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.

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

  • 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.

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

  • 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

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

  • 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.

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

  • 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.

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

  • 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.

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

  • 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

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

  • 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.
  • Add and subtract fractions.
  • Add fractions and write the answer as a mixed number.
  • 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.

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

  • 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.

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

  • 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.

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

  • 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.

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

  • 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.

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

  • 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.

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

  • 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.

Quadratics

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?

  • Define a ‘quadratic’ expression.
  • 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.

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

  • 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.
  • Factorising quadratic expressions.

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.

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

  • Reading and plotting coordinates.
  • 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.
  • 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)

  • 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.
  • Recognise the advantages and disadvantages between measures of average.

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

  • 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°.

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

  • 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.
  • 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?