Module 1
### Volume

### Similarity

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

- What are the properties of similar shapes?

- What is a scale factor?

- How do similar shapes relate to enlargement?

- How do scale factors for length, area and volume compare?

- How do scale factors and similarity relate to percentages and ratio?

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

- Find a scale factor between similar shapes

- Find missing sides using a SF

- Express the ratio of two sides to demonstrate similarity

- Enlarge a shape by a scale factor

- Enlarge a shape by a negative scale factor

- Describe the enlargement of a shape

- Express the similarity of two shapes as ratios and percentages

- Find the scale factors of areas and volumes

- Find the area, surface area and volume of similar shapes

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

- Finding the perimeter and area of 2D shapes

- Recognising proportional relationships

### Pythagoras

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

- What is the hypotenuse of a right-angled triangle?

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

- What is the Pythagoras’ Theorem?

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

- Substitute values in Pythagoras’ forumula

- Find the hypotenuse of a right-angled triangle
- Find a shorter side of a right-angled triangle

- Find the length of a line segment between two coordinates

- Recognise Pythagoras’ theorem within problems

- Use Pythagoras’ theorem in 3D problems.

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

- Square numbers

- Square roots
- Properties of triangles

### Trigonometry

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

- What are the conventional names of the sides of any right-angled triangle?

- How does the ratio of the sides of a triangle relate to similar shapes?

- What do the trigonmetric ratios of sine, consine and tangent relate to?

- What are the three formulae required to calculate a missing side or angle of any right angled triangle?

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

- Name each side of a right-angled triangle

- Identify the correct ratio for problem

- Rearrange the formula to make different parts the subject

- Solve to find a missing side

- Solve to find a missing angle

- Solve multistep problems

- Apply knowledge to 3D problems

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

- Rearranging formulae

- Solving equations

- Properties of triangles

### Volume

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

- What are the properties of cuboids, prisms, cylinders, pyramids, cones and spheres?

- What is the formula for calculating the volumes of cuboid, prism, cylinder, pyramid, cone or sphere?

- What is volume?

- What is surface area and how is different from volume?

- What is for formula for calculating the surface area of a sphere?

- What is the formula for calculating the curved surface area of a cone?

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

- Calculate the volume of a cuboid, prism, cylinder, pyramid, cone and sphere.

- Calculate the surface area of a cuboid, prism, cylinder, pyramid, cone and sphere.

- Calculate the volume of a frustrum.

- Solve problems involving volumes and surface area.

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

- Identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

- Calulate the area of 2D shapes including circles

Module 2

### Equations & Inequalities

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

- What is an equation?

- What is an unknown?

- What is an identity?

- What is the difference between an equality and an identity?

- What are inequalities?

- What is the difference between an inequality and an equation?

- What does it mean for a soluation to satisfy an inequality?

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

- Solve linear equations, with integer coefficients, in which the unknown appears on either side or on both sides of the equation;

- Solve linear equations which contain brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution;

- Solve linear equations in one unknown, with integer or fractional coefficients;

- Set up and solve linear equations to solve to solve a problem;

- Derive a formula and set up simple equations from word problems, then solve these equations, interpreting the solution in the context of the problem

- Solve simple linear inequalities in one variable, and represent the solution set on a number line;

- Solve two linear inequalities in x, find the solution sets and compare them to see which value of x satisfies both solve linear inequalities in two variables algebraically.

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

- The ability to use negative numbers with the four operations and recall and use hierarchy of operations and understand inverse operations;

- Dealing with decimals and negatives on a calculator;
- Using index laws numerically.

- Use algebraic notation and symbols correctly;

- Write an expression;

- Know the difference between a term, expression, equation, formula and an identity;

- Manipulate an expression by collecting like terms;

- Substitute positive and negative numbers into expressions such as 3x + 4 and 2×3 and then into expressions involving brackets and powers;

- Substitute numbers into formulae from mathematics and other subject using simple linear formulae, e.g. l × w, v = u + at;

- Use instances of index laws for positive integer powers;

- Use index notation (positive powers) when multiplying or dividing algebraic terms;

- Use instances of index laws, including use of zero, fractional and negative powers;

- Multiply a single term over a bracket;

- Recognise factors of algebraic terms involving single brackets and simplify expressions by factorising, including subsequently collecting like terms;

### Linear Graphs

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

- What is the gradient of a straight line graph?

- What does the gradient represent (including in a context)?

- What do each of the variables of y=mx+c represent?

- What does c of y=mx+c represent in a given context?

- How do you know if two straight lines are parallel to each other from their equations?

- How do you know if two straight lines are perpendicular to each other?

- What is a negative reciprocal and how does this relate to gradients of straight lines?

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

- Plot and draw graphs of y = a, x = a, y = x and y = –x, drawing and recognising lines parallel to axes, plus y = x and y = –x;

- Find the equation of a straight line from a graph in the form y = mx + c;

- Plot and draw graphs of straight lines of the form y = mx + c with and without a table of values;

- Sketch a graph of a linear function, using the gradient and y-intercept (i.e. without a table of values);

- Find the equation of the line through one point with a given gradient;

- Identify and interpret gradient from an equation ax + by = c;

- Find the equation of a straight line from a graph in the form ax + by = c;

- Plot and draw graphs of straight lines in the form ax + by = c;

- Identify when two lines are perpendicular and state their respective gradients using m and -1/m.

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

As equations and inequalities.

### Quadratics

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

- What is a quadratic?

- What is a cubic/recpircal and circle graph?

- What are the properties of a quadratic graph?

- How does factorisation of a quadratic help us solve?

- What must a quadratic equal to be solved?

- How do the solutions of a quadratic relate to the roots?

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

- Recognise a linear, quadratic, cubic, reciprocal and circle graph from its shape;

- Generate points and plot graphs of simple quadratic functions, then more general quadratic functions;

- Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function;

- Interpret graphs of quadratic functions from real-life problems;

- Factorise quadratic expressions in the form ax2 + bx + c;

- Solve quadratic equations by factorisation;

- Solve quadratic equations that need rearranging;

- Set up and solve quadratic equations;

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

As equations and inequalities.

### Simultaneous Equations

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

- Why do we need to solve multiple equations simulataneously?

- How do the solutions to two simultaneous equations relate to the graphs of the equations?

- Why might some graphical soluations only be estimations of their soluations?

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

- Find the exact solutions of two simultaneous equations in two unknowns;

- Interpret the intersection of two functions as the exact or approximate solutions to the functions;

- Use elimination or substitution to solve simultaneous equations;

- Solve exactly, by elimination of an unknown, two simultaneous equations in two unknowns:

- linear / linear, including where both need multiplying;

- 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

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

As equations and inequalities.

Module 3

### Percentages

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

- What is a percentage?

- What is simple interest?

- What is the effect of compound growth or decay?

- Which is preferable in the short term, simple interest or compound growrth?

- What is the general form of a percentage change? How can this be used to find the original amount?

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

- Work out a percentage increase or decrease, including: simple interest, income tax calculations, value of profit or loss, percentage profit or loss.

- Compare two quantities using percentages, including a range of calculations and contexts such as those involving time or money.

- Find a percentage of a quantity using a multiplier.

- Use a multiplier to increase or decrease by a percentage in any scenario where percentages are used.

- Find the original amount given the final amount after a percentage increase or decrease (reverse percentages), including VAT.

- Use calculators for reverse percentage calculations by doing an appropriate division.

- Use percentages in real-life situations, including percentages greater than 100%.

- Describe percentage increase/decrease with fractions, e.g. 150% increase means times as big.

- Understand that fractions are more accurate in calculations than rounded percentage or decimal equivalents, and choose fractions, decimals or percentages appropriately for calculations.

- Calcuate compound percentages/growth and decay.

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

Convert between fractions, decimals and percentages. Express a given number as a percentage of another number. Express one quantity as a percentage of another where the percentage is greater than 100%. Find a percentage of a quantity. Find the new amount after a percentage increase or decrease.

### Ratio & Proportion

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

- What does a ratio represent?

- Why do we use ratios to share?

- What does a part of a ratio look like?

- How do ratios link to parts of a whole?

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

- Write a ratio as a fraction.

- Divide a given quantity into two or more parts in a given part : part or part : whole ratio.

- Use a ratio to find one quantity when the other is known.

- Write a ratio as a linear function.

- Identify direct proportion from a table of values, by comparing ratios of values.

- Use a ratio to compare a scale model to real-life object.

- Use a ratio to convert between measures and currencies, e.g. £1.00 = €1.36.

- Scale up recipes.

- Convert between currencies.

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

- Express the division of a quantity into a number parts as a ratio.

- Write ratios in form 1 : m or m : 1 and to describe a situation.

- Write ratios in their simplest form, including three-part ratios.

- Know the four operations of number.

- Be able to find common factors.

- Have a basic understanding of fractions as being ‘parts of a whole’.

- Define percentage as ‘number of parts per hundred’.

- Awareness that percentages are used in everyday life.

### Proportion

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

- What does it mean to be proportional?

- What does it mean to be inversely proportional?

- What do the graphical representations of proportion look like?

- What are real life examples of direct and inverse proportion?

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

- Recognise and interpret graphs showing direct and inverse proportion.

- Identify direct proportion from a table of values, by comparing ratios of values, for x squared and x cubed relationships.

- Write statements of proportionality for quantities proportional to the square, cube or other power of another quantity.

- Set up and use equations to solve word and other problems involving direct proportion.

- Use y = kx to solve direct proportion problems, including questions where students find k, and then use k to find another value.

- Solve problems involving inverse proportion using graphs by plotting and reading values from graphs.

- Solve problems involving inverse proportionality.

- Set up and use equations to solve word and other problems involving direct proportion or inverse proportion.

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

As above.

### Compound Measures

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

- What is a compound measure?

- What is the relationship between scalar measurements and compound measurements?

- What are the formulae for speed, density, pressure, rates?

- What do these units refer to: g/cm3 to kg/m3, kg/m2 to g/cm2, m/s to km/h?

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

- Use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate;

- Change freely between related standard units (e.g. time, length, area, volume/capacity, mass) and compound units (e.g. speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

- Change g/cm3 to kg/m3, kg/m2 to g/cm2, m/s to km/h.

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

- Know the units of measurement for length, mass, volume, time etc.

- Substitution into a formula.

- Rearrange a formula

Module 4

### Estimation & Accuracy

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

- What is the purpose of rounding?

- Why does rounding aid estimation?

- What is the effect of rounding too early in a calculation?

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

- Round numbers to the nearest 10, 100, 1000;

- Round to the nearest integer, to a given number of decimal places and to a given number of significant figures;

- Estimate answers to one- or two-step calculations, including use of rounding numbers and formal estimation to 1 significant figure: mainly whole numbers and then decimals.

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

- Knowing place value

- Knowing BIDMAS

### Bounds

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

- Why is it useful to know the maximum or minimum quantity of something given that it was rounded?

- What does it mean to truncate a number? How is this different from rounding?

- What equality symbols do we use for the upper and lower bound of a value?

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

- Calculate the upper and lowers bounds of numbers given to varying degrees of accuracy.

- Calculate the upper and lower bounds of an expression involving the four operations.

- Find the upper and lower bounds in real-life situations using measurements given to appropriate degrees of accuracy.

- Find the upper and lower bounds of calculations involving perimeters, areas and volumes of 2D and 3D shapes.

- Calculate the upper and lower bounds of calculations, particularly when working with measurements.

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

- Know the names and properties of 3D forms.

- Know how to find the perimeter, area or volume of a shapes.

- Be able to substitute numbers into an equation and give answers to an appropriate degree of accuracy.

- Know the various metric units.

### Recurring Decimals

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

- What notation denotes recurring decimals?

- What is the difference between an irrational number and a recurring decimal?

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

- By writing the denominator in terms of its prime factors, decide whether fractions can be converted to recurring or terminating decimals;

- Convert a fraction to a recurring decimal;

- Convert a recurring decimal to a fraction;

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

Know the FDP equivalents.

### Index Laws

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

- What is an axiom?

- Why does X^0=1?

- What is the link between negative exponents, reciprocals and division?

- Why are fractional exponents roots?

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

- Use index notation for integer powers of 10 including negative powers.

- Recognise powers of 2, 3, 4, 5.

- Use the square, cube and power keys on a calculator and estimate powers and roots of any given positive number, by considering the values it must lie between, e.g. the square root of 42 must be between 6 and 7.

- Find the value of calculations using indices including positive, fractional and negative indices.

- Recall that n0 = 1 and n–1 = for positive integers n as well as, = √n and = 3√n for any positive number n.

- Understand that the inverse operation of raising a positive number to a power n is raising the result of this operation to the power.

- Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer powers, fractional and negative powers, and powers of a power.

- Solve problems using index laws.

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

- Know numerical index notation.

- Recognise index notation algebraically

Module 5

### Surds

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

- What is a surd?

- Why are surds useful in calculations?

- Why do we rationalise surds?

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

- Understand surd notation, e.g. calculator gives answer to sq rt 8 as 4 rt 2;

- Simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3).

- Rationalise the denominator involving surds

- Calculate with surds

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

- Recognise roots of square and cube numbers

- Recognise root notation

- Double bracket expansion

- Multiplying fractions

### Standard Form

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

- Why do we use standard form?

- How does it simplify calculations?

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

- Convert large and small numbers into standard form and vice versa.

- Add and subtract numbers in standard form.

- Multiply and divide numbers in standard form.

- Interpret a calculator display using standard form and know how to enter numbers in standard form.

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

- Index law

### Sequences

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

- What is a linear sequence?

- What is a quadratic sequence?

- What is a geometric sequence?

- How can you tell the difference between types of sequence?

- How do sequences link with graphical representations?

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

- Recognise simple sequences including at the most basic level odd, even, triangular, square and cube numbers and Fibonacci-type sequences (including those involving numbers in standard form or index form).

- Generate sequences of numbers, squared integers and sequences derived from diagrams.

- Describe in words a term-to-term sequence and identify which terms cannot be in a sequence.

- Generate specific terms in a sequence using the position-to-term rule and term-to-term rule.

- Find and use (to generate terms) the nth term of an arithmetic sequence.

- Use the nth term of an arithmetic sequence to decide if a given number is a term in the sequence, or find the first term above or below a given number.

- Identify which terms cannot be in a sequence by finding the nth term.

- Continue a quadratic sequence and use the nth term to generate terms.

- Find the nth term of quadratic sequences.

- Distinguish between arithmetic and geometric sequences.

- Use finite/infinite and ascending/descending to describe sequences.

- Recognise and use simple geometric progressions (rn where n is an integer, and r is a rational number > 0 or a surd).

- Continue geometric progression and find term to term rule, including negative, fraction and decimal terms.

- Solve problems involving sequences from real life situations.

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

- The ability to use negative numbers with the four operations and recall and use hierarchy of operations and understand inverse operations;

- Dealing with decimals and negatives on a calculator;

- Using index laws numerically.

Module 6

### Probability

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

- What is probability?

- What does it mean to be random?

- What is the likelihood of winning the lottery?

- What does the probabilities of all possible outcomes sum to?

- What does 0 and 1 represent in probability?

- Is anything certain?

- What regions do the intersection and union represent on a Venn diagram?

- What does mutually exclusive mean?

- What is the difference between experimental and theoretical probability?

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

- Distinguish between events which are impossible, unlikely, even chance, likely, and certain to occur.

- Mark events and/or probabilities on a probability scale of 0 to 1.

- Write probabilities in words or fractions, decimals and percentages.

- Find the probability of an event happening using theoretical probability.

- Use theoretical models to include outcomes using dice, spinners, coins.

- List all outcomes for single events systematically.

- Work out probabilities from frequency tables.

- Work out probabilities from two-way tables.

- Record outcomes of probability experiments in tables.

- Add simple probabilities.

- Identify different mutually exclusive outcomes and know that the sum of the probabilities of all outcomes is 1.

- Using 1 – p as the probability of an event not occurring where p is the probability of the event occurring.

- Find a missing probability from a list or table including algebraic terms.

- Find the probability of an event happening using relative frequency.

- Estimate the number of times an event will occur, given the probability and the number of trials – for both experimental and theoretical probabilities.

- List all outcomes for combined events systematically.

- Use and draw sample space diagrams.

- Work out probabilities from Venn diagrams to represent real-life situations and also ‘abstract’ sets of numbers/values.

- Use union and intersection notation.

- Compare experimental data and theoretical probabilities.

- Compare relative frequencies from samples of different sizes.

- Find the probability of successive events, such as several throws of a single dice.

- Use tree diagrams to calculate the probability of two independent events.

- Use tree diagrams to calculate the probability of two dependent events.

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

- Students should understand that a probability is a number between 0 and 1, and distinguish between events which are impossible, unlikely, even chance, likely, and certain to occur.

- Students should be able to mark events and/or probabilities on a probability scale of 0 to 1.

- Students should know how to add and multiply fractions and decimals.

- Students should have experience of expressing one number as a fraction of another number.

### Statistics

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

- What are the different averages?

- When is it better to use the mean?

- When is it better to use the median?

- When is it better to use the mode?

- What happens to the original mean when one of the numbers is removed?

- When will the mean go up? When will it go down? Why?

- What is quantitative data?

- What is qualitative data?

- What is continuous data?

- What is discrete data?

- What is the difference between univariate data and bivariate data?

- What is an outlier?

- Why do we use scatter diagrams?

- What does the line of best fit allow us to do?

- What does interpolation mean?

- What does extrapolation mean?

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

- Specify the problem and plan an investigation.

- Decide what data to collect and what statistical analysis is needed.

- Consider fairness.

- Recognise types of data: primary secondary, quantitative and qualitative.

- Identify which primary data they need to collect and in what format, including grouped data.

- Collect data from a variety of suitable primary and secondary sources.

- Understand how sources of data may be biased.

- Explain why a sample may not be representative of a whole population.

- Understand sample and population.

- Use statistics found in all graphs/charts in this unit to describe a population;

- Know the appropriate uses of cumulative frequency diagrams;

- Construct and interpret cumulative frequency tables;

- Construct and interpret cumulative frequency graphs/diagrams and from the graph:

estimate frequency greater/less than a given value;

find the median and quartile values and interquartile range;

- Compare the mean and range of two distributions, or median and interquartile range, as appropriate;

- Interpret box plots to find median, quartiles, range and interquartile range and draw conclusions;

- Produce box plots from raw data and when given quartiles, median and identify any outliers;

- Know the appropriate uses of histograms;

- Construct and interpret histograms from class intervals with unequal width;

- Use and understand frequency density;

- From histograms:

complete a grouped frequency table;

understand and define frequency density;

- Estimate the mean from a histogram;

- Estimate the median from a histogram with unequal class widths or any other information from a histogram, such as the number of people in a given interval.

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

- Students should understand the different types of data: discrete/continuous.

- Students should have experience of inequality notation.

- Students should be able to multiply a fraction by a number.

- Students should understand the data handling cycle.

### Constructions

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

- How do you use a compass correctly?

- How do you use a protactor/angle measurer correctly?

- What does it mean to bisect a line/angle?

- What does equidistant mean?

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

- Understand and draw front and side elevations and plans of shapes made from simple solids.

- Given the front and side elevations and the plan of a solid, draw a sketch of the 3D solid.

- Use and interpret maps and scale drawings, using a variety of scales and units.

- Read and construct scale drawings, drawing lines and shapes to scale.

- Estimate lengths using a scale diagram.

- Bisect a given angle.

- Construct a perpendicular to a given line from/at a given point.

- Construct angles of 90°, 45°.

- Construct a perpendicular bisector of a line segment.

- Construct a region bounded by a circle and an intersecting line.

- Construct a given distance from a point and a given distance from a line.

- Construct equal distances from two points or two line segments.

- Construct regions which may be defined by ‘nearer to’ or ‘greater than’.

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

- 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