Embedded Files
The KS3 section breaks each strand of the KS3 Program of Study into a sequence of small steps. Each strand is linked to prior learning from Year 5 and 6 and GCSE content. Further exemplification is provided in the NCETM Secondary PD material.
Strands in green contain a high proportion of content overlap with GCSE.
The core strands (red) contain a Microsoft form with example assessment questions from the KS3 Mastery Assessment Material and Secondary Mastery PD material from the NCETM. These forms can be duplicated, adapted and used to identify gaps in knowledge.
The examples assessment questions and quizzes on this site should be use to assess informally to identify gaps and inform planning.
The secondary Assessment Materials used to create these quizzes will support you and your colleagues in assessing students depth of understanding at KS3. They will enable you to make judgements on the degree to which students have mastered various components of the KS3 maths curriculum. The questions require students to have both procedural and conceptual knowledge.
As a department, complete the table below to identify the questions that require procedural and conceptual knowledge, possible misconceptions/gaps that could be identified and suggest any teaching or intervention strategies. The example below is for KS3 Number Quiz 1.
Procedural and Conceptual Knowledge and Understanding.docxNumber
Number
N1 understand and use place value for decimals, measures and integers of any size
Understand and use place value for decimals and measures.
Understand place value in integers
Understand place value in the context of measure
Understand place value in decimals
Understand place value in decimals, including recognising exponent and fractional representations of the column headings in place value tables
GCSE
N1. order decimals; use the symbols =, ≠, <, >, ≤, ≥
Prior Learning
5NPV–1 Know that 10 tenths are equivalent to 1 one, and that 1 is 10 times the size of 0.1. …
Know that 10 hundredths are equivalent to 1 tenth, and that 0.1 is 10 times the size of 0.01.
5NPV–2 Recognise the place value of each digit in numbers with up to 2 decimal places, and
compose and decompose numbers with up to 2 decimal places using standard and non-standard partitioning.
6NPV–2 Recognise the place value of each digit in numbers up to 10 million, including decimal fractions, and compose and decompose numbers up to 10 million using standard and non-standard partitioning.
N2a order positive and negative integers and decimals; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
Order and compare positive and negative integers, use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
Order a variety of positive and negative fractions and decimals (order unit fractions or fractions with the same denominator)
GCSE
N1. order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥
Prior Learning
PD Bites Negative Integers.pptxN2b order positive and negative integers, fractions and decimals; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥
Understand that 1 can be written in the form n/n (where n is any integer) and vice versa
Understand that fractions of the form a/b where a > b are greater than 1 and use this awareness to convert between improper fractions and mixed numbers
Understand the process of simplifying fractions through dividing both numerator and denominator by common factors
Compare decimals using < and >
Compare and order fractions by converting to decimals
Compare and order fractions by converting to fractions with a common denominator
Order a variety of positive and negative fractions and decimals using appropriate methods of conversion and recognising when conversion to a common format is not required
GCSE
N1. order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥
Prior Learning
6F–1 Recognise when fractions can be simplified, and use common factors to simplify fractions.
5F–1 Find non-unit fractions of quantities.
6F–2 Express fractions in a common denomination and use this to compare fractions that are similar in value.
6F–3 Compare fractions with different denominators, including fractions greater than 1, using reasoning, and choose between reasoning and common denomination as a comparison strategy.
5F–2 Find equivalent fractions and understand that they have the same value and the same position in the linear number system.
5F–3 Recall decimal fraction equivalents for, and, and for multiples of these proper fractions.
N3 use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property
Understand what a multiple is and be able to list multiples of n
Understand the concept of square and cube
Understand the concept of square root and cube root
Identify and explain whether a number is or is not a multiple of a given integer
Understand how to use the keys for squares and other powers and square root on a calculator
Understand and use correct notation for positive integer exponents
Understand what a factor is and be able to identify factors of positive integers
Understand what a prime number is and be able to identify prime numbers
Understand that a positive integer can be written uniquely as a product of its prime factors
Use the prime factorisation of two or more positive integers to efficiently identify the highest common factor
Use the prime factorisation of two or more positive integers to efficiently find their lowest common multiple
GCSE
N4. use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation theorem
N5. apply systematic listing strategies use of the product rule for counting
Prior Learning
5MD–2 Find factors and multiples of positive whole numbers, including common factors and common multiples, and express a given number as a product of 2 or 3 factors.
NCETM Secondary PD material 1.2.1, 1.2.3
N4a Addition and Subtraction
Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
Understand the mathematical structures that underpin addition and subtraction of positive and negative integers
Generalise and fluently use written addition and subtraction strategies, including columnar formats, with decimals (GD – addition and subtraction in different bases)
Understand the mathematical structures that underpin the addition and subtraction of fractions
Generalise and fluently use addition and subtraction strategies to calculate with fractions and mixed numbers
GCSE
N2. apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative; understand and use place value (e.g., when working with very large or very small numbers, and when calculating with decimals)
Prior Learning
3AS–2 Add and subtract up to three-digit numbers using columnar methods.
NCETM Secondary PD material 2.1
N4b Multiplication and division
use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
Understand the mathematical structures that underpin multiplication and division of positive and negative integers
Generalise and fluently use written multiplication strategies to calculate accurately with decimals
Generalise and fluently use written division strategies to calculate accurately with decimals
Understand the mathematical structures that underpin the multiplication of fractions
Understand how to multiply unit, non-unit and improper fractions
Understand the mathematical structures that underpin the division of fractions
Divide a fraction by a whole number
Divide a whole number by a fraction
Divide a fraction by a fraction
Generalise and fluently use strategies to multiply with mixed numbers (e.g. 2 3/4 × 1 2/3)
GCSE
N2. apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative; understand and use place value (e.g., when working with very large or very small numbers, and when calculating with decimals)
Prior Learning
5NF–1 Secure fluency in multiplication table facts, and corresponding division facts, through continued practice.
5MD–3 Multiply any whole number with up to 4 digits by anyone-digit number using a formal written method.
5MD–4 Divide a number with up to 4 digits by a one-digit number using a formal written method, and interpret remainders appropriately for the context.
https://www.ncetm.org.uk/classroom-resources/primm-2-24-division-dividing-by-two-digit-divisors/
https://www.ncetm.org.uk/classroom-resources/primm-2-25-using-compensation-to-calculate/
https://www.ncetm.org.uk/classroom-resources/primm-2-18-using-equivalence-to-calculate/
NCETM Secondary PD material 2.1.1–2.1.4
N5 use conventional notation for the priority of operations,
including brackets, powers, roots and reciprocals use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
Know the commutative law and use it to calculate efficiently
Know the associative law and use it to calculate efficiently
Calculate using priority of operations, including brackets, powers, exponents and reciprocals
Know how to fluently use certain calculator functions and use a calculator appropriately
Know the distributive law and use it to calculate efficiently
Use the associative, distributive and commutative laws to flexibly and efficiently solve problems
GCSE
N3. Priority of operations, including brackets, powers, roots and reciprocals
Prior Learning
5NF–2 Apply place-value knowledge to known additive and multiplicative number facts (scaling facts by 1 tenth or 1 hundredth).
NCETM Secondary PD material 2.1.5
N6 recognise and use relationships between operations including inverse operations
Factorise multiples of 10n in order to simplify multiplication and division of both integers and decimals, e.g. 300 × 7000, 0.3 × 0.007, 0.9 ÷ 0.03, etc
Prior Learning
6AS/MD–1 Understand that 2 numbers can be related additively or multiplicatively, and quantify additive and multiplicative relationships (multiplicative relationships restricted to multiplication by a whole number).
6AS/MD–2 Use a given additive or multiplicative calculation to derive or complete a related calculation, using arithmetic properties, inverse relationships, and place-value understanding.
NCETM Secondary PD material 2.1.1–2.1.2
N7 use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations
Understand the concept of square and cube
Understand the concept of square root and cube root
Understand and use correct notation for positive integer exponents
Understand how to use the keys for squares and other powers and square root on a calculator
GCSE
N6. use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5. estimate powers and roots of any given positive number
N7. calculate with roots, and with integer indices and fractional indices
N8. calculate exactly with fractions and multiples of π.
Simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions)
Simplify surd expressions involving squares and rationalise denominators
NCETM Secondary PD material 1.2.2
PD Bites Positive integer indices and roots.pptxN8 interpret and compare numbers in standard form A × 10n 1 ≤ A < 10, where n is a positive or negative integer or zero
Be able to write any integer in a range of forms, e.g. 53 = 5.3 × 10, 530 × 1/10, 5300 × 0.01, etc.
Understand that very large numbers can be written in the form a × 10n, (where 1 < a ≤ 10) and appreciate the real-life contexts where this format is usefully used
Understand that very small numbers can be written in the form a × 10−n, (where 1 < a ≤ 10) and appreciate the real-life contexts where this format is usefully used
GCSE
N9. calculate with and interpret standard form A x 10^n, where 1 ≤ A < 10 and n is an integer.
N13.use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate
Prior Learning
5MD–1 Multiply and divide numbers by 10 and 100; understand this as equivalent to making a number 10 or 100 times the size, or 1 tenth or 1 hundredth times the size.
6NPV–1 Understand the relationship between powers of 10 from 1 hundredth to 10 million, and use this to make a given number 10, 100, 1,000, 1 tenth, 1 hundredth or 1 thousandth times the size (multiply and divide by 10, 100 and 1,000).
NCETM Secondary PD material 1.3.3
N9 work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.372 and 3/8)
Understand that a fraction represents a division and that performing that division results in an equivalent decimal
Appreciate that any terminating decimal can be written as a fraction with a denominator of the form 10n (e.g. 0.56 = 56/100, 560/1000, etc.)
Know how to convert from fractions to decimals and back again using the converter key on a calculator
Know how to enter fractions as divisions on a calculator and understand the limitations of the decimal representation that results
GCSE
N10.work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7 2 or 0.375 or 3 8)
change recurring decimals into their corresponding fractions and vice versa
Prior Learning
5NPV–5 Convert between units of measure, including using common decimals and fractions.
NCETM Secondary PD material 1.3.1
N10 define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100%
N11 interpret fractions and percentages as operators
Find a fraction of a given amount
Given a fraction and the result, find the original amount
Express one number as a fraction of another
Describe one number as a percentage of another
Find a percentage of a quantity using a multiplier
Calculate percentage changes (increases and decreases)
Calculate the original value, given the final value after a stated percentage increase or decrease
Find the percentage increase or decrease, given start and finish quantities
GCSE
N12.interpret fractions and percentages as operators.
Prior Learning
https://www.ncetm.org.uk/classroom-resources/primm-3-10-linking-fractions-decimals-and-percentages/
NCETM Secondary PD material 3.1.3, 3.1.5
PD Bites Fractions and Percentages of Amounts.pptxN12 use standard units of mass, length, time, money and other measures, including with decimal quantities
Change freely between related standard units (for example time, length, area, volume/capacity, mass)
Use standard units of mass, length, time, money, and other measures, including with decimal quantities.
GCSE
G14.use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money, etc.)
Prior Learning
5NPV–4 Divide 1 into 2, 4, 5 and 10 equal parts, and read scales/number lines marked in units of 1 with 2, 4, 5 and 10 equal parts.
6NPV–4 Divide powers of 10, from 1 hundredth to 10 million, into 2, 4, 5 and 10 equal parts, and read scales/number lines with labelled intervals divided into 2, 4, 5 and 10 equal parts
N13 round numbers and measures to an appropriate degree of accuracy (for example, to a number of decimal places or significant figures)
Round numbers to up to three decimal places
Understand the concept of significant figures
Round integers to a required number of significant figures
Round numbers to any number of decimal places
Round decimals to a required number of significant figures
GCSE
N15. round numbers and measures to an appropriate degree of accuracy (e.g., to a specified number of decimal places or significant figures). Use inequality notation to specify simple error intervals due to truncation or rounding
Prior Learning
5NPV–3 Reason about the location of any number with up to 2 decimals places in the linear number system, including identifying the previous and next multiple of 1 and 0.1 and rounding to the nearest of each.
6NPV–3 Reason about the location of any number up to 10 million, including decimal fractions, in the linear number system, and round numbers, as appropriate, including in contexts.
NCETM Secondary PD material 1.1.2–1.1.3
PD Bites Rounding.pptxN14 use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b
Understand what is meant by a sensible degree of accuracy
Estimate numerical calculations
Estimate and check if solutions to problems are of the correct magnitude
Determine whether calculations using rounding will give an underestimate or overestimate
Understand the impact of rounding errors when using a calculator, and the way that these can be compounded to result in large inaccuracies
Calculate possible errors expressed using inequality notation a < x ≤ b
GCSE
N14 estimate answers; check calculations using approximation and estimation, including answers obtained using technology
N16.apply and interpret limits of accuracy, including upper and lower bounds
NCETM Secondary PD material 1.1.4
PD Bites Estimation.pptxN15 use a calculator and other technologies to calculate results accurately and then interpret them appropriately
Calculate using priority of operations, including brackets, powers, exponents and reciprocals
Use the associative, distributive and commutative laws to flexibly and efficiently solve problems
Know how to fluently use certain calculator functions and use a calculator appropriately
PD Bites: Understanding the Laws of Arithmetic
(Improving Learning in Mathematics, N5) from the STEM e-library (free registration).
Pupils learn to interpret numerical expressions using words and area representations, recognise the order of operations and equivalent expressions and understand the distributive laws of multiplication and division over addition (the expansion of brackets).
NCETM Secondary PD material 2.1.5
N16 appreciate the infinite nature of the sets of integers, real and rational numbers.
Order a variety of positive and negative fractions and decimals using appropriate methods of conversion and recognising when conversion to a common format is not required
Appreciate that, for any two numbers there is always another number in between them
NCETM Secondary PD material 1.1.4
Algebra
Algebra
NCETM Algebra Department Workshop and handouts
PD Bites Part Part Whole Models.pptxA1 use and interpret algebraic notation, including:
• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a² in place of a × a, a³ in place of a × a × a; a²b in place of a × a × b
• a/b in place of a ÷ b
• coefficients written as fractions rather than as decimals
• brackets
Understand that a letter can be used to represent a generalised number
Understand that algebraic notation follows particular conventions and that following these aids clear communication
Understand and recognise that a letter can be used to represent a specific unknown value or a variable
Understand that relationships can be generalised using algebraic statements
GCSE
A1 use and interpret algebraic notation, including:
-ab in place of a × b
-3y in place of y + y + y and 3 × y
-a2 in place of a × a, a3 in place of a × a × a, a2b in place of a × a × b
-a/b in place of a ÷ b
-coefficients written as fractions rather than as decimals
-brackets
Prior Learning
NC PofS Statements Pupils should be taught to express missing number problems algebraically
NCETM Secondary PD material 1.4.1
PD Bites Use and interpret algebraic notation.pptxA2 substitute numerical values into formulae and expressions, including scientific formulae
Understand that substituting particular values into a generalised algebraic statement gives a sense of how the value of the expression changes
GCSE
A2 substitute numerical values into formulae and expressions, including scientific formulae
Prior Learning
NC PofS Statements Pupils should be taught to express missing number problems algebraically
NCETM Secondary PD material 1.4.1
PD Bites Substitution.pptxA3 understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors
Know the meaning of and identify: term, coefficient, factor, product, expression, formula and equation
GCSE
A3. understand and use the concepts and vocabulary of expressions, equations, formulae, identities inequalities, terms and factors
A6 know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
Prior Learning
NC PofS Statements Pupils should be taught to express missing number problems algebraically
NCETM Secondary PD material 1.4.1
PD Bites Expressions and Equations.pptxA4 simplify and manipulate algebraic expressions to maintain equivalence by:
collecting like terms
multiplying a single term over a bracket
taking out common factors
expanding products of two or more binomials
Identify like terms in an expression, generalising an understanding of unitising
Simplify expressions by collecting like terms
Understand how to use the distributive law to multiply an expression by a term such as 3(a + 4b) and 3p2(2p + 3b)
Understand how to use the distributive law to factorise expressions where there is a common factor, such as 3a + 12b and 6p3 + 9p2b
Apply understanding of the distributive law to a range of problem-solving situations and contexts (including collecting like terms, multiplying an expression by a single term and factorising), e.g. 10 – 2(3a + 5), 3(a ± 2b) ± 4(2ab ± 6b), etc.
Use the distributive law to find the product of two binomials
Understand and use the special case when the product of two binomials is the difference of two squares
Find more complex binomial products
GCSE
A4 simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) …
Factorising quadratic expressions of the form x2 + bx + c, including the difference of two squares.
Simplifying expressions involving sums, products and powers, including the laws of indices
Factorising quadratic expressions of the form ax2+bx+c simplifying expressions involving sums, products and powers, including the laws of indices
A8. solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph
NCETM Secondary PD material 1.4.2–1.4.4
PD Bites Simplify and manipulate algebraic expressions.pptxA5 understand and use standard mathematical formulae; rearrange formulae to change the subject
Apply an understanding of inverse operations to a formula in order to make a specific variable the subject (in a wide variety of increasingly complex mix of operations)
GCSE
A5. understand and use standard mathematical formulae; rearrange formulae to change the subject
NCETM Secondary PD material 1.4.5
PD Bites Inverse operations and changing the subject.pptxA6 model situations or procedures by translating them into algebraic expressions or formulae and by using graphs
Understand that relationships can be generalised using algebraic statements
Understand that substituting particular values into a generalised algebraic statement gives a sense of how the value of the expression changes
Read and interpret points from a graph to solve problems
Model real-life situations graphically
Recognise that the point of intersection of two linear graphs satisfies both relationships and hence represents the solution to both those equations
Understand that different types of equation give rise to different graph shapes, identifying quadratics in particular
GCSE
A7. where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function’; interpret the succession of two functions as a ‘composite function’.
Prior Learning
A7 use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)
Recognise that there are many different types of equations of which linear is one type
Understand that in an equation the two sides of the 'equals' sign balance
Understand that a solution is a value that makes the two sides of an equation balance
Understand that a family of linear equations can all have the same solution
Solve a linear equation requiring a single additive step
Solve a linear equation requiring a single multiplicative step
Know that when an additive step and a multiplicative step are required, the order of operations will not affect the solution
Understand that an equation needs to be in a format to be 'ready' to be solved, through collecting like terms on each side of the equation
Recognise that equations with unknowns on both sides of the equation can be manipulated so that the unknowns are on one side
Solve complex linear equations, including those involving reciprocals
Appreciate the significance of the bracket in an equation
Recognise that there is more than one way to remove a bracket when solving an equation
Solve equations involving brackets where simplification is necessary first
GCSE
A17.solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph
A18. solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph
A19.solve two simultaneous equations in two variables
A19.solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph
A20.find approximate solutions to equations numerically using iteration
A21.translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.
A22.solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph
Prior Learning
NC PofS Statements Pupils should be taught to find pairs of numbers that satisfy an equation with two unknowns
https://www.ncetm.org.uk/classroom-resources/primm-1-31-problems-with-two-unknowns/
NCETM Secondary PD material 2.2.1–2.2.4
PD Bites Problems with Unknowns.pptxPD Bites Solve Equations.pptxA8 work with coordinates in all four quadrants
Describe and plot coordinates, including non-integer values, in all four quadrants
Solve a range of problems involving coordinates
Know that a set of coordinates, constructed according to a mathematical rule, can be represented algebraically and graphically (link to sequences)
Understand that a graphical representation shows all of the points (within a range) that satisfy a relationship
NCETM Secondary PD material 4.2.1
A9 recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in x and y and the Cartesian plane
A10 interpret mathematical relationships both algebraically and graphically
A11 reduce a given linear equation in two variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically
A12 use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations
A13 find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs
Describe and plot coordinates, including non-integer values, in all four quadrants
Solve a range of problems involving coordinates
Know that a set of coordinates, constructed according to a mathematical rule, can be represented algebraically and graphically (link to sequences)
Understand that a graphical representation shows all of the points (within a range) that satisfy a relationship
Recognise that linear relationships have particular algebraic and graphical features as a result of the constant rate of change
Read and interpret points from a graph to solve problems
Model real-life situations graphically
Understand that there are two key elements to any linear relationship: rate of change and intercept point
That writing linear equations in the form y = mx + c helps to reveal the structure
Solve a range of problems involving graphical and algebraic aspects of linear relationships
Understand that different types of equation give rise to different graph shapes, identifying quadratics in particular
Recognise that the point of intersection of two linear graphs satisfies both relationships and hence represents the
GCSE
A9. plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient
A10. identify and interpret gradients and intercepts of linear functions graphically and algebraically
A11. identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square
A12. recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1 x with x ≠ 0, exponential functions = x y k for positive values of k, and the trigonometric functions (with arguments in degrees) for angles of any size
A13.sketch translations and reflections of a given function
A14.plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration
A15.calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
A16.recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.
NCETM Secondary PD material 4.2.2–4.2.3
PD Bites Coordinates and Graphs.pptxA14 generate terms of a sequence from either a term-to-term or a position-to-term rule
Appreciate that a sequence is a succession of terms formed according to a rule
Understand that a sequence can be generated and described using term-to-term approaches
Understand that a sequence can be generated and described by a position-to-term rule
GCSE
A23.generate terms of a sequence from either a term-to-term or a position-to-term rule
Prior Learning
NC PofS Statements Pupils should be taught to generate and describe linear number sequences
NCETM Secondary PD material 4.1.1
A15 recognise arithmetic sequences and find the nth term
Understand the features of an arithmetic sequence and be able to recognise one
Understand that any term in an arithmetic sequence can be expressed in terms of its position in the sequence (nth term)
Understand that the nth term allows for the calculation of any term
Determine whether a number is a term of a given arithmetic sequence
GCSE
A25.deduce expressions to calculate the nth term of linear and quadratic sequences.
NCETM Secondary PD material 4.1.2
PD Bites Sequences.pptxRatio and Proportion
Ratio and Proportion
NCETM Department Workshop and handouts
PD Bites Bar Models.pptxPD Bites Multiplicative Reasoning.pptxR1 change freely between related standard units (for example time, length, area, volume/capacity, mass
GCSE
R1. 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
R2 use scale factors, scale diagrams and maps
Use ratio to describe rates (e.g. exchange rates, conversions, cogs, etc.)
GCSE
R2. use scale factors, scale diagrams and maps
NCETM Secondary PD material 3.1.4
R3 express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
Find a fraction of a given amount
Given a fraction and the result, find the original amount
Express one number as a fraction of another
GCSE
R3. express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1
NCETM Secondary PD material 3.1.3
R4 use ratio notation, including reduction to simplest form
R5 divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio
Be able to divide a quantity into a given ratio
Be able to determine the whole, given one part and the ratio
Be able to determine one part, given the other part and the ratio
GCSE
R4. use ratio notation, including reduction to simplest form
R5. divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio; apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)
NCETM Secondary PD material 3.1.4
R6 understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction
R7 relate the language of ratios [and the associated calculations to the arithmetic of fractions and to linear functions]
Appreciate that any two numbers can be connected via a multiplicative relationship
Understand that a multiplicative relationship can be expressed as a ratio and as a fraction
Be able to calculate the multiplier for any given two numbers
Appreciate that there are an infinite number of pairs of numbers for any given multiplicative relationship (equivalence)
Use a double number line to represent a multiplicative relationship and connect to other known representations
Understand the language and notation of ratio and use a ratio table to represent a multiplicative relationship and connect to other known representations
Use a graph to represent a multiplicative relationship and connect to other known representations
Use a scaling diagram to represent a multiplicative relationship and connect to other known representations
GCSE
R6. express a multiplicative relationship between two quantities as a ratio or a fraction
N11. identify and work with fractions in ratio problems
R8. relate ratios to fractions and to linear functions
NCETM Secondary PD material 3.1.1–3.1.4
PD Bites Proportional Reasoning.pptxR8 solve problems involving percentage change, including: percentage increase, decrease and original value problems [and simple interest in financial mathematics]
Describe one number as a percentage of another
Find a percentage of a quantity using a multiplier
Calculate percentage changes (increases and decreases)
Calculate the original value, given the final value after a stated percentage increase or decrease
Find the percentage increase or decrease, given start and finish quantities
GCSE
R9. define percentage as ‘number of parts per hundred’; interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than 100%; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics
NCETM Secondary PD material 3.1.5
R9 solve problems involving direct and inverse proportion, including graphical and algebraic representations
Understand the connection between multiplicative relationships and direct proportion
Recognise direct proportion and use in a range of contexts including compound measures
Recognise and use inverse proportionality in a range of contexts
GCSE
R7. understand and use proportion as equality of ratios
R10.solve problems involving direct and inverse proportion, including graphical and algebraic representations
R13.understand that X is inversely proportional to Y is equivalent to X is proportional to 1 Y; construct and interpret equations that describe direct and inverse proportion
R14.interpret the gradient of a straight-line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
R15.interpret the gradient at a point on a curve as the instantaneous rate of change.
apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts
R16.set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes
NCETM Secondary PD material 3.1.6
R10 use compound units such as speed, unit pricing and density to solve problems.
GCSE
R11.use compound units such as speed, rates of pay, unit pricing, density and pressure
R12.compare lengths, areas and volumes using ratio notation; make links to similarity
Geometry
Geometry
G1 derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)
Use the properties of a range of polygons to deduce their perimeters
Understand that the areas of composite shapes can be found in different ways
Derive and use the formula for the area of a trapezium
Understand the concept of surface area and find the surface area of 3D shapes in an efficient way
Be aware that all prisms have two congruent polygonal parallel faces (bases) with parallelogram faces joining the corresponding vertices of the bases
GCSE
G12.identify properties of the faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres
G16.know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)
Prior Learning
5G–2 Compare areas and calculate the area of rectangles (including squares) using standard units.
NCETM Secondary PD material 6.2.1–6.2.3
PD Bites Area and Perimeter.pptxG2 calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes
Recognise that there is constant multiplicative relationship (π) between the diameter and circumference of a circle
Use the relationship C = πd to calculate unknown lengths in contexts involving the circumference of circles
Understand the derivation of, and use the formula for, the area of a circle
Solve area problems of composite shapes involving whole and/or part circles, including finding the radius or diameter given the area
Use the constant cross-sectional area property of prisms and cylinders to determine their volume
GCSE
G9. identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
G17.know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr2 ; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids
G18.calculate arc lengths, angles and areas of sectors of circles
Prior Learning
6G–1 Draw, compose, and decompose shapes according to given properties, including
dimensions, angles and area, and solve related problems.
NCETM Secondary PD material 6.2.1
PD Bites Circles.pptxG3 draw and measure line segments and angles in geometric figures, including interpreting scale drawings
GCSE
G1. use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries; use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description
G13.construct and interpret plans and elevations of 3D shapes
G15.measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings
Prior Learning
5G–1 Compare angles, estimate and measure angles in degrees (°) and draw angles of a given size.
G4 derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line
Understand a circle as the locus of a point equidistant from a fixed point
Use intersecting circles to construct triangles and rhombuses from given lengths
Be aware that the diagonals of a rhombus bisect one another at right angles
Be aware that the diagonals of a rhombus bisect the angles
Use the properties of a rhombus to construct a perpendicular bisector of a line segment
Use the properties of a rhombus to construct a perpendicular to a given line through a given point
Use the properties of a rhombus to construct an angle bisector
GCSE
G2. use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line
NCETM Secondary PD material 6.4.1–6.4.2
G5 [describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively] and rotationally symmetric
G6 [use the standard conventions for labelling the sides and angles of triangle ABC], and know and use the criteria for congruence of triangles
Recognise rotational symmetry in shapes
Recognise that similar shapes have sides in proportion to each other but angle sizes are preserved
Recognise that for congruent shapes both side lengths and angle sizes are preserved
Understand and use the criteria by which triangles are congruent
GCSE
G1. use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries; use the standard conventions for labelling and referring to the sides and angles of triangles; draw diagrams from written description
G19.apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures
NCETM Secondary PD material 6.1.2
G7 [derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures (for example, equal lengths and angles) using appropriate language and technologies]x
GCSE
G4. derive and apply the properties and definitions of special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus; and triangles and other plane figures using appropriate language
G8 identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
Understand the nature of a translation and appreciate what changes and what is invariant
Understand the minimum information required to describe a translation (vertical and horizontal displacement)
Translate objects from information given in a variety of forms
Understand the nature of reflections and appreciate what changes and what is invariant
Understand the minimum information required to describe a reflection (line of reflection)
Reflect objects using a range of lines of reflection (including non-vertical and non-horizontal)
Understand the nature of rotations and appreciate what changes and what is invariant
Understand the minimum information required to describe a rotation (centre of rotation, size and direction of rotation)
Rotate objects using information about centre, size and direction of rotation
GCSE
G8. describe the changes and invariance achieved by combinations of rotations, reflections and translations
G24.describe translations as 2D vectors
G25.apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs
NCETM Secondary PD material 6.3.1–6.3.3
G9 identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
Understand the nature of enlargements and appreciate what changes and what is invariant
Understand the minimum information required to describe an enlargement (centre of enlargement and scale factor)
Enlarge objects using information about the centre of enlargement and scale factor
GCSE
G5. use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)
G7. identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional and negative scale factors)
G10 [apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles]
G11 understand and use the relationship between parallel lines and alternate and corresponding angles
Solve problems that require use of a combination of angle facts to identify values of missing angles, providing explanations of reasoning and logic used
GCSE
G3. apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles; understand and use alternate and corresponding angles on parallel lines;)
Prior Learning
5G–1 Compare angles, estimate and measure angles in degrees (°) and draw angles of a given size.
NCETM Secondary PD material 6.1.1
G12 derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons
Know and understand proofs that in a triangle, the sum of interior angles is 180 degrees
Know and understand proofs for finding the interior and exterior angle of any regular polygon
Solve problems that require use of a combination of angle facts to identify values of missing angles, providing explanations of reasoning and logic used
GCSE
G3. derive and use the sum of angles in a triangle (e.g., to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)
NCETM Secondary PD material 6.1.1
G13 apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ theorem, and use known results to obtain simple proofs
Be aware that there is a relationship between the lengths of the sides of a right-angled triangle
Use and apply Pythagoras' theorem to solve problems in a range of contexts
GCSE
G6. apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs
G20.know the formulae for Pythagoras’ theorem
NCETM Secondary PD material 6.1.1–6.1.3
G14 use Pythagoras’ theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles
Understand that the trigonometric functions are derived from measurements within a unit circle
Recognise the right-angled triangle within a unit circle and use proportion to scale to similar triangles
Know how the sine, cosine and tangent ratios are derived from the sides of a right-angled triangle
Choose appropriate trigonometric relationships to use to solve problems in right-angled triangles
Use trigonometric ratios to find a missing side in a right-angled triangle
Use trigonometric ratios to find a missing angle in a right-angled triangle
GCSE
G20.know the formulae for the trigonometric ratios; apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures
G21.know the exact values of sinθ and cosθ for θ = 0 , 30, 45, 60 and 90 ; know the exact value of tanθ for θ = 0, 30, 45 and 60.
G22. know and apply the sine rule and cosine rule, find unknown lengths and angles
G23.know and apply Area = 1/2absinC to calculate the area, sides or angles of any triangle.
NCETM Secondary PD material 3.2.1–3.2.2, 6.1.3
G15 [use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D]xii
G16 interpret mathematical relationships both algebraically and geometrically.
GCSE
G10.apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
G11.solve geometrical problems on coordinate axes
Probability
Probability
P1 record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0–1 probability scale
Understand that some outcomes are equally likely, and some are not
Understand that the likelihood of events happening can be ordered on a scale from impossible to certain
Understand that the likelihood of outcomes can be determined by designing and carrying out a probability experiment
GCSE
P1. record describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees
P2. apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
NCETM Secondary PD material 5.3.1
P2 understand that the probabilities of all possible outcomes sum to 1
Understand that probability is a measure of the likelihood of an event happening and that it can be assigned a numerical value
Calculate and use theoretical probabilities for single events
Understand that the probabilities of all possible outcomes sum to one
Calculate and use theoretical probabilities for combined events using a variety of appropriate representations, including Venn diagrams
GCSE
P3. relate relative expected frequencies to theoretical probability, using appropriate language and the 0 - 1 probability scale
P4. apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one
NCETM Secondary PD material 5.3.3
P3 enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams
Systematically find all the possible outcomes for two events using a range of appropriate diagrams
Systematically identify all possible outcomes for more than two events using appropriate diagrams, e.g. lists
Find theoretical probabilities from sets of outcomes organised in a systematic way from a range of appropriate representations
GCSE
P6. enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams
P7. construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
P8. calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
P9. calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams
NCETM Secondary PD material 5.3.2
P4 generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities.
GCSE
P5. understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
NCETM Secondary PD material 5.3.3
Statistics
Statistics
S1 describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)
Understand what the mean is measuring, how it is measuring it and calculate the mean from data presented in a range of different ways
Understand what the median is measuring, how it is measuring it and find the median from data presented in a range of different ways
Understand what the mode is measuring, how it is measuring it and identify the mode from data presented in a range of different ways
Understand what the range is measuring, how it is measuring it and calculate the range from data presented in a range of different ways
Understand that the different measures of central tendency offer a summary of a set of data
Understand how certain statistical measures may change as a result in changes of data
Understand range as a measure of spread, including a consideration of outliers
Understand that the different statistical representations offer different insights into a set of data
Use the different measures of central tendency and spread to compare two sets of data
Use the different statistical representations to compare two sets of data
Recognise relationships between bivariate data represented on a scatter graph
Given a statistical problem, choose what data needs to be analysed to explore that problem
Given a statistical problem, choose appropriate statistical measures to explore that problem
Given a statistical problem, choose appropriate representations to explore that problem
Given a statistical problem, choose appropriate measures and representations to effectively summarise and communicate conclusions
GCSE
S1. infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
Prior Learning
https://www.ncetm.org.uk/classroom-resources/primm-2-26-mean-average-and-equal-shares/
NCETM Secondary PD material 5.1.1, 5.2.1–5.2.2
S2 construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data
Construct bar charts from data presented in a number of different ways
Construct pictograms from data presented in a number of different ways
Construct pie charts from data presented in a number of different ways
Construct scatter graphs from data presented in a number of different ways
GCSE
S2. interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, tables and line graphs for time series data and know their appropriate use
S3. construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use
NCETM Secondary PD material 5.1.2
S3 describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs.
GCSE
S4. interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:
-appropriate graphical representation involving discrete, continuous and grouped data, including box plots
-appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers, quartiles and inter-quartile range)
S5. apply statistics to describe a population
S6. use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing
NCETM Secondary PD material 5.2.1
MATHEMATICAL PROMPTS FOR DEEPER THINKING VIDEOS
MATHEMATICAL PROMPTS FOR DEEPER THINKING VIDEOS
Each video on this page is accompanied by a set of PowerPoint slides which are available below the videos.
The slides and videos combined are designed to act as a catalyst for teachers to work together in identifying and thinking about an aspect of students’ mathematical understanding.
Although the focus of these materials is the maths, you might also notice the ways in which the teacher supports and challenges the students. You may like to reflect with colleagues on how you would respond to students if in a similar situation, what would you do that’s the same, and what would you do differently?
https://www.ncetm.org.uk/classroom-resources/secmm-mathematical-prompts-for-deeper-thinking-videos/
INSIGHTS FROM EXPERIENCED TEACHERS
INSIGHTS FROM EXPERIENCED TEACHERS
This set of videos models the sort of professional conversations that the materials might stimulate. The conversations occur between pairs of teachers (including experienced Secondary Mastery Specialists) and focus on specific sections of the materials, drawing out the ideas and connections that provoked them to think more deeply.
They also discuss how some of the questions might provoke deeper understanding amongst students in the classroom. We encourage these sort of conversations within maths departments and across schools, both in person and online. All the videos feature online conversations as they were recorded in summer 2020.
https://www.ncetm.org.uk/classroom-resources/insights-from-experienced-teachers/
Contents
Moving from number to algebra– Andrea Wickham and Amanda Mckay
Thinking deeply and reasoning with statistics– Kevin Styles and Jayne Watts
Thinking deeply about significant figures– Emily Curtis-Harper and Vicky Wheelhouse
Understanding the structure of number and number operations– Sara Sinnerton and Helen Billinge
Structures of addition and subtraction: negative numbers– Steve Harvey and Matt Panchal
Use of representations in teaching place value, directed number and multiplying fractions– Mark Robson and Ross Garvey
Understanding mathematical structure behind simplifying fractions, and ruler and compass constructions– Vicci Marshall and Nick Bunney
Recorded by experienced teachers, these videos offer advice and ideas for colleagues, especially less experienced colleagues such as NQTs, or non-maths specialists, and TAs/tutors teaching small groups.
Particularly useful for Covid recovery, they cover key learning points, assessing prerequisite knowledge, key questions, language, representations and common misconceptions and pitfalls. They also include lesson ideas and activities for students.
Each video is accompanied by a supporting document of notes, explanations and further resources, and the PowerPoint slides used in the video. These resources are below the video playlist and can be downloaded as zip files.
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