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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 2: Algebra (Core)
Recommended prior knowledge
Students 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.
Students who are following the extended syllabus will move through this faster but need to have all these skills in place before working on the extended units, or
applying them in other areas of mathematics.
Students 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
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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
Learning objectives
Suggested teaching activities
Learning resources
General guidance
The framework document gives an overview of all the steps for the
development in algebra, leading from arithmetic into algebra. It may be useful
in unpicking student misconceptions when working with algebra.
www.counton.org/resources/ks3framework
/pdfs/equations.pdf
The interacting mathematics bank of documents 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.
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
http://people.umass.edu/~clement/pdf/Intui
tive%20Misconceptions%20in%20Algebra
.pdf
www.springerlink.com/content/557502366l
86518p/
The two remaining documents are articles about the misconceptions pupils
have about algebra.
Many 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.
2.7
CCSS:
A-SSE1
Identify terms,
factors, and
coefficients
General guidance
Throughout the unit ensure correct use of vocabulary associated with
algebra.
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
There is an activity in the year 8 booklet Page 6, with the answers and likely
problems students 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
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
students to sort by a rule like all the statements where ‘a’ has a coefficient of
3, or all the expressions with 2 terms.
2.4
Exponents (indices)
Notes and exemplars
Includes rules of indices with negative indices.
Simple examples only,
e.g., q 3 × q –4, 8x 5 ÷ 2x 2
General guidance
Check students 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 students will take an expression
expanded to 2 x 5 x 5 x 5 and will turn it into 2 x 15 = 30. Students 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.
www.counton.org/resources/ks3framework
/pdfs/equations.pdf page 115
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 students 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.
2.5
CCSS:
A-CED5
Rearrangement and
evaluation of simple
formulae
Notes and exemplars
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.
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.
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
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
students 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 exemplars
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
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.
Pupils 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.
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc Year 8 booklet pages 7 and
8
www.mmlsoft.com/index.php?option=com
_content&task=view&id=9&Itemid=10
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 students 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).
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
2.9
Factorization:
common factor only
Notes and exemplars
Past Paper 31 June 2011 Q3
(syllabus 0580)
CCSS:
A-SSE2
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
Continuation of a
sequence of numbers
or patterns; recognize
patterns in
sequences;
generalize to simple
algebraic statements,
including
determination of the
nth term
?
?
4a
2
14a
Notes and exemplars
e.g., find the nth term of:
• 5 9 13 17 21
• 2 4 8 16 32
General guidance
The first link spreadsheet to download generates sequences which can be
modelled. The teacher notes indicate the wrong cells. Once the sheet is
unprotected the arrows give you the two cells.
This is a good route into the ‘differences and check a term’ method as it
allows modelling.
www.teachfind.com/nationalstrategies/algebra-%E2%80%93sequences-functions-and-graphs
www.teachfind.com/national-strategies/ictsupporting-mathematics-fibs-and-truthslesson-notes
www.teachfind.com/nationalstrategies/fibs-and-truths
The second resource fibs and truths has both a lesson plan and the
spreadsheet it looks at Fibonacci.
Teaching activities
Give the students 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
v1 2Y01
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
and to discuss ways of describing the sequence. Some of these obviously
have nth term rules beyond the mathematics of core students 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.
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 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
Create expressions
and create and solve
linear equations,
including those with
fractional expressions
Notes and exemplars
Explain each algebraic step of the solution.
May be asked to interpret solutions to a problem given in context.
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
referred to in the text is on page 124 of the framework document.
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?’
Students 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.
v1 2Y01
Cambridge IGCSE Mathematics (US) (0444)
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc year 9 page 9 booklet
http://nrich.maths.org/2290
Past Paper 32 June 2011 Q10
(syllabus 0580)
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-constructing-and-solving-linearequations-sc
www.counton.org/resources/ks3framework
/pdfs/equations.pdf page 124
www.algebralab.org/lessons/lesson.aspx?f
ile=Algebra_OneVariableWritingEquations
.xml
Past Paper 32 June 2011 Q4a
(syllabus 0580)
Past Paper 31 June 2011 Q3
(syllabus 0580)
6
Syllabus ref
Learning objectives
Suggested teaching activities
a
Learning resources
x 2
+ 3
16
(a + 3) x 2 = 16
2(a + 3) = 16
- 3
5
÷ 2
8
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. Students will naturally start to change by bigger
steps and eventually develop the balancing method naturally.
Explore all the ways of altering the original equation to make a family of equivalent equations
subtracting 1 from each side
multiplying both sides by 2
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
adding b to both sides
3a + 3 = 15
2a + 2 = 14 - a
a + 2 = 14 - 2a
3a + 4 = 16
adding 1 to both sides
subtracting a from both sides
Be aware the number line method though it adds depth to pupil
understanding only works for positive solutions but it is away into modelling
the balance method that has more resonance in these days of digital rather
than balance scales.
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Cambridge IGCSE Mathematics (US) (0444)
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
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’ link gives a bank of problems that can be turned into algebra
to solve.
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.
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 students 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
students.
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