Download Key Ideas in NC algebra 2013

Anne Watson
South West 2013
KEY IDEAS IN THE NATIONAL CURRICULUM:
WHAT’S X GOT TO DO WITH IT?
WHAT IS ALGEBRA?
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What are the pre-algebraic experiences
appropriate for primary children?
HEALTH WARNING
The Secretary of State and Ministers for
Education are legally entitled to make changes
without consultation and without giving
reasons
 My role on the panel and here
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PRIMARY NATIONAL CURRICULUM
STRUCTURE
Aims: fluency, reasoning, problem-solving
 Statements: programme of study (statutory
list of content: mainly things to do)
 Notes and guidance: to support pedagogy
and progression (non-statutory)
 Two-year chunks
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SOURCES FOR TODAY
Draft curriculum
 ACME synthesis of responses from
mathematics education community
 Research (e.g. nuffieldfoundation.org.uk)
 Possible GCSE content
 Mathematics
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ACME COMMENTS (WHAT IS ACME?)
Expectations of algebraic thinking could be
based on reasoning about relations between
quantities, such as patterns, structure,
equivalence, commutativity, distributivity, and
associativity
 Early introduction of formal algebra can lead to
poor understanding without a good foundation
 Algebra connects what is known about number
relations in arithmetic to general expression of
those relations, including unknown quantities
and variables.
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WHERE ARE WE GOING WITH ALGEBRA FOR
EVERYONE? FROM KS4:
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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
TRANSFORMED THINKING ABOUT ALGEBRA
Generalising relations between quantities
 Equivalence: different expressions meaning the
same thing
 Solving equations (finding particular values of
variables for particular states)
 Expressing real and mathematical situations
algebraically (recognising additive,
multiplicative and exponential relations)
 Relating features of graphs to situations (e.g.
gradient of straight line)
 New relations from old
 Standard notation
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KEY IDEAS
Generalise relationships
 Equivalent expressions
 Solve equations
 Express situations
 Relate representations
 New from old
 Notation
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EXPLICIT STATEMENTS ABOUT ALGEBRA YR 6
(HEALTH WARNING)
Programme of study:
 express missing number problems
algebraically
 use simple formulae expressed in words
 generate and describe linear number
sequences
 find pairs of numbers that satisfy number
sentences involving two unknowns.
 enumerate all possibilities of combinations
of two variables
NON-STATUTORY GUIDANCE YR 6
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Pupils should be introduced to the use of symbols
and letters to represent variables and unknowns in
mathematical situations that they already
understand, such as:
missing numbers, lengths, coordinates and angles
 formulae in mathematics and science
 arithmetical rules (e.g. a + b = b + a)
 generalisations of number patterns
 number puzzles (e.g. what two numbers can add up to).
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YOUR IMMEDIATE THOUGHTS AND CONCERNS?
Programme of study:
 express missing number
problems algebraically
 use simple formulae
expressed in words
 generate and describe
linear number sequences
 find pairs of numbers that
satisfy number sentences
involving two unknowns.
 enumerate all possibilities
of combinations of two
variables
Notes and guidance:
Pupils should be introduced to the
use of symbols and letters to
represent variables and unknowns
in mathematical situations that they
already understand, such as:
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missing numbers, lengths,
coordinates and angles
formulae in mathematics and
science
arithmetical rules (e.g. a + b = b + a)
generalisations of number patterns
number puzzles (e.g. what two
numbers can add up to).
MY IMMEDIATE THOUGHTS/CONCERNS

How can this build on what children already
know?
 missing
number problems
 simple formulae expressed in words
 linear number sequences
 number sentences involving two unknowns
 combinations of two variables
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What do you do already? Year 6 is too late!
SEARCHING FOR HIDDEN PRE-ALGEBRA USING
THE KEY IDEAS
Generalise relationships
 Equivalent expressions
 Solve equations
 Express situations
 Relate representations
 New from old
 Notation
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SEARCHING FOR HIDDEN ALGEBRA IN THE
PRIMARY DRAFT CURRICULUM, YRS 1-2
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Year 1
counting as enumerating objects
patterns in the number system
repeating patterns
number bonds in several forms
add or subtract zero.
Generalise
Equivalence
Solve
Express
Representations
New from old
Notation
Year 2
add to check subtraction (inverse)
add numbers in a different order (associativity)
inverse relations to develop multiplicative reasoning
ENUMERATION
12 = 3 lots of 4
 12 = 4 lots of 3
 12 = two groups of 6
 12 = 6 pairs
 12 = 2 lots of 5 plus two extra
c
c
 c= ab = ba = 2( ) = 2( - 1) + 2 etc.
2
2
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DIFFERENT KINDS OF PATTERN
Repeating
a, b, b, a, b, b, ......
(3n+1)th square is red
Continuing (arithmetic, linear ...)
1, 4, 7, 10 ....
Spatial
(nth term is 3n+1)
ADDITIVE REASONING
a+b=c
b+a=c
c–a=b
c–b=a
Generalise
Equivalence
Solve
Express
Representations
New from old
Notation
c=a+b
c=b+a
b=c- a
a=c- b
MULTIPLICATIVE REASONING
a = bc
a = cb
b=a
c
c=a
b
bc = a
cb = a
a=b
c
a=c
b
Generalise
Equivalence
Solve
Express
Representations
New from old
Notation
HIDDEN IN YEARS 3-4
Generalise
Equivalence
Solve
Express
Representations
New from old
Notation
Year 3
 mental methods
 commutativity and associativity
 Year 4
 write statements about the equality of expressions
(e.g. use the distributive law 39 × 7 = 30 × 7 + 9 ×
7 and associative law (2 × 3) × 4 = 2 × (3 × 4))
 write and use pairs of coordinates, e.g. (2, 5)
 one or more lengths have to be deduced using
properties of the shape
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HIDDEN IN YEARS 5-6
Generalise
Equivalence
Solve
Express
Representations
New from old
Notation
perimeter of composite shapes
 order of operations
 relate unit fractions and division.
 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
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WHAT ELSE DO YOU CURRENTLY TEACH THAT
FEEDS IN TO ALGEBRA?