Anne Watson Winchester 2014 Big issues for today Algebra Division What is algebra What are the prealgebraic and algebraic experiences appropriate for primary children ACME comments Expectations of algebraic thinking could be based on reasoning about relations between quantities such a ID: 542067
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Slide1
lurking algebra
Anne Watson
Winchester 2014Slide2
Big issues (for today)
Algebra
DivisionSlide3
What is algebra?
What are the pre-algebraic and algebraic experiences appropriate for primary children?Slide4
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. Slide5
Where are we going with algebra for everyone? from 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 functionsSlide6
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 notationSlide7
Key ideas
Generalise relationships
perimeter of a square of side s is 4s
Equivalent expressions
a + b – a = b
Solve equations
what is y if 11 = 9 – y ?
Express situations
j – a = 5
Relate representations
graphing x + y = 5
New from old
2(a + b)=2a + 2b
Notation
all the aboveSlide8
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 Slide9
Non-statutory Guidance yr 6
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). Slide10
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:
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). Slide11
My 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
What do you do already? Year 6 is too late!
Or too early!Slide12
Searching for hidden pre-algebra using the key ideas
Generalise relationships
Equivalent expressions
Solve equations
Express situations
Relate representations
New from old
NotationSlide13
Searching for hidden algebra in the primary draft curriculum, yrs 1-2
Year 1
counting as enumerating objects
patterns in the number system
repeating patterns
number bonds in several forms
add or subtract zero.
Year 2
add to check subtraction (inverse)
add numbers in a different order (
associativity
)
inverse relations to develop multiplicative reasoning
Generalise
Equivalence
Solve
Express
Representations
New from old
NotationSlide14
enumeration12 = 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=
ab
=
ba
= 2( ) = 2( - 1) + 2 etc.Slide15
different kinds of pattern
a, b, b, a, b, b, ...... (3n+1)
th
square is red
Repeating
Continuing (arithmetic, linear ...)
Spatial
1, 4, 7, 10 .... (nth term is 3n+1) Slide16
Additive reasoning
a + b = c
c
= a + b
b + a = c
c
= b + a
c – a = b
b
= c - a
c – b = a
a
= c - b
Generalise
Equivalence
Solve
Express
Representations
New from old
NotationSlide17
Multiplicative reasoning
a =
bc
bc
= a
a =
cb
cb
= a
b =
a
a
= b
c
c
c =
a
a
= c
b
b
Generalise
Equivalence
Solve
Express
Representations
New from old
NotationSlide18
Hidden in years 3-4
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
Generalise
Equivalence
Solve
Express
Representations
New from old
NotationSlide19
Hidden in years 5-6
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
Generalise
Equivalence
Solve
Express
Representations
New from old
NotationSlide20
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 Slide21
The many faces and places of division
Anne Watson
Winchester 2014Slide22
divisionWhat is division?Slide23
Rods, tubes and sweets
How many logs of length 60cm. can I cut from a long log of length 240 cm?
How many bags of 15 sweets can I make from a pile of 120 sweets?
I have to cut 240 cm. of copper tubing to make 4 equal length tubes. How long is each tube?
I have
t
o share 120 sweets between 8 bags. How many per bag? Slide24
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? Slide25
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? Slide26
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?Slide27Slide28Slide29Slide30Slide31Slide32Slide33
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.Slide34
Division as inverse of area model for multiplication
Long Division – Part 1 on
Vimeo
(15
mins
)
http://vimeo.com/45986110Slide35
Whole school development
Collaboration across years and key stages
Coherent development throughout school
Something relevant every weekSlide36
anne.watson@education.ox.ac.uk