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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 1: Number (Core)
Recommended prior knowledge
Understanding of:
• Integer names of numbers to 106
• Prime numbers under 20 and square numbers under 50
• Multiples and factors of numbers under 30
• Use of four operations with parentheses for integers, and simple decimals and fractions
• Simple fractional parts of a number
• Simple ratios and the ability to scale a recipe
• Simple integer powers in there expanded and index form
• Mental percentage calculations for 50%, 10%, 25% and 1%
• Rounding to nearest unit, ten, hundred and thousand and one decimal place
• Reading times from analogue clocks and converting between am / pm and 24 hour clock
Context
The skills of number underpin algebra and are required for problem solving and in all other strands of mathematics, therefore this unit should come first but may be
broken down and scattered at the beginning of other units. The links are outlined in the other units. However, subsets of work can be created, so block 1 could be
1.1 and 1.3 taught simultaneously to create connections with 1.2 and 1.8 and then be linked to the Unit 2 Algebra. Block 2 could be 1.5, 1.7, 1.4, 1.6, and 1.9 and
would then link to algebra, data handling and trigonometry and enlargement. The notion of a multiplier, found from the modelling of the block of 4 being the link into
all of these. Block 3, 1.10 and 1.11, could be taught separately. Students who are following the extended syllabus will move through this faster but need to have all
these skills in place before working on the extended units, or applying them in other areas of mathematics.
Outline
Most items in this unit should have been met with at various levels of skill development in the past. Within the suggested teaching activities ideas are listed to identify
and remediate misconceptions and to pull learning through to the required standard. It covers an understanding of the real number system, and the symbols for
comparing values, multiples, factors, primes, use of four operations and parentheses, square and square root, fractions, decimals and percentage, exponents of
numbers, standard index form, ratio and proportion, simple and compound interest, scales, estimating and rounding, time and speed distance time problems. The
learning resources give both teaching ideas, summaries of the skills and investigative problems to develop the problem solving skills and a depth of understanding of
the mathematics, through exploration and discussion.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
1
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
General guidance
Check that pupils can name numbers correctly by
changing numbers in words to digits and vice versa
and can write down numbers correctly from their
spoken names.
http://mathmojo.com/interestinglessons/names%20
of%20large%20numbers/names_of_large_numbers
.html
Use of symbols: =, ≠, ≤,≥, <, >
Teaching activities
• A card match resource with spare similar but
incorrect versions
• Researching large and small names and
realising that they are not universal
• Work with number lines and naming the 10
divisions between and a few on either side of
two numbers like 3.4 and 3.5, or 2000 and
3000
General guidance
Ensure that pupils realise 1 is not a prime. The
definition that a prime has only two factors show 1 is
not prime.
Find products of primes through tree diagrams,
expressing the product using powers where
necessary. Page 53 of the document at the head of
this unit.
Past Paper 11 June 2011 Q2a
(syllabus 0580)
www.vex.net/~trebla/numbertheory/eratosthenes.ht
ml
www.bbc.co.uk/schools/gcsebitesize/maths/number
/primefactorsrev1.shtml
www.counton.org/resources/ks3framework/pdfs/pla
ce_value.pdf (page 53)
Teaching activities
Completing the net of Erosthenes (The applet allows
you to change the range of numbers for finding the
primes and can be used to discuss the maximum
number to eliminate.
v1 2Y01
Give definitions for rational, irrational and real
numbers.
http://yourschoolmaster.com/mathematics/mentalpr
oblems/mental_oral_starter1.pdf
Teaching activities
• Students sort a set of numbers under those
headings.
• Create a Target Board (eg a number in each
http://nrich.maths.org/1404
Cambridge IGCSE Mathematics (US) 0444
Past Paper 31 June 2011 Q6
(syllabus 0580)
2
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
hole of a 5 x 4 grid) with numbers that are
square, prime, rational and irrational, positive,
negative and ask them to identify those for a
particular heading on whiteboards. The link is
one that explains the resource although the
particular one is too simple for use here. Try
these numbers, 45, 49, √7, 0.569, 47, ¾, √81,
π, 100, 93, 25, 5, 9, 72, 0.09, -7, 1, 400, 4000,
106
General guidance
Especially work with the positive and negative
numbers in relation to ordering them and noting the
reflective nature either side of zero. Link to 1.5 and
equivalences when covered.
Past Paper 13 June 2011 Q5
(syllabus 0580)
Teaching activities
Set up true and false statements using these: =, ≠, ≤,≥,
<, > between numbers in different forms and ask pupils
to sort them under true and false or ask them to
correct the false statements.
1.3
Multiples and factors, including,
greatest common factor, least
common multiple
Notes and exemplars
GCF and LCM will be used and knowledge of prime
factors is assumed.
General guidance
Check understanding of finding all factors of a number
by checking whether each integer divided into the
number until the quotient is less than the divisor.
Ensure multiple and factor are distinguished.
www.bbc.co.uk/schools/gcsebitesize/maths/number
/factorsmultiplesrev1.shtml
www.counton.org/resources/ks3framework/pdfs/pla
ce_value.pdf (page 55)
http://nrich.maths.org/5468
Past Paper 12 June 2011 Q14
(syllabus 0580)
Teaching activities
• Use target boards with numbers like 12, 15,
75, 5, 66,1, 22, 25, 4, 7, 13, 50, 9, 10, 33, 111,
8, 11, 14, 100, Asking questions like list the
multiples of 25 and the factors of 25 to check
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
3
Syllabus ref
Learning objectives
Suggested teaching activities
•
1.2
Use of the four operations and
parentheses
Learning resources
whether there is confusion.
Find all the factors of numbers less than 30;
note that the Primes only have 1 and the
number as factors and that square numbers
are the only ones with an odd number of
factors.
Notes and exemplars
Applies to integers, fractions, and decimals.
www.counton.org/resources/numeracy/pdfs/y456str
2.pdf
General guidance
Check pupils are able to add, subtract, multiply and
divided, integers, decimals and fractions. Treating
integers and decimals as the same and sliding across
the place value system (do not move the decimal
point) as the link. The two national strategy documents
may be lengthy but they have many teaching ideas as
well as detailed developmental steps.
Work with problems that involve deciding which of four
operations is required.
www.counton.org/resources/ks3framework/pdfs/nu
mber_operations.pdf
www.teachfind.com/nationalstrategies/mathematics-itp-fractions
Teaching activities
For an introduction to addition and subtraction of
fractions use the flash ITP and the word document to
explain how to use it. Use two fraction bars with the
two fractions to be added/subtracted. Two more for
bars that equivalent fractions with the same
denominators and a fifth for the combined answer.
Shading a fraction of a shaded area justifies the
multiplication
The multiplier method in ratio and proportion if worked
with integers and reversed justifies inverting the
fraction for division.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
4
Syllabus ref
Learning objectives
Suggested teaching activities
multiplier to get from 3 to 5
x
5
Learning resources
multiplier to get from 5 to 3
5
x
3
3
5
3
3
x5
÷3
÷5
x3
1
1
but to undo a multiplication you divide by the same number
so ÷
5
3
must be equivalent to x
3
5
Students must know that division can be represented
as a fraction.
1.8
Radicals, calculation of square
root and cube root expressions
General guidance
Pupils need to know the order of operations
(BODMAS) and also to know what order of operations
their calculator will follow if they put things in, in the
sequence written.
http://nrich.maths.org/1013
Teaching activities
Ask them to use 5 numbers (include at least one
negative to practice working with negatives) and ask
them to find as many different answers using all four
operations and brackets. Discuss the outcomes –
initially do not allow repeats of the numbers.
The nrich tasks work with some or all over the
operations and powers.
http://nrich.maths.org/931
Notes and exemplars
e.g. the area of a square is 54.76 cm2.
Work out the length of one side of the square.
Find the value of the cube root of 64.
http://nrich.maths.org/2194
http://nrich.maths.org/769
http://nrich.maths.org/6368
General guidance
Ensure that students understand that 92 is 81 so the
√81 is 9.
Teaching activities
Ask students to guess the √67 to 1 decimal place – get
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
5
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
them to check guesses on a calculator after they have
realised it must be between 8 and 9 and closer to 8 so
trying 8.1 or 8.2 or 8.3.
Students need to know how to find square and cube
roots on a scientific calculator
1.5
Language and notation of
fractions, decimals, and
percentages; recognize
equivalences between decimals,
fractions, ratios, and percentages
and convert between them
Order quantities given in different
forms by magnitude, by first
converting into same form
General guidance
Use number lines to show equivalent forms.
Converting decimals back to fractions should be linked
to naming the decimal as a rational number, including
converting recurring decimals.
Percentages need to be understood as a fraction out
of 100 and because 1/100 is 0.001 so 1% is 0.001.
http://teachfind.com/nationalstrategies/mathematics-interactive-teachingprogram-itp-fractions-0
Teaching activities
• This can be modelled on 10x10 grid of
squares if the link is not known.
Shade a variety of fractions on the 100 square
and note the link between 10 columns or
individual squares and the tenths and
hundredths of the decimal equivalent.
• Conversions can be practiced with sets of
cards with the mixed forms to be matched.
http://nrich.maths.org/1249
Finding Percentages, fractions and decimals of an
amount.
http://teachfind.com/national-strategies/using-ictmathematics-fractions-decimal-percentage-ratioand-proportion
http://nrich.maths.org/1283
http://nrich.maths.org/2086
http://nrich.maths.org/5467
Past Paper 12 June 2011 Q6
(syllabus 0580)
Past Paper 31 June 2011 Q1
(syllabus 0580)
Teaching activities
• practiced on spider diagrams. i.e an amount of
money is in the centre various percentages are
around the outside. Students find these
percentages. This can also be linked to an
activity with one percentage of an amount fact
in the middle of a spider diagram and students
write around the outside other acts that must
be true because the central one is true.
• If practice is required to convert fractions to
decimals, get students to find 1/7, 2/7, 3/7 and
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
6
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
then to see if they can see what is happening
to the digits and to predict the other decimal
equivalences of 4, 5 and 6 sevenths. 13ths
and 17ths work similarly but with more than
one pattern.
1.7
CCSS:
N-RN1
N-RN2
Meaning and calculation of
exponents (powers, indices)
including positive, negative, and
zero exponents.
Rules for exponents
Notes and exemplars
e.g. work out 4–3 as a fraction
e.g. work out 24 × 2–3
Convert numbers in and out of scientific notation.
Calculate with values in scientific notation.
Scientific notation (Standard
Form) a × 10n where 1 ≤ a < 10
and n is an integer
General guidance
To prove meanings first develop the rules for
exponents and then set up examples by working
through statements like
34 = 3 x 3 x 3 x 3 = 81 (check students can multiply a
chain of numbers correctly and don’t simplify it to 4 lots
of 3).
And 34 x 33 = (3 x 3 x 3 x 3) x (3 x 3 x 3) = 37 =....
Demonstrate a number and show the additive rule.
Show division as cancelling in a fraction.
2x2x2x2x2
e.g. 25÷23 =
= 2 x 2 = 22 =.....
2x2x2
Show a number of examples to develop the
subtracting rule.
Then show 23÷25 = 2-2 and show this also as
2x2x2
1
1
=
= 2.
2x2x2x2x2 2x2 2
Similarly expand 52÷52 to show that 50 = 1.
Explain what standard form, convert numbers in and
out of standard form is and look at problems that
involve standard form.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
www.bbc.co.uk/schools/gcsebitesize/maths/number
/powersrootsrev1.shtml
http://nrich.maths.org/6448
Past Paper 12 June 2011 Q6
(syllabus 0580)
7
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
1.4
Ratio and proportion
General guidance
It is important to teach the links between ratio and
proportion and to teach the links between fractions
decimals and ratios.
The thinking blocks website explains the use of blocks
to model ratio questions and has a bank of increasing
difficulty questions. The Multiplicative and proportional
reasoning units, take the blocks into that higher level
thinking.
www.counton.org/resources/ks3framework/pdfs/frac
tions.pdf
Any proportional reasoning question can be displayed
on two sides on a number line i.e. in the typical
example of 3m cost $4 what does 7.5m cost. Lengths
can be one side of the number line and the money the
other. Then the parallel bar model shows one of the
multipliers. The problem is arranged as a block of 4
from the number line and the multiplier used to solve
the problem. Multipliers can also be found for the
vertical links in the block of four.
http://teachfind.com/national-strategies/interactingmathematics-year-8-multiplicative-relationshipsmini-pack
x
3m
7.5m
x
3
3
x7.5
1.6
v1 2Y01
Percentages, including
applications such as interest and
profit
1
7.5
3
3
7.5
4
?
x
?
http://teachfind.com/national-strategies/interactingmathematics-key-stage-3-year-9-proportionalreasoning-mini-pack
7.5
7.5
÷3
$4
www.thinkingblocks.com/ThinkingBlocks_Ratios/TB
_Ratio_Main.html
7.5
3
so
4 x 7.5
3
=10
Notes and exemplars
Excludes reverse percentages.
Includes both simple and compound interest.
Cambridge IGCSE Mathematics (US) 0444
www.counton.org/resources/ks3framework/pdfs/frac
tions.pdf
8
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
General guidance
Teach both
1. Finding the percentage increase (or decrease)
and to add (or subtract) it from the 100% value
2. Adjusting the percentage before find the % of
the amount
i.e. to add 16% you either find 16% of the value and
add to the value or you find 116% of the value.
The difference between simple and compound interest
can be modelled.
Teaching activities
Working with compound interest and simple interest for
each year. An interesting question might be to find the
simple interest required to be an equivalent value to a
particular compound interest over say five years.
1.9
CCSS:
N-Q1
N-Q2
N-Q3
Quantities – choose and interpret
units and scales, define
appropriate quantities (including
money)
Notes and exemplars
Also relates to graphs and geometrical measurement
topics.
Includes converting between units, e.g. different
currencies.
Past Paper 31 June 2011 Q1
(syllabus 0580)
Teaching activities
Set up currency conversion graphs by checking $100
in online currency calculators. Plot (0,0) and (100,?)
and read values off the graph. Convert $150 by adding
$50 value to $100 value etc. Use the proportionality
block of four and multiplier of section 1.4, treat scaling
problems the same way.
Estimating, rounding, decimal
places, and significant figures –
choose a level of accuracy
appropriate for a problem
v1 2Y01
General guidance
Link all types of rounding to a simple model. i.e. draw a
blank number line, have the number it would cut to at
one end and the rounding up number the other and
mark the midpoint. Estimate where the actual number
is and decide whether it is closer to one end or the
Cambridge IGCSE Mathematics (US) 0444
Past Paper 12 June 2011 Q4
(syllabus 0580)
9
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
other to decide which it is.
Deal with the special case of a five ending.
So to round 3.457 to 2 d.p. the two ends would be 3.45
and 3.46 and the midpoint 3.455 so it is nearer 3.46 to
2 d.p. test understanding with the difficult cases like
1.999 or 3.45678 rounded to 2 d.p or 3.45678.
Students need to understand which the most
significant figure is in a number, how to maintain that
figure’s place value under rounding and to see that
significant figure rounding follows the same principle
as any other type of rounding. Test with rounding a
number like 0.03456 to 2d.p and 2s.f. to check the two
types of rounding are understood and also check
numbers like 345678 can also be rounded to 3s.f.
Estimating should be linked to rounding to 1s.f. and
then working with the rounded figures in most cases.
Teaching activities
Work with examples that have been rounded and ask
students to give the range of possible answers noting
when to us ≤ or <
1.10
v1 2Y01
Calculations involving time:
seconds (s), minutes (min), hours
(h), days, months, years including
the relation between consecutive
units
Notes and exemplars
1 year = 365 days.
Includes familiarity with both 24-hour and
12-hour clocks and extraction of data from dials and
schedules.
www.bbc.co.uk/skillswise/worksheet/ma01line-l1-wtime-calculations
General guidance
Demonstrate that because time does not deal with
base ten calculators can only be used if the units are
converted to be the same. Also that if a time works out
as 6.7hrs this is not 6hrs 7 or 70mins (common
misconceptions) but 6 hours and 0.7 x 60mins i.e. 6hrs
42mins.
A variety of calculations should be set up to practice
this.
Past Paper 12 June 2011 Q11
(syllabus 0580)
Cambridge IGCSE Mathematics (US) 0444
www.bbc.co.uk/schools/ks3bitesize/maths/measure
s/time/revise1.shtml
10
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
Time difference on number lines can be an effective
model to support a calculation.
Ensure students are shown both am/pm and 24hour
clock times and know how to move between them.
Teaching activities
Work with time tables and TV or radio schedules to
work out lengths of time for journeys, or the total time a
network shows a set of programmes etc.
1.11
Speed, distance, time problems
General guidance
The websites show the triangle model for solving these
problems. This works for some students but does not
lead to understanding. Use the fact that mph has ‘per’
in it and means division to establish one link and work
with rearranging formulae to give understanding.
www.bbc.co.uk/scotland/learning/bitesize/standard/
maths_i/numbers/dst_rev1.shtml
www.speeddistancetime.info/test.php
Look at problems that require changes of units both of
length and time to solve them.
Look at problems that require interpretation of distance
time graphs to gather information for the solution of the
problem.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
11