There are 226 Challenging Topics
Below are the challenging 226 topics on Foundation (GCSE Grades 4 - 5) and the first step to Higher Level Mastery
You need to be able to do the first 3 sections containing 778 topics to pass at Foundation Level!
You also need to complete the last section containing the hardest 89 topics for Mastery of Higher Level (Grades 6 - 9)
Number
- STEP 07
- Understand that each of the headings in the place value system, to the left of the units column, can be written as a power of ten
- Multiply both sides of an inequality by a negative number
- To understand the difference between squaring a negative number and subtracting a squared number within a more complex calculation
- Find the reciprocal of simple numbers/fractions mentally, e.g. 10 and 1/10, 1/3 and 3 etc.
- Understand the order in which to calculate expressions that contain powers and brackets in both the numerator and denominator of a fraction
- Know that a number multiplied by its reciprocal is 1
- Know that the reciprocal of a reciprocal is the original number
- Use conventional notation for priority of operations, including roots and reciprocals
- Understand and use compound measures: density; pressure; speed
- Round numbers and measures to an appropriate degree of accuracy (dp or sig fig)
- Find HCF and LCM using Prime Factors
- Use prime factorisation to represent a number as a product of its primes using index notation
- Recognise that prime factor decomposition of a positive integer is unique
- Add and subtract fractions (mixed) - positive and negative
- Use the square, cube and power keys on a calculator
- Use the index laws to include negative power answers and understand that these answers are smaller than 1
- Use the laws of indices to multiply and divide numbers written in index notation
- Convert between currencies
- Estimate answers to calculations by rounding numbers to 1 sig fig
- Check reasonableness of answers
- Estimate answers to one-step or two-step calculations
- Write numbers greater than 10 in standard index form
-
- STEP 08
- Multiply and divide simple fractions (mixed) - positive and negative
- Calculate with roots (surds - exact values)
- Write numbers less than 10 in standard index form
- Order numbers written in standard index form
- Convert between large and small numbers into standard form and vice-versa
- Add and subtract in standard form
- Multiply and divide numbers in standard form
-
- STEP 09
- Use inequality notation to specify simple error intervals due to truncation or rounding
- Estimate powers and roots of any given positive number
- Recall that n⁰ = 1 and n⁻¹ = 1/n for positive integers n as well as n½ = √n and n⅓ = ³√n for any positive number n (Note: n½ is n to the power ½ and n⅓ is n to the power ⅓)
The MatheMagician
"Number is about 22-28% of the Foundation exam and 12-18% of the Higher exam."
Algebra
- STEP 07
- Use systematic trial and improvement to find the approximate solution to one decimal place of equations such as x³ = 29
- Construct and solve equations that involves multiplying out brackets by a negative number (e.g. 4(2a - 1) = 32 - 3(2a - 2))
- Derive a simple formula, including those involving squares, cubes and roots
- Multiply out brackets involving positive terms such as (a + b)(c + d) and collect like terms
- Substitute positive and negative integers into linear expressions and expressions involving powers
- Know and understand the meaning of an identity and use the ≠ sign
- Factorise to one bracket by taking out the highest common factors for all terms e.g. 2x²y + 6xy² = 2xy(x + 3y)
- Find an unknown where it is not the subject of the formula and where an equation must be solved.
- Rearrange simple equations
- Know that the gradient of a line is the change in y over change in x.
- Without drawing the graphs, compare and contrast features of graphs such as y = 4x, y = 4x + 6, y = x + 6, y = -4x, y= x - 6
- Identify parallel lines from their equations
- Generate points and plot graphs of simple quadratic functions, then more general functions
- Construct a table of values, including negative values of x for a function such as y = ax²
- Recognise a graph which represents a quadratic function
- Plot the graphs of linear functions in the form y = mx + c and recognise and compare their features
- Recognise that linear functions can be rearranged to give y explicitly in terms of x e.g. rearrange y + 3x - 2 = 0 in the form y = 2 - 3x
- Solve simple linear inequalities in one variable and represent the solution on a number line e.g. 3n + 2 <11 and 2n - 1 >1
- Represent the solution set for inequalities using set notation
- Argue mathematically to show algebraic expressions are equivalent e.g 2x(x + 3) - 4(3x - x²) = 6x(x - 1)
- Find and use the nth term of an arithmetic sequence
- Simplify simple expressions involving index notation
-
- STEP 08
- Find the equation of a straight-line from its graph
- Identify the line of symmetry of a quadratic graph
- Recognise that when the linear and inverse of a linear function such as y = 2x, y = 3x are plotted, they are a reflection in the line y = x
- Interpret distance-time graphs and calculate the speed of individual sections, total distance and total time
- Interpret gradient as rate of change in distance-time and speed-time graphs, containers emptying and filling and unit price graphs
- Identify and interpret roots, intercepts and turning points of a quadratic graph
- Given the coordinates of points A and B, calculate the length of AB
- Plot and draw graphs of straight lines WITHOUT using a table of values (use intercept and gradient)
- Write down the equation of a line parallel to a given line
- Recognise a quadratic function from its equation and explain the shape of it's graph
- Solve more complex linear inequalities in one variable and represent the solution on a number line e.g. -6 < 2n+4 or -9 < 2n + 3 < 7
- Use algebra to support proofs e.g. show that the volume of a cube with side lengths of (2x - 1)cm is (8x³ - 12x² + 6x - 1)cm³
- Use algebra to support and construct arguments
- Generate arithmetic sequences of numbers , squared integers and sequences derived from diagrams
- Identify which terms cannot be in a sequence
- Generate the sequence of triangle numbers by considering the arrangement of dots and deduce that T(n) = 1 + 2 + 3 + .... + n, the sum of the series
- Recognise and use simple geometric progressions (rn where n is an integer and r is a rational number number > 0 or a surd)
- By looking at the spatial patterns of triangular numbers, deduce that the nth term is 1/2n(n + 1)
- Use function machines to find terms of sequence
- Solve exactly, by elimination of an unknown, linear/linear simultaneous equations, including where both need multiplying
- Solve linear/linear simultaneous equations graphically
- Solve simultaneous equation, linear/linear simultaneous equations,where neither or one equation needs multiplying
- Write simultaneous equations to represent a situation
- Set up and solve a pair of simultaneous equations in two variables
- Solve simultaneous equations representing a real-life situation graphically and interpret the solution in the context of the question
-
- STEP 09
- Change the subject of a complex formula that involves fractions, e.g. make u or v the subject of the formula 1/v + 1/u =1/t
- Identify and interpret gradient from an equation ax + by = c
- Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function
- Recognise, sketch and interpret graphs of simple cubic functions
- Recognise, sketch and interpret reciprocal graphs
- Find the coordinates of the midpoint of a line from coordinates using a formula
- Solve linear inequalities in two variables graphically
- Solve two simultaneous inequalities algebraically and show the solution set on a number line
- Answer simple proof and 'show that' questions using consecutive integers (n, n+ 1), squares a², b², even numbers 2n, and odd numbers 2n + 1
- Continue a quadratic sequence and use the nth term to generate terms
- Use finite /infinite and ascending / descending to describe sequences
- Distinguish between arithmetic and geometric sequences
- Continue geometric progression and find term to term rule, including negative, fraction and decimal terms
- Simplify expressions involving brackets and powers e.g. x(x²+x+4), 3(a + 2b) − 2(a + b)
- Square a linear expression and collect like terms
The MatheMagician
"Algebra is about 17-23% of the Foundation exam and 27-33% of the Higher exam."
Geometry &
Measures
- STEP 07
- Mark on a diagram the position of point B given its bearing from the point A
- Use accurate drawing to solve bearings problems
- Use the sum of the interior angles of an n-sided polygon
- Calculate the interior angles of polygons
- Find the size of each interior angle or the size of each exterior angle or the number of sides of a regular polygon
- Calculate volumes of shapes made from cuboids, for lengths given as whole numbers
- Calculate the volume of right prisms
- Calculate the surface area of right prisms
- Calculate the lengths and areas given the volumes in right prisms
- Calculate the lengths, areas and volumes in cylinders
- Use the formulae for the circumference and area of a circle, given the circumference or area, to calculate the radius or diameter
- Find the perimeters and areas of semicircles and quarter circles
- Begin to use congruency to solve simple problems in triangles and quadrilaterals
- Use the information given about the length of sides and sizes of angles to determine whether triangles are congruent, or similar
- Use straight edge and compass to construct the perpendicular from or to a point on a line segment
- Use straight edge and compasses to construct a triangle, given right angle, hypotenuse and side (RHS)
- Draw the locus equidistant between 2 points or from a point
- Produce shapes and paths by using descriptions of loci
- Use construction to find the locus of a point that moves according to a rule
- Understand loci about a point, line and corner.
- Construct angles of 60°, 90°, 30°, 45°
- Know the formula for Pythagoras' theorem and use to find the hypotenuse
- Know that the perpendicular distance from a point to a line is the shortest distance to the line
- Justify if a triangle is right-angled given its three lengths
- Use vector notation for translations
- Use 2D Vector notation for translation
- Understand and use the language and notation associated with enlargement
- Enlarge 2D shapes, given a fractional scale factor
- Find the centre of rotation
- Describe a transformation
- Describe reflections on a coordinate grid
- Colour in missing squares to complete a reflection
- Recognise whether a reflection is correct
- Express points as position vectors
- Understand and use vector notation
-
- STEP 08
- Find the surface area of simple shapes (prisms) using the formulae for triangles and rectangles, and other shapes
- Find the surface area and volumes of compound solids constructed from cubes, cuboids, cones, pyramids, spheres, hemispheres, cylinders
- Recognise the formulae for length of arcs in a circle.
- Recognise the formulae for area of sectors in a circle.
- Solve problems involving angles, triangles and circles
- Use similarity to solve problems in 2D shapes
- Use simple examples of the relationship between enlargement and areas and volumes of simple shapes and solids
- Understand that a locus in 3D can be a plane or curved surface and extend understanding of loci to include 3D problems, e.g. know that all the points equidistant from a single point in space
form the surface of a sphere
- Understand how standard constructions using straight edge and compasses relate to the properties of two intersecting circles with equal radii
- Know the formula for Pythagoras' theorem and use to find a shorter side
- Use and apply Pythagoras' theorem to solve problems
- Use the sine, cosine and tangent ratios to find the lengths of unknown sides in a right-angled triangle, using straight-forward algebraic manipulation, e.g. calculate the adjacent (using cosine),
or the opposite (using sine or tangent ratios)
- Understand the language of planes, and recognise the diagonals of a cuboid
- Derive the fact that base angles of isosceles triangles are equal
- Transform 2-D shapes by simple combinations of rotations, reflections and translations, using ICT (e.g. repeated reflection, rotation or translation, reflections in the x and y axes, rotations
about (0, 0))
- Transform 2D shapes by a more complex combinations of rotations, reflections and translations, e.g. a reflection, followed by a rotation etc
- Add and Subtract vectors
-
- STEP 09
- Prove and use the fact that the angle in a semicircle is a right angle ;
- Prove and use the fact that angles in the same segment are equal
- Prove and use the fact that opposite angles of a cyclic quadrilateral sum to 180°
- Prove and use facts about the angle subtended at the centre and at the circumference;
- Use the sine, cosine and tangent ratios to find the lengths of unknown sides in a right-angled triangle, using more complex algebraic manipulation, e.g. the hypotenuse (using cosine or sine),
or adjacent (using the tangent ratio)
- Use the appropriate ratio to find a length, or angle, and hence solve a two-dimensional problem
- Find angles of elevation and angles of depression
- Know that the tangent at any point on a circle is perpendicular to the radius at that point
- Know that the perpendicular from the centre to the chord bisects the chord
- Complete a formal geometric proof of similarity of two given triangles
The MatheMagician
"Geometry is about 12-18% of the Foundation exam and 17-23% of the Higher exam."
Ratio, proportion &
rates of change
- STEP 07
- Interpret and write ratios to describe a situation
- Understand and use compound measures (density, speed, pressure)
- Solve problems using constant rates and related formulae
- Solve problems involving compound measures
- Write lengths, areas and volumes of two shapes as ratios in simplest form
- Estimate conversions
- Use algebraic methods to solve problems involving variables in direct proportion
- Use expressions of the form y α 1/x
- Interpret the gradient of a straight line graph as a rate of change
- Use calculators to explore exponential growth and decay
- Use compound interest
- Represent repeated proportional change using a multiplier raised to a power
Understand direct proportion as equality of ratios
- Understand direct proportion as equality of ratios
- Use measures in ratio and proportion problems ( currency conversion, rates of pay, best value)
- Express a multiplicative relationship between two quantities as a ratio or a fraction
- Use the unitary method for an inverse operation, e.g. If I know an item was 80% of the original cost in a sale, find the original price
- Use and interpret scale drawings, where scales use mixed units, and drawings aren't done on squared paper, but have measurements marked on them.
- Know that enlargements of 2D shapes produce similar shapes
-
- STEP 08
- Write a ratio as a linear function
- Extend to simple conversions of compound measures (e.g. convert 2 m/s to km/hr)
- Convert imperial units to imperial units
- Convert between metric and imperial measures
- Use graphs to calculate measures including unit price, average speed, distance, time, acceleration
- Use percentages in real-life situations: compound interest, depreciation, percentage profit and loss
- Calculate repeated proportional change
- Find the original amount given the final amount after a percentage change ( reverse percentages)
- Use calculators for reverse percentage calculations by doing an appropriate division
- Understand that the ratio of any two sides is constant in similar right-angled triangles
- Understand the implications of enlargement for perimeter
- Identify the scale factor of an enlargement as the ratio of the lengths of any two corresponding line segments
- Enlarge 2-D shapes and recognise the similarity of resulting shapes
-
- STEP 09
- Use expressions of the form y α x²
- Identify direct proportion from a table of values by comparing ratios of values
The MatheMagician
"Ratio, Proportion & Rates of Change is about 22-28% of the Foundation exam and 17-23% of the Higher exam."
Probability
- STEP 07
- Record outcomes of events in a Venn Diagram
- Use theoretical models to include outcomes using spinners, dice, coins etc.
-
- STEP 08
- Find a missing probability from a list or two-way table including algebraic terms
- Use tree diagrams to calculate the probability of two dependent events
-
- STEP 09
- Topic does not reach Step 09 but does continue at Higher Step 10
The MatheMagician
"Statistics & Probability is about 12-18% of the Foundation exam and 12-18% of the Higher exam."
Statistics
- STEP 07
- Use more complex two way tables
- Construct on paper, and using ICT, frequency diagrams for grouped discrete data
- Find the median, mode and range from a stem and leaf diagram
- Estimate the mean of grouped data using the mid-interval value
- Understand that the frequency represented by corresponding sectors in two pie charts is dependent upon the total populations represented by each of the pie charts
- Recognise the advantages and disadvantages between measures of average
- Criticise questions from a questionnaire
- Understand how sources of data may be biased
- Decide what data to collect and what analysis is needed
- Write questionnaire questions to eliminate bias, on timing and location of survey to ensure sample is representative
- Know the definition of random sampling
- State how reliable their predictions are
- Draw a line of best fit by eye and understand what they represent
- Understand that correlation does not imply causality
- Distinguish between positive, negative and zero correlation using lines of best fit
- Appreciate that correlation is a measure of the strength of the association between two variables and that zero correlation does not necessarily imply 'no relationship' but merely 'no linear
relationship'
- Use a line of best fit, or otherwise, to predict values of one variable given values of the other variable
- Intepret scatter graphs in terms of the relationship between two variables
- Use the line of best fit to make predictions
- Interpolate and extrapolate apparent trends whilst knowing the dangers of doing so
- Interpret correlation in terms of the problem
-
- STEP 08
- Know the appropriate use of a cumulative frequency diagram
- Construct cumulative frequency tables
- Calculate possible values of the set of data given summary statistics
- Interpret box plots to find median, quartiles, range and interquartile range and draw conclusions
- Produce box plots from raw data and identify outliers when given quartiles and median
- Use random numbers to get a sample
-
- STEP 09
- Interpret and analyse information in a range of linear graphs - to describe how one variable changes in relation to another
- Construct cumulative frequency graphs
- Interpret cumulative frequency graphs
- Find the median, quartiles and interquartile range for large data sets with grouped data
- Compare the measures of spread between a pair of box plots/cumulative frequency graphs
- Select and justify a sampling scheme and a method to investigate a population, including random and stratified sampling
The MatheMagician
"Statistics & Probability is about 12-18% of the Foundation exam and 12-18% of the Higher exam."
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