|
|
PROGRAMME OF STUDY
Knowledge, skills and understanding
Teaching should ensure that appropriate connections are made between the sections on number and algebra , shape, space and measures , and handling data . During key stage 4 (higher) pupils take increasing responsibility for planning and executing their work. They refine their calculating skills to include powers, roots and numbers expressed in standard form. They learn the importance of precision and rigour in mathematics. They use proportional reasoning with fluency and develop skills of algebraic manipulation and simplification. They extend their knowledge of functions and related graphs and solve a range of equations, including those with noninteger coefficients. They use short chains of deductive reasoning, develop their own proofs, and begin to understand the importance of proof in mathematics. Pupils use definitions and formal reasoning to describe and understand geometrical figures and the logical relationships between them. They learn to handle data through practical activities, using a broader range of skills and techniques, including sampling. Pupils develop the confidence and flexibility to solve unfamiliar problems and to use ICT appropriately. By seeing the importance of mathematics as an analytical tool for solving problems, they learn to appreciate its unique power. Note
>This programme of study is intended for pupils who have attained a secure level 5 at the end of key stage 3. Teachers are expected to plan work drawing on all the numbered subsections of the programme of study.
>For some groups of pupils, all or part of particular lettered paragraphs may not be appropriate. Note about sections
>There is no separate section of the programme of study numbered Ma1 that corresponds to the first attainment target, using and applying mathematics . Teaching requirements relating to this attainment target are included within the other sections of the programme of study.
Using and applying number and algebra
1) Pupils should be taught to:
Problem solving
a) select and use appropriate and efficient techniques and strategies to solve problems of increasing complexity, involving numerical and algebraic manipulation
b) identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches
c) break down a complex calculation into simpler steps before attempting a solution and justify their choice of methods
d) make mental estimates of the answers to calculations; present answers to sensible levels of accuracy; understand how errors are compounded in certain calculations
Communicating
e) discuss their work and explain their reasoning using an increasing range of mathematical language and notation
f) use a variety of strategies and diagrams for establishing algebraic or graphical representations of a problem and its solution; move from one form of representation to another to get different perspectives on the problem
g) present and interpret solutions in the context of the original problem
h) use notation and symbols correctly and consistently within a given problem
i) examine critically, improve, then justify their choice of mathematical presentation; present a concise, reasoned argument
Reasoning
j) explore, identify, and use pattern and symmetry in algebraic contexts, investigating whether a particular case may be generalised further and understand the importance of a counterexample; identify exceptional cases when solving problems
k) understand the difference between a practical demonstration and a proof
l) show stepbystep deduction in solving a problem; derive proofs using short chains of deductive reasoning
m) recognise the significance of stating constraints and assumptions when deducing results; recognise the limitations of any assumptions that are made and the effect that varying the assumptions may have on the solution to a problem.
Numbers and the number system
2) Pupils should be taught to:
Integers
a) use their previous understanding of integers and place value to deal with arbitrarily large positive numbers and round them to a given power of 10; understand and use negative integers both as positions and translations on a number line; order integers; use the concepts and vocabulary of factor (divisor), multiple, common factor, highest common factor, least common multiple, prime number and prime factor decomposition
Powers and roots
b)
use the terms square, positive square root, negative square root, cube and cube root; use index notation [ for example, 8 2 , 8to the power minus two thirds] and index laws for multiplication and division of integer powers; use standard index form, expressed in conventional notation and on a calculator display
Fractions
c) understand equivalent fractions, simplifying a fraction by cancelling all common factors; order fractions by rewriting them with a common denominator
Decimals
d)
recognise that each terminating decimal is a fraction [for example, 0.137 = 137 one thousandths] ; recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals [for example, one-seventh = 0.142857142857...] ; order decimals
Percentages
e)
understand that 'percentage' means 'number of parts per 100', and interpret percentage as the operator 'so many hundredths of' [for example, 10% means 10 parts per 100 and 15% of Y means 15 one-hundredths * Y]
Ratio
f) use ratio notation, including reduction to its simplest form and its various links to fraction notation.
Calculations
3) Pupils should be taught to:
Number operations and the relationships between them
a) multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1; find the prime factor decomposition of positive integers; understand 'reciprocal' as multiplicative inverse, knowing that any nonzero number multiplied by its reciprocal is 1 (and that zero has no reciprocal, because division by zero is not defined); multiply and divide by a negative number; use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer, fractional and negative powers; use inverse operations, understanding that the inverse operation of raising a positive number to power n is raising the result of this operation to power 1 divided by nG
b) use brackets and the hierarchy of operations
c)
calculate a given fraction of a given quantity, expressing the answer as a fraction; express a given number as a fraction of another; add and subtract fractions by writing them with a common denominator; perform short division to convert a simple fraction to a decimal; distinguish between fractions with denominators that have only prime factors of 2 and 5 (which are represented by terminating decimals), and other fractions (which are represented by recurring decimals); convert a recurring decimal to a fraction [for example, 0.142857142857... = one-seventh]
d)
understand and use unit fractions as multiplicative inverses [for example, by thinking of multiplication by one-fifth as division by 5, or multiplication by six-sevenths as multiplication by 6 followed by division by 7 (or vice versa)] ; multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction
e)
convert simple fractions of a whole to percentages of the whole and vice versa; then understand the multiplicative nature of percentages as operators [for example, a 15% increase in value Y, followed by a 15% decrease is calculated as 1.15 * 0.85 * Y] ; calculate an original amount when given the transformed amount after a percentage change; reverse percentage problems [for example, given that a meal in a restaurant costs ?36 with VAT at 17.5%, its price before VAT is calculated as ? 36 divided by 1.175]
f) divide a quantity in a given ratio
Mental methods
g)
recall integer squares from 2 * 2 to 15 * 15 and the corresponding square roots, the cubes of 2, 3, 4, 5 and 10, the fact that n 0 = 1 and n -1 = 1/n for positive integers n [for example, 10 0 = 1; 9 -1 = E] , the corresponding rule for negative numbers [for example, 5 -2 = D = C], n to the power halfB = square root n and n to the power one-third = 3 to the power one-third n for any positive number n [for example, 25to the power half = 5 and 64to the power one-third = 4]
h)
round to a given number of significant figures; develop a range of strategies for mental calculation; derive unknown facts from those they know; convert between ordinary and standard index form representations [for example, 0.1234 = 1.234 * 10 -1 ] , converting to standard index form to make sensible estimates for calculations involving multiplication and/or division
Written methods
i) use efficient methods to calculate with fractions, including cancelling common factors before carrying out the calculation, recognising that in many cases only a fraction can express the exact answer
j)
solve percentage problems, including percentage increase and decrease [for example, simple interest, VAT, annual rate of inflation] ; and reverse percentages
k)
represent repeated proportional change using a multiplier raised to a power [for example, compound interest]
l) calculate an unknown quantity from quantities that vary in direct or inverse proportion
m)
calculate with standard index form [for example, 2.4 * 10 7 * 5 * 10 3 = 12 *10 10 = 1.2 * 10 11 , (2.4 * 10 7 ) divided_by (5 * 10 3 ) = 4.8 * 10 3 ]
n) use surds and pi in exact calculations, without a calculator; rationalise a denominator such as 1 divided by square root of 3 = square root of 3 divided by 3
Calculator methods
o) use calculators effectively and efficiently, knowing how to enter complex calculations; use an extended range of function keys, including trigonometrical and statistical functions relevant across this programme of study
p) understand the calculator display, knowing when to interpret the display, when the display has been rounded by the calculator, and not to round during the intermediate steps of a calculation
q) use calculators, or written methods, to calculate the upper and lower bounds of calculations, particularly when working with measurements
r) use standard index form display and how to enter numbers in standard index form
s) use calculators for reverse percentage calculations by doing an appropriate division
t)
use calculators to explore exponential growth and decay [for example, in science or geography] , using a multiplier and the power key.
Solving numerical problems
4) Pupils should be taught to:
a) draw on their knowledge of operations and inverse operations (including powers and roots), and of methods of simplification (including factorisation and the use of the commutative, associative and distributive laws of addition, multiplication and factorisation) in order to select and use suitable strategies and techniques to solve problems and word problems, including those involving ratio and proportion, repeated proportional change, fractions, percentages and reverse percentages, inverse proportion, surds, measures and conversion between measures, and compound measures defined within a particular situation
b) check and estimate answers to problems; select and justify appropriate degrees of accuracy for answers to problems; recognise limitations on the accuracy of data and measurements.
Equations, formulae and identities
5) Pupils should be taught to:
Use of symbols
a)
distinguish the different roles played by letter symbols in algebra, using the correct notational conventions for multiplying or dividing by a given number, and knowing that letter symbols represent definite unknown numbers in equations [for example, x 2 + 1 = 82] , defined quantities or variables in formula [for example, V = IR], general, unspecified and independent numbers in identities [for example, ( x + 1) 2 = x 2 + 2 x + 1 for all x ], and in functions they define new expressions or quantities by referring to known quantities [for example, y = 2 7 x ; f( x ) = x 3 ; y = m with x is not equal to 0]
b)
understand that the transformation of algebraic entities obeys and generalises the welldefined rules of generalised arithmetic [for example, a ( b + c ) = ab + ac ] ; expand the product of two linear expressions [for example, ( x + 1)( x + 2) = x 2 + 3 x + 2] ; manipulate algebraic expressions by collecting like terms, multiplying a single term over a bracket, taking out common factors [for example, 9 x - 3 = 3(3 x - 1)] , factorising quadratic expressions including the difference of two squares [for example, x 2 -9 = ( x + 3) ( x -3)] and cancelling common factors in rational expressions [for example, 2( x + 1) 2 /( x + 1) = 2( x + 1)]
c) know the meaning of and use the words 'equation', 'formula', 'identity' and 'expression'
Index notation
d)
use index notation for simple integer powers, and simple instances of index laws [for example, x 3 * x 2 = x 5 ; x squared divided by x cubed = x 1 ; ( x 2 ) 3 = x 6 ] ; substitute positive and negative numbers into expressions such as 3 x 2 + 4 and 2 x 3
Equations
e)
set up simple equations [for example, find the angle a in a triangle with angles a , a + 10, a + 20] ; solve simple equations [for example, 5 x = 7; 11 4 x = 2; 3(2 x + 1) = 8; 2(1 x )= 6(2 + x ); 4 x 2 = 49; 3 = N] by using inverse operations or by transforming both sides in the same way
Linear equations
f) solve linear equations in one unknown, with integer or fractional coefficients, in which the unknown appears on either side or on both sides of the equation; solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Formulae
g)
use formulae from mathematics and other subjects [for example, for area |
