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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 3 pupils take increasing responsibility for planning and executing their work. They extend their calculating skills to fractions, percentages and decimals, and begin to understand the importance of proportional reasoning. They are beginning to use algebraic techniques and symbols with confidence. They generate and solve simple equations and study linear functions and their corresponding graphs. They begin to use deduction to manipulate algebraic expressions. Pupils progress from a simple understanding of the features of shape and space to using definitions and reasoning to understand geometrical objects. As they encounter simple algebraic and geometric proofs, they begin to understand reasoned arguments. They communicate mathematics in speech and a variety of written forms, explaining their reasoning to others. They study handling data through practical activities and are introduced to a quantitative approach to probability. Pupils work with increasing confidence and flexibility to solve unfamiliar problems. They develop positive attitudes towards mathematics and increasingly make connections between different aspects of mathematics. Note
>This programme of study covers the attainment range for this key stage. 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) explore connections in mathematics to develop flexible approaches to increasingly demanding problems; select appropriate strategies to use for numerical or algebraic problems
b) break down a complex calculation into simpler steps before attempting to solve it
c) use alternative approaches to overcome difficulties and evaluate the effectiveness of their strategies
d) select efficient techniques for numerical calculation and algebraic manipulation
e) make mental estimates of the answers to calculations; use checking procedures to monitor the accuracy of their results
Communicating
f) represent problems and solutions in algebraic or graphical forms; move from one form of representation to another to get different perspectives on the problem; present and interpret solutions in the context of the original problem
g) develop correct and consistent use of notation, symbols and diagrams when solving problems
h) examine critically, improve, then justify their choice of mathematical presentation; present a concise, reasoned argument
Reasoning
i) explore, identify, and use pattern and symmetry in algebraic contexts, investigating whether particular cases can be generalised further and understanding the importance of a counterexample; identify exceptional cases when solving problems; make conjectures and check them for new cases
j) show stepbystep deduction in solving a problem; explain and justify how they arrived at a conclusion
k) distinguish between a practical demonstration and a proof
l) recognise the importance of 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 numbers, 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 and negative square root (knowing that the square root sign denotes the positive square root), cube, cube root; use index notation for small integer powers and index laws for multiplication and division of positive integer powers
Fractions
c) use fraction notation; understand equivalent fractions, simplifying a fraction by cancelling all common factors; order fractions by rewriting them with a common denominator
Decimals
d)
use decimal notation and recognise that each terminating decimal is a fraction [ for example, 0.137 = 137 one-thousandths] ; order decimals
Percentages
e)
understand that 'percentage' means 'number of parts per 100' and use this to compare proportions; 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 and proportion
f) use ratio notation, including reduction to its simplest form and its various links to fraction notation
g) recognise where fractions or percentages are needed to compare proportions; identify problems that call for proportional reasoning, and choose the correct numbers to take as 100%, or as a whole.
Calculations
3) Pupils should be taught to:
Number operations and the relationships between them
a)
add, subtract, multiply and divide integers and then any number; 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 [for example, 8000 = 2 6 *5 3 ]
b) use brackets and the hierarchy of operations; know how to use the commutative, associative and distributive laws to do mental and written calculations more efficiently
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
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, 20% discount on ?150 gives a total calculated as ?(0.8 * 150)]
f)
divide a quantity in a given ratio [for example, share ?15 in the ratio 1:2]
Mental methods
g)
recall all positive integer complements to 100 [for example, 37 + 63 = 100] ; recall all multiplication facts to 10 * 10, and use them to derive quickly the corresponding division facts; recall the cubes of 2, 3, 4, 5 and 10, and the fractiontodecimal conversion of familiar simple fractions [for example, one-half, one-quarter, one-fifth, one-tenth, one-hundredth, one-third, two-thirds, one-eighth]
h) round to the nearest integer and to one significant figure; estimate answers to problems involving decimals
i)
develop a range of strategies for mental calculation; derive unknown facts from those they know [for example, estimate square root of85] ; add and subtract mentally numbers with up to two decimal places [for example, 13.76 5.21, 20.08 + 12.4]; multiply and divide numbers with no more than one decimal digit [for example, 14.3 * 4, 56.7 divided_by 7], using factorisation when possible
Written methods
j) use standard column procedures for addition and subtraction of integers and decimals
k)
use standard column procedures for multiplication of integers and decimals, understanding where to position the decimal point by considering what happens if they multiply equivalent fractions [for example, 0.6 * 0.7 = 0.42 since six-tenths * seven-tenths = 42 one-hundredths = 0.42] ; solve a problem involving division by a decimal by transforming it to a problem involving division by an integer
l) 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
m)
solve simple percentage problems, including increase and decrease [for example, simple interest, VAT, discounts, pay rises, annual rate of inflation, income tax, discounts]
n)
solve word problems about ratio and proportion, including using informal strategies and the unitary method of solution [for example, given that m identical items cost # y , then one item costs #_y_divided_by_m and n items cost #( n * x ), the number of items that can be bought for # z is z *m_divided_by_y ]
Calculator methods
o)
use calculators effectively and efficiently: know how to enter complex calculations using brackets [for example, for negative numbers, or the division of more than one term] , know how to enter a range of calculations, including those involving measures [for example, time calculations in which fractions of an hour need to be entered as fractions or decimals]
p) use the function keys for reciprocals, squares, square roots, powers, fractions (and how to enter a fraction as a decimal); use the constant key
q)
understand the calculator display, interpreting it correctly [for example, in money calculations, and when the display has been rounded by the calculator] , and knowing not to round during the intermediate steps of a calculation.
Solving numerical problems
4) Pupils should be taught to:
a) draw on their knowledge of the operations and the relationships between them, and of simple integer powers and their corresponding roots, to solve problems involving ratio and proportion, a range of measures and compound measures, metric units, and conversion between metric and common imperial units, set in a variety of contexts
b) select appropriate operations, methods and strategies to solve number problems, including trial and improvement where a more efficient method to find the solution is not obvious
c) use a variety of checking procedures, including working the problem backwards, and considering whether a result is of the right order of magnitude
d) give solutions in the context of the problem to an appropriate degree of accuracy, recognising 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, knowing that letter symbols represent definite unknown numbers in equations [for example, x 3 + 1 = 65] , defined quantities or variables in formulae [for example, V = IR], general, unspecified and independent numbers in identities [for example, 3 x + 2 x = 5 x , or 3( a + b ) = 3 a + 3 b , or ( x + 1)( x - 1) = x 2 - 1] and in functions they define new expressions or quantities by referring to known quantities [for example, y = 2 - 7 x ]
b)
understand that the transformation of algebraic expressions obeys and generalises the rules of arithmetic; simplify or transform algebraic expressions by collecting like terms [for example, x 2 + 3 x + 5 - 4 x + 2 x 2 = 3 x 2 - x + 5] , by multiplying a single term over a bracket, by taking out single term common factors [for example, x 2 + x = x ( x + 1)], and by expanding the product of two linear expressions including squaring a linear expression [for example, ( x + 1) 2 = x 2 + 2 x + 1, ( x - 3)( x + 2) = x 2 - x -6]; distinguish in meaning between the words 'equation', 'formula', 'identity' and 'expression'
Index notation
c) use index notation for simple integer powers, and simple instances of index laws; substitute positive and negative numbers into expressions such as 3 x 2 + 4 and 2 x 3
Equations
d)
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, 3(2 x +1) = 8, 2(1 - x ) = 6 (2 + x ), 4 x 2 = 36, 3 = z], by using inverse operations or by transforming both sides in the same way
Linear equations
e) solve linear equations, with integer 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
f)
use formulae from mathematics and other subjects [for example, formulae for the area of a triangle, the area enclosed by a circle, density = mass/volume] ; substitute numbers into a formula; derive a formula and change its subject [for example, convert temperatures between degrees Fahrenheit and degrees Celsius, find the perimeter of a rectangle given its area A and the length l of one side]
Direct proportion
g) set up and use equations to solve word and other problems involving direct proportion, and relate their algebraic solutions to graphical representations of the equations
Simultaneous linear equations
h) link a graphical representation of an equation to its algebraic solution; find an approximate solution of a pair of linear simultaneous equations by graphical methods, then find the exact solution by eliminating one variable; consider the graphs of cases that have no solution, or an infinite number of solutions
Inequalities
i) solve simple linear inequalities in one variable, and represent the solution set on a number line
Numerical methods
j)
use systematic trial and improvement methods with ICT tools to find approximate solutions of equations where there is no simple analytical method [for example, x 3 + x = 100] .
Sequences, functions and graphs
6) Pupils should be taught to:
Sequences
a) generate common integer sequences (including sequences of odd or even integers, squared integers, powers of 2, powers of 10, triangular numbers)
b)
find the first terms of a sequence given a rule arising naturally from a context [for example, the number of ways of paying in pence using only 1p and 2p coins, or from a regularly increasing spatial pattern] ; find the rule (and express it in words) for the n th term of a sequence
c) generate terms of a sequence using termtoterm and positiontoterm definitions of the sequence; use linear expressions to describe the n th term of an arithmetic sequence, justifying its form by referring to the activity or context from which it was generated
Functions
d) express simple functions, at first in words and then in symbols; explore the properties of simple polynomial functions
e)
use the conventions for coordinates in the plane; plot points in all four quadrants; recognise (when values are given for m and c ) that equations of the form y = mx + c correspond to straightline graphs in the coordinate plane; plot graphs of functions in which y is given explicitly in terms of x [for example, y = 2 x + 3] , or implicitly [for example, x + y = 7]
f)
construct linear functions arising from reallife problems and plot their corresponding graphs; discuss and interpret graphs arising from real situations [for example, distancetime graph for an object moving with constant speed]
g)
generate points and plot graphs of simple quadratic and cubic functions [for example, y = x 2 , y = 3 x 2 + 4, y = x 3 ]
Gradients
h)
find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c ); investigate the gradients of parallel lines and lines perpendicular to these lines [for example, knowing that y = 5 x and y = 5 x 4 represent parallel lines, each with gradient 5 and that the graph of any line perpendicular to these lines has gradient one-fifth] .
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