Mathematics

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ks3 mathematics

The LAB Maths department’s guiding principle for curriculum intent is as follows:

  • To provide the highest quality education in mathematics that prepares all students for future learning or employment.
  • To ensure that all students receive a knowledge rich education in mathematics rooted in conceptual understanding, reasoning, communication and language and problem solving. 
  • For all students to recognise the interconnectedness of mathematics and to have an appreciation for both its utility and its beauty.
  • The maths curriculum builds iteratively in order for students to build an extensive knowledge base at Key Stage 3 in preparation for Key Stage 4. All students will experience the main pillars of mathematics: number, geometry, algebra, ratio and proportion, probability and statistics.

MYP Assessment Criteria

Criterion A

Knowledge & Understanding

Criterion B

Investigating Patterns

Criterion C

Communication

Criterion D

Real-Life Application

Statement of inquiry

Different systems and forms of representation develop as civilizations evolve and humans interact.

Links to prior learning

  • Primary: KS1 & KS2 Maths
  • Use the number line to display decimals and round decimals to the nearest whole number, to 1 or 2 dp Round whole numbers to the nearest 1000, 100 or 10
  • Mark the approximate position of a number on a number line
  • Read and write decimals with up to 6 digits in figures and words
  • Use approximation to estimate the answers to calculations
  • Relate decimal arithmetic to integer arithmetic

Link to assessment

Criterion A

Core declarative knowledge: What should students know?

  • What is a number?  What is the difference between measuring and counting?
  • Why is using place value helpful?  What is base 10?
    What is the relationship between place value columns?
  • Describe what happens when you multiply by 10, 100 or 1000?  How does rounding help with estimating?
  • When might mental methods be more efficient than written methods?  What is multiplication?
  • Can you use a number line to represent multiplication?  What is division?  Is division commutative?
  • How are multiplication and division linked?
  • What happens if a number does not divide exactly?  What is commutativity, associativity & distributivity?
  • How do arrays and area models help you understand commutativity of multiplication?  How do arrays and area models help you understand associativity and distributivity?
  • What techniques can you use to multiply and divide decimals?
  • How does lining up your decimal numbers help with calculating/problem solving?

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

  • Recognise concrete representations and place value models of integers and decimals
  • Understand decimal notation and place values and identify the values of the digits in a decimal
  • Convert between decimal and fraction where the denominator is a factor of 10 or 100
  • Use correctly the symbols <, > and the associated language
  • Multiply, and divide, any integer or decimal by 10, 100, 1000, or 10,000
  • Mentally add and subtract sets of numbers including decimals
  • Use the commutativity and associativity of addition
  • Understand and use the formal written algorithms for addition and subtraction including decimals
  • Use commutativity, associativity and distributivity to solve calculations efficiently
  • Use column method to multiply integers
  • Use a formal algorithm for division
  • Multiply and divide whole numbers and decimals
  • Find factors and multiples.
  • Recognise and define: prime, square and cube numbers
  • Use the definitions of factors and multiples to find common factors and common multiples
  • Express an integer as a product of its factors

Statement of inquiry

Being able to represent different forms of quantities has helped humans explore and describe our planet.

Links to prior learning

  • Primary KS1 & KS2 Maths
  • Recognise the difference between the four operations.
  • Recognise the relationship between the inverse operations
  • Being able to represent numbers as a position on a number line
  • Knowing the placement of negative numbers Being able to order negative numbers 0>-1

Link to assessment

Criterion A, B and C

Core declarative knowledge: What should students know?

  • Does the order of addition and subtraction matter?  Why might BIDMAS be misleading?
  • Does it make a difference if you multiply or divide first?
  • For worded problems, should we apply operations in the same order that they appear?  How are indices linked to multiplication?
  • What does equal priority mean?  How can multiplying negative numbers help me in dividing negative numbers?
  • How does multiplying and dividing by negatives affect the concept of multiplication as scaling?
  • If I am adding a negative number, does my number want to get more/less positive/negative?
  • If I am subtracting a negative number, does my number want to get more/less positive/negative?  Why do we need to use letters?
  • What can letters in maths represent?  What is the difference between the equal sign and the identity sign?  What is the difference between 3x^2 and (3x)^2?
  • Is ab the same or different to ba?  Is a/b the same as b/a? a+b and b+a? a-b or b-a?

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

  • Define each element of BIDMAS
  • Understand the priority of operations, including equal priority
    Form and identify equivalent calculations based on distributivity, commutativity and the order of operations
  • Interpret negative numbers in a variety of contexts
  • Compare and order positive and negative numbers
  • Use positive and negative numbers to express change and difference
  • Calculate using all four operations with positive and negative values
  • Use number lines to model calculations with negative numbers
  • Explore scaling with negative multipliers
  • Form and manipulate expressions involving negative numbers
  • Develop understanding of algebraic notation
  • Collect like terms to simplify expressions
  • Substitute numerical values into expressions and evaluate
  • Expand and factorise single brackets
  • Develop understanding of the equality and inequality signs
  • Form equations or inequalities from abstract and real life contexts
  • Use different contexts, including sequences, to construct expressions, equations and inequalities.

Statement of inquiry

Generalising relationships between measurements can help explore the formation of human and natural landscapes.

Links to prior learning

  • KS1 & KS2 Maths
  • Relate the word angle to the distance between two intersecting straight lines
    Be able to define acute, obtuse, right angle, straight line in terms of degrees

Link to assessment

Criterion A and D

Core declarative knowledge: What should students know?

  • What can letters in maths represent?
  • What is the difference between the equal sign and the identity sign?
  • What is the difference between 3x^2 and (3x)^2?
  • Is ab the same or different to ba?
  • Is a/b the same as b/a?  a+b and b+a?  a-b or b-a?
  • How would you describe what an angle is?
  • What do they measure?
  • What is a degree?
  • How do you use protractors/angle measurers correctly?
  • What is a point of intersection?
  • How could you define a line of symmetry?

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

  • Develop understanding of algebraic notation
  • Collect like terms to simplify expressions
  • Substitute numerical values into expressions and evaluate
  • Expand and factorise single brackets
  • Develop understanding of the equality and inequality signs
  • Form equations or inequalities from abstract and real life contexts
  • Use different contexts, including sequences, to construct expressions, equations and inequalities.
  • Draw and measure acute and obtuse angles to the nearest degree Estimate the size of a given angle Know and use the angle facts: angles at a point, angles at a point on a straight line, vertically opposite angles Define parallel and perpendicular lines Use angle facts around corresponding, alternate and co interior angles to find missing angles Find unknown angles. Form algebraic expressions and solve equations related to unknown angles

Statement of inquiry

Artistry and creativity are enhanced through an understanding of how measurement helps to define forms.

Links to prior learning

  • KS1 & KS2 Maths:

    • Relate the word angle to the distance between two intersecting straight lines
    • Be able to define acute, obtuse, right angle, straight line in terms of degrees
    • Be able to recognise a triangle
    • Be able to recognise different types of triangle
    • Be able to recognise a circle

     

  • KS1 & KS2 Maths:

    • Be able to define the words horizontal and vertical
    • Recognise a cartesian plane
    • Be able to define perimeter and area, recognising the difference

Link to assessment

Criterion A

Core declarative knowledge: What should students know?

  • How could you define a line of symmetry?
  • What are the possible orders of rotational symmetry for a triangle?
  • What is the difference between scalene, isosceles and equilateral triangles?
  • What is the difference the radius and the diameter of a circle?
  • How do you use a compass correctly?
  • How do you use protractors/angle measurers correctly?
  • What is a point of intersection?
  • How could you define a line of symmetry?
  • What are the possible orders of rotational symmetry for a triangle?
  • What is the difference between scalene, isosceles and equilateral triangles?
  • What is the difference the radius and the diameter of a circle?
  • How do you use a compass correctly?
  • Does the order of the numbers matter?
  • If you know the mid-point, can you find the line segment?

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

  • Define and identify the order of rotational symmetry
  • Identify and count the lines of symmetry
  • Describing the properties of scalene, isosceles and equilateral triangles
  • Know that the interior angles in a triangle sum to 180°
  • Solve problems involving unknown angles in triangles
  • Naming the basic features of circles. Constructing triangles using a pair of compasses and ruler given the length of the sides. Constructing triangles with the same interior angles using a protractor. Constructing triangles given two sides and an angle
  • Reading and writing coordinates of points in all four quadrants. Including non-integer coordinates
  • Finding the mid-point of a line segment or two points
  • Using the midpoint and a point on the line to find the coordinates of another point on the line
  • Recognise and plot horizontal and vertical lines on a coordinate axis
  • Understanding equations of horizontal and vertical lines

Statement of inquiry

Using logic to simplify and manipulate quantities can help us explore human connections within families, communities and cultures.

Links to prior learning

  • KS1 & KS2 Maths:

    • Be able to define the words horizontal and vertical
    • Recognise a cartesian plane
    • Be able to define perimeter and area, recognising the difference
    • Be able to recognise a triangle
    • Be able to recognise different types of triangle
    • Be able to recognise a circle

     

  • From Y7:

    • Find factors (From M1)
    • Find HCF/LCM (From M1)

Link to assessment

Criterion D

Core declarative knowledge: What should students know?

  • Does the order of the numbers matter?
  • If you know the mid-point, can you find the line segment?
  • What shapes can be described as rectilinear?
  • What lengths are multiplied to find the area?
  • What is the difference between area and perimeter?
  • What is a prime number?
  • What is the Lowest Common Multiple? (LCM)
  • What is the Highest Common Factor? (HCF)
  • What does it mean to prime factorise a number?

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

  • Reading and writing coordinates of points in all four quadrants. Including non-integer coordinates
  • Finding the mid-point of a line segment or two points
  • Using the midpoint and a point on the line to find the coordinates of another point on the line
  • Recognise and plot horizontal and vertical lines on a coordinate axis
  • Understanding equations of horizontal and vertical lines
  • Calculating the perimeter of polygons
  • Finding the area of rectilinear shapes
  • Finding the area of other 2-D shapes including triangles, and special quadrilaterals
  • Find the area & perimeter of compound shapes (inc finding missing sides)
  • Translate shapes and describe translations using column vectors
  • Rotate shapes about a point by multiples of 90 degrees, clockwise or anti-clockwise
  • Describe rotations accurately
  • Reflecting shapes by horizontal, vertical and diagonal lines
  • Enlarge a shape by a positive and/or unit fraction scale factor
  • Find factors and multiples.
  • Recognise and define: prime, square and cube numbers
  • Use the definitions of factors and multiples to find common factors and common multiples
  • Express an integer as a product of its factors
  •  

Statement of inquiry

Using a logical process to simplify quantities and establish equivalence can help analyse competition and cooperation.

Links to prior learning

  • KS1 & KS2 Maths:
    • Relationship between fractions, decimals and percentages
    • Algorithms for manipulation of fractions.
  • From Y7:
    • Find factors (From M1)
    • Find HCF/LCM (From M1)

Link to assessment

Criterion A, B and C

Core declarative knowledge: What should students know?

  • What is a prime number?  What is the Lowest Common Multiple? (LCM)  What is the Highest Common Factor? (HCF)
  • What does it mean to prime factorise a number?
  • What is a numerator?  What is a denominator?
  • What is an improper fraction?  What is a proper fraction?
  • What is the relationship between the division of fractions and the multiplication of them?  How do we add fractions with unlike denominators?
  • What does equivalent mean?  What is a bar model?  What is simplifying?

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

  • Find factors and multiples.
  • Recognise and define: prime, square and cube numbers
  • Use the definitions of factors and multiples to find common factors and common multiples
  • Express an integer as a product of its factors
  • Be able to ‘build’ numbers by considering products.
  • Use index notation
  • Find factors and multiples, square numbers, cube numbers, prime number, triangular numbers
  • Write a number as a product of primes
  • Find the common factor and common multiple using the prime factorisation
    Find the highest common factor and lowest common multiple using the prime factorisation
  • Recognise and name equivalent fractions
  • Convert fractions to decimals
  • Convert terminating decimals to fractions in their simplest form
  • Convert between mixed numbers and improper fractions
  • Compare and order numbers (including like and unlike fractions)
  • Find a fraction of a set of objects or quantity
  • Find the whole given a fractional part
  • Multiply and divide fractions by a whole number or fraction
  • Add and subtract fractions with like denominators
  • Add and subtract fractions with unlike denominators
  • Add and subtract fractions mixed numbers and improper fractions
  • Convert between improper fractions and mixed numbers

Statement of inquiry

Producing equivalent forms through simplification can help to clarify, solve and create puzzles and tricks.

Links to prior learning

  • From KS1 & 2:

    • Recognise the inequalities symbols, but will refer to them as crocodiles eating the larger number.

     

  • From Y7 M2:

    • Be able to use letters to represent unknowns or variables
    • Be able to define generalisation in maths
    • Form and solve equations

Link to assessment

Criterion A

Core declarative knowledge: What should students know?

  • What is a sequence?  What does it mean to generalise?  What is the nth term?
  • What is the difference between an equation, expression and inequality?
  • Does an equation always have a solution?
  • What does the word inverse mean?
  • Why do I need to perform the same operations to both sides of my equation?
  • How do I decide what order to perform the inverse operations in?
  • What do inequalities represent?  How do inequalities relate to equations?
  • Are the same methods for solving inequalities the same as equations?
  • What happens when you multiply/divide both sides of an inequality by a negative number?  Can you prove why this happens?

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

  • Identify and generate terms of a sequences
  • Finding a given term in a linear sequence
  • Developing a rule for finding a term in a linear sequence
  • Generalising the position to term rule for a linear sequence (nth term)
  • Form and solve equations including those with unknowns on both sides and those involving algebraic fractions
  • Represent, form and solve inequalities
  • Use number lines and inequality symbols to represent and describe sets of numbers.
  • Use substitution to determine whether values satisfy given inequalities.
  • Solve linear inequalities with the unknown on one side.
  • Form inequalities in geometrical contexts
  • Use bar models to manipulate linear inequalities between two variables.
  • Compare manipulating linear equations and linear inequalities.

Statement of inquiry

Representing patterns of change as relationships can help determine the impact of human decision-making on the environment.

Links to prior learning

  • KS1 & KS2:

    • Round to the nearest whole number, 10, 100, 1000 etc…
    • Year 7 – place value, tenths, hundredths.. etc..

Link to assessment

Criterion A, B and C

Core declarative knowledge: What should students know?

  • How does the word linear relate to general form of y=ax+c
  • What happens as the coefficient of x changes?
  • What happens as the coefficient of x becomes negative?
  • What happens as the y-intercept changes?
  • How do you know if two lines are parallel?
  • How do you round to decimal places?
  • How do you round to significant figures?
  • What is the difference between rounding to d.p and rounding to s.f?
  • How can we use estimation to help solve a problem?
  • How do we estimate an answer to a question?

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

  • Identify the equations of horizontal and vertical lines (from year 7)
  • Plot coordinates from a rule to generate a straight line
  • Recognise y = ax & equations of the form y= ax + c Identify key features of a linear graph including the y-intercept and the gradient
  • Make links between the graphical and the algebraic representation of a linear graph
  • Recognise different algebraic representations of a linear graph Identify parallel lines from algebraic representations
  • Identify whether to round up or down.
  • Round to decimal places and significant figures.
  • Use estimation to solve problems.
  • Make links between fact family questions and using estimation to help

Statement of inquiry

Using a logical process to simplify quantities and establish equivalence can help analyse competition and cooperation.

Links to prior learning

  • KS1 & KS2:

    • Be able to recognise ratio notation
    • Be able to define percent
    • Be able to construct bar models for ratio
    • Understand the terms horizontal and vertical
    • Year 7 – Equation of vertical and horizontal lines

Link to assessment

Criterion A and D

Core declarative knowledge: What should students know?

  • What is a ratio?  Why do we use ratios to share?  What does a part of a ratio look like?
  • What is a coordinate?  What is a gradient?  What does parallel mean?  What is the Y-intercept?
  • What does it mean to be proportional?  What does it mean to be inversely proportional?  What do the graphical representations of proportion look like?

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

  • Understand the concept of ratio and use ratio language and notation
  • Connect ratio with understanding of fractions Compare two or more quantities in a ratio
  • Recognise and construct equivalent ratios Express ratios involving rational numbers in their simplest form
  • Construct tables of values and use graphs as a representation for a given ratio
  • Compare ratios by finding a common total value
  • Explore ratios in different contexts including speed and other rates of change
  • Contrast ratio relationships involving discrete and continuous measures
  • Identify the equations of horizontal and vertical lines (from year 7)
  • Plot coordinates from a rule to generate a straight line
  • Recognise y = ax & equations of the form y= ax + c
  • Identify key features of a linear graph including the y-intercept and the gradient
  • Make links between the graphical and the algebraic representation of a linear graph
  • Recognise different algebraic representations of a linear graph
  • Identify parallel lines from algebraic representations
  • Explore contexts involving proportional relationships
  • Represent proportional relationships using tables and graphs
  • Represent proportional relationships algebraically
  • Recognise graphs of proportional relationships
  • Solve proportion problems
  • Define inverse proportional relationships
  • Represent inverse proportion relationships algebraically

Statement of inquiry

Producing equivalent forms through simplification can help to clarify, solve and create puzzles and tricks.

Links to prior learning

  • KS1 & KS2:
    • Mean, Median, Mode and Range

Link to assessment

Criterion A

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 happens to the original mean when one of the numbers is removed?
  • When will the mean go up? When will it go down? Why?
  • How could you compare the two data sets?
  • When is the mean better to use?  When is the median better to use?  When is the mode better to use?
  • What is continuous data?  What is discrete data?

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

  • Explore contexts involving proportional relationships
  • Represent proportional relationships using tables and graphs
  • Represent proportional relationships algebraically
  • Recognise graphs of proportional relationships
  • Solve proportion problems
  • Define inverse proportional relationships
  • Represent inverse proportion relationships algebraically
  • Find the mean, median mode and range from raw datasets
  • Use the mean, median and mode to compare data sets
  • Use an average plus the range to compare datasets
  • Find the mode, median and mean from tables and graphical representations (not grouped)
  • Explore methods of data collection including surveys, questionnaires and the use of secondary data
  • Appreciate the difference between discrete and continuous data
  • Classify and tabulate data
  • Conduct statistical investigations using collected data

Statement of inquiry

Being able to represent relationships effectively can help justify characteristics and trends uncovered in communities.

Links to prior learning

  • KS1 & KS2:

    • 2D Shapes
    • Drawing a graph

Link to assessment

Criterion A, B, C and D

Core declarative knowledge: What should students know?

  • 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?
  • What are the definitions of the circumference, radius, diameter, a chord, a sector and a segment?
  • Is the circumference proportional to the diameter?
  • What is pi?  What is an irrational number?  What approximation can be used for pi?
  • How many decimal places of pi do you need to calculate the circumference of earth at the equator to accuracy of a hydrogen atom?

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

  • Find the mode, median and mean from tables and graphical representations (not grouped)
  • Explore methods of data collection including surveys, questionnaires and the use of secondary data
  • Appreciate the difference between discrete and continuous data
  • Classify and tabulate data
  • Conduct statistical investigations using collected data
  • Construct scatter graphs
  • Recognise clusters and outliers
  • Analyse the shape, strength and direction to make conjectures for possible bivariate relationships
  • Plot a line of best fit
  • Use a line of best fit to interpolate and extrapolate inferences
  • Explore relationship between circumference and diameter/radius
  • Use the formula for circumference
  • Explore relationship between area and radius
  • Use the formula for area of a circle
  • Find the area and circumference of a semi-circle and other sectors
  • Find the area and perimeter of composite shapes involving sectors of circles
  •  

Statement of inquiry

Generalizing the relationship between measurements can influence decisions that impact the environment.

Links to prior learning

  • KS1 & KS2:

    • 3D shape names

     

  • Year 7:

    • 2D shapes and their characteristics
    • Area and perimeter

Link to assessment

Criterion A

Core declarative knowledge: What should students know?

  • What are the characteristics of 3D shapes?
  • What is volume?
  • What is surface area?
  • What is a cross section?
  • How do you convert between different units of measure?
  • How do you convert between different units of area & volume?
  • How do you use a protractor?
  • Do you know the bearing from A to B will be different from B to A?

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

  • Name prisms, nets of prisms and using language associated with 3-D shapes
  • Finding the volume and surface area of cuboids
  • Finding the volume and surface area of other prisms including cylinders
  • Finding the volume and surface area of composite solids
  • Solving equations and rearranging formulae related to volumes
  • Convert between different units of area and volume
  • Name angles use the associated language with bearings.
  • Find the bearing of one location to the other.
  • After finding the bearing of A to B, what is the relationship of that bearing to the bearing of B to A.

Statement of inquiry

Producing equivalent forms through simplification can help to clarify, solve and create puzzles.

Links to prior learning

  • KS1 & KS2:
    • Basic probability
  • Year 8:
    • Two way tables and data

Link to assessment

Criterion A, B and C

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?
  • What is a sample space?
  • Calculate probabilities from a sample space.
  • What does solving simultaneous equations mean?  When you solve simultaneous equations, what do the solutions mean?  What are they?

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

  • Record, describe and analyse the frequency of outcomes of simple probability experiments
  • Define and use key language terms such as randomness, fairness, equally and unequally likely outcomes Use the 0-1 probability scale
  • Understand that the probabilities of all possible outcomes sum to 1
  • Enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
  • Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities.

Statement of inquiry

Generalisations about complex systems, such as, climate or social and economic organisations, become more logical when consideration is given to the ordered spontaneity nature of nonlinearity.

Links to prior learning

  • KS1 & KS2:
    • Adding and subtracting with negative numbers.
  • Year 7:
    • Solving equations, making the variable the subject
  • Year 8:
    • Manipulating equations
    • Students learnt last year how to plot straight line graphs and the characteristics of intersecting lines.

Link to assessment

Criterion A

Core declarative knowledge: What should students know?

  • Why is using a graph to find a solution sometimes an estimate?  What does using the graph to find a solution physically represent?  What are the characteristics of a linear, exponential and reciprocal graph?
  • How can we manipulate equations to solve simultaneous equations algebraically?  In what situations will there be 2 roots? 3 roots?  What do the roots of the equations mean?  Can you sketch a linear graph? What about a quadratic graph?
  • What is a polygon?  What is a regular polygon?  What are the characteristics of regular polygons?  How do you calculate the interior angles of a polygon?  How do you calculate the exterior angles of a polygon?  How do you find the sum of the angles in a regular polygon both interior and exterior?

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

  • Use linear and quadratic graphs to estimate values of y or x for given values of x or y
  • Find approximate solutions of simultaneous linear equations
  • Find approximate solutions to contextual problems from given graphs of a variety of functions
  • Use linear, exponential and reciprocal graphs to find solutions (including in context)
  • Use algebraic manipulation to solve simultaneous equations to find the root/roots
  • Use knowledge of angles in a triangle and angles in a quadrilateral to find the angles (interior and exterior) of any polygon.

Statement of inquiry

Generalisations about complex systems, such as, climate or social and economic organisations, become more logical when consideration is given to the ordered spontaneity nature of nonlinearity.

Links to prior learning

  • KS1 & KS2:
    • Names of different polygons
  • Year 8:
    • Circles and compound shapes
    • Volume and S.A

Link to assessment

Criterion A and D

Core declarative knowledge: What should students know?

  • What is a polygon?  What is a regular polygon?  What are the characteristics of regular polygons?  How do you calculate the interior angles of a polygon?  How do you calculate the exterior angles of a polygon?  How do you find the sum of the angles in a regular polygon both interior and exterior?
  • Can you bisect a line?  Can you bisect an angle?  How do you use a protractor?
    Do you know the bearing from A to B will be different from B to A?
  • Can you construct a circle with a given radius?  How do you construct a SSS SAS ASA triangle?  How do you construct an equilateral triangle?  How do you construct different quadrilaterals?

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

  • Use knowledge of angles in a triangle and angles in a quadrilateral to find the angles (interior and exterior) of any polygon.
  • Name angles use the associated language with bearings. Find the bearing of one location to the other.  After finding the bearing of A to B, what is the relationship of that bearing to the bearing of B to A.
  • Use different techniques to construct polygons and circles.

Statement of inquiry

Generalising relationships between measurements can help develop principles, processes and solutions.

Links to prior learning

  • KS1 & KS2:
    • Volume of a cuboid.
  • Year 7:
    • Area of 2D shapes
  • Year 8:
    • Circles and compound shapes
    • Volume and S.A

Link to assessment

Criterion A and D

Core declarative knowledge: What should students know?

  • What are the properties of a right angled triangle?  What is the hypotenuse?  How can you identify the hypotenuse or the longest side of any triangle from its angles?
  • What is the Pythagoras Theorem?  What is the difference between an equation, expression and inequality?  Does an equation always have a solution?
  • What does the word inverse mean?  Why do I need to perform the same operations to both sides of my equation?  How do I decide what order to perform the inverse operations in?
  • How do I calculate volume of a prism?  How do I find the area of a cross section?  What is a cross section?  How do you calculate the surface area of a prism?  How is surface area different to volume?
  • How can the nets of shapes help us calculate surface area?  What are the correct units for the answer?  Can you work backwards to find the area of the cross section or a missing length?

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

  • Find the length of the hypotenuse. Find the length of one of the shorter sides. Prove whether a triangle is right angle triangle. Applying knowledge to real life problems around missing sides of right angle triangles.
  • Find the area and circumference of a semi-circle and other sectors Find the area and perimeter of composite shapes involving sectors of circles
  • Name prisms, nets of prisms and using language associated with 3-D shapes Finding the volume and surface area of cuboids Finding the volume and surface area of other prisms including cylinders Finding the volume and surface area of composite solids
  • Solving equations and rearranging formulae related to volumes Convert between different units of area and volume

Statement of inquiry

Producing equivalent forms through simplification can help to clarify, solve and create puzzles.

Links to prior learning

  • KS1 & KS2:
    • Names of 2D shapes.
    • Multiplication & division.
  • Year 7:
    • Name and characteristics of 2D shapes.
  • Year 8:
    • Ratio, proportion and roots.

Link to assessment

Criterion A, B and C

Core declarative knowledge: What should students know?

  • What is a scale factor?  Why does a shape sometimes get smaller when we enlarge it?  What happens to the shape when we use a negative scale factor to enlarge it?
  • What are the characteristics of similar shapes?  How do you prove two or more shapes are similar?  How can you calculate a missing length on a similar shape?
  • What is a surd?  How can you simplify a surd?  Can you add subtract multiply and divide surds?
  • What are the trigonometric ratios?  How can we use the trigonometric ratios to calculate missing lengths and missing angles?

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

  • Describe a single enlargement with the scale factor and centre of enlargement. Enlarge a shape by a fractional scale factor. Enlarge a shape by a negative scale factor. Understand and be able to regurgitate the characteristics of similar shapes. Finding missing lengths and angles.
  • Define a surd and non-examples. Understand how to simplify surds and add & subtract surds. Understand how to multiply and divide surds. Understand how to expand brackets with surds. Understand how to rationalise the denominator of a fraction involving surds.
  • Be able to label the sides of a triangle: Hypotenuse, Adjacent and Opposite. Use the trigonometric ratios to calculate missing lengths and missing angles. Use the trigonometric ratios to calculate accurate values for the angles 30° and 60°. Use the trigonometric ratios to calculate angles of elevation and depression. Apply knowledge to trigonometry in 3 dimensions. Understand and recognise the trigonometric graphs and be able to sketch them. Understand and be able to use the sine and cosine rule. Know and understand how to use the formula for area of a triangle involving sine.
  • Students apply knowledge of pythagoras and trigonometry to their problem solving.

Statement of inquiry

Producing equivalent forms through simplification can help to clarify, solve and create puzzles.

Links to prior learning

  • KS1 & KS2:
    • Axes and graphs.
  • Year 7:
    • Expanding and factorising.
  • Year 8:
    • Plotting linear graphs and identifying points of intersection.

Link to assessment

Criterion A, B and C

Core declarative knowledge: What should students know?

  • What is a quadratic?
  • What characteristics does it have?
  • How do you solve a quadratic equation?
  • What methods are there available to us to solve a quadratic?
  • What is the quadratic formula?
  • Can you sketch a quadratic?
  • Can you factorise a quadratic?
  • Do you know and understand how to use the complete the square method?
  • Can you solve a quadratic equation using iteration?
  • What are the roots of a quadratic equation? What do they mean?
  • Can you the turning point and the line of symmetry?

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

  • Know and understand the different methods for solving a quadratic equations. Understand and recognise when a method is more efficient than the others.
  • Be able to accurately sketch a quadratic graph.
  • Understand and recognise maximum, minimum and turning points.

Click on the links below to view the videos and resources for the extension activities.

Module 1

Module 2

Positive & Negative Number and Algebra

Module 3

Angles

Module 4

Coordinates

Click on the links below to view the videos and resources for the extension activities.

Module 1

Module 2

Equations, Inequalities & Linear Graphs

Watch
In Our Time (also available on Spotify)

Module 3

Accuracy and Estimation

Module 4

Proportion

Listen
In Our Time (also available on Spotify)

Click on the links below to view the videos and resources for the extension activities.

Module 1

Module 2

Simultaneous Equations

Listen
In Our Time (also available on Spotify)

Module 3

Angles in Polygons

Listen
In Our Time (also available on Spotify)

Module 4

Volume

Listen
In Our Time (also available on Spotify)

back to ks4 subjects

ks4 mathematics

link to specification

Number

Link to prior learning

  • 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.

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.

Expressions

Link to prior learning

  • 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.

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.

Equations & Inequalities

Link to prior learning

  • 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.

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

2D Shapes

Link to prior learning

  • 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;

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.

Circles

Link to prior learning

  • 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.

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.

3D Shapes

Link to prior learning

  • 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.

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 calculate 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.

FDP

Link to prior learning

  • 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’.

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

Fractions

Link to prior learning

  • 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.

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.

Percentages

Link to prior learning

  • 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.

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.

Ratio

Link to prior learning

  • 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.

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.

Proportion

Link to prior learning

  • 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.

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?

  • Calculate 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.

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.

Sequences

Link to prior learning

  • 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.

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.

Straight Line Graphs

Link to prior learning

  • 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?

  • 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.

Quadratics

Link to prior learning

  • 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.

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.

Graphs

Link to prior learning

  • Reading and plotting coordinates.
  • Straight line graphs.
  • Calculating the gradient of a line.

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.

Probability

Link to prior learning

  • 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.

Core declarative knowledge: What should students 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.

Statistics

Link to prior learning

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

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.

Constructions

Link to prior learning

  • 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.

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°.

Probability

Link to prior learning

  • 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.

Core declarative knowledge: What should students know?