Transcript Aims - Learning

Aims
• To look at the ways in which the teaching of
mathematics has changed;
• To look at how calculations involving addition,
subtraction, multiplication and division
progress throughout school;
• To look at ways in which parents can help
their children;
Aims
• To look at the ways in which the teaching
of mathematics has changed;
In my day we were
taught just one way of
working a sum out!
In my day…
35
+ 24
59
46
-13
33
How has mathematics
changed?
• Daily mathematics lesson with a clear
structure;
• Emphasis on mental calculations;
• Interactive whole class and group
teaching;
• Enjoyable practical approaches;
• Mathematics with understanding;
•
Teach a variety of methods for
•
each calculation;
Overview
Up to Year 3 the emphasis is on:
o working mentally
o calculations recorded in horizontal
number sentences
o some jottings for more challenging
numbers
o practical models and images
In Year 3-6 children will be gradually taught
more formal written methods of calculation
but they will still use mental methods
and jottings where appropriate.
Progressive Methods
• Addition
Year 4
• Subtraction
Year 3
• Multiplication
Year 5
•
•
Year 6
Division
15 minutes in each classroom
Mathematics at …………
School
Addition
Addition- Progression
o Mental calculations involving
practical work and number lines
o Expanded method using
partitioning
o Compact ‘standard’ method
Laying the
foundations……
*Number lines
*Practical equipment
*Multilink cubes
*Real life contexts
*Number bonds
*Patterns
Addition
• Children are
encouraged to
develop a mental
picture of the
number system in
their heads to use
for calculation.
They develop ways
of recording
calculations using
pictures, etc.
Using number lines
•They use number lines and practical
resources to support calculation and
teachers demonstrate the use of the
number line.
A number line is
just a ‘picture’ of
how we work out
some calculations
in our heads!
5+4=9
1
2
3
4
5
6
7
8
9 10 11 12 13
Always start with the biggest number!
Real life contexts
There were 13 children on the bus,
5 more children got on in Blackpool.
How many children were on the bus
altogether?
Counting up method
25 + 47 =
+20
+3
47
67
+2
70 72
So we start at the biggest number…47
And count up 25 places
What number do we end on???
So….
25 + 47 =
72
• Informal method using
partitioning
47 + 76 = (40 + 70) + (7 + 6)
= 110 + 13
= 123
30
34
4
+
25
20
5
=
59
50
+
=
9
59
Expanded, vertical method
4 7
+ 7 6
1 3
+ 1 1 0
1 2 3
4 7
+ 7 6
1 1 0
+ 1 3
1 2 3
Addition using a compact
method
No ‘Carrying’
4 1
+2 6
6 7
Addition using a compact
method – with carrying
• The final step, when the children
have a sound grasp of place value &
of the whole process…
Carrying
4 7
+ 7 6
1 2 3
1
1
Choose the most
appropriate method…
• 143 children attended a dance
festival…then 274 parents arrived.
How many people were at the dance
festival altogether?
•
•
•
•
•
•
•
•
•
Vocabulary
Add
Plus
Altogether
Addition
Total
Count on
Sum of
Increase
More than
+
Mathematics at …………
School
Subtraction
Subtraction - Progression
• Mental calculations including
practical work and using a number
line
• Expanded method using partitioning
• Compact ‘standard’ method
Laying the
foundations……
*Number lines
*Practical equipment
*Multilink cubes
*Real life contexts
*Number bonds
*Patterns
Subtraction
• Children are
encouraged to develop
a mental picture of
the number system in
their heads to use for
calculation. They
develop ways of
recording calculations
using pictures etc.
Subtraction
There are five frogs if 2 frogs jumped into the lake
how many would be left?
Using number lines
• They use number
lines and practical
resources to
support calculation.
Teachers
demonstrate the
use of the number
line.
A number line is
just a ‘picture’ of
how we work out
some calculations
in our heads!
Number Line Subtraction
‘counting back’ method
0 1
2
3
4
5 6
7 8
9 10
Tom has 10 sweets. If he gives 3 sweets to
his friend John how many sweets will he have
left?
‘Counting on’ method
‘subtraction’ by finding the ‘difference’
84 - 26 =
53 - 26
26 = 27
53
4
20
3
30
50
Round UP to a
multiple of 10
Jump to the
nearest multiple
of 10
4 + 20 + 3
=
27
Expanded, vertical
method
• 84 – 26 =
•
•
•
•
•
•
84
-26
4 (to 30)
+ 5 0 (to 80)
4 ( to 84)
58
Expanded, vertical
method
89 =
- 57
80
50
30
+
+
+
9
7
2 = 32
Expanded, vertical method
with exchange
70
83 – 26
80  1 3
-
20
50


6
7
= 57
Subtraction using a
compact ‘standard’ method
• By decomposition
• Uses children’s understanding of the
number system
•
•
53
185
•
-21
- 37
•
32
148
7
1
Subtraction
•
•
•
•
•
•
•
Take away
Subtract
Find the difference
Minus
Less than
Fewer than
How many are left?
-
Mathematics at …………
School
Multiplication
Multiplication Progression
o Mental calculation including practical work and the
use of number lines
o Understanding of multiplication as:
repeated addition
an array
scaling
o Grid method
o Vertically expanded method
o
Compact ‘standard’ method
Counting in 2s, 10s and 5s
Children will experience equal groups of objects and will
count in 2s and 10s and begin to count in 5s. They will
work on practical problem solving activities
involving equal sets or groups.
Repeated Addition
4+4+4=
12
4 x 3 = 12
Repeated Addition
How many hats are there?
5 + 5 + 5 = 15
or
3 x 5 = 15
Repeated Addition
Repeated addition can be shown easily on a number line:
5x3=5+5+5
5
0
1
2
3
5
5
4
5
6
7
8
9
10 11 12 13 14 15
and on a bead bar:
5x3=5+5+5
5
5
5
Learning Times Tables Facts
·It is vital that children know their times tables as all
the work higher up the school relies on them
knowing their tables.
·Tables should be learned at least 2-3 times a week.
Year 2
2 times table
5 times table
10 times table
Year 3
2 times table
3 times table
4 times table
5 times table
6 times table
10 times table
Year 4
Derive and recall all multiplication facts up to 10 x 10 (5 second
recall)
Years 5 & 6 Derive and recall quickly all multiplication facts
up to 10 x 10 (5 second recall)
Multiplication
3 x 7
0
7
14
21
0
7
14
21
Arrays
Children should be able to model a multiplication
calculation using an array.
This knowledge will support with the development of
the grid method.
3x5
5x3
3+3+3+3+3
5+5+5
Grid method of multiplication
10
6
60
3
18
60 + 18 = 78
so 6 x 13 = 78
Grid method
TU x U
23 x 8
X
HTU x U
346 x 9
X
Grid method of multiplication
TU x TU
72 x 38
X
30
8
HTU x TU
372 x 24
70
2
2100
60
560 16
X
20
300 70
2
6000 1400 40
4
1200 280
8
Vertically expanded method
A vertically expanded method links into the grid method
and is a good way of moving children on to compacted
methods.
38
x 7
2 1 0 (30x7)
+ 5 6 ( 8x7)
266
126
x 3
3 0 0 (100x3)
6 0 (20 x 3)
+ 1 8 (6x3)
378
Standard Method
35
•x 4
1 40
•
2
261
x 6
15 66
3
Long Multiplication
•
57
• x 23
171
+1 1 4 0
1 31 1
2
1
1
Multiplication
•
•
•
•
•
Times
Multiply
Multiplication
Lots of
Find the product of
x
Mathematics at …………
School
Division
Division - Progression
o Mental calculations including practical work and the
use of the number line
o Understanding division as sharing
o Understanding division as grouping
o Visualising division using:
repeated subtraction
arrays
o Division as chunking
o
Standard short method of division
Division as sharing
Can you share the 6 giraffes equally between 2
fields?
Division as grouping
• There are 6 sweets. How many people
can have two sweets each?
Three people
Division by grouping
How can you work out a division sum using groups?
12 ÷ 3
There are 4 groups of 3, so 12 ÷ 3 = 4
Division Facts
• Year 2 – know division facts associated with the 2,
5 and 10 times tables.
• Year 3 – know division facts associated with the 2,
3, 4, 5, 6 and 10 times tables.
• Year 4 – Derive and recall all division facts
associated with all times tables up to 10x10
• Year 5/6 – Derive and recall quickly all division
facts associated with all times tables up to 10x10
Grouping
•I can count on along a number line to work out 153;
0
3
6
9
12
•3 + 3 + 3 + 3 + 3
I need 5 jumps of +3 to reach 15.
Therefore 153=5
15
Division as repeated
subtraction
• You can work out 12 ÷ 3 by taking off
threes until you get to zero.
• Like this:
• 12 - 3 - 3 - 3 - 3 = 0
• That means there are four 3s in 12.
Arrays
• To divide 12 by 3, I can think of an array
of 12 in groups of 3 like this:












• I have 4 groups of 3, so 123=4
Chunking
We need to do the division sum
78 ÷ 6.
We can do this by chunking…
78
- 60 ( 10 x 6 )
18
- 18 ( 3 x 6 )
0
Q. How many 6’s have been
subtracted?
78 ÷ 6 = 13
Chunking with remainders
85 ÷ 7 =
85
- 70
15
- 14
1
12 r.1
(10 x 7)
( 2 x 7)
Division by ‘chunking’ or
‘lots of’
432
-1 5 0 (15 x 10)
282
-1 5 0 (15 x 10)
132
- 6 0 (15 x 4)
72
- 6 0 (15 x 4)
12
28 r 12
Short Division
1 6r3
• 4 67
2
136r2
5 68 2
1
3
Division
•
•
•
•
Divide
Division
Share
Split equally
÷
Key Messages
o Secure mental strategies.
o A solid understanding of the number system.
o Practical, hands on experience including counters and
base 10 apparatus.
o Visual images including number lines and arrays.
o Experience of expanded methods to develop
understanding and avoid rote learning.
o Secure understanding of each stage before moving onto
the next.
o The questions at the forefront of their minds:
‘Can I do it in my head? If not which method will
help me?’
From all of these they learn to construct strategies that they
can apply in many different areas
An everyday tool ……
• Mathematics is something that we
use in our everyday lives, all of the
time. We are here to teach children
and to give them a variety of
strategies to cope. All children may
not all end up at the same point, but
they will all have some understanding
of how to work sums out.
How to help your child with
mathematics!
Visual maths
• Number lines
1
2
3
4
5
6
• Noticing numbers
23
Counting
• Rhymes/Songs
• 10 green bottles
• 1, 2, 3, 4, 5 once I caught a fish alive
• Play number games
• Snakes and Ladders
• Monopoly
Sorting
• Socks
• Cars
• Shoes
Measures
• Keep a record of your child's growth;
• Use weighing scales in cookery;
• Capacity – different containers to
play with in the sink or bath;
Shape and space
• Recognising shapes around them e.g.
doors, windows, cans, boxes etc
• Construction sets, Lego,
• Shapes of cakes, biscuits,
sandwiches.
Money
• Involve children in shopping activities;
• Reading prices, selecting the correct
coins, calculating totals, working out
change;
DfEE 1999 (10 years ago!)
‘Parents who are confident about
maths tend to have children who are
also confident, and these children
are ready to tackle and assimilate
new ideas in a way that is impossible
for children who feel uncertain
about, or even fear, maths.’
Thank you for attending!