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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 1: Number (Extended)
Recommended prior knowledge
All of Core and particularly Core 1. Only those parts of the learning objectives or notes and exemplars not included in the core units are itemised, so this document
should be read alongside the core document.
Context
There is one Core number unit and this is the only Extended number unit. Once Core 1and the other prior experience for Core 1 is completed this unit can be slotted
in at any point. It is probably best taught as a whole but used to revise some of the Core unit 1.
Outline
The unit extends the knowledge of Core 1 be aware that examination questions that relate to aspects of Core 1 may have a greater degree of challenge as they
combine with other areas of mathematics. This unit covers a deeper knowledge of rational numbers, inverse percentage problems, percentiles, fractional exponents,
simplification of square and cube root expressions.
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
1.1
Knowledge of: natural
numbers, integers
(positive, negative,
and zero), prime
numbers, square
numbers, rational and
irrational numbers,
real numbers
Notes and exemplars
Understand that the sum or product of two rational numbers is rational; that
the sum of a rational number and an irrational number is irrational; and that
the product of a non-zero rational number and an irrational number is
irrational.
www.counton.org/resources/ks3framewor
k/pdfs/fractions.pdf page 65
CCSS:
N-RN3
Use of symbols: =, ≠,
≤,≥, <, >
http://nrich.maths.org/2756
http://nrich.maths.org/4717
General guidance
The easiest way to tackle the notes and exemplars part is to look at this in
terms of fractions. i.e. to turn rationals into fractions (definition of a rational)
and note what happens when you multiply by a number that cannot be a
fraction.
The case students may find least convincing is when rounded versions of Π
are discussed which are rational and when both irrational and recurring
decimals are written with ....... to show the pattern continues.
Past Paper 22 June 2011 Q2
(syllabus 0580)
The ‘counton’ website resource page 65 shows one way to do converting
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
recurring decimals to fractions, but also introduces an interesting error
problem to discuss.
Teaching activities
Devise a set of cards with statements that are true or false and ask students
to decide and then justify. If the ‘rationals’ and ‘irrationals’ are written as clues
it can also be both revision and challenging. e.g. three times the hypotenuse
of right angled isosceles triangle whose equal sides are 1 metre is rational.
1.6
Percentages,
including applications
such as interest and
profit
Notes and exemplars
Includes reverse percentages.
Includes percentiles.
www.counton.org/resources/ks3framewor
k/pdfs/fractions.pdf page 75 and 77
http://nrich.maths.org/1375
General guidance
The two ‘counton’ pages provide advice on method and some problems.
Students invariably muddle finding the selling price given the start price and
finding the start price and given the discount. Use of number lines will help to
resolve this when linked to the proportionality block of 4 discussed in Core 4
(1.4). This visualization of the difference might help some students.
0.7
Reducing £30 by 30%
0
RP
reduced price
Past Paper 42 June 2011 Q1b
(syllabus 0580)
£30
start price
RP
£30
70
100
RP = £21
0%
100% - discount
70%
100%
x 0.7
The start price was reduced by 30% to £28
0
£28
reduced price
SP
start price
x
10
7
£28
SP
SP = £40
70
0%
100%-discount
70%
100
100%
x
10
7
Teaching activities
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
Set up a problem where a local boutique buys in 50 pairs designer jeans for
$40 a pair. It sells them initially for $110, then it reduces by 5% then by a
further 20%. The final pairs are reduced again. They sell 10 pairs at full price
and must make 70% profit on the total deal to cover all their costs. Confirm it
is possible and suggest how many pairs to sell before they reduce each time,
the final price and the final discount. How would the numbers differ for a profit
of 50% over the whole deal. Present a report to the class on findings.
There is clearly more than one answer but in exploring the problem the
mathematics is explored.
You might want to make the figures more realistic for a local shop and to
change jeans to something that appeals more to the class at the time.
The figures in the ‘nrich’ resource will need changing.
1.7
CCSS:
N-RN1
N-RN2
Meaning and
calculation of
exponents (powers,
indices) including
fractional exponents
Notes and exemplars
1
e.g. 5 2 = 5
1
2
Evaluate 5–2, 100 , 8
-2
3
General guidance
To some extent this is just a coding change i.e. a power ‘a half’ and ‘square
root’ sign. Convince pupils using examples that can be shown to be true using
the index laws and then practice switching between the codes.
Past Paper 21 June 2011 Q4
(syllabus 0580)
Past Paper 22 June 2011 Q2
(syllabus 0580)
Past Paper 22 June 2011 Q4
(syllabus 0580)
Past Paper 22 June 2011 Q6
(syllabus 0580)
Past Paper 42 June 2011 Q1a
(syllabus 0580)
The usefulness is of course for simplifying using the index rules.
Evaluating the expressions requires practice and students need a little
experience to work out the order of working for a problem like 8
-2
3
Students need practice without calculators to understand the process, but
should also be able to use the calculator as well for non integer solutions.
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
1.8
Radicals, calculation
and simplification of
square root and
cube root
expressions
Notes and exemplars
www.mmlsoft.com/index.php?option=com
_content&task=view&id=9&Itemid=10
e.g., simplify 200 + 18
Write (2 + 3 )2 in the form a + b 3
General guidance
Students need to use their understanding of factors, squares and cubes to
work with this topic effectively. So first remind students of these
www.mmlsoft.com/index.php?option=com
_content&task=view&id=11&Itemid=12
Past Paper 22 June 2011 Q2
(syllabus 0580)
Teaching activities
Ultimately these types of problem require practice and so use the ‘formulator
tarsia’ software to set up a hexagon or domino puzzle to make this practice
more interesting, by matching forms of the same expression.
Revisit frequently as a starter asking students to complete a few examples.
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