Mathematics

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.

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

• Find the exact solutions of two simultaneous equations in two unknowns;
  
• Interpret the intersection of two functions as the exact or approximate solutions to the functions;
  
• Use elimination or substitution to solve simultaneous equations;
  
• Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns:
  
  • linear / linear, including where both need multiplying;
    
• Set up and solve a pair of linear simultaneous equations in two variables, including to represent a situation;
  
• Interpret the solution in the context of the problem

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

Core declarative knowledge: What should students know?

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

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

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

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?

• 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?

Link to prior learning

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

Core declarative knowledge: What should students know?

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

Core declarative knowledge: What should students know?

• What are the properties of similar shapes?
  
• What is a scale factor?
  
• How do similar shapes relate to enlargement?
  
• How do scale factors and similarity relate to percentages and ratio?
  

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

• Use the basic congruence criteria for triangles (SSS, SAS, ASA and RHS);
  
• Solve angle problems involving congruence;
  
• Identify shapes which are similar; including all circles or all regular polygons with equal number of sides;
  
• Understand similarity of triangles and of other plane shapes, use this to make geometric inferences, and solve angle problems using similarity;
  
• Identify the scale factor of an enlargement of a shape as the ratio of the lengths of two corresponding sides;
  
• Understand the effect of enlargement on perimeter of shapes;
  
• Solve problems to find missing lengths in similar shapes;
• Know that scale diagrams, including bearings and maps are ‘similar’ to the real-life examples.
  
• Scale a shape on a grid (without a centre specified);
  
• Understand that an enlargement is specified by a centre and a scale factor;
  
• Enlarge a given shape using (0, 0) as the centre of enlargement, and enlarge shapes with a centre other than (0, 0);
  
• Find the centre of enlargement by drawing;
  
• Describe and transform 2D shapes using enlargements by:
  
  • a positive integer scale factor
  • a fractional scale factor;
• Identify the scale factor of an enlargement of a shape as the ratio of the lengths of two corresponding sides, simple integer scale factors, or simple fractions;
• Understand that distances and angles are preserved under reflections, so that any figure is congruent under this transformation;
  
• Understand that similar shapes are enlargements of each other and angles are preserved – define similar in this unit;
  

Core declarative knowledge: What should students know?

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

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

• Understand, recall and use Pythagoras’ Theorem in 2D, including leaving answers in surd form
  
• Given 3 sides of a triangle, justify if it is right-angled or not
  
• Calculate the length of the hypotenuse in a right-angled triangle, including decimal lengths and a range of units
  
• Find the length of a shorter side in a right-angled triangle
  
• Apply Pythagoras’ Theorem with a triangle drawn on a coordinate grid
  
• Calculate the length of a line segment AB given pairs of points

Core declarative knowledge: What should students know?

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

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

• Understand, use and recall the trigonometric ratios sine, cosine and tan, and apply them to find angles and lengths in general triangles in 2D figures
• Use the trigonometric ratios to solve 2D problems
• Find angles of elevation and depression
• Round answers to appropriate degree of accuracy, either to a given number of significant figures or decimal places, or make a sensible decision on rounding in context of question
• Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°

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.

Core declarative knowledge: What should students know?

• What is the formula for the sum of the angles in any polygon?
  
• How do you use the formula to find the size of one angle in a regular shape?
  
• What is the angle sum of all exterior angles of any shape?
  
• What are the rules for angles in parallel lines?
  
• What are the three rules for measuring bearings?

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

• Find missing angles using properties of corresponding and alternate angles
  
• Understand and use the angle properties of parallel lines.
  
• Recognise and name pentagons, hexagons, heptagons, octagons and decagons
  
• Understand ‘regular’ and ‘irregular’ as applied to polygons
  
• Use the sum of angles of irregular polygons
  
• Calculate and use the sums of the interior angles of polygons
  
• Calculate and use the angles of regular polygons
  
• Use the sum of the interior angles of an n-sided polygon
  
• Use the sum of the exterior angles of any polygon is 360°
  
• Use the sum of the interior angle and the exterior angle is 180°
  
• Identify shapes which are congruent (by eye)
  
• Explain why some polygons fit together and others do not
  
• Use and interpret maps and scale drawings
  
• Estimate lengths using a scale diagram
  
• Make an accurate scale drawing from a diagram
  
• Use three-figure bearings to specify direction
  
• Mark on a diagram the position of point B given its bearing from point A
  
• Give a bearing between the points on a map or scaled plan
  
• Given the bearing of a point A from point B, work out the bearing of B from A
  
• Use accurate drawing to solve bearings problems
  
• Solve locus problems including bearings

Core declarative knowledge: What should students know?

• 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 the radius and the diameter of a circle?
  
• How do you use a compass correctly?
  
• What is a vector?
  
• How does moving the point of rotation affect the image?
  
• What is a rotation?
  
• Does an enlargement always make a shape bigger?
  
• What is an object?
  
• What is an image?
  
• What other translations can be described with a vector?
  
• How does moving the point of rotation affect the image?
  
• What happens when I move the shapes vertices?
  
• What happens to the image if I move the reflection line?
  
• Can a combination of transformations be described by a single transformation?

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

• Identify congruent shapes by eye
  
• Understand clockwise and anticlockwise
  
• Understand that rotations are specified by a centre, an angle and a direction of rotation
  
• Find the centre of rotation, angle and direction of rotation and describe rotations
  
• Describe a rotation fully using the angle, direction of turn, and centre
  
• Rotate a shape about the origin or any other point on a coordinate grid
  
• Draw the position of a shape after rotation about a centre (not on a coordinate grid)
  
• Identify correct rotations from a choice of diagrams
  
• Understand that translations are specified by a distance and direction using a vector
  
• Translate a given shape by a vector
  
• Describe and transform 2D shapes using single translations on a coordinate grid
  
• Use column vectors to describe translations
  
• Understand that distances and angles are preserved under rotations and translations, so that any figure is congruent under either of these transformations.
  
• Understand that reflections are specified by a mirror line
  
• Identify correct reflections from a choice of diagrams
  
• Understand that reflections are specified by a mirror line
  
• Identify the equation of a line of symmetry
  
• Transform 2D shapes using single reflections (including those not on coordinate grids) with vertical, horizontal and diagonal mirror lines
  
• Describe reflections on a coordinate grid

Core declarative knowledge: What should students know?

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

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

• Express a given number as a percentage of another number in more complex situations
  
• Calculate percentage profit or loss
  
• Make calculations involving repeated percentage change, not using the formula
  
• Find the original amount given the final amount after a percentage increase or decrease
  
• Use compound interest

Core declarative knowledge: What should students know?

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

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

• Understand and use compound measures:
  
  • density
    
  • pressure
    
  • speed
    
• Convert between metric speed measures
  
• Read values in km/h and mph from a speedometer
  
• Calculate average speed, distance, time – in miles per hour as well as metric measures
  
• Use kinematics formulae to calculate speed, acceleration (with formula provided and variables defined in the question)
  
• Change d/t in m/s to a formula in km/h, i.e. d/t × (60 × 60)/1000 – with support

Core declarative knowledge: What should students know?

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

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

• Use numbers raised to the power zero, including the zero power of 10;
  
• Convert large and small numbers into standard form and vice versa;
  
• Add and subtract numbers in standard form;
  
• Multiply and divide numbers in standard form;
  
• Interpret a calculator display using standard form and know how to enter numbers in standard form.
  

Core declarative knowledge: What should students know?

• “What does a formula represent?
  
• How is a formula rearranged?
  
• What does it mean to make a variable the subject of a formula?
  
• What process is required to move variables?
  
• What process is required when the subject is in two different terms? “

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

• Know the difference between an equation and an identity and use and understand the ≠ symbol
  
• Change the subject of a formula involving the use of square roots and squares
  

Core declarative knowledge: What should students know?

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

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

• Write simultaneous equations to represent a situation;
  
• Solve simultaneous equations (linear/linear) algebraically and graphically;
  
• Solve simultaneous equations representing a real-life situation, graphically and algebraically, and interpret the solution in the context of the problem;

Core declarative knowledge: What should students know?

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

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

• Round to the nearest integer;
  
• Round to a given number of decimal places;
  
• Round to any given number of significant figures;
  
• Estimate answers to calculations by rounding numbers to 1 significant figure;
  
• Use one calculation to find the answer to another.

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?

• Understand and use column notation in relation to vectors;
  
• Be able to represent information graphically given column vectors;
  
• Identify two column vectors which are parallel;
  
• Calculate using column vectors, and represent graphically, the sum of two vectors, the difference of two vectors and a scalar multiple of a vector.
  

Core declarative knowledge: What should students know?

• What does a formula represent?
  
• How is a formula rearranged?
  
• What does it mean to make a variable the subject of a formula?
  
• What process is required to move variables?
  
• What process is required when the subject is in two different terms?

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

• Change the subject of a simple formula, i.e. linear one-step, such as x = 4y
  
• Change the subject of a formula, including cases where the subject is on both sides of the original formula, or involving fractions and small powers of the subject
  
• Change the subject of a formula, including cases where the subject occurs on both sides of the formula, or where a power of the subject appears
  
• Change the subject of a formula such as , where all variables are in the denominators

Core declarative knowledge: What should students know?

• What is the quadratic formula?
  
• What does it mean to complete the square?
  
• What does a quadratic expressed in the form of (x+p)2+b=0 tell us?
  
• What is the relationship between the general form of a quadratic, completing the square and the quadratic formula?

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

• Set up and solve quadratic equations;
  
• Solve quadratic equations by factorisation and completing the square;
  
• Solve quadratic equations that need rearranging;
  
• Solve quadratic equations by using the quadratic formula;
  
• Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns:
  
  • linear / quadratic;
    
  • linear / x2 + y2 = r2

Core declarative knowledge: What should students know?

• How do you add two fractions together?
  
• How do you maintain the equivalence of a fraction?
  
• How do you multiply/divide two fractions?
  
• How do you eliminate a denominator?
  
• How do you maintain the equivalence of two sides of an equation?

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

• Simplify algebraic fractions
  
• Multiply and divide algebraic fractions
  
• Solve quadratic equations arising from algebraic fraction equations

Core declarative knowledge: What should students know?

• What is a function?
  
• What does the notation represent?
  
• What notation refers to the inverse function?
  
• What is a composite function?

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

• Use function notation
  
• Find f(x) + g(x) and f(x) – g(x), 2f(x), f(3x) etc algebraically
  
• Find the inverse of a linear function
  
• Know that f –1(x) refers to the inverse function
  
• For two functions f(x) and g(x), find gf(x)
  
• Interpret the reverse process as the ‘inverse function’
  
• Interpret the succession of two functions as a ‘composite function’ (the use of formal function notation is expected)
  

Core declarative knowledge: What should students know?

• What is the relationship between the equation of a circle and the Pythagoras’ Theorem?
  
• How do you find the equation of a straight line?
  
• How do you find the gradient of a line perpendicular to another?

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

• Draw circles, centre the origin, equation x2 + y2 = r2
  
• Find the equation of a tangent to a circle at a given point, by:
  
  • Finding the gradient of the radius that meets the circle at that point (circles all centre the origin)
  • Finding the gradient of the tangent perpendicular to it
    using the given point
• Recognise and construct the graph of a circle using x2 + y2 = r2 for radius r centred at the origin of coordinates.
  

Core declarative knowledge: What should students know?

• What is the formula for the sum of the angles in any polygon?
  
• How do you use the formula to find the size of one angle in a regular shape?
  
• What is the angle sum of all exterior angles of any shape?
  
• What are the rules for angles in parallel lines?
  
• What are the three rules for measuring bearings?

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

• Understand and use the angle properties of parallel lines and find missing angles using the properties of corresponding and alternate angles, giving reasons
• Use the angle sums of irregular polygons
• Calculate and use the sums of the interior angles of polygons; use the sum of angles in a triangle and use the angle sum in any polygon to derive the properties of regular polygons
• Use the sum of the exterior angles of any polygon is 360°
• Use the sum of the interior angles of an n-sided polygon
• Use the sum of the interior angle and the exterior angle is 180°
• Find the size of each interior angle, or the size of each exterior angle, or the number of sides of a regular polygon, and use the sum of angles of irregular polygons
• Calculate the angles of regular polygons and use these to solve problems
• Use the side/angle properties of compound shapes made up of triangles, lines and quadrilaterals, including solving angle and symmetry problems for shapes in the first quadrant, more complex problems and using algebra
• Use angle facts to demonstrate how shapes would ‘fit together’, and work out interior angles of shapes in a pattern
• Calculate bearings and solve bearings problems, including on scaled maps, and find/mark and measure bearings
  

Core declarative knowledge: What should students know?

• What are the circle theorems?
  
• How do you construct a proof of a circle theorem?
  
• What are the relationships between various circle theorems?
  

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

Prove and use the facts that:

• The angle subtended by an arc at the centre of a circle is twice the angle subtended at any point on the circumference
• The angle in a semicircle is a right angle
• The perpendicular from the centre of a circle to a chord bisects the chord;
  angles in the same segment are equal
• Alternate segment theorem
• Opposite angles of a cyclic quadrilateral sum to 180°
• Understand and use the fact that the tangent at any point on a circle is perpendicular to the radius at that point
• Find and give reasons for missing angles on diagrams using:
  circle theorems
• Isosceles triangles (radius properties) in circles
• The fact that the angle between a tangent and radius is 90°
• The fact that tangents from an external point are equal in length

Core declarative knowledge: What should students know?

• When do we use the sine and cosine rule?
  
• What are the formula?
  
• What do the variables represent?
  
• How are the two formula derived?
  
• What is the relationship to trigonometric formula for the area of a triangle and the common formula for the area of a triangle?

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

• Know and apply Area = ab sin C to calculate the area, sides or angles of any triangle.
  
• Know the sine and cosine rules, and use to solve 2D problems (including involving bearings).
  
• Use the sine and cosine rules to solve 3D problems

Core declarative knowledge: What should students know?

• How do you plot the trigonometric graphs?
  
• What are the exact values for sin, cos and tan and what is the pattern between them?

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

• Recognise, sketch and interpret graphs of the trigonometric functions (in degrees)
• y = sin x, y = cos x and y = tan x for angles of any size.
• Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45° , 60° and 90° and exact value of tan θ for θ = 0°, 30°, 45° and 60° and find them from graphs.

Core declarative knowledge: What should students know?

• What is the effect of an sum or multiplier on a function?
  
• How does this differ if it inside the bracket or outside?

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

• Apply to the graph of y = f(x) the transformations y = –f(x), y = f(–x) for sine, cosine and tan functions f(x).
  
• Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(x + a)
  for sine, cosine and tan functions f(x).
  
• Interpret and analyse transformations of graphs of functions and write the functions algebraically, e.g. write the equation of f(x) + a, or f(x – a):
  
• Apply to the graph of y = f(x) the transformations y = –f(x), y = f(–x) for linear, quadratic, cubic functions;
  
• Apply to the graph of y = f(x) the transformations y = f(x) + a, y = f(x + a)
  for linear, quadratic, cubic functions;

Core declarative knowledge: What should students know?

• What does the area under a graph represent?
  
• Why are trapezia used to estimate the area?
  
• What does the gradient of a curve represent?
  
• Why is the gradient of the tangent to a curve only an estimate of the gradient of the curve?

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

• Estimate area under a quadratic or other graph by dividing it into trapezia;
  
• Interpret the gradient of linear or non-linear graphs, and estimate the gradient of a quadratic or non-linear graph at a given point by sketching the tangent and finding its gradient
  
• Interpret the gradient of non-linear graph in curved distance–time and velocity–time graphs:
  
  • for a non-linear distance–time graph, estimate the speed at one point in time, from the tangent, and the average speed over several seconds by finding the gradient of the chord
    
  • for a non-linear velocity–time graph, estimate the acceleration at one point in time, from the tangent, and the average acceleration over several seconds by finding the gradient of the chord
    
• Interpret the gradient of a linear or non-linear graph in financial contexts
  
• Interpret the area under a linear or non-linear graph in real-life contexts;
  
• Interpret the rate of change of graphs of containers filling and emptying;
  
• Interpret the rate of change of unit price in price graphs.

Core declarative knowledge: What should students know?

• What is the purpose of iteration?
  
• What does it mean to converge?
  
• What variable needs to be the subject in order to solve using iteration?

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

• Use iteration with simple converging sequences to find approximate solutions to equations numerically.

Core declarative knowledge: What should students know?

• What is the difference between a scalar and a vector?
  
• What do vectors represent?
  
• How do you know if two vectors are parallel?
  
• How do you know if two vectors are collinear?

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

• Understand and use vector notation, including column notation, and understand and interpret vectors as displacement in the plane with an associated direction.
  
• Understand that 2a is parallel to a and twice its length, and that a is parallel to –a in the opposite direction.
  
• Represent vectors, combinations of vectors and scalar multiples in the plane pictorially.
• Calculate the sum of two vectors, the difference of two vectors and a scalar multiple of a vector using column vectors (including algebraic terms).
  
• Find the length of a vector using Pythagoras’ Theorem.
  
• Calculate the resultant of two vectors.
  
• Solve geometric problems in 2D where vectors are divided in a given ratio.
  
• Produce geometrical proofs to prove points are collinear and vectors/lines are parallel.

Core declarative knowledge: What should students know?

• What is the difference between a historgram and a bar chat?
  
• What type of data does a histogram use?
  
• How do you calculate frequency density?
  
• What does the area of a bar represent?
  
• Why are some bars different widths?
  

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

• 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.

Core declarative knowledge: What should students know?

• What does a proof represent?
  

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

• Solve ‘Show that’ and proof questions using consecutive integers (n, n + 1), squares a2, b2, even numbers 2n, odd numbers 2n +1;