Scheme of work – Cambridge IGCSE® [Subject] ([syllabus code])
Scheme of work
Cambridge IGCSE®
Mathematics (US)
0444
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Contents
Contents ..................................................................................................................................................................................................................................................... 2
Overview (Core curriculum and Extended curriculum) ......................................................................................................................................................................... 4
Unit 1: Number – Core curriculum ........................................................................................................................................................................................................... 7
Unit 1: Number – Extended curriculum ................................................................................................................................................................................................. 22
Unit 2: Algebra – Core curriculum ......................................................................................................................................................................................................... 26
Unit 2: Algebra – Extended curriculum ................................................................................................................................................................................................. 35
Unit 3: Functions – Core curriculum ..................................................................................................................................................................................................... 44
Unit 3: Functions – Extended curriculum ............................................................................................................................................................................................. 50
Unit 4: Geometry – Core curriculum ...................................................................................................................................................................................................... 57
Unit 4: Geometry – Extended curriculum .............................................................................................................................................................................................. 64
Unit 5: Transformations and vectors – Core curriculum ..................................................................................................................................................................... 69
Unit 5: Transformations and vectors – Extended curriculum ............................................................................................................................................................. 71
Unit 6: Geometrical measurement – Core curriculum ......................................................................................................................................................................... 76
Unit 6: Geometrical measurement – Extended curriculum ................................................................................................................................................................. 81
Unit 7: Co-ordinate geometry – Core curriculum ................................................................................................................................................................................. 85
Unit 7: Co-ordinate geometry – Extended curriculum ......................................................................................................................................................................... 90
Unit 8: Trigonometry – Core curriculum ............................................................................................................................................................................................... 93
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Unit 8: Trigonometry – Extended curriculum ....................................................................................................................................................................................... 98
Unit 9: Probability – Core curriculum .................................................................................................................................................................................................. 102
Unit 9: Probability – Extended curriculum .......................................................................................................................................................................................... 106
Unit 10: Statistics – Core curriculum................................................................................................................................................................................................... 108
Unit 10: Statistics – Extended curriculum........................................................................................................................................................................................... 113
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Overview (Core curriculum and Extended curriculum)
This scheme of work provides ideas about how to construct and deliver a course. The 2013 syllabus for 0444 has been broken down into ten teaching units with
suggested teaching activities and learning resources to use in the classroom.
Recommended prior knowledge
It is recommended that candidates have followed the curriculum framework for Cambridge Secondary 1 Mathematics
www.cie.org.uk/qualifications/academic/lowersec/cambridgesecondary1/resources or have followed similar courses which cover these learning objectives.
Outline
There are two levels of achievement in Cambridge IGCSE Mathematics, via two separate routes: Core and Extended. In this scheme of work we have differentiated
between what is covered in the Core curriculum and what is covered in the Extended curriculum. There are two separate schemes of work laid out in the following
units:
Core curriculum scheme of work:
Core Units 1–10
Extended curriculum scheme of work: Extended Units 1–10
All the material in the Core Curriculum is covered within the Extended scheme of work. The order of topic coverage is similar in the two schemes, so that it should be
possible to deliver both in parallel if required (for example where a single class contains Core and Extended learners).
The units within the Core and Extended curriculum scheme of work are:
Unit 1: Number
Unit 2: Algebra
Unit 3: Functions
Unit 4: Geometry
Unit 5: Transformations and vectors
Unit 6: Geometrical measurement
Unit 7: Co-ordinate geometry
Unit 8: Trigonometry
Unit 9: Probability
Unit 10: Statistics
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Core units should be read before the related Extended units, and units should be read sequentially as written, i.e. Units 1 to 10. The main focus of these units is on
preparing learners for Cambridge IGCSE Mathematics Core or Extended. The mathematics is a universal component of many mathematics courses although there
are some variations in names and symbols. However, some of the activities have a generic application which will also be of use to teachers when preparing learners
for cross-curricular activities that require mathematics e.g. algebra and functions and science courses.
Common Core State Standards (CCSS)
In each unit the relevant standards are indicated in bold blue lettering (CCSS), in the ‘Syllabus ref’ column. This allows teachers to identify how standards are met in
particular activities.
Teacher support
Syllabus 0444 specimen papers and mark schemes are available for teachers on Teacher Support at http://teachers.cie.org.uk. Past papers and other support
material for Cambridge IGCSE Mathematics syllabus 0580 are referred to in this scheme of work and may be another useful resource. These can also be found on
Teacher Support. Syllabus 0580 examination papers have a different structure from the 0444 syllabus but the content of the questions is similar. We also offer online
and face-to-face training; details of forthcoming training opportunities are posted on the website.
Resources
The up-to-date resource list for the Cambridge IGCSE Mathematics (syllabus 0580) can be found at www.cie.org.uk The resource list for syllabus 0580 includes
textbooks which have been endorsed by Cambridge International Examinations. 'Endorsed by Cambridge' resources have been written to be closely aligned to the
syllabus they support, and have been through a detailed quality assurance process.
Websites:
The particular pages in the ‘Learning resources’ column for the units have been explored but not other aspects of these websites, so only the particular resources
are recommended. There may be other useful materials on these websites but they have not been checked.
Core and Extended curriculum units:
http://teachfind.com/national-strategies/mathematics
www.counton.org/resources/ks3framework/
http://mathmojo.com/interestinglessons/
www.vex.net/~trebla/numbertheory/eratosthenes.html
www.bbc.co.uk/schools/gcsebitesize/maths/
http://yourschoolmaster.com/mathematics/mentalproblems/mental_oral_starter1.pdf
http://nrich.maths.org/public/
www.bbc.co.uk/skillswise/maths
www.bbc.co.uk/schools/ks3bitesize/maths/
www.bbc.co.uk/scotland/learning/bitesize/standard/maths_i/
www.speeddistancetime.info/
www.thinkingblocks.com/ThinkingBlocks_Ratios/TB_Ratio_Main.html
http://people.umass.edu/~clement/pdf/Intuitive%20Misconceptions%20in%20Algebra.pdf
www.springerlink.com/content/557502366l86518p/
www.algebralab.org/lessons/
www.mmlsoft.com/index.php?option=com_content&task=view&id=9&Itemid=10
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www.purplemath.com/
www.royalmail.com/
www.excellencegateway.org.uk/
www.mathleague.com/help/geometry/geometry.htm
www.youtube.com/watch?v=bK53Wn4Jdpc
www.gcsemathstutor.com/
www.mathsnet.net/
http://projects.exeter.ac.uk/csm-survey/files/CSM10_Intro_to_trigonometry.pdf
www.mathplayground.com/
www.mathsisfun.com/
http://rds.censusatschool.org.uk/
www.cimt.plymouth.ac.uk/projects/mepres/allgcse/bkb8.pdf
Extended curriculum units only:
www.algebra-class.com/
http://mash.dept.shef.ac.uk/RearrangingFormulae.html
www.wdeptford.k12.nj.us/high_school/Fish/Honors%20Alg%20worksheets/Direct%20and%20Inverse%20Variation%20Worksheet.pdf
www.gcseguide.co.uk
www.kutasoftware.com/FreeWorksheets/
www.timdevereux.co.uk/maths/geompages/index.html
www.maths4scotland.co.uk
www.mathwarehouse.com/classroom/worksheets/
www.haeseandharris.com.au/samples/igcse_20.pdf
www.korthalsaltes.com/
www.ltscotland.org.uk/Images/pythagoras3d_tcm4-123382.ppt
http://rpmullen.com/standards/geometry/oncore/geounit8_3.PDF
www.mathsteaching.wordpress.com
www.shodor.org/interactivate/lessons
® IGCSE is the registered trademark of Cambridge International Examinations.
© Cambridge International Examinations 2012
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 1: Number – Core curriculum
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 Core 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. Learners 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.
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Syllabus ref
and CCSS
1.1
Learning objectives
Suggested teaching activities
Learning resources
Knowledge of: natural numbers,
integers (positive, negative, and
zero), prime numbers, square
numbers, rational and irrational
numbers, real numbers
General guidance
Check that learners can name numbers correctly by
changing numbers in words to digits and vice versa
and can write down numbers correctly from their
spoken names.
Names of large numbers listed:
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 learners 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. See ‘Counton’ page 53.
Teaching activities
Completing the net of Eratosthenes (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.
Give definitions for rational, irrational and real
numbers.
Teaching activities
 Learners sort a set of numbers under those
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Cambridge IGCSE Mathematics (US) 0444
Past Paper 11 June 2011 Q2a
(syllabus 0580)
Prime factors:
www.bbc.co.uk/schools/gcsebitesize/maths/number
/primefactorsrev1.shtml
‘Counton’ page 53:
www.counton.org/resources/ks3framework/pdfs/pla
ce_value.pdf
Eratosthenes:
www.vex.net/~trebla/numbertheory/eratosthenes.ht
ml
Target boards:
http://yourschoolmaster.com/mathematics/mentalpr
oblems/mental_oral_starter1.pdf
Proof sorter:
8
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities

headings.
Create a ‘Target Board’ (e.g. a number in each
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.
Learning resources
http://nrich.maths.org/1404
Past Paper 31 June 2011 Q6
(syllabus 0580)
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
learners to sort them under true and false or ask them
to correct the false statements.
1.3
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Multiples and factors, including,
greatest common factor, least
common multiple
Notes and examples
GCF and LCM will be used and knowledge of prime
factors is assumed.
Factors and multiples:
www.bbc.co.uk/schools/gcsebitesize/maths/number
/factorsmultiplesrev1.shtml
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.
‘Counton’ page 55:
www.counton.org/resources/ks3framework/pdfs/pla
ce_value.pdf
Cambridge IGCSE Mathematics (US) 0444
Factors and multiples game:
http://nrich.maths.org/5468
9
Syllabus ref
and CCSS
1.2
Learning objectives
Use of the four operations and
parentheses
Suggested teaching activities
Learning resources
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
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.
Past Paper 12 June 2011 Q14
(syllabus 0580)
Notes and examples
Applies to integers, fractions, and decimals.
General guidance
Check learners 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 (see link in Learning resources column)
may be lengthy but they have many teaching ideas as
well as detailed developmental steps.
‘Counton’ page 82:
www.counton.org/resources/ks3framework/pdfs/nu
mber_operations.pdf
The National Strategies:
www.teachfind.com/nationalstrategies/mathematics-itp-fractions
Work with problems that involve deciding which of four
operations is required.
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.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
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.
multiplier to get from 3 to 5
x
5
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
Learners must know that division can be represented
as a fraction.
General guidance
Learners 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.
‘nrich’ tasks:
http://nrich.maths.org/1013
http://nrich.maths.org/769
http://nrich.maths.org/6368
Teaching activities
Ask them to use five 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.
http://nrich.maths.org/931
The nrich tasks (see links in Learning resources
column) work with some or all over the operations and
powers.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Notes and examples
Choose mental or written methods appropriate to the
number or the context.
General guidance
To some extent the choice is dictated by the skill of the
learner. The following are skills need to be acquired
over time.
1. Understanding doubling and halving
strategies.
2. Knowing that repeated doubling halving leads
to factors of 4, 8 etc. and that multiplying by 5
is the same as multiplying by 10 and halving
etc.
3. Develop mental skills for identifying equivalent
integer calculations. E.g. knowing that 74 x 28
is the same as 148 x 14 and 296 x 7
4. Understanding that 7.4 x 2.8 is the same as 74
x 28 ÷ 100
5. Knowing 4 x 25 is a hundred to solve problems
involving 25
6. Looking for common factors in questions even
when there are decimals and fractions
involved.
7. Be able to work mentally with simple decimals
and fractions
Teaching activities
Have a range of problems on cards and ask learners
to work out different ways of finding the solution.
Compare and discuss the methods as a class and look
for shortcuts that minimise the written methods or even
turn it into a mental calculation or a calculation with
jottings.
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Syllabus ref
and CCSS
1.8
Learning objectives
Suggested teaching activities
Learning resources
Radicals, calculation of square
root and cube root expressions
Notes and examples
e.g. the area of a square is 54.76 cm 2.
Work out the length of one side of the square.
‘nrich’ task:
http://nrich.maths.org/2194
Find the value of the cube root of 64.
General guidance
Ensure that learners understand that 92 is 81 so the
81 is 9.
Teaching activities
Ask learners to guess the 67 to 1 decimal place – get
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.
Learners 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
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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.
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
Cambridge IGCSE Mathematics (US) 0444
Fractions:
http://teachfind.com/nationalstrategies/mathematics-interactive-teachingprogram-itp-fractions-0
Fractions, decimal, percentage, ratio and
proportion:
http://teachfind.com/national-strategies/using-ictmathematics-fractions-decimal-percentage-ratioand-proportion
‘nrich’ tasks:
http://nrich.maths.org/1249
http://nrich.maths.org/1283
http://nrich.maths.org/2086
13
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
cards with the mixed forms to be matched.
http://nrich.maths.org/5467
Finding percentages, fractions and decimals of an
amount.
Teaching activities


1.7
CCSS:
N-RN1
N-RN2
Meaning and calculation of
exponents (powers, indices)
including positive, negative, and
zero exponents.
Notes and examples
e.g. work out 4–3 as a fraction
Explain the rules for exponents
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
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Practised on ‘spider diagrams’
(www.learningtolearn.group.shef.ac.uk/.html)
i.e. an amount of money is in the centre
various percentages are around the outside.
Learners 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 learners 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 learners to find 1/7, 2/7, 3/7 and
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.
Past Paper 12 June 2011 Q6
(syllabus 0580)
Past Paper 31 June 2011 Q1
(syllabus 0580)
Powers and roots:
www.bbc.co.uk/schools/gcsebitesize/maths/number
/powersrootsrev1.shtml
e.g. work out 24 × 2–3
‘nrich’ task:
http://nrich.maths.org/6448
General guidance
To prove meanings first develop the rules for
exponents and then set up examples by working
Cambridge IGCSE Mathematics (US) 0444
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
through statements like
34 = 3 x 3 x 3 x 3 = 81 (check learners 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 the difference between 34 x 33 and (34)3
e.g. (34)3 = 34x34x34 to preserve the rules of powers,
and then either applying the multiplication rule or by
expanding demonstrate that this must be 312 or 34x3
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 the definition of radical
exponents as an extension to
integral exponents.
General guidance
Once the rules are established for integer exponents
the radical components can be justified.
e.g.
3
5x 3 5x 3 5 = ( 3 5)3 = 51 = 5
\ (5 ? )3 = 51
\ ?x3 = 1
\ ?=
Once
2
3
1
3
is understood as 2 x
1
3
then the multiplication
2
3
rule applies so 5 = ( 3 5 )2
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
As well as evaluating numbers with fractional
exponents, have a set of cards with fractional, root and
multiplication of both root and non root powers and ask
learners to match equivalent forms. Include some
rogue cards so that learners can only be successful if
they understand the form and will not just match easy
ones and guess the rest.
1.4
Ratio and proportion
Explain what standard form is and convert numbers in
and out of standard form looking at problems that
involve standard form.
Past Paper 12 June 2011 Q6
(syllabus 0580)
General guidance
It is important to teach the links between ratio and
proportion and to teach the links between fractions
decimals and ratios.
‘Counton’ page 60:
www.counton.org/resources/ks3framework/pdfs/frac
tions.pdf
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.
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.
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Cambridge IGCSE Mathematics (US) 0444
Solving ratio word problems:
www.thinkingblocks.com/ThinkingBlocks_Ratios/TB
_Ratio_Main.html
Multiplicative relationships:
http://teachfind.com/national-strategies/interactingmathematics-year-8-multiplicative-relationshipsmini-pack
Proportional reasoning:
http://teachfind.com/national-strategies/interactingmathematics-key-stage-3-year-9-proportionalreasoning-mini-pack
16
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
x
3m
7.5m
7.5
7.5
3
3
x7.5
÷3
$4
x
1.6
Percentages, including
applications such as interest and
profit
1
7.5
3
3
7.5
4
?
x
?
7.5
3
so
4 x 7.5
3
=10
Notes and examples
Excludes reverse percentages.
Includes both simple and compound interest.
‘Counton’:
www.counton.org/resources/ks3framework/pdfs/frac
tions.pdf
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.
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Cambridge IGCSE Mathematics (US) 0444
17
Syllabus ref
and CCSS
1.9
CCSS:
N-Q1
N-Q2
N-Q3
Learning objectives
Suggested teaching activities
Learning resources
Use units to understand problems
and guide the solution to multistep problems
Notes and examples
Also relates to graphs and geometrical measurement
topics.
Past Paper 31 June 2011 Q1
(syllabus 0580)
Quantities – choose and interpret
units and scales, define
appropriate quantities (including
money)
Includes converting between units, e.g. different
currencies.
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.
Look at problems that contain a mixture of units of say
length. i.e. the price of 1m of rope and requesting the
price of 75cm. or money problems with costs
expressed in dollars in part of the problem and cents in
another part of the problem.
Include calculator problems where the display has to
be interpreted as one of the units in a question where
more than one is given at the outset. e.g. weight and
cost or weight in two different forms
Look at problems of conversions between three
currencies where the link between two pairs is given
but the third pair is required. e.g. £1 = $1.55 and 1
Euro = 77p how many Euro’s would you get for $100
Estimating, rounding, decimal
places, and significant figures –
choose a level of accuracy
appropriate for a problem
v2 2Y10
Notes and examples
Use estimation to check answers and consider
whether the answer is reasonable in the context of the
problem.
Cambridge IGCSE Mathematics (US) 0444
Past Paper 12 June 2011 Q4
(syllabus 0580)
18
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
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
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.
Learners 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
learners to give the range of possible answers noting
when to us ≤ or <
Look at problems and estimate the answer using 1
significant figure and then decide whether the actual
answer is reasonable. Look at ways of estimating that
give a range between which the answer must sit and
again check solutions.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
19
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Create a bank of problems with calculations partially
written that when followed through look odd e.g. the
height of a man works out at 4m and ask whether the
data in the question was reasonable or the partial
calculation was incorrect. i.e. a learner’s calculation for
a proportion on a scale drawing that is incorrect
because a pair has been inverted. Ask learners to
correct the problem.
1.10
Calculations involving time:
seconds (s), minutes (min), hours
(h), days, months, years including
the relation between consecutive
units
Notes and examples
1 year = 365 days.
Includes familiarity with both 24-hour and
12-hour clocks and extraction of data from dials and
schedules.
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.
Time difference on number lines can be an effective
model to support a calculation.
Time calculations:
www.bbc.co.uk/skillswise/worksheet/ma01line-l1-wtime-calculations
Time – Introduction:
www.bbc.co.uk/schools/ks3bitesize/maths/measure
s/time/revise1.shtml
Past Paper 12 June 2011 Q11
(syllabus 0580)
Ensure learners 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.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
20
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
1.11
Speed, distance, time problems
General guidance
The two websites in the Learning resources column
show the triangle model for solving these problems.
This works for some learners 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.
Distance, speed and time:
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.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
21
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 1: Number – Extended curriculum
Recommended prior knowledge
All core units and particularly Core Unit 1. Only those parts of the learning objectives or notes and examples not included in the core units are itemised, so this
document should be read alongside Core Unit 1.
Context
There is one core curriculum number unit (Unit 1: Number) and this is the only extended curriculum number unit (Unit 1: Number). Once Core Unit 1 and the other
prior experience for Core Unit 1 are completed, the Extended Unit 1 can be slotted in at any point during the course. It is probably best taught as a whole but used to
revise some of the Core Unit 1.
Outline
This unit extends the knowledge of Core Unit 1 – please note that Past Paper examination questions that relate to aspects of Core Unit 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
and CCSS
1.1
CCSS:
N-RN3
Learning objectives
Suggested teaching activities
Learning resources
Knowledge of: natural
numbers, integers
(positive, negative,
and zero), prime
numbers, square
numbers, rational and
irrational numbers,
real numbers
Notes and examples
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.
‘Counton’ page 65:
www.counton.org/resources/ks3framewor
k/pdfs/fractions.pdf
General guidance
The easiest way to tackle the notes and examples 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 learners may find least convincing is when rounded versions of 
are discussed which are rational and when both irrational and recurring
decimals are written with ....... to show the pattern continues.
http://nrich.maths.org/4717
Use of symbols: =, ≠,
≤,≥, <, >
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
http://nrich.maths.org/2756
Past Paper 22 June 2011 Q2
(syllabus 0580)
22
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
The ‘counton’ website resource page 65
(www.counton.org/resources/ks3framework/pdfs/fractions.pdf ) shows one
way to do converting 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 learners
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 examples
Includes reverse percentages.
Includes percentiles.
‘Counton’ pages 75 and 77:
www.counton.org/resources/ks3framewor
k/pdfs/fractions.pdf
General guidance
The two ‘counton’ pages 75 and 77 provide advice on method and some
problems. (See web link in Learning resources column)
‘nrich’ web link:
http://nrich.maths.org/1375
Learners invariably confuse 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
Unit 4 (1.4). This visualisation of the difference might help some learners.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
Past Paper 42 June 2011 Q1b
(syllabus 0580)
23
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
0.7
Reducing £30 by 30%
0
0%
RP
reduced price
100% - discount
70%
£30
start price
RP
£30
70
100
100%
The start price was reduced by 30% to £28
0
0%
£28
reduced price
100%-discount
70%
SP
start price
100%
RP = £21
x 0.7
x
10
7
£28
SP
70
x
SP = £40
100
10
7
Teaching activities
Set up a problem where a local boutique buys in 50 pairs designer jeans for
$40 a pair. It sells them initially for $110, and 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’ web link learning resource will need changing.
http://nrich.maths.org/1375
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
24
Syllabus ref
and CCSS
1.7
CCSS:
N-RN1
N-RN2
Learning objectives
Suggested teaching activities
Learning resources
Meaning and
calculation of
exponents (powers,
indices) including
positive, negative,
zero and fractional
exponents
Notes and examples
Past Paper 21 June 2011 Q4
(syllabus 0580)
1
2
e.g. 5 = 5
1
2
Evaluate 5–2, 100 , 8
-2
3
Past Paper 22 June 2011 Q2
(syllabus 0580)
General guidance
(See Core Unit 1)
Past Paper 22 June 2011 Q4
(syllabus 0580)
The usefulness is of course for simplifying using the index rules.
Evaluating the expressions requires practice and learners need a little
Past Paper 22 June 2011 Q6
(syllabus 0580)
experience to work out the order of working for a problem like 8
-2
3
Learners need practice without calculators to understand the process, but
should also be able to use the calculator as well for non integer solutions.
1.8
Radicals, calculation
and simplification of
square root and
cube root
expressions
Notes and examples
e.g., simplify 200 + 18
Write (2 +
3 )2 in the form a + b
3
General guidance
Learners need to use their understanding of factors, squares and cubes to
work with this topic effectively. So first remind learners of these
Teaching activities
Ultimately these types of problem require practice and so use the ‘formulator
tarsia’ software (see web links in the Learning resources column) to set up a
hexagon or domino puzzle to make this practice more interesting, by
matching forms of the same expression.
Past Paper 42 June 2011 Q1a
(syllabus 0580)
Formulator tarsia software:
www.mmlsoft.com/index.php?option=com
_content&task=view&id=9&Itemid=10
and
www.mmlsoft.com/index.php?option=com
_content&task=view&id=11&Itemid=12
Past Paper 22 June 2011 Q2
(syllabus 0580)
Revisit frequently as a starter asking learners to complete a few examples.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
25
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 2: Algebra – Core curriculum
Recommended prior knowledge
Learners should already be able to:
 use a letter to represent an unknown and have an understanding that in some cases the piece of algebra is an equation and leads to (a) specific value(s) but
otherwise the letter in an expression stands for multiple values
 spot patterns in sequences of objects and numbers and to describe these in words
 plot coordinates in the first quadrant
 work out the order of operations for a statement written with more than one order of operation and parentheses
Context
Unit 1 should precede this unit as algebra is generalised arithmetic. This unit can be split into three blocks. It is the first of two algebra blocks.
 Block 1 – Language of algebra covering the language and tools for manipulating algebra (2.7, 2.4, 2.5 – evaluation of simple formulae, 2.8, 2.9)
 Block 2 – Sequences (2.13) could be linked into the Functions unit 3
 Block 3 – Solving equations (2.3, 2.5 – turning a formula into an equation by substituting values for all but one unknown and 2.6). This could be split into two
as well to allow the solution of linear equations to be consolidated before 2.6 is taught. 2.6 should also be linked to functions in Unit 3 to show graphical
solutions and could be taught with that unit.
Learners 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.
Learners should already be able to:
 use a letter to represent an unknown and have an understanding that in some cases the piece of algebra is an equation and leads to (a) specific value(s) but
otherwise the letter in an expression stands for multiple values
 spot patterns in sequences of objects and numbers and to describe these in words
 plot coordinates in the first quadrant
 work out the order of operations for a statement written with more than one order of operation and parentheses
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
26
Outline
Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. This unit
develops an understanding of the language of algebra looking at the coding and the manipulation of algebra. It develops the generalisation of sequences and the
creating and solving of equations. 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.
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
General guidance
The framework document on the ‘Counton’ web link, gives an overview of all
the steps for the development in algebra, leading from arithmetic into algebra.
It may be useful in unpicking learner misconceptions when working with
algebra.
‘Counton’ – Algebra:
www.counton.org/resources/ks3framework
/pdfs/equations.pdf
The interacting mathematics bank of documents (see link opposite) is full of
useful information about the teaching of algebra and the steps in developing
equation solving for understanding from intuitive examples through to a full
balancing method. The first document is a training course and the second the
participants booklet the others booklets were training booklets, but the
information about the development of algebra and tasks are invaluable. The
trainer’s booklet needs to be read alongside the year 7, 8 and 9 booklets.
Note particularly the ways of developing equation solving in year 7, ‘clouding
the picture’ in year 8 and year 9, and the pyramid tasks for creating
equations.
Interacting mathematics:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
To read about the misconceptions learners have about algebra see the web
links listed opposite. Many misconceptions stem from the initial introduction of
a letter. It should be for a variable not an item. i.e. a should not stand for an
apple, but for things like the cost of an apple, the weight of an apple, the
circumference of an apple.
Intuitive misconceptions in algebra as a
source of math anxiety:
http://people.umass.edu/~clement/pdf/Intui
tive%20Misconceptions%20in%20Algebra
.pdf
Diagnosing learners’ misconceptions in
algebra (pilot study):
www.springerlink.com/content/557502366l
86518p/
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
27
Syllabus ref
and CCSS
2.7
CCSS:
A-SSE1
Learning objectives
Suggested teaching activities
Learning resources
Identify terms,
factors, and
coefficients
General guidance
Throughout the unit ensure correct use of vocabulary associated with
algebra.
Constructing and solving linear equations
Year 8 booklet (file 0085-2004PDF-EN.pdf
page 6:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
There is an activity in the Constructing and solving linear equations
Year 8 booklet (file 0085-2004PDF-EN.pdf) page 6, with the answers and
likely problems learners will find in the training course page 32, in the
resource given.
Teaching activities
Set up a number of algebraic, statements on separate cards and ask learners
to sort by a rule like all the statements where ‘a’ has a coefficient of 3, or all
the expressions with two terms.
2.4
Exponents (indices)
Notes and examples
Includes rules of indices with negative indices.
Simple examples only,
e.g., q 3 × q –4, 8x 5 ÷ 2x 2
‘Counton’ page 112:
www.counton.org/resources/ks3framework
/pdfs/equations.pdf
General guidance
Check learners can distinguish between, 3a2, (3a)2, and 32a as this is one of
the commonest errors. Check also they understand these statements as an
expanded string and can substitute numbers and complete the calculation to
show that these are different. Often learners will take an expression
expanded to 2 x 5 x 5 x 5 and will turn it into 2 x 15 = 30. Learners should use
calculators to check their substituted expansions are multiplied up correctly,
by using power keys and expanded versions as well as calculating without a
calculator.
Past Paper 11 June 2011 Q8
(syllabus 0580)
Past Paper 12 June 2011 Q9
(syllabus 0580)
This links to Unit 1, 1.7 and develops the algebraic components in the same
way. Check learners can distinguish between, 3a2, (3a)2, and 32a as this is
one of the commonest errors. Check also they understand these statements
as an expanded string and can substitute numbers and complete the
calculation to show that these are different.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
28
Syllabus ref
and CCSS
2.5
CCSS:
A-CED5
Learning objectives
Suggested teaching activities
Rearrangement and
evaluation of simple
formulae
Notes and examples
e.g., make r the subject of:
• p = rt – q
r -t
•w=
y
e.g., when x = –3 and y = 4, find the value of
xy 2.
Learning resources
General guidance
Substituting values into expressions can improve the knowledge of the coding
of algebra. Substituting for all but one of the values in a formula to find the
final one can be one way of practicing this skill and equation solving. This
links to the formulae associated with mensuration in Unit 6.
Making a letter the subject of a formula can be developed using the ‘clouding
the picture’ strategy.
Teaching activities
As well as straight exercises in substitution. Set up a bank of statements e.g.
2a=a2, 1/a is always less than 1, 3a2 is less than 30, a2 = 2 x a, etc. and ask
learners to say whether it is always, sometimes or never true and to ask them
to specify for which values it is true.
2.8
Expansion of
parentheses (simple
examples only
Simplify expressions
Notes and examples
e.g. expand and simplify 4(5c – 3d ) – 7c
General guidance
Work from the grid method to expand expressions. The learning resource
suggested gives some advice on the development of this.
Collecting terms to simplify expressions is a source of many misconceptions.
Not distinguishing between
1. Multiplying by 2 and squaring
2. 3a2, (3a)2, and 32a
3. Multiplying two negatives to make a positive and subtracting one
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
Constructing and solving linear equations
Year 8 booklet (file 0085-2004PDF-EN.pdf
pages 7 and 8:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
Formulator Tarsia:
www.mmlsoft.com/index.php?option=com
_content&task=view&id=9&Itemid=10
29
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
term followed by a second subtraction. 9c-2c-3c resulting in 14c or
10c not 4c. They read it as negative followed by negative means add.
4. Learners also do not remember that they cannot cancel a fraction if
there is addition or subtraction in either the bottom or the top of the
fraction and need to be reminded the fraction line acts as a
parentheses.
Learning resources
www.mmlsoft.com/index.php?option=com
_content&task=view&id=11&Itemid=12
Past Paper 31 June 2011 Q3
(syllabus 0580)
Teaching activities
Create a bank of expressions some of which are simplified versions of others
and ask learners to match them.
Create a hexagon or domino puzzle using the Tarsia software found at
‘mmlsoft’. The first document explains its use and the second allows you to
download (scroll down to find Formulator Tarsia).
2.9
Factorization:
common factor only
CCSS:
A-SSE2
Notes and examples
e.g. 6x 2 + 9x = 3x(2x + 3)
Teaching activities
Approach factorization from the grid method i.e. have a grid that is expanded
and ask what the original, outside might have been. There may not be a
unique answer but the discussion of several solutions will produce the
complete factorization.
e.g.
x
?
2.13
CCSS:
F-BF2
v2 2Y10
Past Paper 31 June 2011 Q3
(syllabus 0580)
Continuation of a
sequence of numbers
or patterns; recognise
patterns in
sequences;
?
?
4a2
14a
Notes and examples
e.g. find the nth term of:
• 5 9 13 17 21
• 2 4 8 16 32
Cambridge IGCSE Mathematics (US) 0444
Fibs and truths lesson notes:
www.teachfind.com/national-strategies/ictsupporting-mathematics-fibs-and-truthslesson-notes
30
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
generalise to simple
algebraic statements,
including
determination of the
nth term
General guidance
The learning resource ‘Fibs and truths lesson notes’ has both a lesson plan
and the spreadsheet it looks at Fibonacci.
Teaching activities
Give the learners two numbers like 3, 9, and ask them to find as many
different sequences or patterns as they can. The obvious ones are add 6,
square the previous term, or multiply by 3, but you can also do a Fibonacci
type, by adding the 3 and 9 etc., or a triangle number type by adding 7 next,
or a two pattern rule like add 6 subtract 1, or 4, 10, 5, 11 and so on there are
many different ones, including just, 3, 9, 3, 9, 3, 9..... Use this to open debate
and to discuss ways of describing the sequence. Some of these obviously
have nth term rules beyond the mathematics of core learners but they can be
described in words by term to term rules which can be refined with
discussion.
Use geometric as well as algebraic ideas to justify nth term rules.
Learning resources
Equations, formulae, expressions and
identities:
www.teachfind.com/nationalstrategies/fibs-and-truths
Constructing and solving linear equations:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc year 9 page 9 booklet
‘nrich’ task:
http://nrich.maths.org/2290
Past Paper 32 June 2011 Q10
(syllabus 0580)
Plot sequences against their term number to show which are linear and which
are not. Discuss why points are not joined up. After finding a sequence from a
geometric pattern, the ‘nrich’ web link learning resource provides some
ideas. What it also allows is a geometric vision of the structure of the algebra.
The first activity is drawn three ways. Generalise the algebra from the
drawings.
2.3
CCSS:
A-CED1
A-REI1
A-REI3
v2 2Y10
Create expressions
and create and solve
linear equations,
including those with
fractional expressions
Notes and examples
Explain each algebraic step of the solution.
May be asked to interpret solutions to a problem given in context.
Construct a viable argument to justify a solution method
General guidance
The interacting mathematics bank of documents shows the steps necessary
for the development of equation solving. The trainer’s booklet needs to be
read alongside the year 7, 8 and 9 booklets. Note particularly the ways of
introducing equation solving in year 7, ‘clouding the picture’ in year 8 and
year 9, and the pyramid tasks for creating equations. The ‘arithmagon’ task
Cambridge IGCSE Mathematics (US) 0444
Interacting mathematics:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
‘Counton’ page 124:
www.counton.org/resources/ks3framework
/pdfs/equations.pdf
Translating word problems into equations:
31
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
referred to in the text is on page 124 of the framework document.
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_OneVariableWritingEquations
.xml
Many learners have a tendency to solve in their heads or to only record parts
of the equation as they change it with the result that they equate sections of
algebra that are not connected and make mistakes. Therefore the emphasis
is to slide from the intuitive of the ‘I think’ problems to recordings that justify
the solution
Teaching activities
A good way into the topic is to start with problems like
‘I think of a number add 3 and multiply my answer by 2 and get 16. What was
my number?’
Learners write this down as a function machine chain and then translate it
into algebra. They solve by reversing the function machine. This is then also
translated into algebra. e.g.
a
x 2
+ 3
Past Paper 32 June 2011 Q4a
(syllabus 0580)
Past Paper 31 June 2011 Q3
(syllabus 0580)
Translating word problems into equations:
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_OneVariableWritingEquations
.xml
16
(a + 3) x 2 = 16
2(a + 3) = 16
5
- 3
8
÷ 2
16
2(a + 3) = 16 divide both sides by 2
a + 3 = 8 subtract 3 from both sides
a = 5
Setting up different ‘I think’ problems can encompass the full range of
equations with the variable on one side of the equation.
The ‘clouding the picture’ technique deals with changing equations by simple
steps to lead to rearranging. Learners will naturally start to change by bigger
steps and eventually develop the balancing method naturally.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
32
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Explore all the ways of altering the original equation to make a family of equivalent equations
multiplying both sides by 2
subtracting 1 from each side
adding a to each side
12a + 8 = 56
3a - 1 = 11
3a = 12
5a + 2 = 14 + 2a
6a + 4 = 28
4a + 2 = 14 + a
3a + 1 = 13
3a + 2 = 14
3a + b+ 2 = b + 14
3a + 2b+ 2 = 2b + 14
3a + 3 = 15
2a + 2 = 14 - a
adding b to both sides
3a + 4 = 16
adding 1 to both sides
a + 2 = 14 - 2a
subtracting a from both sides
Be aware the number line method though it adds depth to learner
understanding only works for positive solutions but it is a way into modelling
the balancing method that has more resonance in these days of digital rather
than balance scales.
7x + 5 = 4x + 17
x
x
x x x x
x x x
x x
5
17
x x x
12
5
5
Hence 3x = 12
The ‘algebralab’ web link gives a bank of problems that can be turned into
algebra to solve.
www.algebralab.org/lessons/lesson.aspx?file=Algebra_OneVariableWritingE
quations.xml
To encourage learners to record process rather work in their heads and write
down the answer, use equations with either decimal or fractional coefficient
for the letter or for the constant (or both) and/or with decimal or fractional
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
33
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
answers. Discuss whether the fraction or the decimal is more accurate
representation of the answer.
2.6
CCSS:
A-CED2
A-REI5
A-REI6
Create and solve
simultaneous linear
equations in two
variables
algebraically
General guidance
‘Clouding the picture’ method can be used to change the coefficients of one
of the unknowns so that they are the same or the negative of one another so
that they can be removed. (i.e. by multiplying by different constants along the
branches of a diagram for each equation. There will eventually be identical
except for coefficients for one of the variables in both equations. As an
introduction the back of the year 9 booklet has a diagram to be used to start
from a solution and to build to sets of equations all of which must intersect at
the common point.
Constructing and solving linear equations:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
Past Paper 13 June 2011 Q16
(syllabus 0580)
Link to a graphical solution to show why it works on graphics calculators or a
graphing package. This can to be revisited for Unit 3.
Ensure learners use a check at the end of their working, by substituting in the
equation not used to obtain the second unknown, and go back over their
working if this doesn’t work. Questions with negatives invariably produce
errors when solving simultaneous equations, as do the checks for some
learners.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
34
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 2: Algebra – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 2. Only those parts of the learning objectives or notes and examples not included in the core units are itemised, so this
document should be read alongside the core document. It is also important that Core Unit 1 and 3 and Extended Unit 1 have been completed and understood.
Context
There are two Core algebra units and this is the first of two Extended algebra units. Once Core Units 1, 2 and 3 and the other prior experience for Core Unit 2, are
completed this unit can be slotted in at any point. It is probably best taught in parts as it would provide a very lengthy spell of algebra to complete. Section 2.11 is
required before Extended Unit 8.
Outline
The unit extends the knowledge of Core Unit 2 so be aware that examination questions that relate to aspects of Core Unit 2 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers inequalities and solving inequalities, fractional exponents, rearranging and evaluating more
complex equations and expressions, interpreting algebraic expressions, squaring a binomial, factorizing difference of squares, trinomial and four term expressions
into a product of two parenthesis, algebraic fractions, creation and solution of quadratics, solving equations including rationals and radicals and direct and indirect
variation.
Syllabus ref
and CCSS
2.1
Learning objectives
Suggested teaching activities
Learning resources
Writing, showing, and
interpretation of
inequalities on the real
number line
General guidance
Learners need to understand that a number to the left of another on the
number line is smaller regardless of which side of zero they are working.
Inequalities and simultaneous equations:
www.bbc.co.uk/schools/ks3bitesize/maths/
algebra/inequalities_simultaneous/revise3.
shtml
If using number lines a convention is required to distinguish between the
inclusive and the exclusive case. The BBC website uses solid blobs and
arrows but other sources use open and closed blobs. If learners give a key
this should clarify for any audience.
‘Counton’ page 131:
www.counton.org/resources/ks3framework
/pdfs/equations.pdf
Learners also need to understand when a problem requires only integers and
when it requires all real numbers. A number line can blur this distinction
leading to misconceptions.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
35
Syllabus ref
and CCSS
2.2
Learning objectives
Create and solve
linear inequalities
CCSS:
A-CED1
A-CED2
A-REI3
Suggested teaching activities
Learning resources
Teaching activities
Practise recoding a few cases on number lines in the preferred style and
some problems where the integers that satisfy a set of inequalities are
required.
Notes and examples
e.g., Solve 3x + 5 < 7
Solve –7 ≤ 3n – 1 < 5
‘Counton’ page 131:
www.counton.org/resources/ks3framework
/pdfs/equations.pdf
General guidance
Once the balancing method is understood for solving equations then the
same steps can be used for inequalities.
Inequality word problems:
www.algebra-class.com/solving-wordproblems-in-algebra.html
Learners need to understand why the inequality is reversed for division by a
negative number.
Solving linear inequalities:
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_OneVariableSolvingInequalitie
s.xml
Teaching activities
Use statements like -3x< 6 and substitute integer values back into the
inequality to convince learners of the need to change the direction of the
inequality.
Past Paper 42 June 2011 Q5a
(syllabus 0580)
Past Paper 23 June 2011 Q10
(syllabus 0580)
Past Paper 41 June 2011 Q9
(syllabus 0580)
2.4
CCSS:
A-SSE3
Exponents (indices)
Notes and examples
Includes rules of indices with negatives and
fractional indices.
Past Paper 41 June 2011 Q3c
(syllabus 0580)
Rules of exponents:
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_ExponentsRules.xml
3
e.g., simplify 2x 2 X 5x -4
Past Paper 22 June 2011 Q17
(syllabus 0580)
General guidance
This is the generalisation of Core Unit 1, and Extended Unit 1 (1.7)
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
36
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Learners should be able to solve or simplify using the rules of indices and to
substitute values in the simplified expressions.
2.5
CCSS:
A-CED4
Rearrangement and
evaluation of formulae
Teaching activities
Practise writing the statements in radical form as well.
Notes and examples
Includes manipulation of algebraic expressions to prove identities.
Formula may include indices or cases where the subject appears twice.
e.g., make r the subject of
 V = 43 Π r 3

p = 2rr +s-s
e.g., y = m2 – 4n2
Find the value of y when m = 4.4 and n = 2.8
General guidance
Learners must understand the rules of ‘Bodmas’ and be able to write the
expressions out in words. Where the letter appears on only one side of the
equation then writing it in ‘function machines’ (Core unit 2 (2.3)) can help this
ordering.
Learners need a lot of experience to hone this skill.
Rearranging formulae:
http://mash.dept.shef.ac.uk/RearrangingFo
rmulae.html
Past Paper 22 June 2011 Q11
(syllabus 0580)
Past Paper 21 June 2011 Q16b
(syllabus 0580)
Past Paper 21 June 2011 Q8
(syllabus 0580)
Teaching activities
Give them an expression and ask them to find as many different variations as
they can using the ‘clouding the picture’ technique from Core unit 2 (2.3)
Repeat with an equation at the centre of a spider diagram and ask learners to
find as many rearranged forms as they can, making all the letters the subject
and with different numbers of terms on each side.
The ‘mash’ resource provides links to a lot of examples, a few require a
password, but there are enough that don’t to make this a useful bank of
resources.
To look at identities set up algebraic statements that are true for all values of
‘a’, some values of ‘a’ and no values of ‘a’ e.g. 2a = a2 or 3a2 = 3 x a x a etc.
and discuss the definition of identity.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
37
Syllabus ref
and CCSS
2.6
CCSS:
A-CED2
A-REI5
A-REI6
2.7
CCSS:
A-SSE1
2.8
v2 2Y10
Learning objectives
Create and solve
simultaneous linear
equations in two
variables graphically
Interpret algebraic
expressions in terms
of a context
Suggested teaching activities
Learning resources
Look at questions where learners are asked to prove an algebraic statement
is true and have to construct and or rearrange algebraic statements to do so.
Notes and examples
See functions 3.2
Past Paper 22 June 2011 Q12
(syllabus 0580)
General guidance
This has already been suggested in the Core units and the time should be
used to practice more complex examples, possibly where the equations need
rearranging to have them in a form where the unknowns can be reduced to
one after substitution
Notes and examples
e.g. interpret P (1 + r )n as the product of P and a factor not depending on P.
General guidance
This needs to be dealt with alongside the creation of algebra from problems
to solve by equations of all forms and not treated in its own right except when
expansion is needed. Once an equation has been created and solved the
answer needs to be related back to the original problem.
Expansion of
parentheses, including
the square of a
binomial. Simplify
expressions
It is unfortunate that P is used here and if used by learners they should
understand the difference between the use of P here and distinguish between
P in a probability statement and as an indication of a function statement.
They will need to look at the context to distinguish.
Notes and examples
e.g. expand (2x – 5)2 = 4x 2 – 20x + 25
Past Paper 42 June 2011 Q5c
(syllabus 0580)
General guidance
Use the grid method but ensure that learners keep the sign with the elements
of the expression.
Cambridge IGCSE Mathematics (US) 0444
38
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
(2a -3)(-3a + 4)
x
-3a
4
2a
-3
-6a2
9a
8a
-12
-6a2 + 9a + 8a -12
2
= -6a +17a -12
2.9
CCSS:
A-SSE2
Use equivalent forms
of an expression or
function to reveal and
explain properties of
the quantities or
function represented
Factorization:
difference of squares
trinomial
four term
Notes and examples
9x 2 – 16y 2 = (3x – 4y )(3x + 4y )
6x 2 + 11x – 10 = (3x – 2)(2x + 5)
xy – 3x + 2y – 6 = (x + 2)(y – 3)
Use the structure of an expression to identify ways to rewrite it, for example,
see x4 – y4 = (x2)2 – (y2)2 thus recognising it as a difference of squares that
can be factored as (x2 – y2)(x2 + y2)
Past Paper 42 June 2011 Q5b,dii
(syllabus 0580)
Past Paper 41 June 2011 Q3b
(syllabus 0580)
General guidance
This is an area of mathematics that requires a lot of practice if learners are to
be successful.
Difference of two squares should come after some practice to become a rule.
For the other cases examining what happens to the coefficients when
parentheses are expanded should help understanding.
The most likely source of error is always the negative and examples that
have a positive constant and a negative coefficient of the linear term should
be practiced alongside those with all positives to note the link and the
difference, before the more complicated versions where the constant is
negative either with a positive or negative coefficient of the linear term.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
39
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Using a grid and trying to work out what could give the key terms will help
find the components to test for the brackets.
e.g.
2
= -6a +17a -12
find all the possible values and test
2a, -2a,
3a, -3a,
a, -a
x -6a, 6a
?
-3, -2, -1,
-6, -4, -12,
1, 2, 3,
4, 6, 12
-6a2
?
-12
Practice a number of factorizations as a quick starter over a period of weeks
to give learners enough experience to become familiar with the likely
components to use.
2.10
v2 2Y10
Algebraic fractions:
simplification,
including use of
factorization
addition or subtraction
of fractions with linear
denominators
multiplication or
division and
simplification of two
fractions
Notes and examples
2
e.g. simplify
3
4 7x 21x
4x - 9
- ,
,
÷
2
8x -10x - 3 2x +1 x 4y 2
8
General guidance
A number of skills must be brought together having been understood and
mastered separately for success with this topic
Learners must be able to:
1. manipulate fractions by the four rules with numbers effectively and
understand the generic principle
2. simplifying fractions
3. simplify algebra
4. understand that a fraction line acts as a parentheses
5. factorize or multiply out parentheses
Cambridge IGCSE Mathematics (US) 0444
Past Paper 22 June 2011 Q15
(syllabus 0580)
Past Paper 23 June 2011 Q16
(syllabus 0580)
40
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Once the topic has been covered practice a number of examples as a quick
starter over a period of weeks to give learners enough experience to become
familiar with the likely components to use.
2.11.
CCSS:
A-SSE3
A-REI3
Create and solve
quadratic equations
by:
inspection
factorization
using the quadratic
formula
completing the square
Notes and examples
e.g. x ² = 49
2x ² + 5x – 3 = 0
3x ² – 2x – 7 = 0
Write x ² – 6x + 9 in the form (x – a)2 + b and state the minimum value of the
function.
Quadratic formula will be given.
Past Paper 42 June 2011 Q5b,diii,iv,e
(syllabus 0580)
Past Paper 43 June 2011 Q3
(syllabus 0580)
Past Paper 21 June 2011 Q16a
(syllabus 0580)
General guidance
Once the 2nd and third cases can be factorized, understanding that one or
both brackets can be zero, leads easily to a solution.
Practicing the methods for completing the square and using the formulae
needs practice.
Learners must distinguish which method to apply to solve the particular case
and to find the clues in questions that could guide that choice.
2.12
CCSS:
A-CED1
A-REI2
2.13
CCSS:
v2 2Y10
Solve simple rational
and radical equations
in one variable,
and discount any
extraneous solutions3.
Functions—
Extended curriculum
Notes / Examples
Continuation of a
sequence of numbers
or patterns; recognise
Teaching activities
Solving a particular equation by factorizing, completing the square and the
formula could promote discussion on this.
Notes and examples
e.g. solve x + 2 = 6, x–3 = 27, 2y 4 = 32
General guidance
Learners should be able to solve simple equations involving exponents with
and without calculators and should know when changing the form will help
them to do so.
Notes and examples
 2 5 10 17 26
Cambridge IGCSE Mathematics (US) 0444
Second differences and quadratic
sequences:
www.cimt.plymouth.ac.uk/projects/mepres/
41
Syllabus ref
and CCSS
F-BF2
A-SSE4
Learning objectives
patterns in
sequences; generalise
to simple algebraic
statements, including
determination of the
nth term
Suggested teaching activities

3 6 12 24 48
General guidance
Use the method of second differences to find the nth term of quadratic series
e.g.
Derive the formula for
the sum of a finite
geometric series and
use the formula to
solve problems
Learning resources
book9/bk9i10/bk9_10i3.html
Geometric sequence definition:
www.softchalkcloud.com/lesson/files/JWS
nYa7QyqLGc8/Sum_Finite_Geometric_Se
ries_print.html
Derivation of sum of finite and infinite
geometric progression:
www.mathalino.com/reviewer/derivationof-formulas/sum-of-finite-and-infinitegeometric-progression
Geometric series formula:
www.youtube.com/watch?v=Q39pDPoL0n
o
Teaching activities
Set up a number of quadratic sequences in a spreadsheet and ask learners
to use the above model to find the nth term.
2.14
v2 2Y10
Express direct and
inverse variation in
algebraic terms and
Geometric series sum to figure out
mortgage payments:
www.youtube.com/watch?v=i05-okb1EJg
The ‘softchalkcloud’ link has a nice summary of geometric series but the
video appears to be missing. The ‘mathalino’ link deals with the less than 1
and greater than 1 cases. The ‘youtube’ video pulls the two together – the
screen is messy at times . Whilst some of this is more advanced than
required a blend and teacher input could make this an interesting
introduction. The final ‘youtube’ video deals with the mortgage case and is
well explained. The four resources complement one another but will require
teacher input and practice for individual developmental steps along the way
Past Paper 42 June 2011 Q9
(syllabus 0580)
Notes and examples
e.g. y ∝ x, y ∝ x , y ∝ 1/x, y ∝ 1/x 2
Inverse variation:
www.algebralab.org/lessons/lesson.aspx?f
ile=algebra_conics_inverse.xml
Cambridge IGCSE Mathematics (US) 0444
Past Paper 43 June 2011 Q11
(syllabus 0580)
Past Paper 41 June 2011 Q10
(syllabus 0580)
42
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
use this form of
expression to find
unknown quantities
General guidance
Learners must understand proportionality before they can tackle variation.
They must also understand how to find the k by substituting coordinates in
the equation.
Learning resources
Direct and inverse variation worksheet:
www.wdeptford.k12.nj.us/high_school/Fish
/Honors%20Alg%20worksheets/Direct%20
and%20Inverse%20Variation%20Workshe
et.pdf
Past Paper 22 June 2011 Q8
(syllabus 0580)
Past Paper 41 June 2011 Q3a
(syllabus 0580)
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
43
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 3: Functions – Core curriculum
Recommended prior knowledge
Unit 1, most of Unit 2 and Unit 7 (7.1 and 7.5) and the symmetry work of Unit 5. Learners should also have some experience of dropping positive and negative
integers into function machine rules and completing a table of values to plot a linear function.
Context
This is the second algebra unit. Since symmetries are required teach after Unit 5. Learners 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. This unit can be split into three blocks:
 Block 1 – 3.1,3.2, 3.3, 3.6 deal with the function notation and the plotting of functions
7.5 of unit 7 needs to be taught before Block 2
 Block 2 – 3.5 general characteristics of functions
 Block 3 – 3.12 the effect of a constant on an existing function
Blocks 2 and 3 can be taught later in the course to provide variety and to allow one set of ideas to embed and then to be revisited at the beginning of the next block
to consolidate.
Outline
Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. 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. This unit covers the vocabulary of functions, plotting functions, recognising types of functions from their graphs
and translating functions. It also links functions to real life problems. The unit could be taught as a whole, but could equally well be taught in small bites scattered
throughout the course, so long as the links are drawn between this and other functions.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
44
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
General guidance
These resources gives developmental steps and ideas for underpinning the
learning which also link functions and mapping diagrams to sequences..
Generate points and pot graphs of
functions:
www.counton.org/resources/ks3framework
/pdfs/graphs.pdf
Generate and describe sequences:
www.counton.org/resources/ks3framework
/pdfs/sequences.pdf
3.1
Use function notation
CCSS:
F-IF1
F-IF2
Knowledge of domain
and range
Mapping diagrams
Notes and examples
Understand that a function assigns to each element of the domain exactly
one element of the range. If f is a function and x is an element of its domain
then f(x) denotes the output of f corresponding to the input of x.
Functions: Domain and range:
www.purplemath.com/modules/fcns2.htm
General guidance
Define domain and range – ensure this meaning of range is distinguished
from range in statistics.
Distinguish between functions that have infinite domains and ones that only
have meaning for positive domains (e.g. problems relating to hire purchase
turned into functions).
Also note when it makes sense to join points with a line/ curve to indicate all
possible values, and those which should be left as points only because the in
between values have no meaning i.e. Car hire where you can only hire for
complete days.
Mapping diagrams can be a means of creating values for co-ordinate pairs or
arranging information for pattern spotting to find functions.
2x
v2 2Y10
+3
2
4
7
5
10
13
6
12
15
10
20
23
Cambridge IGCSE Mathematics (US) 0444
45
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
The function notation, vocabulary and mapping diagrams should be used
throughout the unit.
3.2
CCSS:
A-REI11
A-REI11
Understand that the
graph of an equation
in two variables is the
set of all its solutions
plotted in the coordinate plane
Construct tables of
values for functions of
the form ax + b,
± x 2 + ax + b,
a (x≠0) where a and b
x
are integral constants;
draw and interpret
such graphs
Solve associated
equations
approximately by
graphical methods
General guidance
When setting up tables of values to plot (learners should understand this
often only gives enough information to plot the function or to look at the most
interesting aspects of a function) some learners get lost in moving from x to y
in one step so either encourage them to use mapping diagrams or to add
additional rows below the ones given on the exam paper. They can then sum
back to get y.
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
x2
25
16
9
4
1
0
1
4
9
16
25
2x
-10
-8
-6
-4
-2
0
2
4
6
8
10
Solving quadratic equations: Solving “by
Graphing”:
www.purplemath.com/modules/solvquad5.
htm
Past Paper 32 June 2011 Q7
(syllabus 0580)
Past Paper 31 June 2011 Q5
(syllabus 0580)
y
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
Encourage learners to check there plots or coordinates if the plot does not
produce a line or smooth curve.
Ensure learners know that only linear functions are joined with a line and that
curves are joined with a smooth curve, points as crosses not blobs.
Give sufficient opportunities for learners to see the symmetry properties of
quadratics and to try for additional data to get an exact maxima and minima
by finding the line of symmetry and hence the x value and substituting to find
the y value.
Explore the case where the function is equal to a value that enables the value
to be read from the graph by intersecting with y = constant.
In the special case when y = 0 this can be used to solve a quadratic and
explain why there are two answers.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
46
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Also look at cases where a non-linear is intersected by a linear and link to
simultaneous equations.
3.3
CCSS:
A-REI10
F-IF8
Write a function that
describes a
relationship between
two quantities
Notes and examples
e.g. C(x) = 50,000 + 400x models the cost of producing x wheelchairs. Write
a function that represents the cost of one wheelchair.
Past Paper 32 June 2011 Q4
(syllabus 0580)
Teaching activities
Obtain some different mobile phone tariffs.
e.g. Number of texts against cost, number of calls against cost, monthly
rentals.
Create functions with t for number of texts and m for number of minutes etc.
Compare different components on graphs and against a learner’s likely usage
to decide the best tariff for the individual.
Include in discussion this would be a good deal ...........for but not for....
3.5
CCSS:
F-IF4
F-IF7
F-LE5
Recognition of the
following function
types from the shape
of their graphs:
linear f(x) = ax + b
quadratic f(x) = ax2 +
bx + c
reciprocal f(x) = a
x
Interpret the key
features of the
graphs—to include
intercepts; intervals
where the function is
increasing,
decreasing, positive,
negative; relative
maxima and minima;
v2 2Y10
Notes and examples
Some of a, b, c may be 0.
General guidance
This has a link to Unit 7 (7.5) and should be taught after 7.5.
Teaching activities
Use graphics calculators or graphing packages to generate many linear,
quadratic and inverse graphs so that learners can discuss similarities and
differences and discuss, where they are increasing, decreasing, the maxima,
minima and symmetries, and end behaviour.
Make up packs of graphs, tables and function names and ask learners to
match them (put some rogue ones in so that they have to work out all the
intended solutions and don’t finish off the difficult ones by guesswork).
Cambridge IGCSE Mathematics (US) 0444
47
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Notes and examples
e.g., if the function h(n) gives the number of person-hours it takes to
assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.
Royal mail:
www.royalmail.com/portal/rm/content1?cat
Id=400036&mediaId=400347
symmetries; end
behaviour
3.6
CCSS:
F-IF5
Relate the domain of a
function to its graph
and, where applicable,
to the quantitative
relationship it
describes
Teaching activities
Plotting UK parcel weights for international mail against cost gives a graph
that is different from the functions they have been working with so opens up
the debate. i.e. the plot will be a series of non overlapping ascending
horizontal lines. Discuss how you would code the non-inclusive end of the
lines.
Look at cases where particular ranges are more important than others or
where particular domains are invalid even though the function as a series of
algebraic terms could extend to larger domains and ranges.
Plotting the base of a rectangle against its area could lead to discussion
about the validity of the point (0,0) i.e. is it a rectangle?
Discuss the cases where a sequence from a tiling pattern has only integer
values for x but the nth rule is like a function an infinite set of values, whereas
for the specific case only positive integers make sense.
Look at cases where time can be negative if in the particular case could
describe before a given time or after it and when distance can be negative if it
goes backwards.
3.12
CCSS:
F-BF3
v2 2Y10
Description and
identification, using
the language of
transformations, of the
changes to the graph
of
Notes and examples
Where k is an integer.
Introduction to number plumber:
http://nrich.maths.org/6961&part=
Teaching activities
Use graphic calculators or a graphing package to explore the effects and ask
learners to generalise.
Quadratic transformations:
http://nrich.maths.org/7120
Cambridge IGCSE Mathematics (US) 0444
48
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
y = f(x) when y = f(x) +
k, y = k f(x), y = f(x +
k) for f(x) given in
section 3.5
Give them pre-drawn graphs and ask them to model the function, either
graphically or from the table of values obtained from the coordinates read
from the graph.
Learning resources
Once again Unit 7(7.5) should precede this element of work.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
49
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 3: Functions – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 3. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this
Extended Unit 3 should be read alongside Core Unit 3. It is also important that Core Units 1, 2, 3 and 7 and Extended Units 1, 2 and 8 (8.2) have been completed
and understood.
Context
There are two Core algebra units and this is the second of two Extended algebra units. Once Core Units 1, 2 and 3 and the other prior experience for Core Unit 3,
are completed this unit can be slotted in at any point. It is probably best taught in parts as it would provide a very length spell of algebra to complete. There are links
to Extended Unit 7 and 8 and both could be taught before this unit. Extended 8 (8.2) must be taught first.
Outline
The unit extends the knowledge of Core Unit 3 so be aware that examination questions that relate to aspects of Core Unit 3 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers plotting non-linear graphs, comparing the properties of two functions when one is plotted,
recognition of cubic, exponential and trigonometric functions, average rates of change and estimated rate of change of a graph, characteristics of exponential growth
or decay.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
50
Syllabus ref
and CCSS
3.2
CCSS:
A-REI10
A-REI11
F-IF7
Learning objectives
Construct tables of
values and construct
graphs of functions of
the form axn where a
is a rational constant
and n = –2, –1, 0, 1, 2,
3 and simple sums of
not more than three of
these and for
functions of the type
ax where a is a
positive integer.
Solve associated
equations
approximately by
graphical methods
Suggested teaching activities
Learning resources
The odd pages of the document on the ‘Counton’ web link listed have some
useful problems that could be slotted into this unit.
‘Counton’ – Algebra:
www.counton.org/resources/ks3framework
/pdfs/graphs.pdf
General guidance
The advice for setting up the tables of values is the same as Core 3.2.
When setting up tables of values to plot (learners should understand this
often only gives enough information to plot the function or to look at the most
interesting aspects of a function) some learners get lost in moving from x to y
in one step so either encourage them to use mapping diagrams or to add
additional rows below the ones given on the exam paper. They can then sum
back to get y or f(x).
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
x2
25
16
9
4
1
0
1
4
9
16
25
2x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
Past Paper 43 June 2011 Q5
(syllabus 0580)
Past Paper 41 June 2011 Q7
(syllabus 0580)
Past Paper 42 June 2011 Q4
(syllabus 0580)
y
Encourage learners to check there plots or coordinates if the plot does not
produce a line or smooth curve.
Ensure learners know that only linear functions are joined with a line and that
curves are joined with a smooth curve, points as crosses not blobs.
Give sufficient opportunities for learners to see the symmetry properties of
quadratics and to try for additional data to get an exact maxima and minima
by finding the line of symmetry and hence the x value and substituting to find
the y value.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
51
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
In addition learners need to look at solving other problems from related
functions on the graph. This needs to be linked to Core 3.12.
Teaching activities
Although this objective is about plotting graphs, learners could use graphing
packages or graphics calculators to check that the plot is reasonable and that
the associated solving problems are correct.
3.4
CCSS:
F-IF9
Compare properties of
two functions each
represented in a
different way
(algebraically,
graphically,
numerically in tables,
or by verbal
descriptions)
Notes and examples
e.g. given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
General guidance
The knowledge for this skill is acquired in other sections of this unit and
means that good questioning about the graphs plotted and language used is
honed in those sections.
Almost total inequality:
http://nrich.maths.org/5966
Guessing the graph:
http://nrich.maths.org/6990
Past Paper 43 June 2011 Q9d
(syllabus 0580)
Learners need to:
1. look at the patterns of rise and fall in the function values in tables
and to describe these to know when turning points, asymptotes, etc
are occurring in the table
2. know the shape to expect from a function when it is plotted
3. have good use of the vocabulary of graphs
3.5
CCSS:
F-IF4
F-IF7
F-BF3
F-LE5
v2 2Y10
Recognition of the
following function
types from the shape
of their graphs:
cubic f(x) = ax3 + bx2 +
cx + d
exponential f(x) = ax
with 0 < a < 1 or
a>1
trigonometric f(x) =
asin(bx); acos(bx);
tanx
Notes and examples
Some of a, b, c and d may be 0.
Back fitter problem:
http://nrich.maths.org/6506
Including period and amplitude.
Curve fitter problem:
http://nrich.maths.org/6427
General guidance
Learners need to plot families of graphs using graphing packages or graphics
calculators to note the effect of changing a, b, c and d. Initially keep b, c, d at
zero. Then leep a=1 and two of b, c, d =0 and change the third a step at a
time. Note the effects.
Teaching activities
Using the general guidance ask learners to compare families f(x) = ax3, then
Cambridge IGCSE Mathematics (US) 0444
52
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Interpret the key
features of the
graphs—to include
intercepts; intervals
where the function is
increasing,
decreasing, positive,
negative; relative
maxima and minima;
symmetries; end
behaviour and
periodicity
f(x)= x3 + bx2, f(x) = x3 + cx and f(x) = x3 + d. using graphing packages or
graphics calculators and to report back on their findings. (Some advice about
scale may be needed). Discuss afterwards the number of turning points so
that they realise that f(x) = ax3 is the special case where the turning points
are all together but that a cubic normally has two turning points (maxima and
minima) and a point of inflexion. The effects of the values of a, b, c and d
should be predictable from the work in Core 3.
Learning resources
Complete similar processes for other types of function that are required by
the syllabus.
Use tables of values possibly produced in a spreadsheet to explore these
phenomena in a different way.
Use card matching games of functions, tables and plots.
Repeat a similar exercise with other functions in this objective. When looking
at the Trig functions increase the domain to -360 to 720 or higher to
explore periodicity and amplitude within the discussion about other features.
Discuss this by referring to the turning circle graph in Extended Unit 8 (8.2)
which should already have been covered.
The work can be split between groups, so that each group reports on one
element or type of graph.
Use a final card match or true false type activity with a mixture of all the
functions, and tables and algebraic expressions; include the same function
with different range and domain values.
3.7
CCSS:
F-IF6
v2 2Y10
Calculate and
interpret the average
rate of change of a
function (presented
symbolically or as a
table) over a specified
interval.
Estimate the rate of
Notes and examples
e.g. average speed between two points
e.g. use a tangent to the curve to find the slope
General guidance
Relate the average speed between two points as the gradient of the line
joining those two points for a distance time graph. However, if the graph is
the distance from home rather than distance travelled this can become
Cambridge IGCSE Mathematics (US) 0444
Curve fitter problem 2:
http://nrich.maths.org/6428
Steady free fall problem:
http://nrich.maths.org/4851
Past Paper 21 June 2011 Q19
(syllabus 0580)
53
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
change from a graph.
nonsense. This needs discussing and the distinction being made.
Rates of change as tangents have to be explained as a concept first.
Learning resources
Clearly there is a link to calculus here but, calculus is not in the IGCSE so the
skill is being treated in a pre-calculus way.
Teaching activities
Choose a quadratic curve with a pronounced change that then slows and
explore the gradients of lines between two points where one end is fixed at
the point and the other slides along the curve until it comes closer and closer
to the point. Do this from both directions. Then look at slopes of lines when
the x values for the two points are equidistant from the x value of the point
and again reduce the gap. Note the way the first two sets of lines change
gradient but that the third produces parallel lines.
Use this as a guide for asking learners to plot tangents on a quadratic curve,
and an opportunity to practice ruler and set square constructions of parallel
lines.
Repeat the exercise for a graph like f(x) =x3 + 3x2 for the points -4 to 1 to
show this doesn’t work. Discuss how this might help to estimate the slope of
the tangent at the point.
Clearly there is a link to calculus here but, calculus is not in the Cambridge
IGCSE Maths syllabus so the skill is being treated in a pre-calculus way.
Give learners a number of examples to try and then to check with graphing
packages or graphics calculators.
3.8
CCSS:
F-IF8
F-LE1
F-LE3
v2 2Y10
Behaviour of linear,
quadratic, and
exponential functions
linear f(x) = ax + b
quadratic f(x) = ax2 +
bx + c
exponential f(x) = ax
with 0 < a < 1 or a > 1
Notes and examples
Observe, using graphs and tables, that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more
generally) as a polynomial function.
Exponential trend:
http://nrich.maths.org/2677
Use the properties of exponents to interpret expressions for exponential
functions,
e.g., identify percent rate of change in functions
such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,
Cambridge IGCSE Mathematics (US) 0444
54
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
y = (1.2)t/10, and classify them as representing exponential growth or decay.
General guidance
Clearly there is a link to calculus here but, calculus is not in the IGCSE so the
skill is being treated in a pre-calculus way
Link this to the explorations in 3.7 and teach the specific requirements for
percentage rate of change. i.e. by looking at the percentage change from one
gradient to the next whether found from chords or tangents and looking to
see if this is changing in a particular manner. Again learners will need
experience of a variety of cases to distinguish them.
3.9
CCSS:
F-LE2
3.10
Construct linear and
exponential functions,
including arithmetic
and geometric
sequences, given a
graph, a
description of a
relationship, or two
input-output pairs
(include reading these
from a table).
Notes and examples
e.g. find the function or equation for the
relationship between x and y
x
–2 0
2
4
y
3
5
7
9
Simplification of
formulae for
composite functions
such as f(g(x)) where
g(x) is a linear
expression.
Notes and examples
e.g., f(x) = 6 + 2x, g(x) = 7x,
f(g(x)) = 6 + 2(7x) = 6 + 14x
Composing functions with functions:
www.purplemath.com/modules/fcncomp3.
htm
General guidance
Some learners become confused that ‘x’ becomes g(x). So when writing out
include the intermediate step f(g(x)) = 6 + 2(g(x).
Inverting rational functions:
http://nrich.maths.org/6959
Into the exponential distribution:
http://nrich.maths.org/6141
General guidance
Link to Extended Unit 2 (2.13) and the recognition of shapes of functions.
Teaching activities
This could be a quiz game ‘What’s my function?’ i.e. each team starts with
five points and as increasing clues to a function are revealed the points
scored for a right answer decreases. Play teams off against one another or
get teams to challenge one another. When using this version if clues are
wrong the other team receives double the points they would have had at this
point.
Past Paper 21 June 2011 Q20
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
55
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Ensure that learners know that order matters. e.g. in the example above
g(f(x) = 7(6 +2x) = 42 + 14x.
(syllabus 0580)
Past Paper 43 June 2011 Q9a and c
(syllabus 0580)
Past Paper 22 June 2011 Q19
(syllabus 0580)
3.11
Inverse function f –1.
CCSS:
F-BF4
3.13
CCSS:
A-REI12
v2 2Y10
Graph the solutions to
a linear inequality in
two variables as a
half-plane (region),
excluding the
boundary in the case
of a strict inequality.
Graph the solution set
to a system of linear
inequalities in two
variables as the
intersection of the
corresponding halfplanes.
4. Geometry—
Notes and examples
Find an inverse function.
Solve equation of form f(x) = c for a simple function that has an inverse.
Read values of an inverse function from a graph or a table, given that the
function has an inverse.
Generate and describe sequences:
www.counton.org/resources/ks3framework
/pdfs/sequences.pdf pages 161 and 163
Past Paper 21 June 2011 Q20
(syllabus 0580)
Teaching activities
The mapping diagrams in the ‘framework document page 161 and 163
provide a route into this topic, as do reversing function machines (see core
unit 2 (2.3)).
Past Paper 43 June 2011 Q9b
(syllabus 0580)
Notes and examples
e.g. identify the region bounded by the
inequalities
y > 3, 2x + y < 12, y ≤ x.
Solving systems of inequalities:
www.kutasoftware.com/FreeWorksheets/Al
g1Worksheets/Systems%20of%20Inequali
ties.pdf
General guidance
Learners should plot the equality case using only three points (see Extended
Unit 7 (7.5)) and be given guidance on how to code for the line included or
excluded.
Painting by functions:
http://nrich.maths.org/7021
Past Paper 41 June 2011 Q9
(syllabus 0580)
Shading the correct side of the line can faze some learners who find above
and below the line inadequate descriptions. They should choose a point on
one side of the line but not on it and substitute the x value into the inequality
and compare to their y value to see if they are in the correct region or the
opposite one.
Cambridge IGCSE Mathematics (US) 0444
56
Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 4: Geometry – Core curriculum
Recommended prior knowledge
Learners should have a working knowledge of the vocabulary of 2D shape, simple solids (cube, cuboid, prism, cylinder, pyramid, and sphere) angles, and an
intuitive understanding of parallel and perpendicular lines and be able to identify them in their classroom and surroundings and symmetries.
Context
This is the first of five geometry units. This unit must be taught before Units 5 and 7. If split into blocks some could be taught after Unit 1. It could be split into smaller
blocks and taught between other units. This unit can be split into five blocks:
 Block 1 - 4.1, 4.2 and 4.3
 Block 2 - 4.1, 4,2 and 4.4
 Block 3 - 4.5 which could be could be split so that a few constructions are taught in one or two lessons blocks to break up other areas of mathematics.
 Block 4 - 4.6
 Block 5 - 4.7 which could be taught with proportionality in unit, or at the beginning of Unit 7
Learners who are following the Extended curriculum 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
Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. 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. This unit encourages the correct use of terms to describe shapes, their properties and their symmetries. It looks at
the methods for finding missing angles, and to construct accurate diagrams with compass and straight edge, rulers and angle measures, circle properties and angle
in a semi circle. It also looks at similarity.
Syllabus ref
and CCSS
v2 2Y10
Learning objectives
Suggested teaching activities
Learning resources
General guidance
Documents 1 and 2 give a good overview of the coverage of this unit.
Document three is designed to deliver geometric reasoning. It aims to develop
logical reasoning, deeper understanding and as a stepping stone towards
formal proof through the use of learner explanations. It builds geometry
Shape, space and measures:
www.counton.org/resources/ks3framewor
k/pdfs/geometrical.pdf
Cambridge IGCSE Mathematics (US) 0444
Reflection:
57
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
through a series of overlays. This can be effective, but the whole document
needs to read and understood as a whole. It cannot be cherry picked.
www.counton.org/resources/ks3framewor
k/pdfs/transformations.pdf
Geometrical reasoning:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-year-9-geometrical-reasoningmini-pack
4.1
Vocabulary:
acute, obtuse, right
angle, reflex,
equilateral, isosceles,
congruent, similar,
regular, pentagon,
hexagon, octagon,
rectangle, square,
kite, rhombus,
parallelogram,
trapezoid,
and simple solid
figures
General guidance
Ensure learners use the vocabulary correctly throughout the unit.
Teaching activities:
Card sorts with diagrams of angles or polygons or triangles (with all angles
marked), or quadrilaterals (with angles marked) and relevant vocabulary
cards, can be an effective way of checking understanding.
Include additional that is sorted into a pile of unneeded cards.
Odd one out activity – sets of three, angles/ polygons/ triangles/
quadrilaterals. Either with an obvious odd one out or with no obvious odd one
out but learners can note it one is odd one out because it holds a property
others don’t. See page 51 and 52 of the excellence gateway document (see
Learning resources column).
Teaching and learning functional
mathematics page 51 and 52:
www.excellencegateway.org.uk/pdf/Tand
LMathematicsHT281107.pdf
Past Paper 11 June 2011 Q4
(syllabus 0580)
Past Paper 13 June 2011 Q17
(syllabus 0580)
Use a 4 x 3 pin board and rubber bands or square spotty paper and mark of
blocks of 12 spots (4 x 3). Try to identify as many triangles as possible using
the spots as vertices (should be 20) and identify them by type.
Use a 3 x 3 pin board and rubber bands or square spotty paper and mark of
blocks of 9 spots (3 x 3). Try to identify as many quadrilaterals as possible
using the spots as vertices (should be 16) and identify them by type.
Use a 4 x 4 pin board and rubber bands or square spotty paper and mark of
blocks of 16 spots (4 x 4). Try to identify at least one 3 sided, 4 sided, 5 sided
up to 15 sided polygon using the spots as vertices – it is possible and label
those that have known names.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
58
Syllabus ref
and CCSS
4.2
CCSS:
G-CO1
4.3
CCSS:
G-CO3
Learning objectives
Suggested teaching activities
Definitions:
Know precise
definitions of angle,
circle, perpendicular
line, parallel line, and
line segment, based
on the undefined
notions of point, line,
distance along a line,
and distance around
a circular arc
Line and rotational
symmetry in 2D
General guidance
Give the definitions and ask learners to use the definitions when explaining
their reasoning.
Learning resources
Teaching activities
Matching/sort of definitions and vocabulary with some missing words in the
definitions - also supplied on the cards.
Notes and examples
e.g., know properties of triangles, quadrilaterals, and circles directly related to
their symmetries.
General guidance
Ensure learners can both recognise lines of symmetry and rotational
symmetry and its order and can transform a shape by reflection of rotation.
Rangoli designs:
http://nrich.maths.org/5369
Attractive rotations:
http://nrich.maths.org/6987
Weekly problem 42:
http://nrich.maths.org/6742
Ensure learners know that colour is also preserved as part of symmetry.
Teaching activities
Create a worksheet with a triangular flag on a pole that is rotated about the
base of the pole 30, repeat 12 times. e.g.
Past Paper 11 June 2011 Q3
(syllabus 0580)
Ask learners to colour it in so that, it has rotational
symmetry order 1, 2, 3, 4, 6, and 12 and ask them to explain why these are
the only possibilities.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
59
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Have a set of cut out quadrilaterals available and ask learners to fold them,
turn them and identify the rotational and reflective symmetry.
Ask learners to draw polygons in the cells of a two way grid that has no lines
of symmetry, one line of symmetry, two lines of symmetry, four lines of
symmetry as the headers in one direction and order of rotational symmetry, 1,
2, 4 in the other and then to explain why some cells cannot be filled.
Get pictures of different car alloy wheel designs and identify which have
rotational and which have reflective symmetry.
Put learners in pairs and two pairs work as competing teams. They have a set
of square tiles (you specify the number of tiles they can have). The first pair
arranges the tiles in a pattern which has a line of symmetry or rotational
symmetry (tiles must meet on whole edges). The second pair moves one tile
so that the arrangement has no line of symmetry and the moved tile still
touches at least one other tile on a full side. The first pair tries to restore
symmetry moving the first tile or any other tile (just one) and restores the
symmetry, but not back to its original place. The game continues until one pair
cannot continue without repeating a previous arrangement. Rules can be
changed so that more than one tile can be moved. You may find each pair of
teams needs a fifth learner to act as an adjudicator.
Link to Unit 5 (5.6)
4.4
Angles around a point
CCSS:
G-CO9
G-CO10
Angles on a straight
line and intersecting
straight lines
Vertically opposite
angles
Alternate and
corresponding angles
on parallel lines
v2 2Y10
Notes and examples
Formal proof is not required but candidates will be expected to use reasoned
arguments including justifications, to establish geometric results from given
information.
General guidance
Ensure learners have the facts to learn, and regularly test their knowledge of
them. You might want to model how one fact is deduced from others after
getting learners to explore the idea first. The geometric reasoning pack has
some guidance on how to develop this type of reasoning leading to proof.
Past Paper 32 June 2011 Q5
(syllabus 0580)
Past Paper 33 June 2011 Q6a,b,c
(syllabus 0580)
Geometrical reasoning:
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-year-9-geometrical-reasoningmini-pack
Show how to solve sets of problems for each fact separately but also mix up
Cambridge IGCSE Mathematics (US) 0444
60
Syllabus ref
and CCSS
Learning objectives
Angle properties of
triangles,
quadrilaterals, and
polygons
Interior and exterior
angles of a polygon
Suggested teaching activities
Learning resources
the facts so that learners have to choose the appropriate fact to solve a
problem.
Learners also have difficulty identifying angles at a point when the lines are at
a vertex of one of more meeting polygons. So practice at seeing where
various facts can be applied is required. Distinguish between regular and
irregular polygons.
Teaching activities
Have a proof cut up as separate line statements and a diagram. Ask learners
to reconstruct the proof in a logical order.
Draw seven intersecting lines on a page (no more than four intersecting at
one point). Have at least one pair of lines parallel and one line perpendicular
to another. Give one angle and ask learners to find all the other angles on the
sheet but to identify their route around the diagram and to give reasons to
justify the answers. They will need the ends of the line segments and
intersecting points lettered so that they can refer to different parts of the
diagram, by two letter line segment names and three letter angle names.
Draw polygons and using the method of joining 1 vertex to all the others, to
create (n-2) triangles where n is the number of sides of the polygon, create a
table for the angle sum of the polygons, check answers using the exterior
angle of a regular polygon to find the internal regular angle and hence the
total for the interior angle for the regular polygon. Create tables and ask
learners to generalise.
There is a bank of problems to solve at the end of the geometric reasoning
unit.
4.5
Construction
CCSS:
G-CO12
G-CO13
G-C3
G-C4
Make formal
geometric
constructions with
compass and straight
edge only.
v2 2Y10
General guidance
All of these need to be practised – ensure learners have reliable compasses
and sharp pencils to avoid frustration. Many are shown in the first two
resources given for this unit.
Relate the bisector of an angle, construction of perpendicular lines, etc to the
properties of a rhombus.
Cambridge IGCSE Mathematics (US) 0444
Stars:
http://nrich.maths.org/5357
Past Paper 33 June 2011 Q2
(syllabus 0580)
Past Paper 32 June 2011 Q8
61
Syllabus ref
and CCSS
Learning objectives
Copy a segment;
copy an angle; bisect
a segment; bisect an
angle; construct
perpendicular lines,
including the
perpendicular bisector
of a line segment
Construct an
equilateral triangle, a
square, and a regular
hexagon inscribed in
a circle
Suggested teaching activities
Learning resources
(syllabus 0580)
Use 360 angle measures to link to definition of angle and to make
construction and measurement of reflex angles easier.
After a few have been given to learners, ask them how they would complete
others so that they have a means of remembering how some are done by
building from others.
Link skills to the construction of bearings diagrams and practice some of
these as accurate scale diagrams. (Link to proportionality/ratio models in Unit
1).
Construct the
inscribed and
circumscribed circles
of a triangle.
Construct a tangent
line from a point
outside a given circle
to the circle
4.6
CCSS:
G-C1
v2 2Y10
Angle measurement
in degrees.
Read and make scale
drawings
Vocabulary of circles
Properties of circles:
• tangent
perpendicular to
radius at the point of
contact
• angle in a semicircle
Notes and examples
Formal proof is not required but candidates will be expected to use reasoned
arguments including justifications, to establish geometric results from given
information.
Circle theorums:
http://nrich.maths.org/6007
Past Paper 11 June 2011 Q18
(syllabus 0580)
General guidance
Ensure learners can correctly use the names, radius, diameter,
circumference, centre, arc, sector, chord and segment.
Cambridge IGCSE Mathematics (US) 0444
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Set up a bank of problems on cards and ask learners to sort them according
to the circle property which will allow them to be solved as part of a way of
teaching them to justify the solution. Also insist on the recording of the
calculation of missing angles rather than learners just giving an answer as
part of the justifying of an answer.
4.7
Similarity
CCSS:
G-SRT2
G-SRT3
Calculation of lengths
of similar figures
Notes and examples
Use scale factors and/or angles to check for similarity.
Past Paper 33 June 2011 Q6e
(syllabus 0580)
General guidance
Use the proportionality model in Unit 1 to find missing lengths.
Test for similarity by checking corresponding pairs of lengths have the same
multiplier to get from one to the other. Learners sometimes have difficulty
identifying corresponding pairs and keeping all the ratios the same way
around. This needs practicing as a separate skill first.
Link to Enlargement (dilation) Note that angles are preserved under
enlargement. Link to Unit 5 (5.6).
Set up diagrams and draw a fan of lines from the centre of enlargement to the
object and extend to find the vertices of the image. Using
scaling/understanding of symmetry to find these lengths. Then prove the
object and the image are similar.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 4: Geometry – Extended curriculum
Recommended prior knowledge
All Core units, particularly Core Unit 4. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this
document should be read alongside Core Unit 4.
Context
There are five Core geometry units and this is the first of five Extended geometry units. Once Core Unit 4 and the other prior experience for Core Unit 4 are
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 4.
Outline
The unit extends the knowledge of Core Unit 4 so be aware that examination questions that relate to aspects of Core Unit 4 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers understanding the definitions of vocabulary, symmetry in 3D, the additional circle theorem
properties, similarity as it affects area and volume, and congruence.
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
4.1
Vocabulary
General guidance
The difference between this and the core unit is the ‘Know precise definitions’.
Formulator Tarsia:
www.mmlsoft.com/index.php?option=com
_content&task=view&id=9&Itemid=10
Know precise
definitions of acute,
obtuse, right angle,
reflex, equilateral,
isosceles, congruent,
similar, regular,
pentagon, hexagon,
octagon, rectangle,
square, kite,
rhombus,
parallelogram,
trapezoid, and simple
v2 2Y10
Ensure learners have the definitions and check throughout the unit that they
use vocabulary correctly.
www.mmlsoft.com/index.php?option=com
_content&task=view&id=11&Itemid=12
Teaching activities
Use the ‘Tarsia’ software found at ‘mmlsoft’ web links to create a domino set
of definitions and vocabulary and ask learners to complete it periodically as a
lesson starter.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Notes and examples
Recognize symmetry properties of the prism and the pyramid.
For example, given a rectangle, parallelogram, trapezoid or regular polygon,
describe the rotations and reflections that carry onto itself.
Rotational symmetry of a cube:
www.youtube.com/watch?v=gBg4lJ19Gg&feature=related
solid figures
4.3
Line and rotational
symmetry in 3D
CCSS:
G-GCO3
General guidance
Ensure that reflections and rotations are precisely defined. e.g. a line of
symmetry is defined as the line joining the midpoint of opposite sides of a
rectangle, or a rotational symmetry of order 2 about the centre (the point
where the diagonals cross) for a parallelogram.
Planes of symmetry:
www.youtube.com/watch?v=cEXx_8FWC
sE&feature=related
Some learners find it very hard to visualise the 3D symmetries. The use of
models that can be split or rotated on an axis are vital for them to see what is
happening.
Teaching activities
View these two videos and discuss the implications of moving from symmetry
in 2D to 3D (points to lines, lines to planes).
4.6
Vocabulary of circles
CCSS:
G-C1
G-C2
Properties of circles:
• tangents from a
point
• angles at the centre
and at the
circumference on the
same arc
• cyclic quadrilateral
Use the following
symmetry properties
of a circle:
• equal chords are
equidistant from the
v2 2Y10
Notes and examples
Formal proof is not required but candidates will be expected to use reasoned
arguments including justifications, to establish geometric results from given
information.
General guidance
Showing learners the proofs of the circle properties/theorems will add depth to
their understanding. However the main requirement is to solve problems that
relate to those properties. It is experience and practice that is required, both
to select the required facts and to sequence a justification for an answer.
Teaching activities
Learners can be given jumbled up lines to a solution and asked to order them.
Or in some cases if a series of angles are required learners can be
challenged to provide more than one route to the complete set and asked to
decide which is neatest.
Cambridge IGCSE Mathematics (US) 0444
Circle theorums:
http://nrich.maths.org/6007
Cyclic quadrilaterals:
http://nrich.maths.org/6624
Subtended angles:
http://nrich.maths.org/2845
Lens angle:
http://nrich.maths.org/833
Triangles in circles:
http://nrich.maths.org/public/leg.php?code
=104&cl=3&cldcmpid=2844
65
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
centre
• the perpendicular
bisector of a chord
passes through
the centre
• tangents from an
external point are
equal in length
Learning resources
Dynamic geometry and circle theorums:
www.timdevereux.co.uk/maths/geompage
s/index.html
Past Paper 43 June 2011 Q4
(syllabus 0580)
Past Paper 22 June 2011 Q13
(syllabus 0580)
Past Paper 21 June 2011 Q17
(syllabus 0580)
4.7
CCSS:
G-SRT2
G-SRT3
G-SRT5
Similarity
Area and volume
scale factors
Notes and examples
Use of the relationships between areas of similar figures and Extended to
volumes and surface areas of similar solids.
General guidance
The most difficult aspect for learners to grasp is to decide whether the
particular case is about an area or a volume as the problems can be about 3D
objects but the scaling to do with 2D, either because one of the dimensions is
fixed or because it is the surface of the object that is the crux of the problem,
not its volume.
Teaching activities
Draw a large triangle. Find the midpoints of two sides (vertex to vertex and
pinch the midpoint) and fold the triangle along this line connecting the
midpoints. The vertex should touch the opposite side and model nicely that
the area of the smaller triangle fits into the larger four times. Ask learners to
split the sides into thirds along two sides and ask them to fold the top triangle
over and construct other lines to show the equivalent numbers of triangles.
Ask the general case ‘Can they make diagrams that show this effect for other
polygons?’
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
Past Paper 23 June 2011 Q20
(syllabus 0580)
Line, area and volume scale factors:
www.cimt.plymouth.ac.uk/projects/mepres
/book8/bk8i19/bk8_19i3.htm
Growing rectangles:
http://nrich.maths.org/6923
Area and volume scale factors:
www.maths4scotland.co.uk/GHS%20Exa
m%20Revision/GHS%20Credit/Similar%2
0shapes%20%20area%20&%20volume.swf
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/congruencysimilarityrev4.shtml
66
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Take learners through cases of simple objects like cubes of length 3cm, etc.
to prove the squaring and cubing of lengths principal.
4.8
Congruence
CCSS:
G-GCO6
G-GCO7
G-SRT5
Recognise that two
shapes are congruent
and use this to solve
problems
General guidance
Learners need to prove the equal facts and identify them in the two triangles
using geometric reasoning and then to show that the facts fit one of the four
criteria.
Proving triangle are congruent:
www.mathwarehouse.com/classroom/wor
ksheets/congruent_triangles/Triangle_pro
of_ASA-SAS.pdf
Teaching activities:
Ask learners working in groups to construct a variety of triangles with the
following criteria. Some will leave them with choices, or prove impossible. Do
not tell them they are going to check if they are identical until the end.
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/congruencysimilarityrev3.shtml
1. Sides of 4cm, 5cm, 7cm constructed with any one of those as the
base.
2. A base of 5cm with a line at 75 at one end and a line of 6cm at the
other end.
3. A base of 6cm with a side drawn at 75 to this that is 5cm long.
4. A base of 8cm, another line of 7cm and the angle opposite to the
base of 55.
5. Draw a parallelogram with sides 5cm and 7cm and cut in half to form
two triangles.
6. Draw a right angle and sides forming the right angle of 5cm and 9cm.
7. Draw a base of 6cm, a right angle at one end and the hypotenuse at
the other end of 9cm.
8. A triangle with angles 40, 65, 75, base of any length.
Finally learners cut out the triangles and decide when they are identical (even
if flipped over) and when they are not, and if any are similar. Are there any
other combinations they could invent?
Discuss results and the difficulty of constructing some with or without extra
decisions.
Use banks of problems with triangles that are congruent, but have to be
proved so that missing information can be found.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Explore the properties of quadrilaterals that result from congruent triangles.
Link the discussion to rotational and reflective symmetry.
Extend to looking at shapes in general that have been transformed by rotation
or reflection and note that the angles and lengths of sides have not changed.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 5: Transformations and vectors – Core curriculum
Recommended prior knowledge
Unit 4 transformations, Unit 7 (7.1)
Context
This is the second unit of five geometry units. So long as Unit 4 and Unit 7 (7.1) have been taught this can be taught at any time. Unit 5.6 can also be used for
revision of Unit 4 transformations so a gap between the two units is desirable. Learners 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
Vector notation is introduced. All transformations are looked at in the Cartesian plane and the effect of the transformation on the objects by looking at the
coordinates of both the object and the image. Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning
through to the required standard. 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.
Syllabus ref
and CCSS
5.1
CCSS:
N-VM1
Learning objectives
Suggested teaching activities
Learning resources
Vector Notation:
directed line segment
AB ;
x
component form  
y 
General guidance
Define a vector.
Areas of parallelograms:
http://nrich.maths.org/4890
Teaching activities
Set up two points on a horizontal line on a coordinate grid and ask learners
how they would describe moving from one to the other and challenge them to
find a way of accounting for the left right and right left separately hinting at the
number line for guidance. Do the same for up and down movements. Then
two points on a diagonal. Refine the coding to vector notation.
Past Paper 13 June 2011 Q8
(syllabus 0580)
Set up a set of 10 cards with a vector on each. All cards are visible. Learners
are given a start point but choose a finish point on a grid. Player 1 has to pick
a vector cards from the set that will translate the start point as close to the
finish as possible. (it doesn’t matter if they don’t select the best card). They
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Syllabus ref
and CCSS
5.6
CCSS:
G-CO2
G-CO3
G-CO4
G-CO5
G-SRT1
Learning objectives
Transformations on
the Cartesian plane:
Translation, reflection,
rotation, enlargement
(dilation)
Description of a
translation using
column vectors
Suggested teaching activities
Learning resources
plot on the grid to prove they have achieved it. Meanwhile the other player
picks a card from the same set and sends the end point of the first plot as far
away from the finish point as they can manage. Players alternate using one
card at a time. Each player plots in different colours. Once a card is used it is
set aside so both players can see it and of course check the other has plotted
correctly. Player one wins if they are at the finish point or closer to it after their
final plot, than the point player 2 has reached. Player 2 wins if it is the other
way around. (Player 2 has no choice about their final move so player 1 can
still win if cards are chosen strategically).
Finally ask all the class to work out the vector from the start point to the final
end point of the game. Class should discover they all have the same vector (a
check for the accuracy of plotting) and discuss why.
Notes and examples
Representing and describing transformations.
Transformation game:
http://nrich.maths.org/5457
General guidance
Transformations can be made or described – standard short questions.
Ensure learners realise which of the transformations produces a congruent
image and which produce an image that is only similar to the object.
Teaching activities
Learners look at the effect on coordinates of all the transformations by
constructing sets of each and recording the object and image coordinates and
discussing patterns.
Transformations – page 205 :
www.counton.org/resources/ks3framewor
k/pdfs/transformations.pdf
Past Paper 31 June 2011 Q7
(syllabus 0580)
On page 205 of the framework document there is a grid of L shapes and an
activity that can be used for the transformations that produce congruent
outcomes.
This is an opportunity to revise understandings of transformations.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 5: Transformations and vectors – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 5. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this
Extended unit should be read alongside Core Unit 5. There is a link to Core Unit 7 (7.2) and Pythagoras Core Unit 8 (8.1) so these need to have been covered
particularly.
Context
There are five Core geometry units and this is the second of five Extended geometry units. Once Core Unit 5 and the other prior experience for Core Units 5, 7 and
8 are 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 5.
Outline
The unit extends the knowledge of Core Unit 5 so be aware that examination questions that relate to aspects of Core Unit 5 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers finding a vector, the effects of adding and subtracting vectors, finding the magnitude of a vector,
multiplying a vector by a constant, stretches, and inverse and combined transformations.
Syllabus ref
and CCSS
5.1
CCSS:
N-VM1
N-VM2
5.2
CCSS:
N-VM2
v2 2Y10
Learning objectives
Suggested teaching activities
Learning resources
Vector Notation:
use appropriate
symbols for vectors
and their magnitudes
Notes and examples
e.g. v, |v|
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/vectorshirev1.shtml
Find the components
of a vector by
subtracting the coordinates of an initial
point from the co-
Notes and examples
See also section 5.6, translations using column vectors.
General guidance
This needs practicing throughout the unit rather than being treated as a
separate component. However it is necessary to be rigorous with learner use
of symbols for vectors and to understand the different forms used in text and
handwritten mathematics.
http://nrich.maths.org/2390
http://nrich.maths.org/7453
http://nrich.maths.org/6632
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
ordinates of a
terminal point
Use position vectors
General guidance
Learners will have already met the idea of finding a vector for transformations
in Core 5 (5.6) but make need reminding it applies to points to make the
connection here.
Use of position vectors needs practice particularly when connected with
geometric reasoning problems.
http://nrich.maths.org/4890
Past Paper 22 June 2011 Q16
(syllabus 0580)
Past Paper 43 June 2011 Q10
(syllabus 0580)
Past Paper 42 June 2011 Q8
(syllabus 0580)
5.3
CCSS:
N-VM4
N-VM5
5.4
CCSS:
N-VM4
Calculate the
magnitude of a vector
x
2
2
  as x + y
y
 
General guidance
This can be linked to Core Unit 7 (7.2) and to Core Unit 8 (8.1) and needs
developing as a rule and then practicing.
Add and subtract
vectors
Notes and examples
Both algebraic (component) and geometric (parallelogram rule)
addition/subtraction.
Understand that the magnitude of a sum of two vectors is typically not the
sum of the magnitudes.
Understand vector subtraction v – w as v + (-w), where –w is the additive
inverse of w, with the same magnitude as w and pointing in the opposite
direction.
Teaching activities
The discussion for the game in Core Unit 5 (5.1) can be extended to the
addition of the vector cards of the two players for consecutive moves and
finding the vector from start to each end of a move by either player.
Past Paper 21 June 2011 Q18
(syllabus 0580)
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/vectorshirev1.shtml
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/vectorshirev1.shtml
Adding and subtracting vectors part 1:
www.youtube.com/watch?v=2dHk_yJ9ntQ
A Knight’s Journey – an article:
http://nrich.maths.org/1317
Give a grid with the points of a polygon and ask learners to find the vectors for
moving from any point to the next until they return to the start point. Different
learners can start at different points. They can then add the total set of
vectors and explain the result.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Deal with the magnitudes of vectors by Pythagoras and set up a false
hypothesis, which has to be disproved for the sum of the magnitudes of two
specific vectors being the same as the magnitude of the sum. Ask the
question is there any values for which it is true?
Find vectors in both directions from first principals to show that the reverse of
w is –w and then add them and discuss why the result is zero.
5.5
Multiply a vector by a
scalar
CCSS:
N-VM5
Notes and examples
 4 
e.g., 3   = 3 5 = 15
3 
See questions for 5.2 above.
 x   cx 
c  =  
 y   cy 
If c|v| ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c <
0).
General guidance
Link this to the stepping pattern in Core Unit 7 (7.5) and gradient.
Teaching activities
Once the skill has been practised relate to scaling problems and splitting lines
in ratios on a coordinate grid.
5.6
CCSS:
G-CO2
G-CO3
G-CO4
G-CO5
G-SRT1
G-SRT2
Transformations on
the cartesian plane:
stretch
Notes and examples
Representing and describing transformations.
General guidance
Learners need to understand the difference between an enlargement and a
stretch. They need to understand that enlargement is the special case where
the horizontal and vertical scale factors are the same.
Link to the magnitude of vectors for the effects on the horizontal and vertical
change.
Transformation geometry (section 5.4):
www.haeseandharris.com.au/samples/igc
se_20.pdf
The rescaled map problem:
http://nrich.maths.org/4958
Past Paper 41 June 2011 Q5a and b
(syllabus 0580)
Past Paper 42 June 2011 Q8
(syllabus 0580)
v2 2Y10
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Ask learners to complete a number of stretches recording start and finish
coordinates and to explain the general pattern on the coordinates.
Past Paper 43 June 2011 Q8 ignoring
matrix parts
(syllabus 0580)
Look at the stretches that take squares to rectangles, rectangles to
parallelograms, kites to rhombi etc. and the effects on lengths of diagonals
and sides.
5.7
Inverse of a
transformation
CCSS:
G-CO5
General guidance
Learners need to understand:
1. the meaning of an inverse operation as one that takes you back to
where you started
2. that reflection is self inverse
3. that translations require a negative of the vector
4. that enlargements require 1 over the original scale factor and the
centre doesn’t change and link to the inverses of stretches.
5. Rotations have take the angle back the other way so clockwise to
anticlockwise or vice versa. Discuss the difference between this and
continuing on 360 - the original angle of rotation. i.e. it takes you
back to the original position but doesn’t reverse the movement.
Teaching activities
Learners should find the rules above for themselves by drawing the
transformation and describing the transformation from image to object. This
could be used as revision for describing transformations completely.
5.8
CCSS:
G-CO5
Combined
transformations
Notes and examples
e.g. find the single transformation that can replace a rotation of 180° around
4 
the origin followed by a translation by vector  
 2 
Transformations – page 205:
www.counton.org/resources/ks3framewor
k/pdfs/transformations.pdf
General guidance
Learners need to understand that order matters and to complete several
examples first one way around and then the other to see this in action.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
In Core Unit 5 this activity was used – on page 205 of the framework
document there is a grid of L shapes and an activity that can be used for the
transformations that produce congruent outcomes.
It or something similar can be adapted here to create a competition. Split the
class into groups. Each learner in the group describes transformations
between any two of the shapes by a combination of two transformations. Two
points for each one correct as judged by the rest of the group, bonus one
point for any that do not include translation as one of the moves. Minus three
points for any incorrect.
Discussion with whole class at the end. A translation and vertical / horizontal
reflection will generally be described by using the axes but any line parallel to
the axes will work with different translations. Finding several of these could be
another challenge and will revise naming vertical and horizontal lines.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 6: Geometrical measurement – Core curriculum
Recommended prior knowledge
Learners should be able to:
to understand definitions of length, area and volume and how to find the area by counting squares and the volume by counting cubes
to know the definitions of solids
to be able to multiply and divide by 10, 100 and 1000
to have made solids from nets.
Context
This is the third geometry unit of five. This unit can be taught as a whole or be broken down into small bits and spread throughout the course. The only unit that
needs to precede this is Unit 1. Learners 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. It may be useful to have three-dimensional models both solid and skeleton framed to support the learning.
Outline
Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. By the end of
this unit learners should have good understanding of how to find a variety of perimeters, areas and (surface areas), volumes, of simple and compound shapes and
be able to express them in appropriate units and convert between units. The learning resources give both teaching ideas, summaries of the skills and their
sequencing and investigative problems to develop the problem solving skills and a depth of understanding of the mathematics, through exploration and discussion.
v2 2Y10
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Syllabus ref
and CCSS
Learning objectives
CCSS:
N-RN1
6.1
Units: mm, cm, m, km
mm2, cm2, m2, ha,
km2
mm3, cm3, ml, cl, l, m3
g, kg
Suggested teaching activities
Learning resources
General guidance
This resource gives a good overview of the developmental steps within the
unit. Specific pages are referred to at the relevant point. It has a variety of
interesting problems to use.
Notes and examples
All units will be metric; conversion between units is expected.
Units of time as given in Unit 1.10.
‘Counton’ – units of measurement:
www.counton.org/resources/ks3framework
/pdfs/measures.pdf
General guidance
Learners need to
1. be able to multiply and divide by 10, 100 and 1000 thinking of this as
sliding left and right across the place value system not moving the
decimal point.
2. know the connection between the units and to think ‘milli’ and ‘Kilo’ as
relating to 1000 and ‘centi’ as 100.
3. be aware of the relative sizes so to know that there will be more
millimetres than cm enabling them to realise they will need to multiply
when converting cm to mm and so on
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/measuresact.shtml
Measures – introduction:
www.bbc.co.uk/schools/ks3bitesize/maths/
measures/use_of_measure/revise1.shtml
‘Counton’ - pages 228 and 230
www.counton.org/resources/ks3framework
/pdfs/measures.pdf
There is often confusion about the 1000cm 3 as 1 litre and 1 m3 as 106 cm3 so
ensure the area and metric units are devised from first principles.
Ensure learners know which units are for length, area, volume, mass and
capacity.
Teaching activities
Once learners know the definitions and connections a quick mental starter on
regular occasions can consolidate the conversions by putting a variety of
measures (of say length) as headers and values scattered in the table in their
appropriate columns. Completing the rows against the clock.
This can also reinforce standard index form if all the units have to be
expressed in that format too.
When setting problems in other sections of this unit ensure that problems are
expressed in a mixture of units requiring conversion to a single unit.
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Syllabus ref
and CCSS
6.2
Learning objectives
Suggested teaching activities
Learning resources
Perimeter and area of
rectangle, triangle,
and compound
shapes derived from
these
Notes and examples
Formula will be given for area of triangle.
Deriving area formulas:
www.youtube.com/watch?v=bK53Wn4Jdp
c
Changing areas, changing perimeters:
http://nrich.maths.org/7534
Area of trapezoid and
parallelogram
General guidance
Learners need to:
1. Understand the difference between perimeter and area
2. Know how the areas of parallelograms, triangles and trapezoids are
linked to their formulae
3. Practice at cutting compound shapes into rectangles and triangles,
finding missing measurements and finding the areas, or completing a
rectangle around a shape and subtracting the unwanted parts
4. Link to substitution in formula (2.5) unit 2
Teaching activities
Work with problems to finding lengths given areas (or perimeters) and one of
the dimensions or in the case of the square none of the dimensions, to assess
understanding.
Perimeter expressions:
http://nrich.maths.org/7283
Threesomes:
http://nrich.maths.org/1841
Adding triangles:
http://nrich.maths.org/1883
Tilted squares:
http://nrich.maths.org/2293
Golden thoughts:
http://nrich.maths.org/271
Dividing the field:
http://nrich.maths.org/498
‘Counton’ pages 234 and 236
www.counton.org/resources/ks3framework
/pdfs/measures.pdf
6.3
CCSS:
G-C5
v2 2Y10
Circumference and
area of a circle
Arc length and area of
sector
Notes and examples
Formulae will be given for circumference and area of a circle.
From sector angles in degrees and simple examples only.
‘Counton' – pages 235 and 237
General guidance
Learners have difficulty with area and circumference even when they are
given the formulae as they mix squaring a number with multiplying by 2 and
do not always correctly identify whether the given information in a problem
states the diameter or the radius. Problems should be set that challenge and
identify whether learners are prone to these misconceptions and remediation
‘Counton’ – page 19 and bottom of page 3
www.counton.org/resources/ks3framework
/pdfs/applying.pdf
Cambridge IGCSE Mathematics (US) 0444
www.counton.org/resources/ks3framework
/pdfs/measures.pdf
78
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
put in place.
Arc length and area of sector should be linked to the proportionality model in
Unit 1 (1.4).
Teaching activities
Draw around a number of circular objects on cm squared paper and cut out.
Fold in half to find the diameter. Count the squares for the area and put string
around the edge and measure the string for the circumference. Record in a
table and let learners notice the ratio of diameter to circumference is
approximately 3 and that the area divided by radius squared is also
approximately 3 as an introduction to pi.
Find diameters and radii, given areas and circumferences to test
understanding.
Look at problems with a practical context. e.g. distance travelled by 20 wheel
turns, or the number of wheel turns required to travel a given distance.
6.4
CCSS:
G-GMD3
Surface area and
volume of prism (in
particular cuboid, and
cylinder)
Surface area and
volume of sphere
v2 2Y10
If the average head circumference is 54.47cm and a witch’s hat is made by
rolling a sector of a circle, what size circle is need for if 2, 3, or 4 hats are to
be made from the circle. Which is the best option to go for? Work out the
area of a brim 5 cm wide for all the hats.
Notes and examples
Formulae will be given for the lateral surface area of cylinder and sphere, and
the volume of prism, cylinder and sphere.
General guidance
Learners need to:
1. have experienced folding nets into solids
2. link the area of nets to areas of compound shapes
3. think of wallpapering the outside of solids to find their surface areas
and to be aware of the polygon faces for each the solids
4. be able to work out the dimensions of the solid from the net of the
solid
5. be aware of the way the a definition of a prism and links to where the
slices would be cut in order to indentify the perpendicular height and
the cross section
Cambridge IGCSE Mathematics (US) 0444
Past Paper 33 June 2011 Q9
(syllabus 0580)
‘Counton’ – pages 239–241
www.counton.org/resources/ks3framework
/pdfs/measures.pdf
Changing areas, changing volumes:
http://nrich.maths.org/7535
All wrapped up – problem:
http://nrich.maths.org/4919
Plutarch’s boxes – problem:
http://nrich.maths.org/749
79
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
6. to be able to make links between the definitions of letters in the
formulas and to identify the relevant lengths from diagrams
7. know which units to use for area and volume
8. be aware that the formulae given may not cover all of the surface
area and how to sum the parts calculated separately.
Learning resources
Efficient cutting – problem:
http://nrich.maths.org/2664
Teaching activities
1. Collect a variety of tins and work out the dimensions of a carton to
pack 40 tins (2 x 4 x 5 tins)
2. Fix a volume and the height and ask for possible dimensions for the
other two dimensions of triangular prisms, cuboids, or the radius of
the cylinder. Find the surface areas and try to maximise
3. Find the volume of icing (0.5cm thick) to cover the top and sides of a
20cm round cake 8cm high. Give the dimensions of a pack of ready
icing and the weight and ask them to work out the number of packs
needed to ice the cake. Then ask the learners to create a table for
cakes of different diameters
6.6
CCSS:
G-MG1
Use geometric
shapes, their
measures, and their
properties to describe
objects
7. Co-ordinate
geometry—Core
curriculum Notes /
Examples
Notes and examples
e.g. modelling a tree trunk or a human torso as a cylinder.
Teaching activities
Decide the minimum quilt size to go over people of different circumferences
and heights.
Estimate the volume of air inside buildings – based on a brick size or a door
height of 2m.
Volume of vases, jugs and then check by filling with water and pouring into
measuring jugs. (some sort of estimate from a maximum and a minimum
model as a range created by surrounding with a cylinder or cuboid or a
combinations of two of these for separate parts of the shape).
Paint tins often give an area of coverage. Research a number of different
qualities of emulsion and their coverage and work out which is the cheapest
and dearest for emulsioning the walls and ceiling of the classroom.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 6: Geometrical measurement – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 6. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this unit
should be read alongside Core Unit 6.
Context
There are five Core geometry units and this is the third of five Extended geometry units. Once Core Unit 6 and the other prior experience for Core Unit 6 are
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 6. It may be useful to have threedimensional models both solid and skeleton framed to support the learning.
Outline
The unit extends the knowledge of Core Unit 6 so be aware that examination questions that relate to aspects of Core Unit 6 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers surface area and volume of pyramids and cones, areas and volumes of compound shapes.
Syllabus ref
and CCSS
6.3
Learning objectives
Suggested teaching activities
Learning resources
Same as Core
curriculum
Notes and examples
From sector angles in degrees only – the phrase ‘simple examples only is
removed.
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/circles2hirev2.shtml
CCSS:
G-C5
Two shapes & printer ink problem:
http://nrich.maths.org/4959
6.4
CCSS:
G-GMD3
v2 2Y10
Surface area and
volume of pyramid (in
particular, cone)
Notes and examples
Formulae will be given for the lateral surface area of cone, and the volume of
pyramid and cone.
Surface area and
volume of sphere
General guidance
As with other formula for surface area and volumes identifying the correct
aspect of a figure to put into the formula is difficult for some learners who find
visualising 3D shapes from 2D representations difficult. Making skeleton
Cambridge IGCSE Mathematics (US) 0444
Paper models of three pyramids that form
a cube:
www.korthalsaltes.com/model.php?name_
en=three%20pyramids%20that%20form%
20a%20cube
Pyramids and cones:
www.algebralab.org/lessons/lesson.aspx?f
81
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
solids can help (Straws joined on vertices with inch long bent sections of pipe
cleaner is a cheap way of avoiding the cost of commercial sets)
ile=geometry_3dpyramidcone.xml
Ensure learners can find all of the sections of the surface area and are aware
that the formula’s given may only represent part of the surface area. This can
be linked to questions that use pythagorus and trigonometry to find the
dimensions needed to solve a problem.
6.5
Areas and volumes of
compound shapes
CCSS:
G-GMD3
Past Paper 23 June 2011 Q21
(syllabus 0580)
Past Paper 42 June 2011 Q7
(syllabus 0580)
Teaching activities
The ‘korthalsaltes’ website gives a net of a pyramid that if made three times
can be turned into a cube – a nice way of looking at the formulae.
Past Paper 31 June 2011 Q6
(syllabus 0580)
Notes and examples
Involving combinations of the shapes in section 6.4. core and extension
Peeling the apple … – problem:
http://nrich.maths.org/4979
General guidance
Once again it is visualising the separation of the shapes that will cause some
learners a problem. This can also mean realising that a truncated solid is the
whole solid minus the top of the solid to leave the truncated portion.
Teaching activities
Trying suggesting some shapes made from combinations of solids and ask
learners to draw a 2D representation with the measurements required to find
the volume and surface area clearly identified – not as easy as it sounds.
6.7
CCSS:
G-GMD4
v2 2Y10
Identify the shapes of
two dimensional cross
sections of threedimensional objects,
and identify threedimensional objects
generated by
rotations of twodimensional objects
General Guidance
After experimenting with solids, learners need to be aware that a cut in a face
of a 3D object produces the edge of a 2D object so counting the faces cut
helps work out the number of edges of the cross section of the cut. Many
learners will find the 3D thinking difficult and need models to manipulate.
Cross sections:
www.learner.org/courses/learningmath/ge
ometry/session9/part_c/index.html
Teaching Activities
Look at the jewels of platonic solids. Use straw outline solids to help the
visualisations and discuss the faces, edges and vertices of both and the links.
Use firm modeling clay and make shapes that are cut up and again discuss
edges, faces and vertices of the original cut to make the new shapes.
Cambridge IGCSE Mathematics (US) 0444
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Use resources in interactive geometry packages to look at these the cuts on
interactive white boards.
The lerner.org resource allows a cube to be rotated and shows the outline of
the 2D outline.
6.8
CCSS:
G-MG2
Apply concepts of
density based on
area and volume in
modeling situations.
Notes and examples
e.g. persons per square mile, BTUs per cubic foot
General Guidance
Ensure learners understand that the ‘per’ means divide the first by the second
to help the ratio to be dealt with the correct way around.
Teaching Activities
Make problems up around utility bills for heating, electricity and gas. Look at
the ratios needed to select boilers for heating and calculations involved.
6.9
CCSS:
G-MG3
Apply geometric
methods to solve
design problems.
Notes and examples
e.g., design an object or structure to satisfy physical constraints or minimise
cost; working with typographic grid systems based on ratios.
A brief history of grids:
www.graphics.com/modules.php?name=S
ections&op=viewarticle&artid=620
General Guidance
The tasks set must be based on real life problems with which learners can
empathise. Working with the design /ICT departments to create a joint project
could be one way forward.
Grid systems:
http://designingfortheweb.co.uk/book/part5
/part5_chapter23.php
Teaching Activities
Take an A3 sheet and cut it in half to make to A4 sheets. Stick one of the A4
sheets in the bottom left corner of the A3 sheet with the same orientation.
Split the spare A4 in half to make to A5 and again stick one on top of the A4
piece. Continue and note that a diagonal can be drawn through all the
corners, from the bottom left. Investigate the ratios. Note that A1 paper has an
area of 1m2 so work out the length of its sides using the information from the
first task.
Five simple steps to designing grid
systems:
www.markboulton.co.uk/journal/comments
/five-simple-steps-to-designing-gridsystems-part-1
Typography lesson:
www.youtube.com/watch?NR=1&v=Zyhu7
gZfu-Q&feature=endscreen
Look at the Golden Ratio and its use for focal points in Art and Design.
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83
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
The first three web links could be used for a research project about
typographic grid systems based on ratios.
The U- tube video is an introduction to terminology. Create some large
images of letters in the same font and ask learners to investigate the ratios
between the lines used in the design.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 7: Co-ordinate geometry – Core curriculum
Recommended prior knowledge
Learners should know how to plot points in all four quadrants.
Context
This is the fourth and fifth geometry units but it could equally be the third of three algebra units. There are clearly overlaps between this unit and Unit 2 (2.13), and
Units 3, 4, 5, 6, 8 and 10. There are links between Unit 8 and 7 and the order is a choice. However, if Unit 7 is taught before Unit 8 then a return to Unit 7 to link
Pythagoras and tangent to 7.3, 7.4, and 7.5 is a possibility. 7.2 and 7.3 can be used to revise aspects of Units, 4, 5, and 6. 7.4 should be linked to the correlation
name types used for scatter graphs in Unit 10. The skills here need to be taught but could be taught in the linked units rather than treating this as a unit in its own
right, or this unit could be used to revisit those other skills in the other units. This unit can be split into three blocks:
 Block 1 – 7.1 should be taught early on in the course probably after Unit 1 to make the skill available for all other units
 Block 2 – 7.2 and 7.3 could be taught separately from the rest of the unit after Unit 8
 Block 3 – 7.4 to 7.6 could be taught after Unit 2 or 3
Learners 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
This unit deals with the technical skills of
 plotting and reading coordinates in all four quadrants,
 finding the distance between points,
 gradient of a line,
 midpoint of a line
 finding the equations of a line by y= mx + c
 understanding lines that are parallel or at right angles to a given line
To facilitate the modelling graphic calculators or graphing packages should be used. The latter skills are delivered through observation of patterns in sets of graphs
so the use of graphics calculators or graphing packages is expected. Within the suggested teaching activities ideas are listed to identify and remediate
misconceptions and to pull learning through to the required standard. 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.
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Syllabus ref
and CCSS
7.1
Learning objectives
Suggested teaching activities
Learning resources
Plotting of points and reading
from a graph in the Cartesian
plane
General guidance
Learners should be able to plot points in all four quadrants but there may
be some who still plot points the wrong way around. The two resources
will remediate.
Maths game – locate the coordinates:
www.mathplayground.com/spaceboyres
cue.html
Reading coordinates can be practised with any graphing activity.
Draw a T-Rex picture:
www.mathsisfun.com/t_rex.html
Learners need to realise that though a graph may have been plotted
using a range of values for x that are integers x= 1.5 or 5.7 can also be
found from the graph. Link to Unit 3 (3.3) functions.
Teaching activities
Give three pairs of coordinates for a quadrilateral and ask learners to
find and name the missing one. This can be a review of past knowledge
of quadrilaterals and does not require learners to have completed Unit 4.
7.2
Distance between two points
CCSS:
G-GPE7
Notes and examples
Questions on this topic would be structured via diagrams.
General guidance
There are several different possible skills involved in this.
Horizontal and vertical distances can be dealt with as subtraction
Diagonal distances can be measured and linked to work on scale
This can be linked to Pythagoras by considering horizontal and vertical
changes and the difference between the x-coordinates and the
y-coordinates. This therefore needs to be linked to Pythagoras in Unit 8.
Teaching activities
Give learners quadrilaterals to plot and ask for the areas of shapes
either as a whole or as two triangles (revision of areas Unit 6).
7.3
CCSS:
G-GPE6
v2 2Y10
Midpoint of a line segment
Notes and examples
Questions on this topic would be structured via diagrams.
General guidance
Learners should explore this by finding the midway points and then
examining the coordinates so that they deduce they find the midway (or
Cambridge IGCSE Mathematics (US) 0444
86
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
average) of the x-coordinates and then the y-coordinates.
Teaching activities
Give learners quadrilaterals to plot and ask them to find the coordinates
of the points where diagonals cross (revision of properties of
quadrilaterals Unit 4) or give coordinates to plot regular polygons and
look at where the lines of symmetry cross edges revision of Unit 5).
Record the co-ordinates of the vertices and intersection to establish a
pattern for finding the midpoint. Similarly look at the ways diagonals
intersect for some quadrilaterals and again record the coordinates
vertices and intersection of diagonals and establish a pattern for the
midpoint.
7.4
Slope of a line segment
General guidance
This should be defined as the horizontal distance divided by the vertical
distance and can be linked to tangent (trigonometry Unit 8).
Learners need to appreciate the difference between positive and
negative slopes and to see the link to the descriptions used for
correlation in scatter graphs (Unit 10).
7.5
Interpret and obtain the equation
of a straight line as y = mx + c
Notes and examples
e.g. obtain the equation of a straight line graph given a pair of coordinates on the line.
General guidance
This is best developed using either graphics calculators or graphing
packages – there are some free ones that can be downloaded.
Diamond collector game:
http://nrich.maths.org/5725
Past Paper 31 June 2011 Q12
(syllabus 0580)
Teaching activities
Ask learners to plot sets of graphs on a graphics calculator or in a
graphing package that have no constant and a positive gradient. Link
back to 7.4 and ask learners to read off values and to realise that for
each increase of 1 in the x-coordinate the y-coordinate increases by the
coefficient of x in the equation of the line- link to Unit 2 (2.13). If different
groups of learners are given different sets it makes the conclusion more
powerful.
v2 2Y10
Cambridge IGCSE Mathematics (US) 0444
87
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Get learners to plot pairs of graphs like y=2x and y=-2x on a graphics
calculator or in a graphing package and to realise the y axis is the line of
symmetry.
Get learners to plot sets of graphs on a graphics calculator or in a
graphing package with varying gradients but the same constant so that
they deduce that c is the intercept with the y axis.
Encourage learners to deduce the rules for y=mx + c for themselves.
Give learners sets of plotted graphs and equations and ask them to
match them.
Ask them to find the equations of lines from graphs.
7.6
Slope of parallel line
CCSS:
G-GPE5
Find the equation of a line
parallel to a given line that
passes through a given point
General guidance
Learners need to realise that parallel lines have the same gradient
Learners need to realise the link between the gradients of perpendicular
lines.
Perpendicular lines:
http://nrich.maths.org/5610
Learners need to be able to find an equation of a line that is parallel to
another line or perpendicular to it that goes through a particular point, by
deducing the gradient and substituting the point to find the constant.
Teaching activities
Ask learners to plot sets of graphs on a graphics calculator or in a
graphing package with the same gradient but different intercepts /
constants. Ask learners to deduce that parallel lines have the same
gradient.
Give learners a pairs of lines that intersect and ask them to suggest two
other lines that would enclose a parallelogram. Collect in results from
learners to get a number of different possible answers, but show that all
the results contain two sets of parallel lines.
Ask learners to plot a number of rhombi and kites and to find the
equations of the diagonals – these will need to be extended to cross the
v2 2Y10
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
y axis. Ask them to deduce what happens to the equations of lines that
are perpendicular to one another to establish the rule.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 7: Co-ordinate geometry – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 7. Only those parts of the learning objectives or notes and examples not included in the core units are itemised, so Extended
Unit 7 should be read alongside Core Unit 7. As there are links to it, Extended Unit 1 should be completed too.
Context
There are five Core geometry units and this is the fourth of five Extended geometry units. Once Core Unit 7 and the other prior experience for Core Unit 7 and
Extended Unit1 are 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 7.
Outline
The unit extends the knowledge of Core Unit 7 so be aware that examination questions that relate to aspects of Core Unit 7 not listed here may have a greater
degree of challenge as they combine with other areas of mathematics. This unit covers how to find a point on a line split in a given ratio, looking at linear equations
of the form ax + by = d and the slope of a perpendicular to a line passing through a given point.
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
7.2
See Core curriculum
Notes and examples
e.g. use coordinates to compute the perimeters of polygons and areas of
triangles using the distance formula.
CCSS:
G-GPE7
Learning resources
General guidance
In the Core unit the skill was explored and linked to Pythagoras. It should
now be linked to work with radicals – Extended Unit 1 (1.8) when
summing the perimeter of polygons and finding areas of triangles.
7.3
CCSS:
G-GPE6
v2 2Y10
Find the point on a directed
line segment between two
given points that partitions the
segment in a given ratio
General guidance
Learners need to understand this in geometry before they understand
this in coordinates. i.e. if two lines in a triangle are divided in the same
ratio then the line joining the two points is parallel to the third side of the
triangle. So if a right angled triangle is formed either the ratio on the
hypotenuse of the triangle is the same as on the x or y height.
Cambridge IGCSE Mathematics (US) 0444
90
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
10
8
splitting AB in ratio 1:2 is the same as
splitting AD or BD in the ratio 1:2
6
x coordinate of p
AD is 14-2
= 12
4
1:2 = 4:8
so x coordinate is 4 more than 2 = 6
B
2
p
5
5
A
2
4
E
10
15
y coordinate of p
BD is 5- 1 = 6
1:2 = 2:4
so y coordinate of p is 2 more than
20
D
C
1 = 1
6
7.5
Interpret and obtain the
equation of a straight line as
ax + by = d (a, b, and d are
integers)
The only remaining idea is for learners to know whether to start at A or B
when working out the split. e.g. If the line had to be split the other way it
would have been called BA and the x and y distances subtracted from
the B coordinates.
Notes and examples
e.g. obtain the equation of a straight line graph given a pair of coordinates on the line.
General guidance
Obtaining the equation of a line and plotting them when b = 0 and y is
alone on one side of the equation has been tackled in Core Unit 7.
The most common way b≠ 0 comes about when the equation is multiplied
by the denominator of a fractional slope and terms are rearranged so that
there are no negatives. Learners need some practice to see this
connection first, and obtaining lines for this form can be tackled this way.
Past Paper 41 June 2011 Q9
(syllabus 0580)
Past Paper 23 June 2011 Q14
(syllabus 0580)
Linear programming: word problems:
www.purplemath.com/modules/linprog4
.htm
To draw a line given in this form challenges learners who want to create
a table of values for x and y and have difficulty rearranging the equation.
However, at this level they should know that only two points are needed
to draw a line, but that it is better to plot 3 so that there is a check for
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
errors. Therefore learners also need to realise that this should be a
straight line. The two obvious pairs of coordinates to plot are when x = 0
to obtain the point the line crosses the y axis, and when y = 0 to find the
point where the line crosses the x-axis. The third point is more
problematic. Using x=1 works for most cases, but when all three points
are close they need to choose a value for x of 5 or 10. This last step is
the one that requires practice so that learners develop sufficient
experience to choose a realistic value.
Teaching activities
Give learners three lines to draw and ask them to find the coordinates of
the intersections (tie to solving simultaneous equations Core Unit 2
(2.6)).
7.6
Slope of perpendicular line
CCSS:
G-GPE5
Find the equation of a line
perpendicular to a given line
that passes through a given
point
Look at problems where the solution is in the space enclosed by the
three lines by also looking at inequalities.
Notes and examples
Understand and explain how the slopes of parallel and perpendicular
lines are related.
General guidance
This has been introduced in the Core unit and simply requires some
formalisation.
Enclosing squares:
http://nrich.maths.org/763
Painting between the lines:
http://nrich.maths.org/7031
Teaching activities
To combine several of the parts of this unit ask learners to plot a
rectangle, given one line, one vertex off the line, and the opposite vertex
on the line. They must give the equations of the other three lines. They
will have to pull together knowledge about parallel and perpendicular
lines (slopes) going through a given point, even if they manage to find the
fourth point by eye. It may be necessary for some to give them three
coordinates and no lines and ask them to find the four lines and then look
at the parallel and perpendicular relationships. This task works well on
graphics calculators
Similar problems with Kites and Rhombi can be produced given the
equation of the diagonal and some of the vertices.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 8: Trigonometry – Core curriculum
Recommended prior knowledge
Units 1 and 4 must have been completed. Learners need a good understanding of where right angles facts occur, (quadrilaterals, diagonals intersecting, lines of
symmetry and edges intersections and angle in a semi circle), the link been square and square root and to find both on a calculator, and similarity.
Context
This is the fifth or five geometry units. Units 1 and 4 must have been completed. Both Pythagoras and Trigonometry are topics that require practice and experience
for learners to use effectively. Delivering the entire unit as a block and not returning to the topic would not be recommended. The final section on identification of
question type can be a revision topic leading to the examination. The overlaps between Units 7 and 8 are such that the order of planning for both units needs to be
thought about simultaneously with respect to the choices outlined in Unit 7 but especially link to the slope of graphs in Unit 7 and the m of y=mx + c to Tangent.
Learners 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
This unit covers the development of Pythagoras as a pattern, from diagrams and suggestions for approaching problems. Trigonometry is introduced from a set of
similar triangles. Learning to identify where right angles occur and selecting the right area of mathematics to solve missing angles and sides is also covered. Within
the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. 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.
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Syllabus ref
and CCSs
8.1
CCSS:
G-SRT6
G-SRT8
Learning objectives
Suggested teaching activities
Use trigonometric ratios and
the Pythagorean Theorem to
solve right-angled triangles in
applied problems
Notes and examples
Problems involving bearings may be included.
Know angle of elevation and depression.
Learning resources
General guidance
Know all the places where right angles occur in rectangles, squares, kites
and rhombi, equilateral and isosceles triangles and where lines of
symmetry bisect odd sided regular polygons, angles in semicircle and
tangents to radii. (review of aspects of Unit 4).
Teaching activities
Set up a two way grid, ‘right angle(s)’ ‘no right angles’ along the top and
‘at a vertex’, ‘where diagonals cross’, ‘where lines of symmetry cross an
exterior line’ down the side and ask learners to put as many polygons as
they can in the spaces.
General guidance
Develop understanding of Pythagoras rule and its use in finding missing
sides in right angled triangles. Ensure time is given to checking that
learners can distinguish between problems that require the hypotenuse as
the answer and those that require one of the other two sides.
Lesson notes for ‘Exploring 2 proofs of
Pythagoras’ theorem:
www.teachfind.com/nationalstrategies/notes-exploring-two-proofspythagoras-therom
Teaching activities
Set up a worksheet with half a dozen right angled triangles with the
squares drawn on their edges, ask learners to find the areas of the
squares and record in a table, so that the largest square (on the
hypotenuse) is in third column, smallest in first column and middle one in
the second column, ask what they can deduce. Use an interactive
geometry model to show it works for many cases. Ask how this would help
to find a missing side and model both for finding the hypotenuse and for
finding one of the non-hypotenuse sides. Use this as revision of square
and square root and finding both on a calculator.
Exploring a geometric proof of
Pythagoras’ theorem:
www.teachfind.com/nationalstrategies/exploring-geometric-proofpythagoras-therom
The ‘teachfind’ web link resource is a lesson plan and two interactive
spreadsheets (view at 100%) to find the next button and enable the
macro.
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Exploring an algebraic proof of
Phythagoras’ theorem:
www.teachfind.com/nationalstrategies/exploring-algebraic-proofpythagoras-theorem
Pythagorean triples1:
http://nrich.maths.org/1309
94
Syllabus ref
and CCSs
Learning objectives
Suggested teaching activities
Learning resources
Find a bank of problems which require Pythagoras to solve them. If the
right angled triangle is not shown i.e. ask learners to find the area of an
isosceles triangle given the lengths of all three sides.
Pythagoras proofs:
http://nrich.maths.org/6553
Learners should draw the diagrams, identify the right angle(s) and then to
sort into two piles – finding the hypotenuse, finding a non-hypotenuse then
solve them.
Are you kidding – problem:
http://nrich.maths.org/851
Past Paper 13 June 2011 Q11
(syllabus 0580)
General guidance
Key skills for Trigonometry
Identifying the sides of the triangle correctly
Knowing the ratios
Identifying which ratio to use
Knowing whether to use the trig function or the inverse of the trig. function
and how these are related to button presses on a calculator
Where is the dot – problem:
http://nrich.maths.org/5615
Teaching activities
To develop trigonometry draw a right angled triangle that fills a page of
squared paper. Drop verticals inside the larger triangle between the
hypotenuse and the base to form a nest of similar right angled triangles.
Create a table with the base, heights and hypotenuse measured for each
of the six triangles. In a further three columns ask them to divide both the
adjacent and the opposite by the hypotenuse and the opposite by the
adjacent (you could do all six ratios possible if you want and there is time).
Discuss the fact that the ratios are almost identical going down a column
for the six triangles – you can go around the room and suggest to some
that you know that various answers/lengths need checking without telling
them how you know. Finally show learners how to do a sine-1, cos-1 and
tan-1 on their calculators for the rough average value of each column (just
give them the button presses without telling them why) to discover the
same answer (approximately).
Past Paper 33 June 2011 Q6d
(syllabus 0580)
Introduction to trigonometry:
http://projects.exeter.ac.uk/csmsurvey/files/CSM10_Intro_to_trigonomet
ry.pdf
Past Paper 31 June 2011 Q10
(syllabus 0580)
Next measure the angle. After realising the angle and the results from
button pressing were the same discuss what has happened and why by
linking to similar triangle work if learners haven’t realised that that is why it
works.
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Learning objectives
Suggested teaching activities
Learning resources
Finally give learners the three ratios as fractions.
Ask learners to invent a Mnemonic to help them to remember the ratios.
e.g.
Silly Old Harry Caught A Herring Trawling Off America
Sine, opposite Hypotenuse.....
Ensure learners understand that the angle has to be known to identify the
adjacent and the opposite. Give them a set of triangles in different
orientations with the right angle and one other angle identified. And ask
them to label the side opp, hyp adj or O, A H etc.
Teach one method for solving all problems:
1. Label triangle (O,A,H)
2. Identify the three facts (two given, one to find) on the diagram
3. Decide which trig ratio it is because two sides are identified on the
diagram even if one is the?
4. Write down the statement in fraction form using the two given
facts with one unknown
5. Rearrange if necessary to get the unknown on one side of
equation and the two knows on the other
6. Decide whether to use the trig key or the inverse trig key on the
calculator
7. Solve and round to 3 significant figures.
Give learners plenty of practice of a mixed bank of problems rather than
sets of sine, then sets of cosine etc.
It might be a good idea to ask learners to sort a pile of problems into, ones
to find the angle, ones to find the hypotenuse and ones to find one of the
other sides at some stage. However, steps 1–7 are identical for all
problems.
Draw a 10cm circle on a coordinate grid, (centre the origin), marking off
10 angles from the origin to intersect with the circumference and noting
their coordinates, Plotting the x-coordinate divided by 10 against angle,
the y-coordinate divided by 10 against the angle, and the x-coordinate
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Learning objectives
Suggested teaching activities
Learning resources
divided by the y-coordinate against the angle either for the first quadrant
or for all 360 to give a different view of trigonometry. This isn’t essential
but gives breadth.
General guidance
Choosing the tool to solve the problem.
Learners can mix up four types of questions, finding a side from an area of
triangle, trigonometry and Pythagoras and missing angle questions that
can be solved by other angle properties so give learners experience of
identifying the question type.
Teaching activities
Print a mixture of questions, and ask learners to sort them into the four
types before they try solving them. They may have to do a little work on
each problem to sort them and the discussion afterwards could be to
identify how they decided the type.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 8: Trigonometry – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 8. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this
document should be read alongside Core Unit 8. It is also necessary for learners to have understood Extended Unit 2 (2.11).
Context
There are five Core geometry units and this is the fifth of five Extended geometry units. Once Core Unit 8, the other prior experience for Core Unit 8 and Extended
Unit 2 (2.11) are 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 8.
Outline
The unit extends the knowledge of Core Unit 8 so be aware that examination questions that relate to aspects of Core Unit 8 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers trigonometry in all four quadrants, the special case ratios for some angles, Sine Rule, Cosine
Rule and Area of Triangle using an angle.
Syllabus ref
and CCSS
8.1
CCSS:
G-SRT6
G-SRT8
Learning objectives
Suggested teaching activities
Know the exact values for the
trigonometric ratios of 0°, 30°,
45°, 60°, 90°
Teaching activities
In Core Unit 8, this task was recommended. Returning to this task can
show the case 0°, 30°, 45°, 60°, 90°. Draw a 10cm circle on a coordinate
grid, (centre the origin), marking off 10 angles from the origin to intersect
with the circumference and noting their coordinates, Plotting the xcoordinate divided by 10 against angle, the y-coordinate divided by 10
against the angle, and the x-coordinate divided by the y-coordinate against
the angle either for the first quadrant.
Learning resources
Using the special triangles below gives the values a different way.
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Learning objectives
Suggested teaching activities
Learning resources
isosceles triangle equal side of unit length
 x = 45°
hypotenuse =
1
x
2
 sin(45°) = cos(45°) =
1
1
2
equilateral triangle sides 2 units
 y = 60° and z = 30°
z
2
base of right angled triangle is 1
2
height is
3
 sin(30°) = cos(60°) =
y
sin(60°) = cos(30°) =
1
2
3
2
Both the visualizations will help learners reconstruct diagrams to remind
themselves which is which if they have difficulty learning these.
8.2
CCSS:
G-SRT7
Extend sine and cosine
values to angles between 0°
and 360°
Explain and use the
relationship between the sine
and cosine of complementary
angles
Graph and know the
properties of trigonometric
functions
Teaching activities
This task has already been recommended in Core 8 and for 8.1 above
completing the full circle and using co-ordinates will show the positive and
negative values in the correct places.
Draw a 10cm circle on a coordinate grid, (centre the origin), marking off 10
angles from the origin to intersect with the circumference and noting their
coordinates, Plotting the x-coordinate divided by 10 against angle, the ycoordinate divided by 10 against the angle, and the x-coordinate divided by
the y-coordinate against the angle either for the first quadrant or for all 360
Drawing any right angled triangle and labelling the lengths a, b, c the
angles α and ɵ and then writing out statements of the trig functions for α
and ɵ should convince learners about the equivalence of sine and cosine of
complementary angles.
8.3
CCSS:
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Sine Rule
Notes and examples
Formula will be given. ASA, SSA (ambiguous case included where the
angle is obtuse).
Cambridge IGCSE Mathematics (US) 0444
Law of sines:
www.youtube.com/watch?v=APNkWrDU1k
99
Syllabus ref
and CCSS
Learning objectives
G-SRT11
Suggested teaching activities
Learning resources
General guidance
Learners need to know the conventions of labelling a triangle to be able to
apply the formula - with the lower case letter for the length of the side
opposite the upper case angle. Some learners find it hard to find opposite
sides so instead describe it as the side that isn’t the two arms of the angle.
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/mat
hs/shapes/furthertrigonometryhirev1.sht
ml
Teaching activities
Use the video to help you construct the proof using white or blackboard
more sequentially and completing the trio of equivalences.
Past Paper 41 June 2011 Q1bii
(syllabus 0580)
Ask learners to solve missing side and angle problems that require Sine
Rule including bearings problems.
8.4
Cosine Rule
CCSS:
G-SRT11
Notes and examples
Formula will be given. SAS, SSS.
General guidance
Learners need to know the conventions of labelling a triangle to be able to
apply the formula - with the lower case letter for the length of the side
opposite the upper case angle. Some learners find it hard to find opposite
sides so instead describe it as the side that isn’t the two arms of the angle.
Teaching activities
From the video for Sine Rule there is a link to the Cosine Rule. Link the
proof solving Quadratic Equations using the formula Extended Unit 2 (2.11)
Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/mat
hs/shapes/furthertrigonometryhirev2.sht
ml
Past Paper 41 June 2011 Q1bi
(syllabus 0580)
Past Paper 42 June 2011 Q3c
(syllabus 0580)
Ask learners to solve missing side and angle problems that require Cosine
Rule including bearings problems.
Finally give learners a bank of mixed problems so that they can distinguish
when to use Sine Rule and when to use Cosine rule, i.e. distinguishing
between cases where you have the included angle and the case where you
don’t have an angle from the others.
8.5
Area of triangle
Notes and examples
Formula will be given.
CCSS:
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Geometry and measures:
www.bbc.co.uk/schools/gcsebitesize/mat
hs/shapes/furthertrigonometryhirev3.sht
100
Syllabus ref
and CCSS
G-SRT9
Learning objectives
Suggested teaching activities
Learning resources
General guidance
Learners need to know the conventions of labelling a triangle to be able to
apply the formula - with the lower case letter for the length of the side
opposite the upper case angle. Some learners find it hard to find opposite
sides so instead describe it as the side that isn’t the two arms of the angle.
ml
You can prove the rule if this is productive but learners need practice
applying the rule and distinguishing this rule from the sine and cosine rule.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 9: Probability – Core curriculum
Recommended prior knowledge
Learners should be able to:
 understand the definition of the probability of an event occurring as the number of times the event can occur divided by the total number of events. That
a probability can only be greater than or equal to 0 and less than or equal to 1 (100%)
 manipulate fractions, decimals and percentage and convert between them
Context
This is the first of two statistics units. This must be taught after Unit 1 and could be used to reinforce fraction decimal and percentage skills. It could be taught early in
the course but should then be revisited. This could be taught as a complete unit or as two blocks (9.1, 9.2 and 9.3) and 9.5. The second block could therefore be
taught later in the course. Learners 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 to problems.
Outline
The content allows discussion of the difference between the probability of an event occurring and what actual happens, the difference between experimental and
theoretic probability and some tools to work out probability. Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to
pull learning through to the required standard. 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.
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Syllabus ref
and CCSS
9.1
CCSS:
S-CP1
Learning objectives
Probability P(A) as a fraction,
decimal, or percentage
Significance of its value,
including using probabilities to
make fair decisions
Suggested teaching activities
Learning resources
This gives a comprehensive guide to probability and the steps of
development and is full of activities and ideas for teaching the topic.
‘Counton’ – probability:
www.counton.org/resources/ks3framewor
k/pdfs/probability.pdf
Notes and examples
Includes an understanding that the probability of an event occurring is
1 – the probability of the event not occurring.
Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections,
or complements of other events (“or,” “and,” “not”).
Flippin’ discs problem:
http://nrich.maths.org/4304
The better bet problem:
http://nrich.maths.org/4334
The knowledge and use of set notation is not expected.
Teaching activities
Shuffle a pack of 0-9 cards and reveal the top card. Ask the class to vote
whether the next card will be higher or lower. And have a recorder note the
outcome versus the class decision. Continue through the whole pack.
Discuss briefly number of times class is correct – you want this to be
incorrect so rig if necessary. Then give out a recording sheet so they
learners can record what has already gone and play again. e.g. The
numbers 0 -9 repeated in 8 rows. So they can cross of the numbers that
have already been used and ring the card currently being held up. Rig it from
time to time so that either higher or lower is impossible and so that the
strongest possibility isn’t the next card that appears and discuss. Most
learners will record the fractions for higher and lower and compare
instinctively but the activity gives an opportunity to discuss certainty and
impossibility and whether the event with the highest probability has to win.
Have a bag containing a total of 10 cubes of two or three different colours.
Pull one out, reveal it and return it and repeat 20 times (learners should
record the results in a frequency table). Ask class to estimate the number of
each colour in the bag. Then reveal the contents or make another twenty
recordings to see if the result refines better. Discuss the number of repeats
needed to give accurate results.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Scattered throughout the learning resource listed are examples where fair
and unfair can be discussed.
9.2
Relative frequency as an
estimate of probability
CCSS:
S-IC2
Notes and examples
Decide if a specified model is consistent with results from a given datagenerating process.
e.g. using simulation, e.g. a model says a spinning coin falls heads up with
probability 0.5.
Would a result of 5 tails in a row cause you to question the model?
Teaching activities
This first task gives a good visual image of some of the conundrums of
probability.
Relative frequency:
www.bbc.co.uk/schools/ks3bitesize/maths
/handling_data/relative_frequency/revise1.
shtml
‘Counton’:
www.counton.org/resources/ks3framewor
k/pdfs/probability.pdf page 283
Probability Art – Use cm squared paper. A square will be coloured red for a
H and Green for a Tail. Start in the top left hand corner of the paper with the
page turned landscape. Toss the coin and colour the first square. Toss again
and colour the next square. Continue until at least one row is complete.
Discuss with the class whether to snake to the next row or to go to the left
hand side. Discuss the total red and greens (hopefully approximately 50%)
but with no pattern in the reds and greens. Discuss the fact the coin has no
memory and the probability does not tell you which event will occur next as
each is independent.
Relative Probability is best demonstrated with things that cannot calculated
by theoretic probability. Tossing a drawing pin (in a sealed jar) see the p283
of the KS3 framework document, or using a page of text and working out the
relative probability for a chosen vowel or consonant. Changing to a different
type or age range text can then be compared.
9.3
CCSS:
S-IC2
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Expected number of
occurrences
General guidance
A discussion is needed to distinguish between the probability of an event
occurring given the probability and it actually occurring – e.g. the science of
weather forecasting. However, it is also necessary to teach that if the
probability of an event is 0.3 and the experiment is repeated 500 times then
you would expect the event to happen 150 times, in spite of the fact that for
each instance 0.3 is less than 0.7 and so the event is less likely.
Cambridge IGCSE Mathematics (US) 0444
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and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Set up variety of probability experiments that have a theoretic probability.
Get each group of five pairs to work on one experiment. Work out the
theoretic probability and the experimental probability for 100 goes. Compare
the theoretic and the experimental probability and pool the results of the
group to get 500 results. Discuss the outcomes for the different experiments
and when the theoretic and experimental converge.
9.5
Possibility diagrams
CCSS:
S-CP1
Tree diagrams including
successive selection with or
without replacement
Notes and examples
Simple cases only.
Teaching activities
Work out the range of outcomes in two way tables with two objects involved.
E.g. possible outcomes when dice are added, or menu options when there
are three main courses and three desserts etc.
Use tree diagrams to show how events combine, noting that you add the
ends of branches but multiply along branches to get probabilities for
combined events. Note where branches add to 1 (100%). Encourage
learners not to simplify fractions until the end but to leave denominators the
same to simplify working and provide checks that the correct parts of the
diagram add to 1(100%).
Probability – combined events:
www.bbc.co.uk/schools/ks3bitesize/maths
/handling_data/probability/revise7.shtml
Statistics and probability:
www.bbc.co.uk/schools/gcsebitesize/math
s/data/probabilityhirev1.shtml
Interactive spinners:
http://nrich.maths.org/6033
Non-transitive dice:
http://nrich.maths.org/7541
Learners find it difficult to decide what to put on the tree diagram so a variety
of problems that requires them to choose is essential, rather than questions
which give pre-labelled branches.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 9: Probability – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 9. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this
document should be read alongside Core Unit 9.
Context
There are two Core statistics units and this is the first of two Extended statistics units. Once the Core Unit 9 and the other prior experience for Core Unit 9 is
completed this unit can be slotted in at any point. It is probably best taught as a whole as there is a flow to the content.
Outline
The unit extends the knowledge of Core Unit 9 so be aware that examination questions that relate to aspects of Core Unit 9 may have a greater degree of challenge
as they combine with other areas of mathematics. This unit covers combined probabilities.
Syllabus ref
and CCSS
9.4
CCSS:
S-CP1
S-CP2
S-CP7
Learning objectives
Suggested teaching activities
Learning resources
Combining events:
Apply the addition rule
P(A or B) = P(A) + P(B) –
P(A and B)
Notes and examples
Understand that two events are independent if the probability of A and B
occurring together is the product of their probabilities and use this
characterization to determine if they are independent.
Statistics and probability:
www.bbc.co.uk/schools/gcsebitesize/math
s/data/probabilityhirev1.shtml
Apply the multiplication
rule P(A and B) = P(A) ×
P(B).
General guidance
This is easiest to model on tree diagrams where the ends of branches are
added and the route along a branch is multiplied. The statement read along
a string of branches leads to use of ‘of’ and therefore an understanding of
why we multiply along the branches. Checking the sub branch additions to 1
also confirms when to add (and check for errors particularly in the case
where items are removed and not returned for the second go).
Teaching activities
Setting up a tree diagram for the game below is interesting and has a
surprising result. Player A always starts and throws a dice. If the dice lands
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Conditional probability and probability of
simultaneous events:
www.shodor.org/interactivate/lessons/Con
ditionalProb/
Chances are – problem:
nrich.maths.org/920
Past Paper 41 June 2011 Q2
(syllabus 0580)
Past Paper 43 June 2011 Q7
106
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
on 1 player 1 wins. If player A does not win player B has a go and wins if
they roll a 2 or a 3, if player B does not win then player three has a go and
wins if a 4, 5 or 6 is rolled. If C doesn’t win return to player A and continue.
Work out the probabilities of each player winning. Who will be the most likely
winner and the likely length of the game. (Theoretical probabilities of each
player winning are P(A)= 3/13. P(B)= 5/13 and P(C) = 5/13 The task could
continue by changing the rules to see the effect of different rules) (This is
adapted from an EDEXCEL coursework task for GCSE).
(syllabus 0580)
Create a set of problems on cards and ask learners to sort if the events are
independent or not. Whether the probabilities are added or multiplied for
solutions etc.
Use a diagram like
this to discuss
probabilities of each
sector of each circle
being selected if a
spinner was placed
at the centre
(measuring angles
at centre). Discuss
independence and
dependence of
combined events on
the two circles.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 10: Statistics – Core curriculum
Recommended prior knowledge
Most of these skills will have been met before but will need developing further. An understanding of proportionality and percentage is required for pie charts in
particular, so Unit 1 must have been covered. Learners also need to know how to plot on graph paper and to measure angles accurately.
Context
This is the second of two statistics units. Ideally this unit should be taught as a whole to link the areas of statistics together. It could also be tied to probability as well.
Using charts to create theoretic probabilities of events occurring. It could therefore follow Unit 9 or be taught adjacent to it. Although the unit is split into separate
skills groups there is an overlap between them. 10.1 and 10.3 go together and 10.1 can be used as a source data for 10.3, 10.4 and 10.8. Learners 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
This unit is about the technical skills of statistics, but ideas are given to tie it to the data handling cycle so that the tasks do not appear meaningless. Although
learners only require technical skills for the syllabus teaching these without showing that charts, graphs, tables averages and range are a means of summarizing
data to answer questions, would limit the rationale for the mathematics so sources have been given to allow statistics to be calculated to answer questions. Within
the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. 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.
Syllabus ref
and CCSS
v2 2Y10
Learning objectives
Suggested teaching activities
Learning resources
All of the skills for developing statistics are summarized in this document
with banks of problems and ideas for covering the full range both with IT
and practically. It’s philosophy is to tie together the data handling cycle
of gathering data to answer a question, summarizing data in charts,
averages measures of dispersion, interpreting data and concluding an
answer to the question. This unit is about the technical skills from the
middle stages of that cycle.
‘Counton’ – statistical methods:
www.counton.org/resources/ks3frame
work/pdfs/specifying.pdf
The ‘censusatschool’ web link gives a large bank of data for use in
creating statistics to compare to other learners school data.
Random data selector:
http://rds.censusatschool.org.uk/
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Syllabus ref
and CCSS
Learning objectives
CCSS:
S-ID1
10.1
Reading and interpretation of graphs
or tables of data
Suggested teaching activities
Learning resources
General guidance
A bank of questions and explanations.
Table and timetables:
www.cimt.plymouth.ac.uk/projects/me
pres/allgcse/bkb8.pdf
General guidance
Learners need to be able to read information off charts, graphs and
tables. The general problem is that they do not read the chart or try to
understand it before approaching problems.
Past Paper 12 June 2011 Q11a
(syllabus 0580)
Teaching activities
Ask learners to work in pairs and ask one to read a chart, table, graph to
another learner. They should read labels in charts and axes, read
scales and work out what one square on graph paper represents.
Describe the shape and detail. Summarise impressions. Pairs feedback
to the class to see what they have found.
Have a selection of charts and graphs from the press and ask learners
to write a paragraph for the paper on the chart (possible give them a
headline i.e. if a chart has information about ages make a sweeping
statement like older people .....). Pairs refine their descriptions. Class
choose the best. They must back their case with comparative figures
drawn from the data source.
10.2
Discrete and continuous data
General guidance
Learners need to learn to set up frequency tables with discrete labels
like shoe sizes or grouped data tables, with equal class intervals and
non overlapping ends.
Past Paper 31 June 2011 Q8a
(syllabus 0580)
They should know how to Add data to frequency tables using tally
marks, but understanding that if you have a list you don’t count up all the
instances of one thing and put all the tally marks that instance at once,
but go through the list as given, and put a tally mark and cross off.
Connect this skill to 10.1 and 10.3.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Give some practical cases, like collecting shoe sizes going around the
room. Learners tally as sizes are called out and check with hands
showing after the tallying. Discuss discrepancy’s and the likelihood of
this happening in any data gathering, which is why large data sets are
needed to increase the validity of conclusions.
Complete frequency tables from bar charts, pictograms scatter
diagrams, and line graphs.
10.3
CCSS:
S-ID1
Compound bar chart, dot plots, line
graph, pie chart, simple frequency
distributions, scatter diagram
General guidance
Learners need experience of constructing all of these.
Ensure
 Axes have titles, and are labelled appropriately for either
discrete (with gaps) or continuous and charts have titles
 Colour is used appropriately on bar charts. Two colours only for
comparable bars on the same chart. i.e. male and female
 Learners realise that pie charts do not give exact numbers for
each portion unless the total is known. Link to proportionality
Unit 1
 Learners know scatters can only be drawn if the paired data is
available as pairs and not from frequency tables
Link back to 10.1
CensusAtSchool:
www.censusatschool.org.uk/getdata/results/phase10
Past Paper 11 June 2011 Q16
(syllabus 0580)
Past Paper 32 June 2011 Q6b
(syllabus 0580)
Teaching activities
Give learners a bank of pie and bar charts and ask if any could be the
same data. Rig it so that two or three bar charts have the same
proportions to one pie chart to emphasis the proportionality of pies.
Learners can invent questions that can be answered from a set of charts
of all types and then swap them. Ask learners to add some that cannot
be answered to give greater depth to the task.
10.4
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Mean, mode, median, and range
from lists of discrete data
General guidance
Ensure learners know how to calculate/find all three averages.
Learners often mix up the non mathematical version of range with the
Cambridge IGCSE Mathematics (US) 0444
Past Paper 31 June 2011 Q8b
(syllabus 0580)
110
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
mathematical and read the ‘subtract’ as ‘to’ so take 9-3 as 9 to 3 not as a
range of 6. The other muddle is with the range of domain and range.
Learners need to understand the statistics version of range as a
separate entity.
Past Paper 32 June 2011 Q6a
(syllabus 0580)
Learners need to understand why we have three averages and that
Mode is the tallest bar on a bar chart or the largest slice of a pie chart.
For median ensure learners remember to order the data first and can
deal with both odd and even numbers of items of data.
Teaching activities
Give learners problems like the mean age of 8 people is 15 one more is
added to the group the mean becomes 17 what was the new person’s
age? This will assess understanding of mean.
Or tell learners the median of four numbers is.... the range is.... and ask
them to find possible solutions. This will test understanding of median
and range.
10.8
CCSS:
S-ID6
Understanding and description of
correlation (positive, negative, or
zero) with reference to a scatter
diagram
Straight line of best fit (by eye)
through the mean on a scatter
diagram
General guidance
Without seeing lines of best fit that are accurate learners find it very
difficult to see what they are aiming at achieving even if a definition is
given, so ensure they have seen examples for correlation work (although
of course a line of best fit is not needed to make that judgement so not
all the examples should have a line of best fit).
Past Paper 32 June 2011 Q6c
(syllabus 0580)
Learners also need to be able to interpret the correlation into a
relationship.
Teaching activities
This can be linked to slope and intercept work as well to join
mathematics up.
Their science experiments may produce some data that can be treated
this way.
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Syllabus ref
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Learning objectives
Suggested teaching activities
Learning resources
Take scatter graphs from the press and ask them to describe the
correlation and write a headline for an article featuring the graph. e.g.
Ice-cream sales against months - Negative correlation. Headline ‘Ice
cream sales plummet in cold snap’ etc. A discussion of the veracity of
headlines and actual data can also be added here.
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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 10: Statistics – Extended curriculum
Recommended prior knowledge
All Core units and particularly Core Unit 10. Only those parts of the learning objectives or notes and examples not included in the Core units are itemised, so this
document should be read alongside Core Unit 10.
Context
There are two Core statistics units and this is the second of two Extended statistics units. Once the Core Unit 10 and the other prior experience for Core Unit 10 is
completed this unit can be slotted in at any point. It is probably best taught as a whole as there is a flow to the content.
Outline
The unit extends the knowledge of Core Unit 10 so be aware that examination questions relate even the aspects of Core Unit 10 may have a greater degree of
challenge as they combine with other areas of mathematics. This unit specifically covers, averages for grouped/continuous data, histograms, cumulative frequency
and using it to find interquartile range and median, box and whisker plots and comparing and justifying decisions based upon data.
Syllabus ref
and CCSS
10.1
Learning objectives
See Core curriculum
Suggested teaching activities
Learning resources
A useful source covering most of the unit with exercises.
Cumulative frequency:
www.cimt.plymouth.ac.uk/projects/mepre
s/book9/bk9_16.pdf
Notes and examples
Make inferences to support or cast doubt on initial conjectures, relate
results and conclusions to the original context.
General guidance
Work alongside the extension and core 10.3, 10.4 and 10.5 once the
skills of drawing or calculating have been mastered to make data
handling meaningful.
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Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
e.g. The taller you are the longer the index finger, or the length of the
index finger is related to the circumference of the wrist. Collect class
data and test.
Find a local issue in the press and use data to verify or otherwise an
unsubstantiated claim.
10.4
Mean, modal class, median,
and range from grouped and
continuous data.
Notes and examples
The term estimated mean may be used in questions involving grouped
continuous data.
Statistics and probability:
www.bbc.co.uk/schools/gcsebitesize/mat
hs/data/measuresofaveragerev1.shtml
General guidance
Modal Class
Past Paper 43 June 2011 Q6 bii
(syllabus 0580)
As this is only a name for the collective of a group rather than single
label data, if learners understand mode as the tallest bar, biggest slice
or item with the highest frequency in a frequency table they will not have
difficulty about identifying it just using the new label.
Past Paper 42 June 2011 Q6 a
(syllabus 0580)
Past Paper 41 June 2011 Q8 a
(syllabus 0580)
Medians from grouped data.
This relies on learners being able to identify the group in which the
median will occur.
Teaching activities
Ask learners to hold up 9 fingers and count in to find the middle finger
and then to find the 5 from the 9 as a rule. Then do the same with 10
fingers and find a rule for the middle pair. Doing it with the hands gives a
way of checking they have the right rule in an examination. As the
similarity between the rules for finding the middle data with odd and
even total numbers is one of the problems with the topic.
The next difficulty is convincing learners that even if they know the two
middle items of data are 49th and 50th in the list that the frequency table
has sorted the data in order. Asking learners to recreate the list from the
frequency table for single item data might be a way to convince them.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
Means from frequency tables.
Learners need to understand that the mean is the sum of the total data
and is divided by the total items. Therefore the first teaching point is to
ask learners to identify from list of data the number of items of data, the
sum of the data and hence the mean. The data is then collected into a
frequency tables and they are reminded of their previous calculation, but
have to work out how to get the two totals from the table. Show that the
answer is complete rubbish if the totals of either column are divided by
the number of rows.
Midpoint means (estimated means) for grouped data.
The technique is of course similar and the extra step is only to find the
midpoints. It might be interesting to do some from the frequency tables
and then to give them the raw data to see how good the estimate was.
Once learners are familiar with the techniques for calculating means
from tables give out some that have been miscalculated with a variety of
misconceptions and errors and ask learners to correct them. This may
include correcting labels and terms used in the statements.
10.5
CCSS:
S-ID1
Histograms with frequency
density on the vertical axis
Notes and examples
Includes histograms with unequal class intervals.
General guidance
The most difficult step is to distinguish between a bar chart and a
histogram. Some would debate whether a bar chart of continuous data
(with equal class intervals) isn’t just a special case of a histogram and
this might be one way into the topic.
The second question is why anyone would collect data that isn’t in equal
class intervals so why would you need a histogram? Often though
statisticians use historic data that may not have been collected with a
new analysis in mind. Census data collected every ten years doesn’t
always ask the same questions. So sometimes there is a twenty year
gap, in some data streams. The second resource deals with these
issues.
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Histograms:
mathsteaching.wordpress.com/2008/01/0
7/histograms/
Histograms and bar graphs:
www.shodor.org/interactivate/lessons/His
togramsBarGraph/
Statistics and probability:
www.bbc.co.uk/schools/gcsebitesize/mat
hs/data/representingdata3hirev3.shtml
Histograms:
www.gcsemathstutor.com/histograms.ph
p
115
Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
The key teaching is:
1. difference between bar charts and histograms
2. learning frequency density
3. plotting histograms
4. retrieving information from histograms
Past Paper 43 June 2011 Q6 biii
(syllabus 0580)
Past Paper 41 June 2011 Q8 bii
(syllabus 0580)
The ‘mathsteaching’ link provides four links the first and fourth are free
and have some examples of histogram questions.
The ‘shodor’ resource has a model lesson and the other two sites have
the facts.
Between them there are plenty of examples to try.
10.6
CCSS:
S-ID1
S-ID2
Cumulative frequency table and
curve and box plots
Median, quartiles, percentiles,
and inter-quartile range
General guidance
It seems trivial but when learners create the cumulative frequencies prior
to plotting they need to change the group names from the grouped
frequency table. This skill in itself will help them understand cumulative
frequency.
Ensure learners understand how to label the horizontal axis correctly for
the cumulative frequency and know it is a curve not a series of points
joined by straight lines. Seeing the elongated ‘S’ shape is not easy for
some learners.
Statistics and probability:
www.bbc.co.uk/schools/gcsebitesize/mat
hs/data/representingdata3hirev4.shtm
CensusAtSchool:
www.censusatschool.org.uk/getdata/results/phase10
Past Paper 42 June 2011 Q6 b and c
(syllabus 0580)
Learners need to understand it is the total of the data that is split into 4
equal sections not the end number written on the vertical axis. They
must also realise the necessity of making the lines across the graph
exactly horizontal to find the point to drop down to the horizontal axis.
They must also understand that it is the points on the horizontal axis that
they read off not the coordinates of the point on the graph or the vertical
percentile points.
Box and Whisker plots are best drawn below the horizontal axis on the
same scale to make the links clear.
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Syllabus ref
and CCSS
Learning objectives
Suggested teaching activities
Learning resources
If learners are comparing two distributions and not drawing on the same
axes then they should make them the same scale.
Cumulative frequency is straight, its associated measures of median and
interquartile range are straightforward if learners learn the steps and
follow them through – interpreting and understanding the values is more
difficult.
Making a statement about height or weight and using the large data
sources at ‘census at school’, selecting year groups for comparison and
drawing cumulative frequency graphs with box and whisker plots can
give some purpose to the activities. It could then be compared to school
data. Use it to select samples by a legitimate sampling technique.
Though this isn’t in the syllabus it gives further meaning to the topic.
10.7
CCSS:
S-ID2
S-ID3
Use and interpret statistics
appropriate to the shape of the
data distribution to compare
centre (median, mean) and
spread
(inter-quartile range) of two or
more different data sets
Teaching activities
Making a statement about height or weight and using the large data
sources at ‘census at school’, selecting year groups for comparison and
drawing cumulative frequency graphs with box and whisker plots can
give some purpose to the activities. It could then be compared to school
data. Use it to select samples by a legitimate sampling technique.
CensusAtSchool:
www.censusatschool.org.uk/getdata/results/phase10
Have a purposeful (meaningful for learners) question to answer that can
use secondary data sources to provide the data. Learners select the
graphs, and other stats to write a convincing argument. A group project
rather than individual ones would be better. Justifying choice of
graphical and statistical presentation should be included. The resulting
reports are presented to the class and the other groups, question or
make a critique of the presented case.
Learners create a display to explain the stats included in this unit to
another cohort of learners.
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