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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 4 (foundation) pupils consolidate their understanding of basic mathematics, which will help them to tackle unfamiliar problems in the workplace and everyday life and develop the knowledge and skills they need in the future. They become more fluent in making connections between different areas of mathematics and its application in the world around them. They become increasingly proficient in calculating fractions, percentages and decimals, and use proportional reasoning in simple contexts. Building on their understanding of numbers, they make generalisations using letters, manipulate simple algebraic expressions and apply basic algebraic techniques to solve problems. They extend their use of mathematical vocabulary to talk about numbers and geometrical objects. They begin to understand and follow a short proof, and use geometrical properties to find missing angles and lengths, explaining their reasoning with increasing confidence. They collect data, learn statistical techniques to analyse data and use ICT to present and interpret the results. Note
>This programme of study is intended for those pupils who have not 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 suitable problemsolving strategies and efficient techniques to solve numerical and algebraic problems
b) break down a complex calculation into simpler steps before attempting to solve it
c) use algebra to formulate and solve a simple problem identifying the variable, setting up an equation, solving the equation and interpreting the solution in the context of the problem
d) make mental estimates of the answers to calculations; use checking procedures, including use of inverse operations; work to stated levels of accuracy
Communicating
e) interpret and discuss numerical and algebraic information presented in a variety of forms
f) use notation and symbols correctly and consistently within a given problem
g) use a range of strategies to create numerical, 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
h) present and interpret solutions in the context of the original problem
i) review and justify their choice of mathematical presentation
Reasoning
j)
explore, identify, and use pattern and symmetry in algebraic contexts [ for example, using simple codes that substitute numbers for letters] , investigating whether particular cases can be generalised further, and understanding the importance of a counterexample; identify exceptional cases when solving problems
k) show stepbystep deduction in solving a problem
l) distinguish between a practical demonstration and a proof
m) 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 positive numbers, both as positions and translations on a number line; order integers; use the concepts and vocabulary of factor (divisor), multiple and common factor
Powers and roots
b) use the terms square, positive square root, cube; use index notation for squares, cubes and powers of 10; express standard index form both 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)
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] ; use percentage in reallife situations [for example, commerce and business, including rate of inflation, VAT and interest rates]
Ratio
f)
use ratio notation, including reduction to its simplest form and its various links to fraction notation [for example, in maps and scale drawings, paper sizes and gears] .
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
b) use brackets and the hierarchy of operations
c)
calculate a given fraction of a given quantity [for example, for scale drawings and construction of models, down payments, discounts] , 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 fraction by an integer, and multiply a fraction by a unit fraction
e)
convert simple fractions of a whole to percentages of the whole and vice versa [for example, analysing diets, budgets or the costs of running, maintaining and owning a car] , then understand the multiplicative nature of percentages as operators [for example, 30% increase on ?150 gives a total calculated as ?(1.3 * 150) while a 20% discount gives a total calculated as ?(0.8 * 150)]
f)
divide a quantity in a given ratio [for example, share ?15 in the ratio of 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 the commutative, associative, and distributive laws and factorisation where possible, or place value adjustments
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; solve a problem involving division by a decimal (up to two places of decimals) 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, VAT, 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 * y_divided_by_m), 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 and use function keys for reciprocals, squares and powers
p)
enter a range of calculations, including those involving standard index form and measures [for example, time calculations in which fractions of an hour must be entered as fractions or as decimals]
q)
understand the calculator display, interpreting it correctly [for example, in money calculations, or 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, interpreting the solution shown on a calculator display, and 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, 5 x + 1 = 16] , 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 , ( x + 1) 2 = x 2 + 2 x + 1 for all values of x ] and in functions they define new expressions or quantities by referring to known quantities [for example, y = 2 x ]
b)
understand that the transformation of algebraic expressions obeys and generalises the rules of arithmetic; manipulate algebraic expressions by collecting like terms, by multiplying a single term over a bracket, and by taking out single term common factors [for example, x + 5 2 x 1 = 4 x ; 5(2 x + 3) = 10 x +15; x 2 + 3 x = x ( x + 3)] ; 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
Inequalities
d) solve simple linear inequalities in one variable, and represent the solution set on the number line
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 expressed initially in words and then using letters and symbols [for example, formulae for the area of a triangle, the area enclosed by a circle, wage earned = hours worked * rate per hour] ; 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, use V = IR to generate a formula for R in terms of V and I].
Sequences, functions and graphs
6) Pupils should be taught to:
Sequences
a) 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
Graphs of linear functions
b)
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]
c) construct linear functions from reallife problems and plot their corresponding graphs; discuss and interpret graphs arising from real situations; understand that the point of intersection of two different lines in the same two variables that simultaneously describe a real situation is the solution to the simultaneous equations represented by the lines; draw line of best fit through a set of linearly related points and find its equation
Gradients
d) 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
Interpret graphical information
e)
interpret information presented in a range of linear and nonlinear graphs [for example, graphs describing trends, conversion graphs, distancetime graphs, graphs of height or weight against age, graphs of quantities that vary against time, such as employment] .
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