Transcript Lurking algebra 2014

Anne Watson
Winchester 2014
LURKING ALGEBRA
BIG ISSUES (FOR TODAY)
Algebra
 Division
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WHAT IS ALGEBRA?
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What are the pre-algebraic and algebraic
experiences appropriate for primary children?
ACME COMMENTS
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
NOT X
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
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Generalise relationships
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Equivalent expressions
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graphing x + y = 5
New from old
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j–a=5
Relate representations
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what is y if 11 = 9 – y ?
Express situations
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a+b–a=b
Solve equations
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perimeter of a square of side s is 4s
2(a + b)=2a + 2b
Notation
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all the above
EXPLICIT STATEMENTS ABOUT ALGEBRA YR 6
Programme of study:
 use simple formulae
 generate and describe linear number
sequences
 express missing number problems
algebraically
 find pairs of numbers that satisfy equations
involving two unknowns.
 enumerate possible 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:
 use simple formulae
 generate and describe linear
number sequences
 express missing number
problems algebraically
 find pairs of numbers that
satisfy equations involving
two unknowns.
 enumerate 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 THOUGHTS/CONCERNS
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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!
 Or too early!
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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
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Year 3
 mental
methods
 commutativity and associativity
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Generalise
Equivalence
Solve
Express
Representations
New from old
Notation
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
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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ALGEBRA YR 6
Programme of study:
 use simple formulae
 generate and describe linear number
sequences
 express missing number problems
algebraically
 find pairs of numbers that satisfy an
equation involving two unknowns.
 enumerate possibilities of combinations of
two variables
Anne Watson
Winchester 2014
THE MANY FACES AND PLACES OF
DIVISION
DIVISION
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What is division?
RODS, TUBES AND SWEETS
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How many logs of length 60cm. can I cut from a long log
of length 240 cm?
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How many bags of 15 sweets can I make from a pile of
120 sweets?
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I have to cut 240 cm. of copper tubing to make 4 equal
length tubes. How long is each tube?
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I have to share 120 sweets between 8 bags. How many
per bag?
Three equal volume bottles of wine have to
be shared equally between 5 people. How
can you do this and how much will each get?
 Three equal sized sheets of gold leaf have to
be shared equally between 5 art students,
and larger sheets are more useful than small
ones. How can you do this and how much
will each get?
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98 equal volume bottles of wine have to be
shared equally between 140 people. How
can you do this and how much will each get?
 98 equal sized sheets of gold leaf have to be
shared equally between 140 art students,
and larger sheets are more useful than small
ones. How can you do this and how much
will each get?
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A piece of elastic 10 cm. long with marks at
each centimetre is stretched so that it is now
50 cm. long. Where are the marks now?
 A piece of elastic is already stretched so that it
is 100 cm. long and marks are made at 10 cm.
intervals. It is then allowed to shrink to 50 cm.
Where are the marks now?
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Sharing out by counting, as we do with chocolate
buttons (and eating the spares)
Sharing out by cutting up congruent shapes, as we do
with a cake or pizza
Sharing out by counting and cutting, as we do if sharing
three cup cakes between five people
Sharing by pouring, as with wine
Folding and cutting, as with ribbon
Folding and cutting, as with a piece of paper
Finding how many of X ‘go into’ Y with physical objects
by fitting
Finding how many of X ‘go into’ Y with linear measures
(e.g. how many centimetres in a metre?)
Finding how many Xs ‘go into’ Y with numbers by
counting, such as counting in 2s, 3s, 10s and so on
Grouping objects in 2s, 3s, 5s and so on.
DIVISION AS INVERSE OF AREA MODEL FOR
MULTIPLICATION
Long Division – Part 1 on Vimeo (15 mins)
 http://vimeo.com/45986110
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WHOLE SCHOOL DEVELOPMENT
Collaboration across years and key stages
 Coherent development throughout school
 Something relevant every week
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