Transcript Working algebraically 2012

Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Working algebraically 5-19
Anne Watson
South West, 2012
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
What is algebra?
• What are the pre-algebraic experiences appropriate for primary
children?
Advisory
Committee on
Mathematics
Education
Hypothetical ......
....... but still a valuable exercise
www.acme-uk.org
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Sources
• Draft curriculum
• ACME synthesis of responses from mathematics education
community
• Research (e.g. nuffieldfoundation.org.uk)
• Glimpse of possible KS4 content
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Where are we going with algebra for
everyone (KS4)?
• arithmetic sequences (nth term)
• algebraic manipulation including expanding products,
factorisation and simplification of expressions
• solving linear and quadratic equations in one variable
• application of algebra to real world problems
• solving simultaneous linear equations and linear inequalities
• gradients
• properties of quadratic functions
• using functions and graphs in real world situations
• transformation of functions
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
To summarise
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Generalising
Notation
Equivalent expressions
Solving equations (finding values of variables)
Expressing real and mathematical situations algebraically
(recognising additive, multiplicative and exponential relations)
• Relating features of graphs to properties of functions and
situations (e.g. gradient of straight line)
• New relations from old
Advisory
Committee on
Mathematics
Education
Key ideas
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Generalise relationships
Notation
Equivalent expressions
Solve equations
Express situations
Relate representations
New from old
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Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Additive reasoning
a+b=c
b+a=c
c–a=b
c–b=a
c=a+b
c=b+a
b=c- a
a=c- b
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Multiplicative reasoning
a = bc
a = cb
b=a
c
c=a
b
bc = a
cb = a
a=b
c
a=c
b
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Draft primary national curriculum
structure
• Aims
• Statements: programme of study (left: statutory)
• Notes and guidance: to support pedagogy (right: non-statutory
but ....)
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Explicit statements about algebra
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POS year 6
Pupils should be taught to:
– solve linear missing number problems, including those involving decimals
and fractions, and find pairs of number that satisfy number sentences
involving two unknowns [280]
– use simple formulae expressed in words [281]
– generate and describe linear number sequences including those involving
negative and decimal numbers, and proper fractions e.g. 1.4, 1.1, 0.8. [282]
Notes and Guidance year 6
– Ensure pupils write some known arithmetical rules algebraically, such as a
+ b = b + a, and known relations such as p = 4s for the perimeter of a
square. They should also interpret word problems as statements about
number and record as a mathematical statement. [283]
– Pupils should also write missing number problems algebraically; for
example, 2x – 4 = 8 therefore 2x = 12 therefore x = 6 or finding missing
lengths in perimeters and missing angles at a point. Pupils should also find
possible solutions for equations with two unknown variables, for example
x + y = 5 includes solutions x = 1 and y = 4, x = 2 and y = 3. [284]
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Your immediate thoughts/concerns?
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
My immediate thoughts/concerns
• solving linear equations with non-integer solutions is hard and
algebraic methods take time to learn
• why are sequences here?
• word formulae have occurred throughout the rest of the document, so
what is extra here and where is the progression?
• does this build on what children already know?
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Searching for hidden pre-algebra using the
key ideas
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Generalise relationships
Notation
Equivalent expressions
Solve equations
Express situations
Relate representations
New from old
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Searching for hidden algebra in the primary
draft curriculum, yrs 1-2
• number bonds in the three forms (e.g. 9 +
7 = 16; 16 – 7 = 9; 16 – 9 = 7)
• using related facts to perform calculations
(e.g. using 3 + 7 = 10, 10 - 7 = 3 and 10 – 3
= 7 to calculate 30 + 70 = 100, 100 - 70 = 30
and 100 – 30 = 70)
• check calculations, including by adding to
check subtraction and adding numbers in
a different order to check addition; for
example 5 + 2 + 1 = 1 + 5 + 2 = 1 + 2 + 5
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Hidden in years 3-4
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commutativity (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240) and
multiplication and division facts (e.g. using 3 x 2 = 6, 6 ÷ 3 = 2 and
2 = 6 ÷ 3 to calculate 30 x 2 = 60, 60 ÷ 3 = 20 and 20 = 60 ÷ 3).
count in multiples of 2, 3, 4, 5, 6, 7, 8, 9, 10, 25, 50, 100 and 1000
from any given number
complete number sequences
derive facts, for example 300 x 2 = 600 into 600 ÷ 3 = 200
use the distributive law to derive facts, for example, 30 x 7 + 9 x 7=
39 x 7.
count forwards and backwards with positive and negative whole
numbers through zero
write and use pairs of coordinates, e.g. (2, 5)
practise recognising line symmetry in a variety of diagrams
one or more lengths have to be deduced using properties of the
shape.
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
Hidden in years 5-6
• calculation of perimeter of composite shapes
• order of operations include the use of brackets, for example: 2 + 1 x 3
= 5 and (2 + 1) x 3 = 9.
• relationship between unit fractions and division to work backwards;
for example, if ¼ of a length is 36cm then the whole length is 36 x 4 =
144cm.
• derive unknown angles and lengths from known measurements.
• use all four quadrants, including the use of negative numbers
• quadrilaterals specified by coordinates in the four quadrants
Advisory
Committee on
Mathematics
Education
www.acme-uk.org
What else do you currently teach that
feeds in to algebra?
Advisory
Committee on
Mathematics
Education
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ACME comments (what is ACME?)
PoS does not provide a coherent progression towards formal
algebra
Procedural approach discourages thinking before acting
Expectations of algebraic thinking could be even more challenging
if they were based on reasoning about relations between
quantities, such as patterns, structure, equivalence,
commutativity, distributivity, and associativity, and models and
representations of these
Early introduction of formal algebra before secondary school can
lead to poor understanding without a good foundation
Algebra in primary connects what is known about number
relations in arithmetic to general expression of those relations,
including unknown quantities and variables. Operations need to
be thoroughly understood in order to make this connection.
The only generalising currently explicit in the PoS relates to
pattern sequences and not other forms of relations
A coherent developmental strand for algebra should be made
explicit, making clear the connections between knowledge of
number, mental methods, generalizing, and representing relations
between quantities and unknowns