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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 2: Algebra (Extended)
Recommended prior knowledge
All of Core and particularly Unit 2. Only those parts of the learning objectives or notes and exemplars not included in the core units are itemised, so this document
should be read alongside the core document. It is also important that Core 1 and 3 and Extended 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 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 Unit 8 Extended.
Outline
The unit extends the knowledge of Core 2 so be aware that examination questions that relate to aspects of Core 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
Learning objectives
Suggested teaching activities
Learning resources
2.1
Writing, showing, and
interpretation of
inequalities on the real
number line
General guidance
Students 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.
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 students give a key
this should clarify for any audience.
www.counton.org/resources/ks3framework
/pdfs/equations.pdf page 131
Students 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.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
1
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
Teaching activities
Practice 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.
2.2
Create and solve
linear inequalities
CCSS:
A-CED1
A-CED2
A-REI3
Notes and exemplars
e.g., Solve 3x + 5 < 7
Solve –7 ≤ 3n – 1 < 5
General guidance
Once the balancing method is understood for solving equations then the
same steps can be used for inequalities.
Students need to understand why the inequality is reversed for division by a
negative number
Teaching activities
Use statements like -3x< 6 and substitute integer values back into the
inequality to convince students of the need to change the direction of the
inequality.
2.4
CCSS:
A-SSE3
Exponents (indices)
Notes and exemplars
Includes rules of indices with negatives and
fractional indices.
3
e.g., simplify 2x 2 X 5x -4
www.counton.org/resources/ks3framework
/pdfs/equations.pdf page 131
www.algebra-class.com/solving-wordproblems-in-algebra.html
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_OneVariableSolvingInequalitie
s.xml
Past Paper 42 June 2011 Q5a
(syllabus 0580)
Past Paper 23 June 2011 Q10
(syllabus 0580)
Past Paper 41 June 2011 Q9
(syllabus 0580)
Past Paper 41 June 2011 Q3c
(syllabus 0580)
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_ExponentsRules.xml
Past Paper 22 June 2011 Q17
(syllabus 0580)
General guidance
This is the generalization of Core unit 1, and Extended unit 1 (1.7)
Students should be able to solve or simplify using the rules of indices and to
substitute values in the simplified expressions.
Teaching activities
Practice writing the statements in radical form as well.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
2
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
2.5
Rearrangement and
evaluation of
formulae.
Notes and exemplars
Formula may include indices or cases where the subject appears twice.
e.g., make r the subject of
• V = 43 Π r 3
http://mash.dept.shef.ac.uk/RearrangingFo
rmulae.html
CCSS:
A-CED4
•
p=
2r -s
r +s
e.g., y = m2 – 4n2
Find the value of y when m = 4.4 and n = 2.8
Past Paper 22 June 2011 Q11
(syllabus 0580)
Past Paper 21 June 2011 Q16b
(syllabus 0580)
Past Paper 21 June 2011 Q8
(syllabus 0580)
General guidance
Students 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.
Students need a lot of experience to hone this skill.
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 students
to find as many rearranged forms as they can, making all the letters the
subject and with different numbers of terms on each side.
2.6
CCSS:
A-CED2
A-REI5
A-REI6
2.7
Create and solve
simultaneous linear
equations in two
variables graphically
Interpret algebraic
expressions
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.
Notes and exemplars
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 exemplars
e.g. interpret P (1 + r )n as the product of P and a factor not depending on P.
CCSS:
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
3
Syllabus ref
Learning objectives
A-SSE1
2.8
Suggested teaching activities
Learning resources
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 write except when
expansion is needed.
Expansion of
parentheses, including
the square of a
binomial. Simplify
expressions
It is unfortunate that P is used here and if used by pupils they should
understand the difference between the use of P here, in a probability
statement and as an indication of a function statement. They will need to look
at the context to distinguish.
Notes and exemplars
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 students keep the sign with the elements
of the expression.
(2a -3)(-3a + 4)
2a
-3
-3a
-6a2
9a
4
8a
-12
x
-6a2 + 9a + 8a -12
2
= -6a +17a -12
2.9
CCSS:
A-SSE2
Factorization:
difference of squares
trinomial
four term
Notes and exemplars
9x 2 – 16y 2 = (3x – 4y )(3x + 4y )
6x 2 + 11x – 10 = (3x – 2)(2x + 5)
xy – 3x + 2y – 6 = (x + 2)(y – 3)
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 students 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.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
4
Syllabus ref
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
?
?
2.10
Algebraic fractions:
simplification,
including use of
factorization
addition or subtraction
of fractions with linear
denominators
multiplication or
division and
simplification of two
fractions
-3, -2, -1,
-6, -4, -12,
1, 2, 3,
4, 6, 12
-6a2
-12
Practice a number as a quick starter over a period of weeks to give students
enough experience to become familiar with the likely components to use.
Notes and exemplars
2
4x - 9
3
4 7x 21x
÷
e.g. simplify
,
- ,
8
8x 2 -10x - 3 2x +1 x 4y 2
Past Paper 22 June 2011 Q15
(syllabus 0580)
Past Paper 23 June 2011 Q16
(syllabus 0580)
General guidance
A number of skills must be brought together having been understood and
mastered separately for success with this topic
Students 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
Teaching activities
Once the topic has been covered practice a number of examples as a quick
starter over a period of weeks to give students enough experience to become
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
5
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
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 exemplars
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.
Students 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:
F-BF2
v1 2Y01
Solve simple rational
and radical equations
in one variable,
and discount any
extraneous solutions
3. Functions—
Extended curriculum
Notes / Exemplars
Continuation of a
sequence of numbers
or patterns; recognise
patterns in
sequences; generalize
to simple algebraic
Teaching activities
Solving a particular equation by factorizing, completing the square and the
formula could promote discussion on this.
Notes and exemplars
e.g. solve x + 2 = 6, x–3 = 27, 2y 4 = 32
General guidance
Students 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 exemplars
•
www.cimt.plymouth.ac.uk/projects/mepres/
book9/bk9i10/bk9_10i3.html
2 5 10 17 26
General guidance
Use the method of second differences to find the nth term
Cambridge IGCSE Mathematics (US) 0444
Past Paper 42 June 2011 Q9
(syllabus 0580)
6
Syllabus ref
2.14
Learning objectives
Suggested teaching activities
Learning resources
statements, including
determination of the
nth term
e.g.
Past Paper 43 June 2011 Q11
(syllabus 0580)
Past Paper 41 June 2011 Q10
(syllabus 0580)
Express direct and
inverse variation in
algebraic terms and
use this form of
expression to find
unknown quantities
Teaching activities
Set up a number of quadratic sequences in a spreadsheet and ask students
to use the above model to find the nth term.
Notes and exemplars
e.g. y ∝ x, y ∝ x , y ∝ 1/x, y ∝ 1/x 2
General guidance
Students must understand proportionality before they can tackle variation.
They must also understand how to find the k by substituting coordinates in
the equation.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
www.algebralab.org/lessons/lesson.aspx?f
ile=algebra_conics_inverse.xml
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)
7