# KS4 Maths (Foundation)

Module 1

### Number

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

• What are the four operations?
• What are multiples and factors
• What are indices and roots?
• What is a prime number?
• What is the relationship between place value columns?
• How to apply the place value table to identify the value of any digit.
• Understanding the additive and multiplicative number properties such as commutativity.

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

• Order positive and negative integers, decimals and fractions.
• Apply the four operations to integers, decimals, simple fractions and mixed numbers both positive and negative.
• To solve problems involving roots and indices.
• To prime factorise and giving the solution in index form.
• Understand decimal notation and place values and identify the values of the digits in a decimal.

• Read, write and interpret mathematical statements involving addition, subtraction and equals signs.
• Represent and use number bonds and related subtraction facts.
• Add and subtract one-digit and two-digit numbers to including zero.
• Solving one-step problems that involve addition and subtraction.
• Using concrete objects and pictorial representations and missing number problems.

### Expressions

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

• What is an expression?
• What is an equation?

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

• Write an expression.
• Use and interpret algebraic notation.
• Substitute numerical values into formulae and expressions, including scientific formulae.
• Understand and use the concepts and vocabulary of expressions, equations, formulae, identities, inequalities, terms and factors.
• Simplify and manipulate algebraic expressions.
• Use index notation and laws in algebra.
• Expand and factorise with single brackets.
• Expand and factorise with double brackets.

• Using simple formulae.
• Use and interpret algebraic notation.
• Simplify and manipulate algebraic expressions.
• Collecting like terms.
• Adding and subtracting negative numbers.
• Order positive and negative integers.

### Equations & Inequalities

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

• The difference between an equation and an identity.
• To solve an equation is to find the only value (or values) of the
unknown that make the mathematical sentence correct.
• For every unknown an equation is needed.
• Knowledge of <,>, ≤ & ≥ notation.
• Numbers which are less or greater than but not equal to are represented on a number line with an unshaded circle.
• Shaded circles are used when an inequality can be equal to a number.
• Inequations have a set of solutions whereas equations have distinct
solutions.
• Inequations can be solved using the balance method.
• When dividing or multiplying both sides of an inequality by a negative number the sign is reversed.

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

• Solve linear equations in one unknown algebraically.
• Solve linear equations with unknowns on both sides of the equation.
• Solve linear inequalities with one variable.
• Represent the solution of linear inequalities on a number line.
• List sets of numbers for an inequality

• Using simple formulae.
• Use and interpret algebraic notation.
• Simplify and manipulate algebraic expressions.
• Collecting like terms.
• Adding and subtracting negative numbers.
• Order positive and negative integers.

### 2D Shapes

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

• What are the properties of different quadrilaterals?
• What are the properties of different triangles?
• What is the formula for the area of a triangle?
• What is the formula for the area of a rectangle/square/parallelogram?
• What is the formula for the area of a trapezium?
• How do work out the area of a compound shape?
• What are the units of measurement for length and area?
• How mm in a cm? How many cm in a m? How many m in a km?
• How mm2 in a cm2? How many cm2 in a m2? How many m2 in a km2?

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

• Indicate given values on a scale, including decimal value;
• Know that measurements using real numbers depend upon the choice of unit;
• Convert between units of measure within one system, including time;
• Convert metric units to metric units;
• Make sensible estimates of a range of measures in everyday settings;
• Measure shapes to find perimeters and areas using a range of scales;
• Find the perimeter of rectangles and triangles;
• Find the perimeter of parallelograms and trapezia;
• Find the perimeter of compound shapes;
• Recall and use the formulae for the area of a triangle and rectangle;
• Find the area of a rectangle and triangle;
• Find the area of a trapezium and recall the formula;
• Find the area of a parallelogram;
• Calculate areas and perimeters of compound shapes made from triangles and rectangles;
• Convert between metric area measures.

• Use geometric language appropriately;
• Use letters to identify points, lines and angles;
• Use two-letter notation for a line and three-letter notation for an angle;
• Identify a line perpendicular to a given line;
• Mark perpendicular lines on a diagram and use their properties;
• Identify parallel lines;
• Mark parallel lines on a diagram and use their properties;
• Recall the properties and definitions of special types of quadrilaterals, including symmetry properties;
• List the properties of each special type of quadrilateral, or identify (name) a given shape;
• Draw sketches of shapes;
• Name all quadrilaterals that have a specific property;
• Identify quadrilaterals from everyday usage;
• Distinguish between scalene, equilateral, isosceles and right-angled triangles;
Module 2

### Circles

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

• What are the different parts of a circle?
• What is the formula for the area of a circle?
• What is the formulae for the circumference of a circle?
• What is pi? What is a useful approximation of pi?

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

• Recall the definition of a circle;
• Identify, name and draw parts of a circle including tangent, chord and segment;
• Recall and use formulae for the circumference of a circle and the area enclosed by a circle circumference of a circle = 2πr = πd, area of a circle = πr2;
• Find circumferences and areas enclosed by circles;
• Use π ≈ 3.142 or use the π button on a calculator;
• Give an answer to a question involving the circumference or area of a circle in terms of π;
• Find radius or diameter, given area or perimeter of a circles;
• Find the perimeters and areas of semicircles and quarter-circles;
• Calculate perimeters and areas of composite shapes made from circles and parts of circles;
• Calculate arc lengths, angles and areas of sectors of circles;
• Round answers to a given degree of accuracy.

• Know and apply formulae to calculate rectangles.
• Know and apply formulae to calculate composite shapes.
• Know and apply formulae to calculate area of triangles.
• Know and apply formulae to calculate parallelograms.
• Know and apply formulae to calculate trapeziums.

### 3D Shapes

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

• What are the properties of cubes and cuboids?
• What are the properties of prisms?
• How are prisms and cylinders different?
• How do you calulate the volume of any prism?
• What is the difference between surface area and volume?
• How mm3 in a cm3? How many cm3 in a m3? How many m3 in a km3?

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

• Identify and name common solids: cube, cuboid, cylinder, prism, pyramid, sphere and cone;
• Sketch nets of cuboids and prisms;
• Recall and use the formula for the volume of a cuboid;
• Find the volume of a prism, including a triangular prism, cube and cuboid;
• Calculate volumes of right prisms and shapes made from cubes and cuboids;
• Find the surface area of a cylinder;
• Find the volume of a cylinder;
• Find the surface area and volume of spheres, pyramids, cones and composite solids;
• Estimate volumes etc by rounding measurements to 1 significant figure;
• Convert between metric volume measures;
• Convert between metric measures of volume and capacity e.g. 1ml = 1cm3;
• Estimate surface areas by rounding measurements to 1 significant figure;
• Find the surface area of a prism;
• Find surface area using rectangles and triangles.
• Round answers to a given degree of accuracy.

• Draw 2-D shapes.
• Recognising 3-D shapes in different orientations and describing them.
• Recognise angles as a property of shape or a description of a turn.
• Comparing and classifying geometric shapes including quadrilaterals
and triangles based on their properties and sizes.
• Identifying acute and obtuse angles.
• Identifying lines of symmetry in 2-D shapes presented in different
orientations.

### FDP

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

• The four operations of number.
• What common factors are
• Basic understanding of fractions as being ‘parts of a whole’.
• Define percentage as ‘number of parts per hundred’.
• Number complements to 10 and multiplication tables.

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

• Convert between frations and decimals.
• Convert between deimals and percentages.
• Convert between percentages and fractions.
• Recognise recurring decimals and convert fractions into recurring decimals.
• Compare and order fractions, decimals and integers, using inequality signs.
• Express a given number as a percentage of another
• Order fractions, decimals and percentages using lists and inequalities

• Students should be able to use the four operations of number.
• Students should be able to find common factors.
• Students have a basic understanding of fractions as being ‘parts of a whole’.
• Students should be able to define percentage as ‘number of parts per hundred’.

### Fractions

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

• What is the numerator?
• What is the denominator?
• What is the vinculum?
• Can identify an improper fraction.
• Define the word ‘reciprocal’.
• The four operations of number.
• Basic understanding of fractions as being ‘parts of a whole’.
• Define factors of a number.
• Define multiples of a number.

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

• Use diagrams to find equivalent fractions or compare fractions.
• Write fractions to describe shaded parts of diagrams.
• Express a given number as a fraction of another, using very simple numbers and where the fraction is both < 1 and > 1.
• Write a fraction in its simplest form and find equivalent fractions.
• Order fractions, by using a common denominator.
• Compare fractions, use inequality signs, compare unit fractions.
• Convert between mixed numbers and improper fractions.
• Multiply and divide an integer by a fraction.
• Multiply and divide a fraction by an integer, including finding fractions of quantities or measurements, and apply this by finding the size of each category from a pie chart using fractions.
• Understand and use unit fractions as multiplicative inverses.
• Multiply fractions: simplify calculations by cancelling first.
• Divide a fraction by a whole number.
• Divide fractions by fractions.

• Recognise, find and name a half as one of two equal parts of an
object, shape or quantity.
• Recognise, find and name a quarter as one of four equal parts of
an object, shape or quantity.
• Interpret fractions as operators.
• Ordering positive and negative decimals.
Module 3

### Percentages

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

• The four operations of number.
• Define percentage as ‘number of parts per hundred’.
• Percentage is a fraction out of 100.
• Factors of a number.
• Multiples of a number.
• How to draw bar models.
• Use the place value table to illustrate the equivalence between
fractions, decimals and percentages.
• To calculate a percentage of an amount without a calculator i.e to calculate 10% of any number by dividing by 10.
• To calculate a percentage of an amount with a calculator.
• Convert percentages to decimals.
• Understanding 100% as the original amount. E.g 10% decrease represents 10% less than 100% = 0.9.

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

• Express a given number as a percentage of another number.
• Find a percentage of a quantity without a calculator: 50%, 25% and multiples of 10% and 5%.
• Find a percentage of a quantity or measurement.
• Calculate amount of increase/decrease.
• Use percentages to solve problems, including comparisons of two quantities using percentages.
• Calculate percentages over 100%.
• Use percentages in real-life situations, including percentages greater than 100%.
• Calculate price after VAT (not price before VAT).
• Calculate simple interest.
• Income tax calculations.
• Use decimals to find quantities.
• Find a percentage of a quantity, including using a multiplier.
• Use a multiplier to increase or decrease by a percentage in any scenario where percentages are used.
• Understand the multiplicative nature of percentages as operators.

• Multiply and divide by powers of ten.
• Understand that per cent relates to ‘number of parts per hundred’.
• Write one number as a fraction of another.
• Calculate equivalent fractions.

### Ratio

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

• How to draw bar models.
• How to simplify fractions.
• How to find factors, common factors and highest common factor of numbers.
• How to use scale factors.
• How to use standard units of mass, length, time, money and other measures.
• It is important to apply equivalent ratios when solving problems
involving proportion including the use of the unitary method.
• To share amount given a ratio it is necessary to find the value of a single share.
• Use ratio notation, including reduction to simplest form.
• Express a multiplicative relationship between two quantities as a
ratio.

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

• Understand and express the division of a quantity into a of number parts as a ratio.
• Write ratios in their simplest form.
• Write/interpret a ratio to describe a situation.
• Share a quantity in a given ratio including three-part ratios.
• Solve a ratio problem in context.
• Use a ratio to find one quantity when the other is known.
• Use a ratio to convert between measures and currencies.
• Compare ratios.
• Write ratios in form 1 : m or m : 1.
• Write a ratio as a fraction.
• Write a ratio as a linear function.
• Write lengths, areas and volumes of two shapes as ratios in simplest form.
• Express a multiplicative relationship between two quantities as a ratio or a fraction.

• Problems involving the relative sizes of two quantities where
missing values can be found by using integer multiplication and
division facts.
• Problems involving the calculation of percentages.
• Problems involving unequal sharing and grouping using
knowledge of fractions and multiples.

### Proportion

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

• Define and state the difference between direct and inverse proportion.
• Understand direct proportion as: as x increase, y increases.
• Understand inverse proportion as: as x increases, y decreases.
• The constant of proportionality, k, is used to define the rate at which two or more measures change.
• Recognising the graphical representations of direct and indirect proportion is vital to understanding the relationship between two
measurements.

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

• Calulcate direct proportion.
• Calculate inverse proportion.
• Solve word problems involving direct and inverse proportion.
• Convert between currencies.
• Solve proportion problems using the unitary method.
• Recognise when values are in direct proportion by reference to the graph form;
• Understand inverse proportion: as x increases, y decreases.
• Recognise when values are in direct proportion by reference to the graph form.
• Understand direct proportion relationship y = kx.

• Solve problems involving the relative sizes of two quantities
where missing values can be found by using integer.
• Multiplication and division facts.
• Solve problems involving the calculation of percentages.
• Solve problems involving unequal sharing and grouping using
knowledge of fractions and multiples.

### Coordinates

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

• Identify which is the x and y ordinate.
• Know how to draw a coordinate grid (cartesian plane)
• State coordinate points on a grid.

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

• Use axes and coordinates to specify points in all four quadrants in 2D.
• Identify points with given coordinates and coordinates of a given point in all four quadrants.
• Find the coordinates of points identified by geometrical information in 2D (all four quadrants).
• Find the coordinates of the midpoint of a line segment.
• Draw, label and scale axes.
• Complete the shape after being given some information about a shape on coordinate axes.
Module 4

### Sequences

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

• Generate terms of a sequence from either a term-to-term or a
position-to-term rule.
• Recognise and use sequences of triangular, square and cube
numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric
progressions.
• Deduce expressions to calculate the nth term of linear and quadratic sequences.
• Quadratic sequences have a constant second difference.
• Linear sequences have a constant first difference.
• Geometric sequences share common multiplying factor rather than
common difference.

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

• Recognise sequences of odd and even numbers, and other sequences including Fibonacci sequences.
• Use function machines to find terms of a sequence.
• Write the term-to-term definition of a sequence in words.
• Find a specific term in the sequence using position-to-term or term-to-term rules.
• Generate arithmetic sequences of numbers, triangular number, square and cube integers and sequences derived from diagrams.
• Recognise such sequences from diagrams and draw the next term in a pattern sequence.
• Find the next term in a sequence, including negative values.
• Find the nth term for a pattern sequence.
• Find the nth term of a linear sequence.
• Find the nth term of an arithmetic sequence.
• Use the nth term of an arithmetic sequence to generate terms.
• 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 over a certain number.
• Use the nth term of an arithmetic sequence to find the first term greater/less than a certain number.
• Continue a geometric progression and find the term-to-term rule, including negatives, fraction and decimal terms.
• Continue a quadratic sequence and use the nth term to generate terms.
• Distinguish between arithmetic and geometric sequences.

• Using symbols and letters to represent
variables and unknowns in mathematical situations.
• Using simple formulae.
• Generate and describe linear number sequences.
• Express missing number problems algebraically.

### Straight Line Graphs

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

• Interpret simple expressions as functions with inputs and outputs.
• How to work with coordinates in all four quadrants.
• Plot graphs of equations that correspond to straight-line graphs in
the coordinate plane.
• How to use the form y = mx + c to identify parallel lines.
• Find the equation of the line through two given points, or through
one point with a given gradient.
• How to identify and interpret gradients and intercepts of linear functions
graphically and algebraically.
• Gradient is a measure of rate of vertical change divided by
horizontal change.
• Parallel lines have the same gradient.
• The intercept always has the x value equal zero.

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

• Use function machines to find coordinates (i.e. given the input x, find the output y).
• Plot and draw graphs of y = a, x = a, y = x and y = –x.
• Recognise straight-line graphs parallel to the axes.
• Recognise that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane.
• Plot and draw graphs of straight lines of the form y = mx + c using a table of values.
• Sketch a graph of a linear function, using the gradient and y-intercept.
• Identify and interpret gradient from an equation y = mx + c.
• Identify parallel lines from their equations.
• Plot and draw graphs of straight lines in the form ax + by = c.
• Find the equation of a straight line from a graph.
• Find the equation of the line through one point with a given gradient.
• Find approximate solutions to a linear equation from a graph.

• Describe positions on a 2-D grid as coordinates in the first quadrant.
• Describe positions on the full coordinate grid (all four quadrants).
• Recognise and describe linear number sequences, including those
involving fractions and decimals, and find the term-to-term rule.
• Generating and describing linear number sequences.

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

• Square negative numbers.
• Substitute into formulae.
• Plot points on a coordinate grid.
• Expand single brackets and collect ‘like’ terms.
• Draw a coordinate plane.

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

• Multiply together two algebraic expressions with brackets.
• Square a linear expression, e.g. (x + 1)2.
• Factorise quadratic expressions of the form x2 + bx + c.
• Factorise a quadratic expression x2 – a2 using the difference of two squares.
• Solve quadratic equations by factorising.
• Find the roots of a quadratic function algebraically.
• Generate points and plot graphs of simple quadratic functions, then more general quadratic functions.
• Identify the line of symmetry of a quadratic graph.
• Find approximate solutions to quadratic equations using a graph.
• Interpret graphs of quadratic functions from real-life problems.
• Identify and interpret roots, intercepts and turning points of quadratic graphs.

• Simplify expressions.
• Expanding products of two or more binomials.
• Factorising simple expressions including the difference of two squares.
• Simplifying expressions involving sums, products and powers,
including the laws of indices.

### Graphs

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

• Distance time graphs show distance away from a point.
• A speed-time graph tells us how the speed of an object changes over time.
• The steeper the gradient of the line, the greater the acceleration.
• Graphs can be used to represent a number of real life situations.

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

• Read values from straight-line graphs for real-life situations.
• Draw straight line graphs for real-life situations, including ready reckoner graphs, conversion graphs, fuel bills graphs, fixed charge and cost per unit.
• Draw distance–time graphs and velocity–time graphs.
• Work out time intervals for graph scales.
• Interpret distance–time graphs, and calculate: the speed of individual sections, total distance and total time.
• Interpret information presented in a range of linear and non-linear graphs.
• Interpret graphs with negative values on axes.
• Find the gradient of a straight line from real-life graphs.
• Interpret gradient as the rate of change in distance–time and speed–time graphs, graphs of containers filling and emptying, and unit price graphs.

• Straight line graphs.
• Calculating the gradient of a line.
Module 5

### Probability

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

• The terms outcome, event and probability are key to describing the likelihood of an event occurring.
• Outcome is the result of an experiment.
• An event is a set of outcomes of a probability experiment.
• Probability describes the likelihood of an event occurring.
• A probability can be given as fraction, decimal or percentage.
• An event which is impossible has a probability of zero.
• An event which is certain to occur has a probability of one.
• When listing all the permutations of two or more events students need a logical and exhaustive systematic method.
• When working with experimental data a probability can only be estimated as contextual factors are likely to be a factor.

#### 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.

• Compare and order fractions, including fractions > 1.
• Use common factors to simplify fractions; use common multiples to express fractions in the same denomination.
• Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions.

### Statistics

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

• What quantitative and qualitative data is.
• What continous and discrete data is.
• The different types of averages.
• State the median, mode, mean and range from a small data set.
• Extract the averages from a stem and leaf diagram.
• Estimate the mean from a table.
• Frequency tables must not have overlapping categories.
• Diagrams must not imply bias e.g equal width bar charts.
• Pictograms need to use suitable symbols to illustrate fractional
amounts.

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

• 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.
• Calculate the mean, mode, median and range for discrete data.
• Can interpret and find the median, mean and range from a (discrete) frequency table.
• Can interpret and find the range, modal class, interval containing the median, and estimate of the mean from a grouped data frequency table.
• Can interpret and find the mode and range from a bar chart.
• Can interpret and find the median, mode and range from stem and leaf diagrams.
• Can interpret and find the mean from a bar chart.
• Understand that the expression ‘estimate’ will be used where appropriate, when finding the mean of grouped data using mid-interval values.
• Compare the mean, median, mode and range (as appropriate) of two distributions using bar charts, dual bar charts, pictograms and back-to-back stem and leaf.

• Students should be able to calculate the midpoint of two numbers.
• Students will have used inequality notation.
• Complete, read and interpret information in tables.

### Constructions

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

• The standard conventions for labelling and referring to the sides
and angles of triangles.
• Draw diagrams from written descriptions.
• Use the standard ruler and compass constructions (perpendicular
bisector of a line segment, constructing a perpendicular to a given
line from/at a given point, bisecting a given angle).
• Know that the perpendicular distance from a point to a line is the
shortest distance to the line

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

• Use straight edge and a pair of compasses to do standard constructions:
• Understand, from the experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA triangles are not.
• Construct the perpendicular bisector of a given line.
• Construct the perpendicular from a point to a line.
• Construct the bisector of a given angle.
• Construct angles of 90°, 45°.

• Identify and construct a radius, diameter, circumference, area,
chord, tangent and arc.
• Measure and begin to record lengths and heights.
• Identify acute and obtuse angles and compare and order angles up
to two right angles by size.
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?