GCSE Maths Revision Course

Key learning outcomes for this video:

• To be able to use the order of operations (BIDMAS).

• To be able to + - x ÷ both positive and negative numbers.

• To be able to round numbers to a given number of significant figures.

• To be familiar with different methods for + - x ÷ large numbers.

Cue card notes and useful reminders…

BIDMAS (this is the order of mathematical operations that we follow)

• Brackets

• Indices

• Division

• Multiplication

• Addition

• Subtraction

Rounding

• If the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up.

• If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down.

• Significant figures are counted from the left hand side after the first digit which is greater than zero.

• Non-zero digits are always significant.

• Zeros between non-zero digits are always significant.

• Leading zeros are never significant.

• Trailing zeros are only significant if the number contains a decimal point.

Adding & Subtracting

See the download resources for more tips

Key learning outcomes for this video:

• To understand and apply the vocabulary and concepts of factors, multiples, prime numbers, prime factors, Highest Common Factor (HCF) and Lowest Common Multiple (LCM).

Cue card notes and useful reminders…

• Factors of a number multiply together to give that number. Always think of factors in pairs.

• The HCF of two numbers is the highest number that divides into both the numbers that you are considering.

• Multiples of a number are the numbers in the times table of that number.

• The LCM of two numbers is the lowest number that appears in both the times tables of the numbers that you are considering.

• A prime number must have two different factors - 1 and itself. (1 is not a prime number, 2 is the lowest prime number).

• When doing prime factorisation, always divide by the lowest prime number that you can.

• For big numbers, use prime factorisation and then a venn diagram for finding HCF and LCM.

HCF ➤ multiply all the numbers in the overlapping part of venn diagram

LCM ➤ multiply all the numbers that you can see in the venn diagram

Key learning outcomes for this video:

• To be confident using the following mathematical vocabulary when working with fractions: numerator, denominator, proper and improper fractions, equivalent fractions, canceling down.

• To be able to find equivalent fractions by multiplying or canceling down.

• To be able to use the fraction button on a calculator.

• To be able to write one number as a fraction of another

• To be able to find a fraction of an amount.

• To be able to change between proper and improper fractions.

Cue card notes and useful reminders…

• The numerator is the top number of a fraction.

• The denominator is the bottom number of a fraction.

• To find a fraction of an amount - divide the amount by the denominator and then multiply by the numerator.

• A proper fraction has a numerator smaller than the denominator and is equal to a decimal less than 1.

• An improper fraction has a numerator greater than the denominator and is equal to a number greater than 1. We can change improper fractions to mixed numbers.

Key learning outcomes for this video:

• To be able to add and subtract fractions.

• To be able to multiply fractions.

• To be able to use the KEEP CHANGE FLIP method for dividing fractions.

• To be able to cancel fractions up and down and diagonally across a multiplication sign.

Cue card notes and useful reminders…

• The numerator is the top number of a fraction.

• The denominator is the bottom number of a fraction.

• The LCM of two numbers is the lowest number that appears in both the times tables of the numbers that you are considering.

• To add or subtract fractions they must have the same denominator. Once they have the same denominator, keep this denominator in your answer and only add or subtract the numerators.

• To multiply fractions:

> Multiply the numerators together to get the numerator in your answer and then multiply the denominators together to get the denominator in your answer.

> Then cancel down your answer if you can.

> If your answer is an improper fraction, change it to a proper fraction.

> Sometimes when multiplying fractions with large numbers you can cancel diagonally across the multiplication sign before multiplying.

> If the fractions in the question are mixed numbers, change them to improper fractions before you multiply.

• To divide fractions:

> Remember to use the KEEP, CHANGE, FLIP method and then continue as above for multiplying.

>If the fractions in the question are mixed numbers, change them to improper fractions before you multiply.

• A proper fraction has a numerator smaller than the denominator and is equal to a decimal less than 1.

• An improper fraction has a numerator greater than the denominator and is equal to a number greater than 1. We can change improper fractions to mixed numbers.

Key learning outcomes for this video:

• To be able to convert between Fractions, Decimals and Percentages

Cue card notes and useful reminders…

• Remember 100% means 1 whole.

• Use the FDP triangle in the downloadable materials to help you with your conversions:

  
  

Key learning outcomes for this video:

• To be able to use and understand place value and rounding when working with decimals.

• To be able to + - x ÷ with decimals.

Cue card notes and useful reminders…

• See the place value table in downloadable materials:

  
  

• Rounding

> If the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up.

> If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down.

> Significant figures are counted from the left hand side after the first digit which is greater than zero.

> Non-zero digits are always significant.

> Zeros between non-zero digits are always significant.

> Leading zeros are never significant.

> Trailing zeros are only significant if the number contains a decimal point.

• When estimating an answer to a calculation, round all numbers to 1s.f.

• When adding and subtracting decimals - line up the decimal points and transfer into your answer in the same position.

• For long multiplication with decimals - you do not need to line up the decimals, instead line the numbers up to the right hand side.

• For long multiplication and grid method with decimals - to decide the position of the decimal point in your answer - count the number of decimal places in both your original numbers and count this many decimal places into your answer from the right hand side.

• When using bus stop method for dividing by a decimal - multiply the decimal on the outside of the bus stop by 10,100, 1000… so that it is an integer. Then multiply the number in the bus stop by the same number and continue with division as normal.

Key learning outcomes for this video:

• To be able to find percentages without a calculator.

• To be able to find percentages with a calculator.

• To be able to find one number as a percentage of another.

Cue card notes and useful reminders…

• We've shown some key percentages in the download materials, which you should be able to find without a calculator.

  Remember you can combine these key percentages above to find more complex percentages.

• To find one number as a percentage of another without a calculator, write the numbers as a fraction of each other and then find an equivalent fraction with 100 as the denominator.

Key learning outcomes for this video:

• To be able to calculate a percentage increase and a percentage decrease.

• To be able to find the original amount after a percentage change.

Cue card notes and useful reminders…

• Multiplier method - INCREASE

100% + percentage change - change into decimal multiplier.

• Multiplier method - DECREASE

100% - percentage change - change into decimal multiplier.

• Reverse percentage changes - always draw a diagram or write an equation of what is happening.

Key Learning Outcomes for this video:

• To be able to use index laws to simplify expressions and make calculations.

• To be able to use the index and root buttons on a calculator.

Cue card notes and useful reminders…

• If you are EVALUATING an expression then you are finding the value of it and your answer should be a number.

• If a base number has no power then it is to the power of 1.

• Any number to the power of zero is equal to 1.

• Squared means a ‘power of 2’, cubed means a ‘power of 3’.

• See index laws in the download materials.

Key learning outcomes for this video:

• To be able to convert ordinary numbers greater and less than 1 to standard form and back again.

• To be able to use the index laws to solve problems using numbers in standard form.

Cue card notes and useful reminders…

• STANDARD FORM: a x 10n where a must be equal to or greater than 1 but less than 10.

> If the power n is positive it moves the decimal point to the right making the number LARGER.

> If the power n is negative it moves the decimal point to the left making the number SMALLER.

• See simple Index laws in download materials

• When multiplying and dividing numbers in standard form - keep in standard form and use the index rules above.

• When adding and subtracting numbers in standard form - change into ordinary numbers, add or subtract and then change back into standard form if necessary.

Key learning outcomes for this video:

• To be able to simplify an algebraic expression by collecting like terms.

• To be able to use substitution to evaluate an algebraic expression.

Cue card notes and useful reminders…

• BIDMAS (this is the order of mathematical operations that we follow)

Brackets

Indices

Division

Multiplication

Addition

Subtraction

• An equation contains an equals sign, an expression does not.

• A variable is an unknown quantity, usually denoted by a letter.

• A coefficient is the number multiplying a variable.

• Terms of an expression are separated by addition or subtraction signs.

• When collecting like terms, you can only collect together the terms that contain exactly the same set of variables.

• In an expression or equation, the sign (+ or -) in front of a term is attached to that term.

• A fraction line means the same as divide.

• If you use a calculator to square a negative always put a bracket around it .

Key Learning Outcomes for this video:

• To be able to change the subject of an equation or a formula by rearranging.

Cue card notes and useful reminders…

• BIDMAS (this is the order of mathematical operations that we follow)

Brackets

Indices

Division

Multiplication

Addition

Subtraction

• When rearranging an equation you work through BIDMAS backwards using inverse (opposite) operations.

Key learning outcomes for this video:

• To be able to solve equations with 1 variable/unknown.

• To be able to form and solve equations.

Cue card notes and useful reminders…

• An equation contains an equals sign and can be solved.

• An expression does not contain an equals sign and cannot be solved.

• A formula is an equation used to solve a real life problem.

• A variable is an unknown quantity, usually denoted by a letter.

• BIDMAS (this is the order of mathematical operations that we follow)

Brackets

Indices

Division

Multiplication

Addition

Subtraction

• When solving an equation you work through BIDMAS backwards using inverse (opposite) operations.

• When solving an equation your final answer will be a number. Remember you can substitute this number back into the original equation to check if it is the correct answer.

• If the variable appears on both sides of the equation, collect like terms first before you start to solve.

Key learning outcomes for this video:

• To be able to factorise and expand an expression.

Cue card notes and useful reminders…

• Once you have expanded brackets, look for like terms to collect.

• Factorising into brackets is the opposite of expanding brackets. Remember you can check your factorised answer by expanding and seeing if it gets you back to the original expression.

Key learning outcomes for this video:

• To be able to substitute values of x into a linear equation and find corresponding values of y.

• To be able to plot the graph of a linear equation.

• To be able to find the equation of a line from a graph.

Cue card notes and useful reminders…

• Remember the x-axis is HORIZONTAL, the y-axis is VERTICAL.

• Coordinates are written (x, y) - ‘along the corridor, up the stairs’.

• Horizontal lines pass through the y-axis and are written y = …

• Vertical lines pass through the x-axis and are written x = …

• The simplest form of a linear equation is y = mx + c

> m is the gradient or slope of the line.

> c is the y-intercept of the line (this is where x = 0)

• The coefficient of x is the number multiplying x.

• A line with a positive gradient slopes up to the right, a line with a negative gradient slopes down to the right.

Key learning outcomes for this video:

• To be able to solve quadratic equations by factorising.

• To be able to solve quadratic equations by using the quadratic formula.

Cue card notes and useful reminders…

• The general form of a quadratic equation is

• To expand two linear brackets use the FOIL method or grid method.

• Remember factorising a quadratic is the opposite of expanding - you are trying to put it back into two brackets ( )( ) multiplied together. Once you have your answer you can expand the brackets to check that it gives you the original quadratic.

• If a quadratic is equal to zero, once you have factorised it into the two brackets you can solve it by letting each individual bracket equal zero and solve - therefore you end up with two solutions from the two brackets (occasionally these solutions can be the same and are called a repeated solution).

• If a quadratic doesn’t have the middle ‘bx’ term and can still be factorised then it factorises by the difference of two squares.

• A clue in the question to use the quadratic formula is when you are asked to give your answer rounded to a certain number of decimal places or significant figures.

• When substituting into the quadratic formula TAKE CARE especially when substituting in negative numbers (remember to square a negative on a calculator you must put a bracket around it.

Key learning outcomes for this video:

• To be able to generate terms of an arithmetic sequence.

• To be able to calculate the nth term of an arithmetic sequence.

• To be able to calculate the nth term of a quadratic sequence.

• To be able to check if a given value is a term of a sequence.

Cue card notes and useful reminders…

• The 'nth' term is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next.

• ‘n’ is the position in the sequence (n = 1 is the first term, n = 2 is the second term…).

• Each number in a sequence is called a term.

• An arithmetic sequence is a sequence which increases or decreases by the same amount between each term. This amount is called the common difference and is added or subtracted from one term to the next.

• A quadratic sequence has a second common difference (i.e. the difference of the differences is always the same).

• To generate the first few terms of a sequence, substitute n = 1, 2, 3, 4, etc… into the nth term formula.

• Remember once you have calculated an nth term you can check it is correct by substituting in a value of n and see if your nth term generates the correct term in the sequence.

• To find if a value is a term in a sequence, let the nth term equal that number and solve - if the solution is an integer then the value is a term in the sequence.

Key learning outcomes for this video:

• To be able to generate the coordinates from an equation to draw a graph.

• To be able to find the mid-point of a line segment.

• To be able to find equations of perpendicular and parallel lines.

• To be able to graph inequalities and find the regions that satisfy these inequalities.

• To be able to find solutions and intersection points from a graph.

Cue card notes and useful reminders…

• Remember that the simplest form of a linear equation is y = mx + c

> m is the gradient or slope of the line.

> c is the y-intercept of the line (this is where x = 0)

• To find the coordinates of a mid-point between two points, find the average of the x and y coordinates.

                                                                   (See example in downloads)

• If a line has a gradient m, then the perpendicular gradient is . This is called the negative reciprocal. Here are some examples:

                                                                   (See example in downloads)

• If a line has a gradient m, then the parallel gradient is also m.

• When graphing inequalities, think of the inequality sign as an equals sign.

• Remember when you square a negative number it gives you a positive answer.

• When you plot a quadratic it should have a curved symmetrical parabola shape and have no straight parts.

• The solutions of any graph are when it crosses the x-axis (when y = 0).

• The intersection points of two graphs occur when they cross.

• A tangent to a curve is a line that touches the curve at one point.

Key learning outcomes for this video:

• To be able to write simultaneous equations from a word problem.

• To be able to solve two linear simultaneous equations by ELIMINATION.

• To be able to solve one linear and one quadratic simultaneous equation by SUBSTITUTION.

Cue card notes and useful reminders…

• When you are solving simultaneous equations you are finding solutions that work for both equations.

• A coefficient is the number multiplying a variable.

• To use the ELIMINATION method you either add or subtract the two equations. You need matching coefficients for one of the variables. If you do not have matching coefficients then multiply one or both of the equations to make one set of coefficients match.

> If the signs of the matching coefficient are the SAME ( ++ or --) then SUBTRACT.

> If the signs of the matching coefficient are DIFFERENT ( +- or -+) then ADD.

• To use the SUBSTITUTION method you might need to rearrange equations and then substitute one into the other.

• Remember when solving a quadratic equation it must be equal to zero, then you can use factorising or the quadratic formula to solve. When you get two solutions for ‘x’ you need to work out two corresponding solutions for ‘y’

Key learning outcomes for this video:

• To be able to find equivalent ratios and to simplify ratios.

• To be able to solve ratio questions including finding the value of a single share, sharing an amount into a given ratio and using comparative ratios.

Cue card notes and useful reminders…

• Remember, the HCF of two numbers is the highest number that divides into both the numbers that you are considering.

• Usually when simplifying ratios we stop dividing when there are no more common factors to divide by. The only time this is not the case is when you are asked to write the ratio in the form 1 : n. In this case you keep dividing to get 1 on one side of the ratio and this means you may get a decimal on the other side.

• When solving a ratio question it is often helpful to find out how much one share is worth.

Key learning outcomes for this video:

• To be able to use a scale factor to change the size of shapes and quantities.

• To be able to work with scale factors for changes in lengths, areas and volumes.

Cue card notes and useful reminders…

• If two shapes are SIMILAR, all their sides have been enlarged from one shape to the other by the same scale factor and all their angles remain the same.

• If two shapes are CONGRUENT, they are exactly the same size and shape.

• If you are enlarging a 2-D shape; the scale factor for area is the square of the scale factor for the sides.

• If you are enlarging a 3-D shape; the scale factor for volume is the cube of the scale factor for the sides.

Key learning outcomes for this video:

• To be able to form and solve equations that represent proportionality.

• Recognise the shape of the graphs for direct and indirect/inverse proportion.

Cue card notes and useful reminders…

• We always use the letter ‘k’ to represent the constant of proportionality.

> You calculate this constant by using given values.

> Then write an equation with your calculated value of k.

> Then often you are asked to substitute in a new value to your equation.

• Indirect proportion and inverse proportion are used to mean the same thing* - one variable increases in proportion as the other decreases

Key learning outcomes for this video:

• To be able to use percentages to calculate growth and decay through the use of a multiplier.

Cue card notes and useful reminders…

• Compound interest and depreciation is looking at how percentages can be used to calculate the growth or decay of something over time.

• Simple interest is calculated on the original amount of money.

• Compound interest is calculated on the original amount and the accumulated interest of previous periods, and therefore can be regarded as “interest on interest.”

• See download for the general formula for calculating compound interest.

• Compound interest is an INCREASE and will have a decimal multiplier greater than 1.

• Depreciation is a DECREASE and will have a decimal multiplier less than 1.

Key learning outcomes for this video:

• To be able to convert from one measure to another using a conversion graph.

• To be able to use formulas to calculate compound measures.

• To be able to interpret Distance - Time graphs.

Cue card notes and useful reminders

• A compound measure is made up of two other measures e.g. speed, distance and time or density, mass and volume or pressure, force and surface area.

• Remember to check your values are in the correct units or to convert your answer to the required units at the end.

Key learning outcomes for this video:

• To be familiar with the different types of angles.

• To be familiar with angle pairs on parallel lines.

• To be able to answer a question by mathematical reasoning, using the appropriate language.

Cue card notes and useful reminders…

• Remember an angle is a turning motion and there are 360° in a full rotation/turn.

• Parallel lines never meet and remain the same distance apart (think of train tracks!). We denote a pair of parallel lines with a little arrow on each line:

                                                         (missing graphic)

• Perpendicular lines meet at 90° (a right angle):

                                                        (missing graphic)

                                                        (missing graphic)

• When you are asked to give a reason for your answer in an angle question, below is a summary of the most common reasons:

> Angles around a point add up to 360°.

> Angles on a straight line add up to 180°.

> Angles in a triangle add up to 90°.

> Vertically opposite angles are equal.

> Corresponding angles are equal.

> Alternate angles are equal

> Co-interior angles add up to 180°.

Key learning outcomes for this video:

• To understand and use appropriate mathematical language when describing different polygons and their properties.

• To be familiar with the properties of different triangles and quadrilaterals.

• To be able to find internal and external angles of polygons.

In this video our GCSE Maths Tutor provides an overview of Polygons and their properties, including internal and external angles. The video begins by defining what a polygon is and how it relates to two-dimensional shapes.

The word "regular" is introduced, which means that all the sides and angles of a polygon are equal. The video goes on to explain the properties of triangles, including isosceles, equilateral, and scalene triangles. The internal angle of 180 degrees for all triangles is also noted.

The video then covers the properties of quadrilaterals, including rectangles, squares, parallelograms, rhombuses, trapeziums, and kites. The properties of each shape, including angles and sides, are explained, and examples are given.

Finally, the video provides tips for solving exam questions that involve polygons.

Cue card notes and useful reminders…

• A polygon is any 2-D shape with straight lines for its sides. A regular polygon has all equal sides and angles.

• The angles in a triangle add up to 180° and the angles in a quadrilateral add up to 360°.

• See download for important formulae that you need to learn

A polygon is a closed shape with straight sides that do not cross each other. In other words, it is a two-dimensional figure made up of three or more straight line segments connected end to end. Polygons can be classified based on the number of sides they have:

• A triangle is a polygon with three sides.

• A quadrilateral is a polygon with four sides.

• A pentagon is a polygon with five sides.

• A hexagon is a polygon with six sides.

• A heptagon is a polygon with seven sides.

• An octagon is a polygon with eight sides.

• A nonagon is a polygon with nine sides.

• A decagon is a polygon with ten sides.

Polygons can also be classified based on the angles between their sides:

A regular polygon has all sides of equal length and all angles of equal measure.

An irregular polygon has sides and angles of different lengths and measures.

It's important to note that polygons do not have to be regular. For example, a rectangle is a quadrilateral with two pairs of parallel sides, but its angles are not all equal.

In addition to the number of sides and angles, polygons can also have other properties. For example, a convex polygon is a polygon where all interior angles are less than 180 degrees, while a concave polygon has at least one interior angle greater than 180 degrees.

In GCSE Maths, students are expected to be able to:

Identify and name polygons based on their number of sides and angles. This includes recognizing and naming polygons such as triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, and decagons. Students may also be asked to identify whether a given shape is a polygon or not.

Calculate the interior and exterior angles of regular polygons. For a regular polygon with n sides, the measure of each interior angle is given by (n-2) x 180 / n. The measure of each exterior angle is 360 / n. Students may be asked to find the measure of a specific interior or exterior angle, or to find the sum of all interior angles in a polygon.

Apply the properties of polygons to solve problems. This may include using the properties of specific polygons (such as rectangles, squares, or trapeziums) to find missing side lengths or angles, or using the properties of polygons to determine whether a given statement is true or false.

Draw and construct polygons using a ruler, protractor, and compass. This includes constructing polygons with specific side lengths and angles, as well as constructing polygons with specific properties (such as a regular hexagon).

In summary, GCSE Maths students are expected to have a solid understanding of polygons, including their properties, angles, and construction. They will be tested on their ability to identify and name polygons, calculate angles, and apply their knowledge to solve problems.

Key learning outcomes for this video:

• To be familiar with different metric units for length, weight and capacity.

• To be able to convert between different units within either length, weight and capacity.

• To be able to estimate using a known measurement.

• To be able to convert between different units of time.

Cue card notes and useful reminders…

• Remember if you are converting from a smaller to a LARGER unit you will DIVIDE and the number will get SMALLER.

• Remember if you are converting from a larger to a SMALLER unit you will MULTIPLY and the number will get BIGGER.

• When estimating an answer to a calculation, round all numbers to 1s.f.

• To convert minutes into a fraction or decimal part of an hour divide by 60.

Measurement and conversions are an important part of Geometry and Measure in GCSE Maths.

Measurement is the process of determining the size, amount, or degree of something. In Geometry and Measure, this typically involves measuring the length, width, or height of objects, as well as measuring angles, areas, and volumes.

Conversions are the process of changing units from one system to another. This is useful when working with measurements in different units, or when comparing measurements across different systems.

For example, let's say we have a rectangle that is 5 cm long and 3 cm wide. To find the area of the rectangle, we would multiply the length by the width:

Area = length x width

Area = 5 cm x 3 cm

Area = 15 cm^2

Note that the unit of area is cm^2 (square centimeters), which means that the measurement represents an area, not a length or a width.

Now, let's say we want to convert the area from cm^2 to m^2 (square meters). To do this, we need to know that 1 m = 100 cm (since there are 100 centimeters in one meter). Therefore, 1 m^2 = (100 cm)^2 = 10,000 cm^2.

To convert 15 cm^2 to m^2, we divide by 10,000:

15 cm^2 ÷ 10,000 = 0.0015 m^2

So the area of the rectangle is 0.0015 m^2.

In GCSE Maths, students will be expected to perform a variety of measurement and conversion tasks. Here are some examples of what they might encounter:

Converting between units: Students may be asked to convert measurements from one unit to another, for example from meters to centimeters or from kilograms to grams. To do this, they will need to know the conversion factors between the units and use them to perform the conversion.

Calculating area and volume: Students will need to be able to calculate the area of 2D shapes (such as rectangles, triangles, and circles) and the volume of 3D shapes (such as cubes, prisms, and cylinders). They will need to know the relevant formulas for each shape and be able to apply them to different situations.

Estimating measurements: Students may be asked to estimate the length, area, or volume of an object based on its dimensions or a given scale. They may also be asked to compare different measurements and estimate which one is larger or smaller.

Solving problems involving measurements: Students will need to be able to use measurements to solve real-world problems, such as calculating the amount of paint needed to cover a room or the cost of materials for a construction project. They will need to be able to interpret measurements and use them to make calculations and decisions.

Overall, students will need to have a good understanding of measurement and conversion concepts, as well as strong mathematical skills in areas such as multiplication, division, and fractions. Practice with different types of measurement and conversion problems can help students develop these skills and prepare for their GCSE Maths exam.

Key Learning Outcomes for this video:

• To be able to calculate the area and perimeter of 2-D shapes, including compound shapes.

• To use the correct units for perimeter and area.

Cue card notes and useful reminders…

- See down for some some useful area formulae

• A compound shape is made up of other shapes. For a compound shape, split it into familiar shapes and find the area and perimeter of them and then carefully add together at the end.

• Units for areas are squared (mm2, cm2, m2, km2)

Our GCSE maths tutor explains perimeter and area in the context of the Geometry and Measure topic of GCSE Maths.

Perimeter is a measurement of the distance around the outside of a two-dimensional shape. It is the sum of the lengths of all the sides of the shape. For example, the perimeter of a square with sides of length 5 cm would be 20 cm (5 cm + 5 cm + 5 cm + 5 cm).

Area is a measurement of the amount of space inside a two-dimensional shape. It is measured in square units, such as square centimeters (cm²) or square meters (m²). The area of a square with sides of length 5 cm would be 25 cm² (5 cm × 5 cm).

It's important to note that the units for perimeter and area are different. Perimeter is measured in units of length (such as centimeters, meters, or feet), while area is measured in units of area (such as square centimeters, square meters, or square feet).

In general, the perimeter of a shape is used to describe the length of its sides, while the area is used to describe how much space the shape takes up.

In the Geometry and Measure topic of GCSE Maths, students are expected to be able to calculate the perimeter and area of a range of two-dimensional shapes, including rectangles, squares, triangles, parallelograms, trapeziums and circles.

When finding the perimeter of a shape, students are typically asked to add up the lengths of all the sides of the shape. For example, they may be asked to find the perimeter of a rectangle with a length of 8 cm and a width of 5 cm. In this case, they would add up the lengths of all four sides of the rectangle: 8 cm + 8 cm + 5 cm + 5 cm = 26 cm. So the perimeter of the rectangle is 26 cm.

When finding the area of a shape, students are typically asked to use a formula that relates to the specific shape they are dealing with. For example, the formula for the area of a rectangle is length × width, while the formula for the area of a triangle is ½ × base × height. Students are expected to know these formulas and be able to apply them to find the area of a given shape.

It's worth noting that students may also be asked to find the perimeter or area of composite shapes, which are made up of two or more simpler shapes. In this case, they may need to break the composite shape down into its individual components and add up the perimeters or areas of each part to get the total perimeter or area of the composite shape.

Finally, students may be asked to use their knowledge of perimeter and area to solve real-world problems, such as finding the amount of fencing needed to enclose a garden or the amount of carpet needed to cover a room.

Key Learning Outcomes for this video:

• To be able to calculate the volume of a cube, cuboid and other prisms.

• To use the correct units for volume.

Cue card notes and useful information…

• A prism is a 3-D shape with the same cross-sectional face running through it (think of a Toblerone box - this is a triangular prism as it has a triangle running all the way through).

• See downloads for some useful volume formulae.

• Units for volume are cubed (mm3, cm3, m3, km3). Capacity is another way of describing volume (often used for liquid), the units for this a ml, cl, l, kl. To convert between volume and capacity use the conversions below to help you:

1ml = 1cm3

1l = 1000cm3

Key learning outcomes for this video:

• To be familiar with the mathematical properties of a range of 3-D shapes.

• To be able to draw accurate plans and elevations of 3D shapes.

• To be able to draw the nets of 3-D shapes.

Cue cards notes and useful information…

• A prism is a 3-D shape with the same cross-sectional face running through it (think of a Toblerone box - this is a triangular prism as it has a triangle running all the way through).

• A pyramid has a polygon as a base and 3 or more straight sides that meet at a point at the top called the apex.

• An elevation is the view of a 3D shape when it is looked at from the side or from the front.

• A plan is the view of a 3D shape when looked at from above.

• A net is the 2-D shape made when a 3-D shape is laid out flat showing each face of the figure. A 3-D shape may have more than one net.

In geometry and measure, 3D shapes refer to objects that have three dimensions: length, width, and height. These shapes are often called solid shapes or simply solids.

Some common examples of 3D shapes include:

• Cubes: A cube is a solid shape with six square faces of equal size. Each face is perpendicular to the others, and all edges and corners are equal in length.

• Rectangular Prisms: A rectangular prism is a solid shape with six rectangular faces of equal size. It is similar to a cube, but the edges and corners are not all equal in length.

• Spheres: A sphere is a solid shape with a curved surface and no corners or edges. All points on the surface of a sphere are the same distance from its center.

• Cylinders: A cylinder is a solid shape with two circular bases and a curved surface. The bases are parallel and congruent to each other, and the height is perpendicular to both bases.

• Pyramids: A pyramid is a solid shape with a polygonal base and triangular faces that meet at a common vertex. The height of a pyramid is the perpendicular distance from the vertex to the base.

These 3D shapes can be measured using various formulas, such as the volume, surface area, or perimeter of the shape. For example, the volume of a cube can be found by multiplying the length, width, and height of the cube, while the surface area of a sphere can be found by using the formula 4πr², where r is the radius of the sphere.

Key learning outcomes for this video:

• To understand what the surface area of a 3-D shape is and to be able to calculate the surface area from the net of a 3-D shape.

• To use the correct units for surface area.

Cue cards notes and useful information…

• A net is the 2-D shape made when a 3-D shape is laid out flat showing each face of the figure. A 3-D shape may have more than one net.

• For a reminder of some area formulae, see downloads

• The net of a cylinder is made up from 2 identical circles and a rectangle. The length of the rectangle is equal to the circumference of the circles. The circumference of a circle is found by using this formula in downloads

• To find the surface area of a cone; you will be given the formula for finding the conical part of the cone (πrl) and need to add this to the area of the circle at the base of the cone.

• Units for surface area are squared.

Surface area is the measure of the total area that the surface of a three-dimensional object occupies. In simpler terms, it is the sum of the areas of all the faces or surfaces of a solid object.

For example, imagine a cube. A cube has six square faces, and to find its surface area, you would add up the area of all six faces. The formula for finding the surface area of a cube is:

Surface area of a cube = 6 × (length of one side)²

Similarly, other three-dimensional objects such as rectangular prisms, cylinders, spheres, cones, and pyramids also have specific formulas for calculating their surface areas.

It is important to remember that when calculating surface area, it is essential to include all faces or surfaces, even those that are not visible. For example, in a cube, the top and bottom faces are visible, but the four side faces also contribute to the surface area.

In a GCSE Maths exam, a student may be asked to calculate the surface area of a three-dimensional object given its dimensions, or they may be given a problem involving surface area that requires them to use their understanding of the concept to solve it.

For example, a question might present a rectangular prism with given dimensions and ask the student to calculate its surface area. The student would need to identify the individual faces of the rectangular prism, calculate the area of each face, and then add the areas together to find the total surface area of the prism.

Alternatively, a question might present a word problem that requires the student to use their understanding of surface area to solve it. For example, a question might describe a scenario where a company wants to cover the surface of a cylindrical container with a specific type of material, and the student would need to calculate how much material they would need to cover the entire surface of the container.

In either case, the student would need to be familiar with the formulas for calculating the surface areas of various three-dimensional objects, as well as how to apply those formulas to solve problems.

Key learning outcomes for this video:

• To be able to describe and apply the 4 transformations: Translation, Enlargement, Reflection and Rotation.

Cue card notes and useful information…

• Remember the x-axis is HORIZONTAL, the y-axis is VERTICAL.

• Coordinates are written (x, y) - ‘along the corridor, up the stairs’.

• A translation is a SLIDING movement. To describe a translation fully use a vector

• The top number describes the horizontal movement of every point on the shape (positive is to the right, negative is to the left) and the bottom number describes the vertical movement of every point on the shape (positive is up, negative is down).

• An enlargement changes the size of a shape (bigger or smaller) but all its angles remain the same and its sides stay in proportion. To describe an enlargement fully you must give a centre of enlargement and a scale factor.

• A reflection is a flipping movement. To describe a reflection fully you must give the equation of the line of reflection.

> Horizontal lines pass through the y-axis and are written y = …

> Vertical lines pass through the x-axis and are written x = …

> The simplest form of a linear equation is y = mx + c

m is the gradient or slope of the line.

c is the y-intercept of the line (this is where x = 0)

• A rotation is a turning movement. To describe a rotation fully you must give the centre of rotation, the direction and the angle of rotation.

In geometry, a transformation is a way of changing the position, size or shape of a geometric figure. There are four basic types of transformations: translation, rotation, reflection and enlargement.

1. Translation: A translation is a type of transformation that moves a figure without changing its size, shape or orientation. This means that every point of the figure is moved by the same distance and in the same direction.

2. Rotation: A rotation is a type of transformation that turns a figure around a fixed point called the center of rotation. The size and shape of the figure remain the same, but its orientation changes.

3. Reflection: A reflection is a type of transformation that flips a figure over a line of reflection. The size and shape of the figure remain the same, but its orientation is reversed.

4. Enlargement: An enlargement is a type of transformation that changes the size of a figure. The shape and orientation of the figure remain the same, but it becomes larger or smaller.

Transformations are important in geometry because they can be used to study the properties of geometric figures, such as congruence, symmetry, and similarity. They also have many real-world applications, such as in computer graphics, engineering, and architecture.

In a GCSE Maths exam, students may be asked to demonstrate their understanding of transformations in a variety of ways. Some common questions on transformations include:

1. Describing transformations: Students may be given a geometric figure that has been transformed and asked to describe the transformation. For example, they may be given a figure that has been translated 3 units to the right and 2 units down, and asked to describe the transformation using words or notation.

2. Identifying transformations: Students may be given two geometric figures and asked to identify the transformation that maps one onto the other. For example, they may be given a triangle and its image after a reflection across the y-axis, and asked to identify the transformation.

3. Performing transformations: Students may be given a geometric figure and asked to perform a specific transformation. For example, they may be asked to rotate a square 90 degrees clockwise around a given point.

4. Combining transformations: Students may be asked to combine two or more transformations to produce a final image. For example, they may be asked to translate a figure 4 units to the right, and then reflect it across the x-axis.

It's important for students to have a solid understanding of the different types of transformations and how to perform them, as well as the effects that transformations have on geometric figures. Practice with these types of questions can help students to become more confident in their abilities and improve their performance on GCSE Maths exams.

Key learning outcomes for this video:

• To be able to use Pythagoras’ Theorem to find missing lengths in right-angled triangles.

Cue card notes and useful information…

• The hypotenuse is the longest side of a right-angled triangle, it is always the side opposite the right angle.

• See download for Pythagoras’ Theorem

• Remember to square root at the end of all calculations to find the missing side.

• If you are finding one of the shorter sides (A or B) you must rearrange the equation and subtract.

• A Pythagorean Triple is a set of three whole numbers (integers) that satisfy Pythagoras’ Theorem. e.g. 3, 4, 5 or 5, 12, 13.

• Remember an isosceles triangle can always be split down the middle to make two identical right-angled triangles.

Pythagoras is a famous mathematician who is best known for his theorem, the Pythagorean Theorem, which relates to the sides of a right-angled triangle. In Geometry and Measure, right-angled triangles are important shapes, and the Pythagorean Theorem can be used to find the length of the sides of a right-angled triangle.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In mathematical notation, this can be written as:

a^2 + b^2 = c^2

Where 'a' and 'b' are the lengths of the two shorter sides of the right-angled triangle, and 'c' is the length of the hypotenuse.

This theorem is really useful in practical applications, for example, in construction, where it can be used to ensure that buildings are structurally sound. It is also used in navigation, surveying, and even in the design of computer graphics and video games.

In GCSE Maths, students will be expected to apply the Pythagorean Theorem to solve problems involving right-angled triangles.

For example, they may be given a right-angled triangle and asked to find the length of one of the sides. To do this, they would use the Pythagorean Theorem and solve for the missing side. Here's an example problem:

A right-angled triangle has sides of length 3cm and 4cm. What is the length of the hypotenuse?

To solve this problem, the student would use the Pythagorean Theorem:

a^2 + b^2 = c^2

Where 'a' and 'b' are the lengths of the two shorter sides of the right-angled triangle, and 'c' is the length of the hypotenuse. In this case, 'a' is 3cm and 'b' is 4cm, so we have:

3^2 + 4^2 = c^2

9 + 16 = c^2

25 = c^2

c = 5

So the length of the hypotenuse is 5cm.

Students may also be asked to apply the Pythagorean Theorem to solve more complex problems, such as finding the length of a diagonal of a rectangular prism or the distance between two points on a coordinate plane.

In summary, in GCSE Maths, students will be asked to apply the Pythagorean Theorem to solve problems involving right-angled triangles, and they will need to be able to use the formula to find the length of one of the sides given the lengths of the other two sides.

Key learning outcomes for this video:

• To be able to use trigonometric ratios for sine, cosine and tangent with right-angled triangles to calculate missing angles and lengths.

Cue card notes and useful reminders…

• The hypotenuse is the longest side of a right-angled triangle, it is always the side opposite the right angle.

• The position of the opposite and adjacent sides always depend on which angle you are using in the question.

• SOH CAH TOA reminds us which sides of the triangle are used with which trigonometric ratios.

• Use the inverse button on your calculator when finding missing angles.

• Rearrange your trigonometric equation when finding missing sides.

Key learning outcomes for this video:

• To be familiar with the correct mathematical terminology and properties of circles.

• To be able to calculate the circumference and area of a circle.

• To be able to calculate arc lengths and areas of sectors of circles.

Cue card notes and useful reminders…

• You should be familiar with all these parts of a circle - see download

• Remember a semi-circle is half of a circle. A sector is a part of a circle, like a pizza slice.

• You should know the formulae (in the downloads) for finding the area and circumference of circle (r is the radius, d is the diameter)

In geometry, a circle is a closed shape consisting of all points in a plane that are the same distance away from a fixed point called the center. The distance from the center to any point on the circle is called the radius.

Circles are a fundamental concept in geometry and have many important properties, including:

• Diameter: The diameter of a circle is a line segment that passes through the center of the circle and whose endpoints are on the circle. It is twice the length of the radius.

• Circumference: The circumference of a circle is the distance around the circle. It is given by the formula C = 2πr, where r is the radius of the circle and π is a mathematical constant approximately equal to 3.14.

• Area: The area of a circle is the amount of space enclosed by the circle. It is given by the formula A = πr^2.

• Chords: A chord is a line segment whose endpoints are on the circle. The longest chord in a circle is the diameter.

• Tangents: A tangent is a line that intersects the circle at exactly one point. At the point of intersection, the tangent is perpendicular to the radius.

In GCSE Maths, students will be expected to have a good understanding of the properties of circles, and to be able to apply that understanding to solve problems. Here are some of the types of questions that students might encounter on a GCSE Maths exam:

• Finding the circumference or area of a circle: Students will be given the radius or diameter of a circle and will be asked to find the circumference or area. They will need to use the appropriate formula (C = 2πr or A = πr^2) and correctly substitute the given values.

• Finding the radius or diameter of a circle: Students may be given the circumference or area of a circle and will be asked to find the radius or diameter. They will need to rearrange the formula (C = 2πr or A = πr^2) and solve for the unknown variable.

• Using the properties of circles to solve problems: Students may be given a diagram involving circles and asked to find an unknown length or angle. They will need to use the properties of circles, such as the fact that angles subtended by the same arc are equal, to set up and solve equations.

• Drawing circles and identifying their properties: Students may be asked to draw circles of a certain size and label them with their properties, such as the radius, diameter, and center.

• Intersecting circles: Students may be given two or more circles that intersect and asked to find the lengths of segments or angles of intersection. They will need to use the properties of circles and basic trigonometry to solve the problem.

These are just a few examples of the types of questions that students might encounter on a GCSE Maths exam. It's important for students to practice applying their knowledge of circles to a variety of problems in order to build their understanding and develop their problem-solving skills.

Key learning outcomes for this video:

• To be familiar with the 8 different circle theorems below.

• When answering circle theorem questions, be able to give reasons to support your answers using the correct mathematical language.

Cue card notes and useful reminders…

• For parts of a circle, see downloads...

• When answering circle theorem questions, if you are asked to give reasons for your answers you must state which circle theorems you have used throughout your working.

• Remember a triangle that is formed by two radii of a circle is an isosceles triangle - this is very common in circle theorem questions.

Circle theorems are a set of rules or principles that apply to circles and can be used to solve problems related to circles. Here are some of the most commonly used circle theorems in GCSE Maths:

1. The angle at the center of a circle is twice the angle at the circumference that subtends the same arc.

• • This theorem is useful when you need to find the measure of an angle at the center of a circle. To apply it, simply draw a line from the center of the circle to each end of the arc and use the theorem to solve for the unknown angle.

1. Angles in the same segment of a circle are equal.

• • This theorem applies when you have two angles that both lie on the same arc of a circle, but are not necessarily adjacent to each other. To solve for the unknown angle, use the fact that the angles in the same segment of a circle are equal.

1. The perpendicular from the center of a circle to a chord bisects the chord.

• • This theorem applies when you need to find the midpoint of a chord in a circle. Simply draw a line from the center of the circle perpendicular to the chord, and use the fact that it bisects the chord to find the midpoint.

1. The angle between a tangent and a radius at the point of contact is 90 degrees.

• • This theorem applies when you need to find the measure of an angle between a tangent and a radius of a circle at the point of contact. The angle is always 90 degrees, so you can use this theorem to solve for the unknown angle.

These are just a few of the most commonly used circle theorems in GCSE Maths. There are several others that you might encounter, but these should give you a good starting point for understanding how to use circle theorems to solve problems related to circles.

In a GCSE Maths exam, students may be asked to apply circle theorems to solve problems related to circles. These problems can range from finding missing angles in a circle to calculating the length of a chord or radius. Here are some examples of the types of questions students may encounter:

1. Find the measure of angle AOB in the circle. In this question, the student would need to apply the first circle theorem, which states that the angle at the center of a circle is twice the angle at the circumference that subtends the same arc. The angle subtended by minor arc AB is angle ADB, so we can use this theorem to solve for angle AOB: angle AOB = 2 x angle ADB.

2. Find the length of chord AB in the circle. In this question, the student would need to apply the third circle theorem, which states that the perpendicular from the center of a circle to a chord bisects the chord. To solve for the length of chord AB, we would need to find the midpoint of the chord, which can be done by drawing a line from the center of the circle perpendicular to chord AB. Once we have the midpoint, we can use the Pythagorean theorem to calculate the length of AB.

3. In the circle, tangents PA and PB meet at point P. Find the measure of angle APB. In this question, the student would need to apply the fourth circle theorem, which states that the angle between a tangent and a radius at the point of contact is 90 degrees. Since the tangent lines PA and PB both intersect radius OP at 90 degree angles, we can conclude that angle APB is also 90 degrees.

These are just a few examples of how circle theorems might be applied in a GCSE Maths exam. Students may encounter a variety of different problems that require them to use circle theorems, so it's important to be familiar with all of the theorems and how to apply them.

Key learning outcomes for this video:

• To be able to draw, complete and interpret a two-way table.

• To be able to draw a comparative bar chart

• To be able to draw and interpret a pie chart.

Reminders that you might find useful…

• Remember in a two-way table add the total of the rows and the total of the columns to check they both add up to the total.

• A bar chart must have both axes labelled and an even scale up the vertical axes starting at zero.

• Remember a full turn in a circle is made up of 360°.

• To draw a pie chart you need to calculate the angle (number of degrees) for ONE piece of data, this is the multiplier.

• The angles of all the parts of the pie chart (that look like slices of pizza) will add up to 360°.

Key Learning Outcomes for this video:

• To be able to calculate and interpret the mean, median, mode and range of a set of data.

• To be able to draw and interpret a stem and leaf diagram.

• To be able to find averages from frequency tables and grouped frequency tables.

Cue card notes and useful reminders…

• Remember the definitions seen in downloads for averages and range.

• You can have more than one mode.

• A stem & leaf diagram must be ordered and have a key.

• If you have grouped data, you take the mid-point of each group and use this as the data point for that group. Therefore when you use these mid-points to calculate the mean, it is an ESTIMATION.

Our GCSE Maths tutor explains presenting data in the context of averages and ranges for GCSE students.

The video explains how to calculate mean, median, and mode averages of a set of data. It also covers how to calculate the range of the data and use it to interpret the accuracy of the averages. The video goes on to discuss how to draw and interpret an ordered stem and leaf diagram. Lastly, it explains how to calculate averages from tables and grouped data charts.

When we have a set of data, we often want to summarize it in a way that is easy to understand and gives us a general idea of what the data looks like. One way we can do this is by using averages and ranges.

An average is a measure of the central tendency of a set of data, which means it represents the typical or average value in the data set. There are several different types of averages that you may come across in your GCSE Maths course:

Mean: The mean is the most commonly used average. It is calculated by adding up all the values in the data set and then dividing by the number of values. For example, if we have the data set 1, 2, 3, 4, 5, the mean would be (1 + 2 + 3 + 4 + 5) ÷ 5 = 3.

Median: The median is the middle value in a data set when the values are arranged in order. If there are an even number of values, the median is the average of the two middle values. For example, if we have the data set 1, 2, 3, 4, 5, the median would be 3. If we have the data set 1, 2, 3, 4, the median would be (2 + 3) ÷ 2 = 2.5.

Mode: The mode is the value that appears most frequently in a data set. For example, if we have the data set 1, 2, 3, 2, 4, 2, the mode would be 2.

Range: The range is the difference between the largest and smallest values in a data set. For example, if we have the data set 1, 2, 3, 4, 5, the range would be 5 - 1 = 4.

Stem-and-leaf diagrams are another way to present data. They are especially useful for showing the distribution of a data set. A stem-and-leaf diagram is a visual representation of the data where each number is split into a stem (the first one or two digits) and a leaf (the last digit). The stems are listed vertically and the leaves are listed horizontally next to them.

Frequency tables and grouped frequency tables are also useful for presenting data. A frequency table shows the number of times each value appears in a data set. A grouped frequency table is similar, but it groups the data into intervals or "bins" and shows the number of values that fall into each interval. From these tables, we can calculate the mean, median, mode and range.

We explain below how presenting data, averages, and ranges might be tested in a GCSE Maths exam.

In a GCSE Maths exam, you might be given a set of data and asked to calculate the mean, median, mode, or range. For example, you might be given the following data set:

2, 3, 5, 7, 7, 8, 9, 11

You might be asked to calculate the mean, which you would do by adding up all the values and dividing by the number of values:

(2 + 3 + 5 + 7 + 7 + 8 + 9 + 11) ÷ 8 = 6.5

You might also be asked to calculate the median, which you would do by ordering the values and finding the middle value (or the average of the two middle values, if there are an even number of values):

2, 3, 5, 7, 7, 8, 9, 11

The median is 7, because it is the middle value.

You might be asked to calculate the mode, which you would do by finding the value that appears most frequently:

2, 3, 5, 7, 7, 8, 9, 11

The mode is 7, because it appears twice, which is more than any other value.

You might also be asked to calculate the range, which you would do by finding the difference between the largest and smallest values:

11 - 2 = 9

In addition to calculating these measures of central tendency and dispersion, you might be asked to interpret them in the context of the data. For example, you might be asked which measure of central tendency best represents the data, or how the range relates to the spread of the data.

You might also be asked to present the data in a stem-and-leaf diagram, frequency table, or grouped frequency table. For example, you might be given a set of data and asked to create a frequency table showing how many values fall into each of several intervals. Or you might be given a stem-and-leaf diagram and asked to use it to calculate the median or range.

Overall, you can expect to see a variety of questions related to presenting data, averages, and ranges on a GCSE Maths exam, and it's important to be familiar with the different types of calculations and representations you might encounter.

Key learning outcomes for this video:

• To be able to create a stratified sample.

• To be able to plot a scatter graph and draw and interpret a line of best fit.

• To recognise and interpret different types of correlation on a scattergraph.

Cue card notes and useful reminders

• Stratified sampling is used to select a sample that is representative of different groups. If the groups in the original data are of different sizes, the number of items selected from each group will be proportional to the number of items in that group.

• A scattergraph can be used to represent data where we are looking for a relationship between the two variables.

• A variable is any characteristic, number, or quantity that can be measured or counted. They can be qualitative (a category) or quantitative (numeric). Quantitative variables can be discrete (take specific values, e.g. shoe size) or continuous (take any value depending how accurately you are measuring, e.g. foot length).

• See different types of correlation in downloads

• A line of best fit roughly follows the pattern of the data points. It does NOT have to go through any points or start on the axes. You can use a line of best fit to make further predictions.

• The closer the data is grouped around the line of best fit the STRONGER the correlation. The more spread out the data is from the line of best fit the WEAKER the correlation.

The video discusses creating a stratified sample to represent a larger group of data, using fractions to calculate how many students to select from each course. The video then shows how to plot a scatter graph and draw a line of best fit to recognize and label a correlation between two variables, using the example of test scores and hours revised.

The video explains that a positive correlation suggests an increase in test scores as hours revised increases and how to predict test scores based on hours revised. The video also covers the topic of outliers and the limitations of predicting outcomes beyond the bounds of a dataset. The video ends by discussing how to describe the strength of a correlation using scatter graphs and lines of best fit.

Key learning outcomes for this video:

1. To be able to create a stratified sample.

2. To be able to plot a scatter graph and draw and interpret a line of best fit.

3. To recognise and interpret different types of correlation on a scattergraph.

Sampling refers to the process of selecting a representative subset of data from a larger population. A stratified sample involves dividing the population into subgroups, or strata, based on a specific characteristic or variable, and then selecting a random sample from each stratum. This ensures that the sample is representative of the entire population and can help to reduce bias in the data.

A scatter graph is a type of graph used to display data that involves two numerical variables. Each data point is plotted as a point on the graph, with one variable plotted on the x-axis and the other variable plotted on the y-axis. A line of best fit can be drawn through the data points to show the general trend of the data. The line of best fit can be used to make predictions or estimate values for one variable based on the value of the other variable.

When interpreting a scatter graph, it's important to look for patterns and relationships between the two variables. Different types of correlation can be observed on a scatter graph, such as positive correlation (when an increase in one variable is associated with an increase in the other variable), negative correlation (when an increase in one variable is associated with a decrease in the other variable), or no correlation (when there is no clear relationship between the two variables).

In the GCSE Maths exam, students may be asked to demonstrate their understanding of sampling and scatter graphs in a few different ways.

With regards to sampling, students may be asked to:

1. Identify the advantages and disadvantages of different sampling methods (such as random sampling, stratified sampling, or quota sampling).

2. Design and carry out a survey or experiment using a particular sampling method.

3. Analyze and interpret data from a sample, and draw conclusions about the larger population based on the sample data.

With regards to scatter graphs, students may be asked to:

1. Plot data points on a scatter graph, and draw a line of best fit through the data.

2. Calculate and interpret correlation coefficients to determine the strength and direction of the relationship between the two variables.

3. Use the line of best fit to make predictions or estimate values for one variable based on the value of the other variable.

4. Interpret real-world scenarios represented by scatter graphs, and draw conclusions or make predictions based on the data.

It's important for students to understand not just how to perform these operations, but also the reasoning behind them. For example, students should understand why a stratified sample is often more representative than a simple random sample, or why a strong correlation on a scatter graph doesn't necessarily imply causation. By demonstrating a deep understanding of the concepts, students will be better equipped to succeed on the GCSE Maths exam and in their future studies.

Key learning outcomes for this video:

• To be able to calculate cumulative frequency and draw a cumulative frequency graph.

• To be able to estimate the median and quartiles from a cumulative frequency graph and calculate the interquartile range.

• To be able to draw and interpret a boxplot from a set of data.

• To be able to compare boxplots.

• To be able to calculate frequency density to create a histogram.

Cue card notes and useful reminders...

• To find cumulative frequency you ADD UP the frequencies. When you plot it on a graph, always plot the cumulative frequencies at the UPPER BOUND of each class interval. The graph should never be decreasing.

• To find the median from a cumulative frequency curve, read across from half of the total cumulative frequency and then down.

• To find the lower quartile from a cumulative frequency curve, read across from a quarter of the total cumulative frequency and then down.

• To find the upper quartile from a cumulative frequency curve, read across from three-quarters of the total cumulative frequency and then down.

• The interquartile range = upper quartile - lower quartile.

• The is a general example of a boxplot in downloads

• When comparing boxplots focus on the median and the intequartile range.

• A variable is any characteristic, number, or quantity that can be measured or counted. They can be qualitative (a category) or quantitative (numeric). Quantitative variables can be discrete (take specific values, e.g. shoe size) or continuous (take any value depending how accurately you are measuring, e.g. foot length).

• When completing a histogram with unequal class intervals you must calculate the frequency density (see downloads)

• There should be no gaps between the bars on a histogram.

When we talk about presenting data, we mean displaying information in a way that makes it easier to understand and interpret. This is important in statistics because it allows us to make sense of the information we have collected and draw meaningful conclusions from it.

One way we can present data is through the use of graphs. There are different types of graphs that we can use depending on the type of data we have. One type of graph that we use frequently in statistics is the cumulative frequency graph.

A cumulative frequency graph shows the cumulative frequency of a set of data, which is the total frequency up to a certain point. To draw a cumulative frequency graph, we first need to calculate the cumulative frequency for each value in our data set. We can then plot these cumulative frequencies against the corresponding values on a graph. The resulting graph will show us how many values are less than or equal to a certain value.

From a cumulative frequency graph, we can estimate the median and quartiles, which are measures of central tendency and spread respectively. The median is the middle value of a set of data, and the quartiles divide the data into four equal parts. To estimate the median and quartiles from a cumulative frequency graph, we can read the values off the graph and use interpolation if necessary.

Another way we can present data is through the use of boxplots. A boxplot is a diagram that shows the distribution of a set of data by displaying the median, quartiles, and extreme values. To draw a boxplot, we first need to calculate the median and quartiles of our data set. We can then draw a box that spans from the lower to upper quartile, with a line at the median. We also draw "whiskers" from the box to the minimum and maximum values of the data set. Boxplots are useful for comparing the distribution of different data sets.

Finally, we can create a histogram by calculating the frequency density of our data and plotting it on a graph. Frequency density is calculated by dividing the frequency of each interval by the width of the interval. Histograms are useful for showing the distribution of data and identifying any patterns or trends.

In the GCSE maths exam, students at the higher level will be expected to demonstrate their understanding and application of the topics related to presenting data in the statistics and probability topic. Here are some typical types of questions that may be asked in the assessment:

1. Cumulative frequency graphs: Students may be asked to interpret a given cumulative frequency graph by estimating the median, quartiles, and interquartile range, or by making comparisons between different data sets. They may also be asked to draw a cumulative frequency graph from a given set of data and use it to answer questions about the data set.

2. Boxplots: Students may be asked to draw a boxplot from a given set of data, or to interpret a given boxplot by identifying the median, quartiles, and outliers. They may also be asked to compare two or more boxplots and make inferences about the data sets.

3. Histograms: Students may be asked to draw a histogram from a given set of data, or to calculate the frequency density and use it to draw a histogram. They may also be asked to interpret a given histogram by identifying the shape, center, and spread of the data set.

4. Interpreting data: Students may be asked to interpret a given data set by calculating measures of central tendency (e.g. mean, median, mode) and measures of spread (e.g. range, interquartile range, standard deviation), or by making comparisons between different data sets. They may also be asked to draw conclusions from the data set and explain their reasoning.

5. Statistical analysis: Students may be asked to carry out a statistical analysis of a given data set, which could involve calculating measures of central tendency and spread, drawing graphs and diagrams to represent the data, and making inferences about the data set.

In addition to these types of questions, students may also be asked to solve problems or answer open-ended questions that require them to apply their knowledge of presenting data in the statistics and probability topic. These questions may involve real-life scenarios or data sets, and may require students to use their problem-solving and critical-thinking skills to arrive at a solution.

Key learning outcomes for this video:

• To be familiar with and use the language and terminology of probability.

• To be able to find a probability from known facts.

• To be able to complete a sample space diagram.

• To be able to complete a probability table to calculate possible outcomes.

• To be able to use relative frequency to find the probability of future events.

Cue card notes and useful reminders…

• The probability scale goes from 0 to 1 - see download

• Probabilities can be written as fractions, decimals and percentages. Remember to use the FDP triangle below to help with conversions - see download

• Usually the easiest way of writing a probability is as a fraction - see download

  
  

• A sample space diagram is the set of all possible outcomes.

• Remember probabilities of all outcomes will always add up to 1. You can never have a probability greater than 1 (or 100%).

• Relative frequency or experimental probability is calculated from the number of times an event happens, divided by the total number of trials in an actual experiment.

Probability Tables:

A probability table is a way of organizing and displaying information about the likelihood of different outcomes in a probability experiment. In a probability table, each possible outcome is listed along with its corresponding probability. For example, let's say we have a bag containing 3 red marbles and 2 blue marbles. A probability table for selecting a marble from the bag might look like this - see download

The probability of selecting a red marble is 3/5, while the probability of selecting a blue marble is 2/5. We can use probability tables to calculate the probability of more complex events by multiplying the probabilities of the individual outcomes. For example, the probability of selecting two red marbles in a row would be (3/5) x (2/4) = 3/10.

Probability Trees:

A probability tree is a visual tool that helps to calculate the probabilities of different outcomes in a probability experiment. Probability trees are often used for situations where there are multiple stages to the experiment, and the outcome of one stage affects the outcome of subsequent stages.

For example, let's say we have a bag containing 3 red marbles and 2 blue marbles, and we want to calculate the probability of selecting a red marble followed by a blue marble (without replacing the first marble). We can use a probability tree to visualize the different possible outcomes - see download

The probability of selecting a red marble followed by a blue marble is the product of the probabilities along the path from the beginning to the end of the tree: (3/5) x (2/4) x (2/3) = 2/15. Probability trees can be very helpful in calculating the probabilities of more complex events, especially when there are multiple stages involved.

In GCSE Maths, students will be expected to understand the concept of probability and be able to apply it in various scenarios. They may be asked to use probability tables and trees to solve problems related to probability. Here are a few examples of the types of questions they might encounter:

Probability Tables:

1. Given a probability table, calculate the probability of a specific event or combination of events. For example: A bag contains 4 red balls and 2 blue balls. If you pick two balls at random without replacement, what is the probability that both balls are red?

2. Use a probability table to calculate the probability of an event given that another event has already occurred. For example: A dice is rolled twice. If the first roll is even, what is the probability that the sum of the two rolls is greater than 5?

Probability Trees:

1. Construct a probability tree to represent a multi-stage probability experiment and use it to calculate the probability of a specific event or combination of events. For example: A box contains 3 blue balls and 2 red balls. If you pick two balls at random without replacement, what is the probability that you pick a blue ball first and a red ball second?

2. Use a probability tree to calculate the probability of an event given that another event has already occurred. For example: A fair coin is flipped twice. If the first flip is heads, what is the probability that the second flip is tails?

In general, students will be expected to be able to set up probability tables and trees, calculate probabilities using them, and interpret the results in a real-world context. It's important that they understand the basic principles of probability and how to apply them to solve problems.

Key learning outcomes for this video:

• To be able to record and calculate probability using frequency trees and probability trees.

• To understand and apply the different concepts of independent events and conditional probability.

• To be able to calculate the probability of a sequence of events.

Cue card notes and useful reminders…

• In a probability tree outcomes are written at the end of branches and probabilities are written on the branches. If you are moving along branches you MULTIPLY the probabilities. If there is more than one route along the tree diagram, multiply along (AND) and then add (OR) these products..

• Remember probabilities of all outcomes will always add up to 1.

• Two events are independent if the probability of the first event happening has no impact on the probability of the second event happening. If the probability of one event happening affects the probability of other events happening, then the two events are not independent.

• The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred.

Key learning outcomes for this video:

• To be able to complete a Venn Diagram using a given set of data.

• To be able to calculate probabilities from a Venn Diagram.

Cue card notes and useful reminders…

• See venn Diagram notation in downloads

Probability Venn diagrams are a useful tool for visualizing and understanding probabilities. They consist of one or more circles, each representing a set of possible outcomes. The size of each circle represents the proportion of outcomes in that set, and the area of the overlapping regions represents the proportion of outcomes that are shared between the sets.

For example, suppose we have two sets of outcomes: set A and set B. We can draw a Venn diagram with two circles, one for set A and one for set B, and an overlapping region representing the outcomes that belong to both sets. The size of each circle represents the proportion of outcomes in that set, and the area of the overlapping region represents the proportion of outcomes that are shared between the sets.

To calculate probabilities using a Venn diagram, we can use the following formula:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

where P(A) is the probability of event A occurring, P(B) is the probability of event B occurring, and P(A ∩ B) is the probability of both A and B occurring. The symbol ∪ means "union" and represents the outcomes that are in either set A or set B or both.

By using this formula, we can calculate the probability of different events occurring and represent them on the Venn diagram. This can help us to understand how different events are related and to make predictions about the likelihood of certain outcomes.

In GCSE maths exams, students may be asked to use probability Venn diagrams to answer questions about the likelihood of certain events occurring.

For example, a typical question might give a Venn diagram showing the probabilities of two events, such as "A" and "B", occurring, and ask the student to calculate the probability of various combinations of events. The student would need to use the formula I mentioned earlier:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

to calculate the probability of different events. They would then shade in the appropriate regions on the Venn diagram to represent the probabilities of different outcomes.

Another type of question might give the probabilities of multiple events, such as "A", "B", and "C", and ask the student to calculate the probability of a specific combination of events occurring. To solve these types of questions, the student would need to draw a Venn diagram with three circles representing the sets of possible outcomes for each event, and then calculate the probability of different combinations using the same formula as before.

In addition to calculating probabilities, students may also be asked to interpret and analyze probability Venn diagrams to answer questions about conditional probabilities, independent events, and other concepts related to probability and statistics.

Overall, the ability to use and interpret probability Venn diagrams is an important skill for students studying probability and statistics at the GCSE level.

Leading up to the exam

• Make a revision timetable in the three months leading up to the exam and stick to it as best you can. Have a list of all the topics you need to cover and tick them off as you have revised them. Ask a teacher at school to help you make a revision timetable if you don’t know where to start.

• Once you have covered all the topics, it is time to do lots of practice papers in the lead up to the exams. Your school should be able to provide you with these along with the mark schemes so that you can check your answers.

• To make progress and prepare well, you need to be doing some extra Maths each week in between your school Maths lessons.

• Make sure you have your own calculator well in advance of your exams and are familiar with using it.

In the exam

• Read the questions carefully.

• Allow one minute per mark of the paper, if you are getting stuck on a question and it is taking too much time then leave it and go back to it at the end if you have time.

• Show all steps of your working when you are answering questions as you can pick up marks for your working.

• Check your answers after each question if you are able, think if they are reasonable in the context of the question.