will be questions that ask you to state why two triangles are identical. Here you will need to examine the side lengths and angles of the triangle and see which of the four categories they fit into. Side, Side, Side (SSS), Side, Angle, Side (SAS), Angle, Side, Angle (ASA), Right Angle, Hypotenuse, Side (RHS).
You may have to flip/rotate the triangles in your head (or use tracing paper) to see if they match up and identify which criterion to use.
Today we look at the popular topic of Transformations. There are four types of transformations to be aware of 1) Reflections (flipping a shape) 2) Rotations (turning a shape) 3) Translations (sliding a shape) and 4) Enlargements (altering the lengths of a shape). With each type of transformation, there are certain statements that need to be mentioned in order to secure full marks. A good indicator of how many things you need to state can be seen by the number of marks on offer.
A little tip: Ask for tracing paper with any transformation question so you can double check by tracing the shape and seeing if it matches up!
In today's blog, we are taking a look at Similarity and Congruence. Starting with congruence, the key thing to remember is that congruent shapes are IDENTICAL all sides are the same length and all angles are the same. The shapes may have been altered slightly so you might have to rotate, flip them so that they match up.
With similar shapes, it is important to understand that all angles remain the same and it is the side lengths that have been altered (essentially, similar shapes are ENLARGEMENTS of each other). You will have to calculate the scale factor in order to find missing values between the shapes.
When confronted with area and volume problems of similar shapes you must square or cube the scale factor to get the correct answer.
On day 58, we take a look at Approximations and Error intervals. When approximating calculations we tend to round each of the numbers to one significant figure. This should then leave us with a simpler calculation to carry out. Approximating your answers is good practice (not just when asked to) so you know if your answer is in the right region and you haven't made any big mistakes.
With error intervals, we are looking to find out what the maximum and minimum value of a rounded number might be. For continuous values, we halve the accuracy level (rounded to) and add it on for the Upper bound and subtract for the Lower bound. You will need to be a bit more careful with discrete values (people on a bus for example)
In this blog, we take a look at the volume of shapes. Volume is the space taken up by a 3D object and there are a few 3D-shapes you should be aware of. When it comes to calculating volume, you will need to establish if the shape is a prism ( the same area running the length of the shape) or a pyramid (faces meeting at a point). To calculate the volume of a prism you need to calculate the area of the face (that extends the whole depth of the shape) and multiply it by the depth. To calculate the volume of a pyramid, multiply the area of the base by the height then divide it by 3.
That last thing we look at is Spheres. You will be given the formula for this so there is no need to memorise it. Just remember to substitute the radius into the formula to get your answer.
In today's countdown, we focus on the topic of area and circumference of circles. As with all area formulae, you will need to memorise these for the area and circumference of a circle. Make sure you are dealing with the element of the circle (diameter or radius) before you substitute into the formula. Always double check your calculations to see if your answer is reasonable (circumference is about three times the length of the diameter).
When calculating sector areas and arc lengths remember to work out the proportion of the circle you are dealing with (amount of degrees given over 360). Then it is a case of using the original formulae to calculate your answer.
In this quick countdown, we take a look at all the definitions of the parts of the circle. These will become useful when dealing with circle theorems, area problems involving circles and the equation of a circle. Make sure you are confident in the terms and where they are situated on a circle.
In today's countdown, we take a look at the more Advanced areas that you will come across. We start off with a Parallelogram which is base times height (perpendicular). Just imagine a 'tilted' rectangle. The formula for the trapezium is more complex but just follow the three steps of 1) Add the parallel sides, 2)Halve it, 3)Multiply by height (perpendicular). You might be required to use this formula to find the area under a graph as well as just a shape.
We finish off with the sine rule to find the area of a triangle when you have two sides and the included angle (no height). All area formulae need to be memorised so make sure you can recall them all.
Today our focus is on Perimeter and Area. The Perimeter of a shape is the distance around a shape and will require you to add up all of the side lengths together. You may have to deal with numbers and algebraic terms so make sure you are confident in collecting like terms. You also need to be aware of the properties of shapes so you can fill in missing lengths because they are equal etc.
Area is the space inside a 2D-shape. It is normally calculated by multiplying two lengths together (a horizontal and a vertical length).
You must take care of the units you have been given in the question and convert them all to the same if required. You are not given any formula for an area so make sure memorise them all.
On day 52 of the countdown, we approach the more complex topic of the Equation of a circle. There are a few areas of prior knowledge required to tackle the whole topic completely. You will need to be confident in calculating missing sides using Pythagoras' theorem (to find the radius), calculating the gradient (to find the equation of the radius and then the tangent). Knowledge of perpendicular lines will be required as well as memorising the general rule for the equation of a circle (x2+y2=r2)
In today's countdown, we take a look at the graphs of Reciprocal and Exponential functions. It is very important that you are able to distinguish between the shapes of the two graphs as you may be asked to fill in a table of values and plot the graph. If you know the shape, then you can check if your calculations are correct.
Make sure you know the key points of each graph and in an exam, check your answers so you do not lose simple marks.
Today we take a look at Quadratic and Cubic graphs. It is important to understand the shape of these graphs and identify where they cross the x and y axes as you will be asked questions about this. Sometimes you will be required to fill in a table of values and then plot the graph. Take care plotting the points and always check the shape of your graph. If it is a quadratic, is it symmetrical?
It is always worthwhile double/triple checking your work here to ensure accuracy.
In today's blog post, we take a look at everyday Contextual graphs and how to interpret the data within them. We take a look at Distance-time graphs initially where the gradient of the graph is calculated to be the speed. If your graph is non-linear then you ay have to use a tangent to work out the speed of an object at a specific point. You will also need to use the Speed/Distance/Time formula triangle to help with calculations. We then move onto Velocity-time graphs where the same principles occur except the gradient of the line is now acceleration and the area under the graph can be calculated to give the distance traveled. Finally, we finish off on financial graphs where we can do cost comparison or currency conversions by reading off the graph.