Transcript Key Stage 3 National Strategy Mathematics

Key Stage 3
National Strategy
Foundations of
algebra
© Crown copyright 2004
Developing algebra in Key Stage 3:
overall objectives
• To explore the teaching of constructing and
solving linear equations by:
– considering the rationale for teaching algebra
– developing a range of effective teaching approaches
– reflecting on research evidence about the teaching of
algebra
• To plan progression from Y7 to Y9 in constructing
and solving linear equations
Key Stage 3 National Strategy Mathematics
© Crown copyright 2004
1.1
Objectives
• To consider pupils’ perceptions of algebra
• To consider the distinctive features of algebra
in KS3
• To consider the ‘why’ and the ‘how’ of teaching
algebra
Key Stage 3 National Strategy Mathematics
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1.2
As many ways as you can
Perimeter = 12 + 7 + 12 + 7
Walking round the shape
Perimeter = 12 + 12 + 7 + 7
Pairing up lengths and widths
Perimeter = 12 × 2 + 7 × 2
Length-doubled plus
width-doubled
Perimeter = 2 × 12 + 2 × 7
Double-length plus
double-width
Perimeter = (12 + 7) × 2
Length-plus-width, doubled
Perimeter = 2 × (12 + 7)
Two lots of length-plus-width
Key Stage 3 National Strategy Mathematics
© Crown copyright 2004
1.3
An opportunity to generalise
Perimeter = l + w + l + w
Walking round the shape
Perimeter = l + l + w + w
Pairing up lengths and widths
Perimeter = l × 2 + w × 2
Length-doubled plus
width-doubled
Perimeter = 2 × l + 2 × w
Double-length plus
double-width
Perimeter = (l + w) × 2
Length-plus-width, doubled
Perimeter = 2 × (l + w)
Two lots of length-plus-width
Key Stage 3 National Strategy Mathematics
© Crown copyright 2004
1.4
Generating expressions and equations
Key Stage 3 National Strategy Mathematics
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Pupils’ perceptions of algebra
Pupils sometimes perceive algebra as difficult.
What do your pupils struggle with most?
Key Stage 3 National Strategy Mathematics
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Why do we teach algebra?
Mason and Sutherland argue that:
• Algebra is fundamental to our way of life
– It provides a mathematical language to express
economic activity
– It underpins models that inform national policy
decisions
– It underpins databases and spreadsheets
• Algebra allows individuals to discover, establish and
express generality
Key Stage 3 National Strategy Mathematics
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1.7
Adding two odd numbers
Key Stage 3 National Strategy Mathematics
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Distinctive features of algebra in KS3
• Developing understanding that algebra is a way of
generalising from arithmetic, from particular cases
or from patterns and sequences
• Providing regular opportunities to construct
algebraic expressions and formulae and to
transform one expression into another – collecting
like terms, taking out common factors, working with
inverses, solving linear equations
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Distinctive features of algebra at KS3
• Using opportunities to represent a problem and its
solution in tabular, graphical or symbolic form, using
a graphical calculator or a spreadsheet where
appropriate, and to relate solutions to the context of
the problem
• Developing algebraic reasoning, including an
appreciation that while a number pattern may
suggest a general result, a proof is derived from the
structure of the situation being considered
Key Stage 3 National Strategy Mathematics
© Crown copyright 2004
1.10
Four-digit numbers
Write down a four-digit number.
Reverse the digits and add the two numbers together.
Is the answer divisible by 11?
Does this work for any four-digit number?
Can you prove it using algebra?
Key Stage 3 National Strategy Mathematics
© Crown copyright 2004
1.11
Symbol sense
‘Symbol sense’ includes:
• an aesthetic feel for the power of symbols –
understanding how and when symbols can and
should be used to display relationships,
generalisations and proofs which otherwise are
hidden and invisible
• an ability to manipulate and to read symbolic
expressions as two complementary aspects of
solving algebraic problems
(Adapted from Arcavi, 1994, p. 31)
Key Stage 3 National Strategy Mathematics
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Key principles in teaching algebra
1. Providing opportunities for pupils to express
generality
2. Asking pupils to ‘find as many ways as you can’
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Principle 1: Providing opportunities for
pupils to express generality
Benefits:
• Pupils appreciate the purpose of algebra
• Pupils are better able to understand the meaning of
expressions if they have generated some for themselves
• Knowing how expressions are built up helps to clarify the
process of ‘undoing’, needed when solving equations
Key Stage 3 National Strategy Mathematics
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Principle 2: Asking pupils to ‘find as many
ways as you can’
Benefits:
• Pupils appreciate that the same general relationship can be
expressed in more than one way
• They manipulate expressions to demonstrate that one
expression is equivalent to another
• They experience forming and transforming expressions in
different ways
• They have opportunities to discuss which transformations are
the most efficient to use in a particular context, e.g. when
solving an equation
Key Stage 3 National Strategy Mathematics
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1.15
True or false?
If 6x + 8 = 4x + 11, which of the following statements
are true?
14x = 15x
6x + 4x = 11 + 8
6x = 4x + 3
2x + 8 = 11
2x = 3
–3 = –2x
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