# 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

• 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.

• 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

• 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.

• 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.

• 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.

As equations and inequalities.

#### Core declarative knowledge: What should students know?

• 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;

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

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.

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.

• 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.

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.

• 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.

• 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.

• 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;

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.

• 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

• 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.

• 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.

• 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.
• 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.

• 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.

• 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’.