There are 89 Mastery Topics
Below are the toughest 89 topics which make it GCSE Higher Level (GCSE Grades 6 - 9)
You need to be able to do all 4 sections containing 867 topics at Higher Level!
Number
- STEP 10
- Use the product rule for counting (i.e. if there are m ways of doing one task and for each of these, there are n ways of doing another task, then the total number of ways the two tasks can be
done is m × n ways)
- Convert a recurring decimal to a fraction in simple cases
- Understand a recurring decimal to fraction proof
- Find the value of calculations using indices including fractional and negative indices
- Understand that the inverse operation of raising a positive number to a power n is raising the result of this operation to the power 1/n
- Simplify surd expressions involving squares ( e.g. √12 =√ (4 × 3) = 2 √3)
- Use fractions, surds and pi in exact calculations, without a calculator
-
- STEP 11
- Calculate the upper and lower bounds of 2-D measurements e.g. area
- Calculate the upper and lower bounds of other compound measurements e.g. density
- Write (3 - √3)² in the form a + b√3
- Rationalise a denominator
-
- STEP 12
- Topic does not reach Step 12
The MatheMagician
"Number is about 22-28% of the Foundation exam and 12-18% of the Higher exam."
Algebra
- STEP 10
- Use function notation
- Deduce turning points by completing the square
- Sketch a graph of a quadratic by factorising, identifying roots and y-intercept, turning point
- Find the equation of the line through two given points
- Find the equation of the line through one point with a given gradient
- Know that a line perpendicular to the line y = mx + c, will have a gradient of -1/m
- Write down the equation of a line perpendicular to a given line
- Interpret and analyse a straight line graph and generate equations of lines parallel and perpendicular to the given line
- Solve quadratic inequalities in one variable, by factorising and sketching the graph to find critical values
- Simplify and manipulate algebraic expressions involving surds and algebraic fractions
- Solve exactly, by elimination of an unknown, linear/quadratic simultaneous equations
- Find approximate solutions to simultaneous equations formed from one linear function and one non-linear (quadratic or circle) function using a graphical approach
-
- STEP 11
- Apply to the graph of y = f(x) the transformations:
y = -f(x)
y = f(-x)
y = -f(-x) for linear, quadratic, cubic, sine and cosine functions
- Apply to the graph of y = f(x) the transformations:
y = f(x) + a
y = f(ax)
y = f(x + a)
y = af(x) for linear, quadratic, cubic, sine and cosine functions f(x)
- Construct the graphs of simple loci including the circle x² + y² = r² for a circle of radius r centred at the origin of the coordinate plane
- Find the gradient of the radius that meets the circle at a given point
- Find the nth term of a quadratic sequence of the form
n²
an²
an² ± b and
an² ± bn ± c
- Solve exactly, by elimination of an unknown, linear/x² + y² = r² simultaneous equations
-
- STEP 12
- Interpret the gradient of linear or non-linear graphs, and estimate the gradient of a quadratic or non-linear graph at a given point by sketching the tangent and finding its gradient
- Find the equation of a tangent to a circle at a given point
- Interpret coodinates for trigonometric graphs
- Plot graphs of the exponential function y = abx for integer values of x and simple positive values of a and b.
- Use iteration with simple converging sequences
The MatheMagician
"Algebra is about 17-23% of the Foundation exam and 27-33% of the Higher exam."
Geometry &
Measures
- STEP 10
- Solve problems including examples of solids in everyday use
- Prove and use the alternate segment theorem
- Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations
- Understand, recall and use Pythagoras' theorem in 3-D problems
- Calculate the length of a diagonal of a cuboid
- Enlarge 2D shapes, given a negative, fractional scale factor
- Know and apply the sine rule a/sin A = b/sin B = c/sin C to find unknown lengths and angles
- Know and apply the cosine rule a² = b² + c² – 2bc cos A to find unknown lengths
- Calculate the area of a triangle given the length of two sides and the included angle
- Work out the magnitude of a vector
- Calculate, and represent graphically, the sum of two vectors, the difference of two vectors and a scalar multiple of a vector
- Calculate the resultant of two vectors
- Solve geometrical problems in 2-D using vector methods
-
- STEP 11
- Use the formulae for length of arcs and area of sectors of circles to solve problems
- Give reasons for angle sizes using mathematical language
- Give reasons for angle and length calculations involving the use of tangent theorems
- Understand and use the fact that tangents from an external point are equal in length
- Know and apply the cosine rule a² = b² + c² – 2bc cos A to find unknown angles
- Know and apply Area = 1/2 ab sin C to calculate the sides or angles of any triangle
- Prove lines are parallel/colinear
-
- STEP 12
- Solve problems involving more complex shapes and solids, including segments of circles and frustums of cones
- Find the area of a segment of a circle given the radius and length of the chord
- Solve problems for areas and volumes of similar shapes and solids
- Use the trigonometric ratios to solve 3-D problems
- Find the angle between a line and a plane (but not the angle between two planes or between two skew lines)
- Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°
- Use the sine and cosine rules to solve 2-D and 3-D problems
- Apply vector methods for simple geometrical proofs
The MatheMagician
"Geometry is about 12-18% of the Foundation exam and 17-23% of the Higher exam."
Ratio, proportion &
rates of change
- STEP 10
- Solve problems involving inverse proportion using graphs by plotting and reading values from graphs
- Solve problems involving inverse proportionality, including problems where y is inversely proportional to the square of x
- Calculate an unknown quantity from quantities that vary in direct or inverse proportion
- Set up and use equations to solve word and other problems involving direct or inverse proportion
- Calculate the new area of a shape after enlargement
-
- STEP 11
- Recognise sketch and interpret graphs of exponential functions y = kx for positive values of k and integer values of x
- Find points that divide a line in a given ratio, using the properties of similar triangles
-
- STEP 12
- Calculate the new volume of a shape after enlargement
The MatheMagician
"Ratio, Proportion & Rates of Change is about 22-28% of the Foundation exam and 17-23% of the Higher exam."
Probability
- STEP 10
- Use a two-way table to calculate conditional probability
- Use a tree diagram to calculate conditional probability
- Use Venn diagrams to calculate conditional probability
- Understand conditional probabilities and decide if two events are independent
- Understand selection with or without replacement
- Use a tree diagram to calculate conditional probability
-
- STEP 11
- Topic does not reach Step 11
-
- STEP 12
- Topic does not reach Step 12
The MatheMagician
"Statistics & Probability is about 12-18% of the Foundation exam and 12-18% of the Higher exam."
Statistics
- STEP 10
- Know the appropriate use of Histograms
- Compare the mean, median, mode and range as appropriate of two distributions
- Use a spreadsheet to calculate mean and range and find median and mode
- Compare distributions and make inferences, using the shapes of distributions and measures of average and spread, including median and quartiles
- Compare median and inter-quartile range of two distributions
- From a cumulative frequency graph estimate frequency greater/less than a given value
- Estimate the mean from a histogram
- Stratified sampling - know the definition and state in terms of proportion, fraction, percentage or ratio
-
- STEP 11
- Use and understand frequency density
- Construct and interpret histograms from class intervals with unequal width
- From a histogram complete a grouped frequency table
- From a histogram understand and define frequency density
- Estimate the median (or other information) from a histogram with unequal class width
-
- STEP 12
- Topic does not reach Step 12
The MatheMagician
"Statistics & Probability is about 12-18% of the Foundation exam and 12-18% of the Higher exam."
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