KS4 Maths (Higher)

Module 1

Similarity

Core declarative knowledge: What should students know?

  • What are the properties of similar shapes?
  • What is a scale factor?
  • How do similar shapes relate to enlargement?
  • How do scale factors for length, area and volume compare?
  • How do scale factors and similarity relate to percentages and ratio?

Core procedural knowledge: What should students be able to do?

  • Find a scale factor between similar shapes
  • Find missing sides using a SF
  • Express the ratio of two sides to demonstrate similarity
  • Enlarge a shape by a scale factor
  • Enlarge a shape by a negative scale factor
  • Describe the enlargement of a shape
  • Express the similarity of two shapes as ratios and percentages
  • Find the scale factors of areas and volumes
  • Find the area, surface area and volume of similar shapes

Links to prior learning (to be made explicit and tested)

  • Finding the perimeter and area of 2D shapes
  • Recognising proportional relationships

Pythagoras

Core declarative knowledge: What should students know?

  • What is the hypotenuse of a right-angled triangle?
  • What is the relationship between the shorter sides of a right-angled triangle and the hypotenuse?
  • What is the Pythagoras’ Theorem?

Core procedural knowledge: What should students be able to do?

  • Substitute values in Pythagoras’ forumula
  • Find the hypotenuse of a right-angled triangle
  • Find a shorter side of a right-angled triangle
  • Find the length of a line segment between two coordinates
  • Recognise Pythagoras’ theorem within problems
  • Use Pythagoras’ theorem in 3D problems.

Links to prior learning (to be made explicit and tested)

  • Square numbers
  • Square roots
  • Properties of triangles

Trigonometry

Core declarative knowledge: What should students know?

  • What are the conventional names of the sides of any right-angled triangle?
  • How does the ratio of the sides of a triangle relate to similar shapes?
  • What do the trigonmetric ratios of sine, consine and tangent relate to?
  • What are the three formulae required to calculate a missing side or angle of any right angled triangle?

Core procedural knowledge: What should students be able to do?

  • Name each side of a right-angled triangle
  • Identify the correct ratio for problem
  • Rearrange the formula to make different parts the subject
  • Solve to find a missing side
  • Solve to find a missing angle
  • Solve multistep problems
  • Apply knowledge to 3D problems

Links to prior learning (to be made explicit and tested)

  • Rearranging formulae
  • Solving equations
  • Properties of triangles

Volume

Core declarative knowledge: What should students know?

  • What are the properties of cuboids, prisms, cylinders, pyramids, cones and spheres?
  • What is the formula for calculating the volumes of cuboid, prism, cylinder, pyramid, cone or sphere?
  • What is volume?
  • What is surface area and how is different from volume?
  • What is for formula for calculating the surface area of a sphere?
  • What is the formula for calculating the curved surface area of a cone?

Core procedural knowledge: What should students be able to do?

  • Calculate the volume of a cuboid, prism, cylinder, pyramid, cone and sphere.
  • Calculate the surface area of a cuboid, prism, cylinder, pyramid, cone and sphere.
  • Calculate the volume of a frustrum.
  • Solve problems involving volumes and surface area.

Links to prior learning (to be made explicit and tested)

  • Identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres
  • Calulate the area of 2D shapes including circles
Module 2

Equations & Inequalities

Core declarative knowledge: What should students know?

  • What is an equation?
  • What is an unknown?
  • What is an identity?
  • What is the difference between an equality and an identity?
  • What are inequalities?
  • What is the difference between an inequality and an equation?
  • What does it mean for a soluation to satisfy an inequality?

Core procedural knowledge: What should students be able to do?

  • Solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation;
  • Solve linear equations which contain brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution;
  • Solve linear equations in one unknown, with integer or fractional coefficients;
  • Set up and solve linear equations to solve to solve a problem;
  • Derive a formula and set up simple equations from word problems, then solve these equations, interpreting the solution in the context of the problem
  • Solve simple linear inequalities in one variable, and represent the solution set on a number line;
  • Solve two linear inequalities in x, find the solution sets and compare them to see which value of x satisfies both solve linear inequalities in two variables algebraically.

Links to prior learning (to be made explicit and tested)

  • The ability to use negative numbers with the four operations and recall and use hierarchy of operations and understand inverse operations;
  • Dealing with decimals and negatives on a calculator;
  • Using index laws numerically.
  • Use algebraic notation and symbols correctly;
  • Write an expression;
  • Know the difference between a term, expression, equation, formula and an identity;
  • Manipulate an expression by collecting like terms;
  • Substitute positive and negative numbers into expressions such as 3x + 4 and 2×3 and then into expressions involving brackets and powers;
  • Substitute numbers into formulae from mathematics and other subject using simple linear formulae, e.g. l × w, v = u + at;
  • Use instances of index laws for positive integer powers;
  • Use index notation (positive powers) when multiplying or dividing algebraic terms;
  • Use instances of index laws, including use of zero, fractional and negative powers;
  • Multiply a single term over a bracket;
  • Recognise factors of algebraic terms involving single brackets and simplify expressions by factorising, including subsequently collecting like terms;

Linear Graphs

Core declarative knowledge: What should students know?

  • What is the gradient of a straight line graph?
  • What does the gradient represent (including in a context)?
  • What do each of the variables of y=mx+c represent?
  • What does c of y=mx+c represent in a given context?
  • How do you know if two straight lines are parallel to each other from their equations?
  • How do you know if two straight lines are perpendicular to each other?
  • What is a negative reciprocal and how does this relate to gradients of straight lines?

Core procedural knowledge: What should students be able to do?

  • Plot and draw graphs of y = a, x = a, y = x and y = –x, drawing and recognising lines parallel to axes, plus y = x and y = –x;
  • Find the equation of a straight line from a graph in the form y = mx + c;
  • Plot and draw graphs of straight lines of the form y = mx + c with and without a table of values;
  • Sketch a graph of a linear function, using the gradient and y-intercept (i.e. without a table of values);
  • Find the equation of the line through one point with a given gradient;
  • Identify and interpret gradient from an equation ax + by = c;
  • Find the equation of a straight line from a graph in the form ax + by = c;
  • Plot and draw graphs of straight lines in the form ax + by = c;
  • Identify when two lines are perpendicular and state their respective gradients using m and -1/m.

Links to prior learning (to be made explicit and tested)

As equations and inequalities.

Quadratics

Core declarative knowledge: What should students know?

  • What is a quadratic?
  • What is a cubic/recpircal and circle graph?
  • What are the properties of a quadratic graph?
  • How does factorisation of a quadratic help us solve?
  • What must a quadratic equal to be solved?
  • How do the solutions of a quadratic relate to the roots?

Core procedural knowledge: What should students be able to do?

  • Recognise a linear, quadratic, cubic, reciprocal and circle graph from its shape;
  • Generate points and plot graphs of simple quadratic functions, then more general quadratic functions;
  • Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function;
  • Interpret graphs of quadratic functions from real-life problems;
  • Factorise quadratic expressions in the form ax2 + bx + c;
  • Solve quadratic equations by factorisation;
  • Solve quadratic equations that need rearranging;
  • Set up and solve quadratic equations;

Links to prior learning (to be made explicit and tested)

As equations and inequalities.

Simultaneous Equations

Core declarative knowledge: What should students know?

  • Why do we need to solve multiple equations simulataneously?
  • How do the solutions to two simultaneous equations relate to the graphs of the equations?
  • Why might some graphical soluations only be estimations of their soluations?

Core procedural knowledge: What should students be able to do?

  • Find the exact solutions of two simultaneous equations in two unknowns;
  • Interpret the intersection of two functions as the exact or approximate solutions to the functions;
  • Use elimination or substitution to solve simultaneous equations;
  • Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns:

    • linear / linear, including where both need multiplying;
  • Set up and solve a pair of linear simultaneous equations in two variables, including to represent a situation;
  • Interpret the solution in the context of the problem

Links to prior learning (to be made explicit and tested)

As equations and inequalities.

Module 3

Percentages

Core declarative knowledge: What should students know?

  • What is a percentage?
  • What is simple interest?
  • What is the effect of compound growth or decay?
  • Which is preferable in the short term, simple interest or compound growrth?
  • What is the general form of a percentage change? How can this be used to find the original amount?

Core procedural knowledge: What should students be able to do?

  • Work out a percentage increase or decrease, including: simple interest, income tax calculations, value of profit or loss, percentage profit or loss.
  • Compare two quantities using percentages, including a range of calculations and contexts such as those involving time or money.
  • Find a percentage of a quantity using a multiplier.
  • Use a multiplier to increase or decrease by a percentage in any scenario where percentages are used.
  • Find the original amount given the final amount after a percentage increase or decrease (reverse percentages), including VAT.
  • Use calculators for reverse percentage calculations by doing an appropriate division.
  • Use percentages in real-life situations, including percentages greater than 100%.
  • Describe percentage increase/decrease with fractions, e.g. 150% increase means times as big.
  • Understand that fractions are more accurate in calculations than rounded percentage or decimal equivalents, and choose fractions, decimals or percentages appropriately for calculations.
  • Calcuate compound percentages/growth and decay.

Links to prior learning (to be made explicit and tested)

Convert between fractions, decimals and percentages. Express a given number as a percentage of another number. Express one quantity as a percentage of another where the percentage is greater than 100%. Find a percentage of a quantity. Find the new amount after a percentage increase or decrease.

Ratio & Proportion

Core declarative knowledge: What should students know?

  • What does a ratio represent?
  • Why do we use ratios to share?
  • What does a part of a ratio look like?
  • How do ratios link to parts of a whole?

Core procedural knowledge: What should students be able to do?

  • Write a ratio as a fraction.
  • Divide a given quantity into two or more parts in a given part : part or part : whole ratio.
  • Use a ratio to find one quantity when the other is known.
  • Write a ratio as a linear function.
  • Identify direct proportion from a table of values, by comparing ratios of values.
  • Use a ratio to compare a scale model to real-life object.
  • Use a ratio to convert between measures and currencies, e.g. £1.00 = €1.36.
  • Scale up recipes.
  • Convert between currencies.

Links to prior learning (to be made explicit and tested)

  • Express the division of a quantity into a number parts as a ratio.
  • Write ratios in form 1 : m or m : 1 and to describe a situation.
  • Write ratios in their simplest form, including three-part ratios.
  • Know the four operations of number.
  • Be able to find common factors.
  • Have a basic understanding of fractions as being ‘parts of a whole’.
  • Define percentage as ‘number of parts per hundred’.
  • Awareness that percentages are used in everyday life.

Proportion

Core declarative knowledge: What should students know?

  • What does it mean to be proportional?
  • What does it mean to be inversely proportional?
  • What do the graphical representations of proportion look like?
  • What are real life examples of direct and inverse proportion?

Core procedural knowledge: What should students be able to do?

  • Recognise and interpret graphs showing direct and inverse proportion.
  • Identify direct proportion from a table of values, by comparing ratios of values, for x squared and x cubed relationships.
  • Write statements of proportionality for quantities proportional to the square, cube or other power of another quantity.
  • Set up and use equations to solve word and other problems involving direct proportion.
  • Use y = kx to solve direct proportion problems, including questions where students find k, and then use k to find another value.
  • Solve problems involving inverse proportion using graphs by plotting and reading values from graphs.
  • Solve problems involving inverse proportionality.
  • Set up and use equations to solve word and other problems involving direct proportion or inverse proportion.

Links to prior learning (to be made explicit and tested)

As above.

Compound Measures

Core declarative knowledge: What should students know?

  • What is a compound measure?
  • What is the relationship between scalar measurements and compound measurements?
  • What are the formulae for speed, density, pressure, rates?
  • What do these units refer to: g/cm3 to kg/m3, kg/m2 to g/cm2, m/s to km/h?

Core procedural knowledge: What should students be able to do?

  • Use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate;
  • Change freely between related standard units (e.g. time, length, area, volume/capacity, mass) and compound units (e.g. speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts
  • Change g/cm3 to kg/m3, kg/m2 to g/cm2, m/s to km/h.

Links to prior learning (to be made explicit and tested)

  • Know the units of measurement for length, mass, volume, time etc.
  • Substitution into a formula.
  • Rearrange a formula
Module 4

Estimation & Accuracy

Core declarative knowledge: What should students know?

  • What is the purpose of rounding?
  • Why does rounding aid estimation?
  • What is the effect of rounding too early in a calculation?

Core procedural knowledge: What should students be able to do?

  • Round numbers to the nearest 10, 100, 1000;
  • Round to the nearest integer, to a given number of decimal places and to a given number of significant figures;
  • Estimate answers to one- or two-step calculations, including use of rounding numbers and formal estimation to 1 significant figure: mainly whole numbers and then decimals.

Links to prior learning (to be made explicit and tested)

  • Knowing place value
  • Knowing BIDMAS

Bounds

Core declarative knowledge: What should students know?

  • Why is it useful to know the maximum or minimum quantity of something given that it was rounded?
  • What does it mean to truncate a number? How is this different from rounding?
  • What equality symbols do we use for the upper and lower bound of a value?

Core procedural knowledge: What should students be able to do?

  • Calculate the upper and lowers bounds of numbers given to varying degrees of accuracy.
  • Calculate the upper and lower bounds of an expression involving the four operations.
  • Find the upper and lower bounds in real-life situations using measurements given to appropriate degrees of accuracy.
  • Find the upper and lower bounds of calculations involving perimeters, areas and volumes of 2D and 3D shapes.
  • Calculate the upper and lower bounds of calculations, particularly when working with measurements.

Links to prior learning (to be made explicit and tested)

  • Know the names and properties of 3D forms.
  • Know how to find the perimeter, area or volume of a shapes.
  • Be able to substitute numbers into an equation and give answers to an appropriate degree of accuracy.
  • Know the various metric units.

Recurring Decimals

Core declarative knowledge: What should students know?

  • What notation denotes recurring decimals?
  • What is the difference between an irrational number and a recurring decimal?

Core procedural knowledge: What should students be able to do?

  • By writing the denominator in terms of its prime factors, decide whether fractions can be converted to recurring or terminating decimals;
  • Convert a fraction to a recurring decimal;
  • Convert a recurring decimal to a fraction;

Links to prior learning (to be made explicit and tested)

Know the FDP equivalents.

Index Laws

Core declarative knowledge: What should students know?

  • What is an axiom?
  • Why does X^0=1?
  • What is the link between negative exponents, reciprocals and division?
  • Why are fractional exponents roots?

Core procedural knowledge: What should students be able to do?

  • Use index notation for integer powers of 10 including negative powers.
  • Recognise powers of 2, 3, 4, 5.
  • Use the square, cube and power keys on a calculator and estimate powers and roots of any given positive number, by considering the values it must lie between, e.g. the square root of 42 must be between 6 and 7.
  • Find the value of calculations using indices including positive, fractional and negative indices.
  • Recall that n0 = 1 and n–1 = for positive integers n as well as, = √n and = 3√n for any positive number n.
  • Understand that the inverse operation of raising a positive number to a power n is raising the result of this operation to the power.
  • Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers, fractional and negative powers, and powers of a power.
  • Solve problems using index laws.

Links to prior learning (to be made explicit and tested)

  • Know numerical index notation.
  • Recognise index notation algebraically
Module 5

Surds

Core declarative knowledge: What should students know?

  • What is a surd?
  • Why are surds useful in calculations?
  • Why do we rationalise surds?

Core procedural knowledge: What should students be able to do?

  • Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2;
  • Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3).
  • Rationalise the denominator involving surds
  • Calculate with surds

Links to prior learning (to be made explicit and tested)

  • Recognise roots of square and cube numbers
  • Recognise root notation
  • Double bracket expansion
  • Multiplying fractions

Standard Form

Core declarative knowledge: What should students know?

  • Why do we use standard form?
  • How does it simplify calculations?

Core procedural knowledge: What should students be able to do?

  • Convert large and small numbers into standard form and vice versa.
  • Add and subtract numbers in standard form.
  • Multiply and divide numbers in standard form.
  • Interpret a calculator display using standard form and know how to enter numbers in standard form.

Links to prior learning (to be made explicit and tested)

  • Index law

Sequences

Core declarative knowledge: What should students know?

  • What is a linear sequence?
  • What is a quadratic sequence?
  • What is a geometric sequence?
  • How can you tell the difference between types of sequence?
  • How do sequences link with graphical representations?

Core procedural knowledge: What should students be able to do?

  • Recognise simple sequences including at the most basic level odd, even, triangular, square and cube numbers and Fibonacci-type sequences (including those involving numbers in standard form or index form).
  • Generate sequences of numbers, squared integers and sequences derived from diagrams.
  • Describe in words a term-to-term sequence and identify which terms cannot be in a sequence.
  • Generate specific terms in a sequence using the position-to-term rule and term-to-term rule.
  • Find and use (to generate terms) the nth term of an arithmetic sequence.
  • Use the nth term of an arithmetic sequence to decide if a given number is a term in the sequence, or find the first term above or below a given number.
  • Identify which terms cannot be in a sequence by finding the nth term.
  • Continue a quadratic sequence and use the nth term to generate terms.
  • Find the nth term of quadratic sequences.
  • Distinguish between arithmetic and geometric sequences.
  • Use finite/infinite and ascending/descending to describe sequences.
  • Recognise and use simple geometric progressions (rn where n is an integer, and r is a rational number > 0 or a surd).
  • Continue geometric progression and find term to term rule, including negative, fraction and decimal terms.
  • Solve problems involving sequences from real life situations.

Links to prior learning (to be made explicit and tested)

  • The ability to use negative numbers with the four operations and recall and use hierarchy of operations and understand inverse operations;
  • Dealing with decimals and negatives on a calculator;
  • Using index laws numerically.
Module 6

Probability

Core declarative knowledge: What should students know?

  • What is probability?
  • What does it mean to be random?
  • What is the likelihood of winning the lottery?
  • What does the probabilities of all possible outcomes sum to?
  • What does 0 and 1 represent in probability?
  • Is anything certain?
  • What regions do the intersection and union represent on a Venn diagram?
  • What does mutually exclusive mean?
  • What is the difference between experimental and theoretical probability?

Core procedural knowledge: What should students be able to do?

  • Distinguish between events which are impossible, unlikely, even chance, likely, and certain to occur.
  • Mark events and/or probabilities on a probability scale of 0 to 1.
  • Write probabilities in words or fractions, decimals and percentages.
  • Find the probability of an event happening using theoretical probability.
  • Use theoretical models to include outcomes using dice, spinners, coins.
  • List all outcomes for single events systematically.
  • Work out probabilities from frequency tables.
  • Work out probabilities from two-way tables.
  • Record outcomes of probability experiments in tables.
  • Add simple probabilities.
  • Identify different mutually exclusive outcomes and know that the sum of the probabilities of all outcomes is 1.
  • Using 1 – p as the probability of an event not occurring where p is the probability of the event occurring.
  • Find a missing probability from a list or table including algebraic terms.
  • Find the probability of an event happening using relative frequency.
  • Estimate the number of times an event will occur, given the probability and the number of trials – for both experimental and theoretical probabilities.
  • List all outcomes for combined events systematically.
  • Use and draw sample space diagrams.
  • Work out probabilities from Venn diagrams to represent real-life situations and also ‘abstract’ sets of numbers/values.
  • Use union and intersection notation.
  • Compare experimental data and theoretical probabilities.
  • Compare relative frequencies from samples of different sizes.
  • Find the probability of successive events, such as several throws of a single dice.
  • Use tree diagrams to calculate the probability of two independent events.
  • Use tree diagrams to calculate the probability of two dependent events.

Links to prior learning (to be made explicit and tested)

  • Students should understand that a probability is a number between 0 and 1, and distinguish between events which are impossible, unlikely, even chance, likely, and certain to occur.
  • Students should be able to mark events and/or probabilities on a probability scale of 0 to 1.
  • Students should know how to add and multiply fractions and decimals.
  • Students should have experience of expressing one number as a fraction of another number.

Statistics

Core declarative knowledge: What should students know?

  • What are the different averages?
  • When is it better to use the mean?
  • When is it better to use the median?
  • When is it better to use the mode?
  • What happens to the original mean when one of the numbers is removed?
  • When will the mean go up? When will it go down? Why?
  • What is quantitative data?
  • What is qualitative data?
  • What is continuous data?
  • What is discrete data?
  • What is the difference between univariate data and bivariate data?
  • What is an outlier?
  • Why do we use scatter diagrams?
  • What does the line of best fit allow us to do?
  • What does interpolation mean?
  • What does extrapolation mean?

Core procedural knowledge: What should students be able to do?

  • Specify the problem and plan an investigation.
  • Decide what data to collect and what statistical analysis is needed.
  • Consider fairness.
  • Recognise types of data: primary secondary, quantitative and qualitative.
  • Identify which primary data they need to collect and in what format, including grouped data.
  • Collect data from a variety of suitable primary and secondary sources.
  • Understand how sources of data may be biased.
  • Explain why a sample may not be representative of a whole population.
  • Understand sample and population.
  • Use statistics found in all graphs/charts in this unit to describe a population;
  • Know the appropriate uses of cumulative frequency diagrams;
  • Construct and interpret cumulative frequency tables;
  • Construct and interpret cumulative frequency graphs/diagrams and from the graph:
    estimate frequency greater/less than a given value;
    find the median and quartile values and interquartile range;
  • Compare the mean and range of two distributions, or median and interquartile range, as appropriate;
  • Interpret box plots to find median, quartiles, range and interquartile range and draw conclusions;
  • Produce box plots from raw data and when given quartiles, median and identify any outliers;
  • Know the appropriate uses of histograms;
  • Construct and interpret histograms from class intervals with unequal width;
  • Use and understand frequency density;
  • From histograms:
    complete a grouped frequency table;
    understand and define frequency density;
  • Estimate the mean from a histogram;
  • Estimate the median from a histogram with unequal class widths or any other information from a histogram, such as the number of people in a given interval.

Links to prior learning (to be made explicit and tested)

  • Students should understand the different types of data: discrete/continuous.
  • Students should have experience of inequality notation.
  • Students should be able to multiply a fraction by a number.
  • Students should understand the data handling cycle.

Constructions

Core declarative knowledge: What should students know?

  • How do you use a compass correctly?
  • How do you use a protactor/angle measurer correctly?
  • What does it mean to bisect a line/angle?
  • What does equidistant mean?

Core procedural knowledge: What should students be able to do?

  • Understand and draw front and side elevations and plans of shapes made from simple solids.
  • Given the front and side elevations and the plan of a solid, draw a sketch of the 3D solid.
  • Use and interpret maps and scale drawings, using a variety of scales and units.
  • Read and construct scale drawings, drawing lines and shapes to scale.
  • Estimate lengths using a scale diagram.
  • Bisect a given angle.
  • Construct a perpendicular to a given line from/at a given point.
  • Construct angles of 90°, 45°.
  • Construct a perpendicular bisector of a line segment.
  • Construct a region bounded by a circle and an intersecting line.
  • Construct a given distance from a point and a given distance from a line.
  • Construct equal distances from two points or two line segments.
  • Construct regions which may be defined by ‘nearer to’ or ‘greater than’.

Links to prior learning (to be made explicit and tested)

  • Students should be able to measure and draw lines and angles.
  • Identify and construct a radius, diameter, circumference, area,
    chord, tangent and arc.
  • Identify acute and obtuse angles