Types of data and Sampling is the foci in today's countdown. You need to be able to recall the types of data and be able to categorise data when it is given to you. Note that there are several categories of data and your information may fit into two or more of these categories. The idea behind sampling is to get a selection of people that will accurately represent the whole population when statistical analysis is carried out. There are two types to be aware of, random (members given a number and then randomly chosen) and stratified (proportionate amount of each group is selected). When dealing with data collection it is always important to think about Bias and whether the data will be collected in a 'fair' manner.
We focus on the topic of Trigonometric graphs in today's countdown blog. You will be expected to recall the shapes of the three trigonometric functions and be able to highlight the key points (x-axis intercepts, Maximum and minimum turning points.). Note that the x-axis is measured in degrees and that the trigonometric graphs are periodical (they repeat every 360 degrees). You should then be able to read off the graphs to give the angles for specific sin/cos/tan values (notice the symmetry and that there will be more than one angle).
In today's blog post, we take a look at the Sine and Cosine rules. These rules are used when we are asked to find missing lengths or angles in non-right-angled triangles. The Sine rule is best used when you have an angle and its opposite length and the Cosine rule when you have two sides and the included angle. It is then a case of substituting know values into the formulae and rearranging to generate your answer.
You will not be given these formulae in the exam so makes sure you commit them to memory.
In today's blog, we attempt to tackle two fairly complex topics in the GCSE course. Firstly, we look at Estimating gradients.Calculating the average gradient from beginning to end does not give a good representation of what is happening in the graph. What can be done is to break the graph down into smaller sections and the average gradient of those are calculated to give a better idea of what is happening. Another method is to find the gradient at specific points on the graph, this involves constructing a series of tangents by eye and calculating the gradient at each point. This is where an estimate comes in because each tangent may vary slightly.
Secondly, we take a look at the area under a curve. The area under a velocity-time graph (most common question) will give you the distance that the object/particle has travelled. You will have to break the graph up into equal widths to give you a triangle and a series of trapezia. Calculate the area of each shape and that will tell the distance travelled.
Today's blog looks at the topic of Using graphs to find solutions. You will normally come across this topic when you need to solve simultaneous equations graphically. With this topic it will be important to construct the graphs accurately to get the correct information (if they are already done for you, no problems!) You should be looking for the point of intersection as this will the point where x and y have the same value for each equation.
Today's focus in our countdown to exams is the topic of Translations and reflections of graphs. You will be given a function (graph) and be asked to manipulate the function and sketch a new graph sometimes labelling the key points on the graphs. Starting with translations, when you add to the function f(x)+a or f(x+a), this causes the graph to move up/down or left/right.
f(x)+a is a movement up/down by a units. f(x+a) is a movement left/right by a units (+=left, -=right).
Reflections are caused by making the function negative or the variable negative.
-f(x) is a reflection in the x axis (the outputs are reversed). f(-x) is a reflection in the y axis (the inputs are reversed)
In today's blog, we look at the topic of Rates of change. With linear functions (straight line graphs) the rate of change can be interpreted from the gradient of the function. It is interpreted as an amount of y per amount of x (e.g. Dollars per hour, Metres per second). When dealing with non-linear functions there are two rates of change you could calculate.
The average rate of change; here you create a chord between two intervals and then calculate the gradient of the cord and interpret as a rate of change. The disadvantage of this is that it doesn't truly reflect the nature of the graph.
The other rate of change is an instantaneous rate of change; here you are working out the rate of change at a specific point. Create a tangent at the point, calculate and interpret the gradient as a rate of change. This will give a more accurate representation of what is happening but more tangents will be required to deliver the bigger picture.
Today, we take a look at the topic of Functions. It is important to think of functions as a machine where a value is put into the function f(x) and an output is calculated. When using functions the key process is substituting the value into the function and yielding an output (imagine completing a table of values for a quadratic). You may be asked to calculate inverse functions which is the process of working backwards from the output to find the value of the number that has been put into the function. Composite functions is where you combine two or more functions to create a new (super) function.
In today's blog post, we focus on the topic of Algebraic fractions. In this higher level topic, it is important to remember the fundamentals of the operations involving fractions. If adding/subtracting, remember you need a common denominator so may have to multiply your fractions (with numbers or algebraic terms). When multiplying multiply across the numerators then multiply across the denominators (cross cancel where possible). When dividing remember to multiply by the reciprocal of the second fraction (keep change flip).
On day 69 of our countdown, we focus on the topic of Standard Form. Standard form is a system of writing very large or very small numbers as a power of ten. Make sure you understand the basic structure of standard form (a number between 1 and 10 then times ten to a power --> a×10b). You will be asked to write numbers in standard form as well as carry out operations using standard form. If you are asked to add or subtract Standard form; take the numbers out of standard form, add/subtract them, convert back to standard form. When asked to multiply/divide standard form; calculate the numbers first, apply the laws of indices to the powers of ten. combine and convert to standard from if necessary.
In today's blog, we extend to diagrams used in probability, these include; Venn diagrams and Probability trees.With Venn diagrams, we organise data into Sets which are then contained within overlapping circles. Only elements of data that share two properties are situated in the overlap. If asked to fill in a Venn diagram, try and start with the overlaps where possible. It is important to understand the key terms surround Venn diagrams such as Union, Intersection and Complement.
With probability trees, you will be looking at two or more events happening. Remember that the probability of each event must add up to one. When the events are combined, we multiply along the branches (do not simplify at this point) to calculate the probability that these events will happen. When answering a question you may need to find the combinations that satisfy the question and then add those probabilities together. Be aware of repeated events where something is NOT replaced. This will mean that the denominator will be reduced in the second event.
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.
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.