Transcript Mental Calculations students slides

PGCE
Arithmetic: Mental Calculation
Learning Objectives:
• Be familiar with the models of the four operations
• Be aware of the properties of the four operations
• Be aware of the relevant vocabulary
• Become familiar with a range of mental strategies
• Consider the importance of structured jottings
What do we mean by addition,
subtraction, multiplication and division?
SHOW ROSS AND HIS CUBES CLIP
Understanding addition, subtraction,
multiplication and division
In order to be able calculate using the four
operations a child needs to know:
• The different models of addition, subtraction,
multiplication and division
• The properties of the four operations
• Vocabulary
Models of addition
• Combining
1, 2, 3
and
1, 2
together makes
1, 2, 3,
4, 5
• Counting on
3
and
2
together makes
3,
4, 5
Models of subtraction
• Taking away
• Counting back
• Difference
Can you find an example for each?
The Singapore Bar Method
Addition - Aggregation
There are 3 footballs in the red basket 2
footballs in the blue basket. How many
footballs are there altogether?
Addition - Augmentation
• Peter has 3 marbles. Harry gives Peter 1
more marble. How many marbles does
Peter have now?
Concrete
Abstract
Subtraction - Comparison Model
• Peter has 5 pencils and 3 erasers. How
• many more pencils than erasers does he
• have?
Moving to the abstract
• Peter has 5 pencils and 3 erasers. How
many more pencils than erasers does he
have?
Generalisation
Giving meaning to calculations
Number stories for 5+3=8
Which models would you use?
• I have 5 sweets and my friend gave me 3
more. How many do I have altogether?
• My sister is 5 years old. How old will she be
in 3 years time?
Number stories for 8-3=5
Which models would you use for children?
• There are 8 apples on a tree. The squirrel
ate 3. How many were left?
• I am 8 and my sister is 3. How many years
older am I than my sister?
• I have 3 conkers and my friend has 8. How
many more has he got than me?
Models for multiplication
What is multiplication?
• Repeated addition
• Lots of / groups of
• Arrays
• Scaling (n times as many, as long, as heavy…)
Models for division
What is division?
• Sharing (equal)
• Grouping, linked to:
• Repeated subtraction
ITPs available to support
• Arrays
• Grouping & number line models
Vocabulary
• Addition
• Multiplication
• Subtraction
• Division
Properties
3
4
7
12
Write some number sentences
using the numbers above.
Use all four operations
eg 3 + 4 = 7
Property 1: Inverse
4+3=7
So …
7–3=4
4 x 3 =12
So …
12 ÷ 3 = 4
3+4=7
So…
7–4= 3
3 x 4 = 12
So …
12 ÷ 4 = 3
Property 2:Commutativity
Does it matter which way round the two
operations are done?
4+3= 3+4=
Does it matter in which order you add these
numbers together?
15 + 7 + 5 + 3 =
Addition is commutative
Which of the other operations are commutative?
Property 3: Associativity
How would you do this, which pair would you start
with?
• 18+36+4=
Does … (18+36)+4 = 18+(36+4)
Is addition associative?
Now try these, put the brackets in different places
so that you start with different pairs.
• 12 – 7 - 2
• 2x3x4
• 12 ÷ 6 ÷ 3
Which are associative?
Property 4: Distributive
7 x 13 = 7 x (10 + 3) = (7 x 10) +(7 x 3)
10
7
Is division?
3
Summary 1
Models, Links and Properties
Addition
Models
● combining sets
Subtraction
● taking away from a set
● counting on or back
● counting on (number line)
(number line)
● difference between
Links
INVERSES
Properties
Commutative
Associative
Neither
Summary 2
Models, Links and Properties
Multiplication
Models
● repeated addition
● lots of / groups of
● arrays
● scaling
Division
● repeated subtraction
● sharing
● groups of
Links
INVERSES
Properties
Commutative
Neither
Associative
Distributive over addition
EYFS 2013 and National Curriculum
2014
EYFS Early Learning Goal:
• Use quantities of objects to add and subtract two
single digit numbers
• Count on and count back to find answers
National Curriculum 2014:
• Solve problems using concrete objects, pictorial
representations and arrays (year 1/2)
• Use inverse relationships to check addition and
subtraction calculations (year 2)
Cont….
• Show that addition of two numbers can be done
in any order (commutative) and subtraction
cannot (year 2)
• Show that multiplication of two numbers can be
done in any order (commutative) and division
cannot (year 2)
• Estimate the answer and use inverse operations
to check answer (year 3)
• Solve problems with scaling (year 3)
• Use commutativity in mental calculations (year 4)
• Solve problems using distributive law (year 4)
Rover has left his bone on the other side of the road. He
can only get there by treading on boxes with an answer
that is 7 (number bonds)
3-1
6+2
6-2
1+1
5-1
5+7
0+7
9-2
1+6
1+3
4+2
5+2
6+6
2+5
6+2
3+4
8-1
2+3
4+3
6+1
1+1
4+4
4+0
8-3
7-6
Mental Strategy 1: Number Bonds
• Not just number bonds for 10
• Extend to number bonds for numbers up to
20
• Make it visual for understanding
Useful apparatus:
Cuisenaire / Colour rods
8
6
Numicom
+
2
https://www.ncetm.org.uk/resources/40533
3+4
7
Mental strategy 2: Partitioning
Use arrow cards to help children deconstruct
numbers and combine multiples of hundred /
ten / ones. (Later on introduce exchanging)
4 3
2 1
43 + 21 = (40 + 20) + (3 + 1) Links to associativity
https://www.ncetm.org.uk/resources/40534
Mental strategy 3: Bridging through 10
To use this strategy children first need to know number bonds to
10 and partitioning.
Start visually using apparatus such as Nubicom / Colour Rods:
8
8
Children need to know number bond
of 8 to make 10
5
2
3
5 is partitioned into 2 + 3
This is then extended to 2 digit numbers and the
apparatus is replaced by empty number line jotting
(covered later)
Mental strategy 4: adding /
subtracting 9
29 + 9
Strategy 5: Complementary Addition
Or subtraction by addition.
200 – 56
How could you model this?
In this calculation you are looking at
the size of the gap between the two
numbers.
4
56
40
60
100
100
200
Other Mental Strategies:KS1
•
•
•
•
•
counting on
using known facts e.g. doubles
derive facts e.g. near doubles
counting on/back in ones and tens
adding or subtracting 9 by adding or subtracting 10
and adjusting by 1
• looking for number bonds to 10
Other Mental Strategies: KS2
• partitioning numbers and dealing with the
multiples of 10 first
• adjusting numbers e.g. up or down to the
nearest 10
• using known facts to derive new facts
• looking for number bonds of 10 or 100,
especially when adding together more than
two numbers
• subtraction using complimentary addition and
compensation methods
Try these….using mental strategies
•
•
•
•
•
47 + 32 =
39 + 18 =
401 + 395 =
57 - 29 =
1000 - 989 =
Discuss your strategies
Skills needed before introducing Multiplication
and Division Mental Strategies
• Working out 3x4 by counting out three
groups/sets of four
• Counting in equal jumps along the number
line – ‘five, ten, fifteen, twenty’
• Starting with tables for 1,2 and 10, knowing
by heart facts such as ‘four tens’&
progressing to facts in 5x table, then others
• Recognising that multiplication can be done
in any order – eg realising that 5x2 is the
same as 2x5 – commutativity
Learning Multiplication Tables
1. Children need to understand what
multiplication tables are.
5 x 3 is displayed in stamps
2.They need to understand the
commutative law
Counting in 2s
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
10
0
Counting in 4s
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9 10
19 20
29 30
39 40
49 50
59 60
69 70
79 80
89 90
99 100
Developing Multiplication and
Division Mental strategies
1. Building on known facts:
If you know 3x5=15… what else do you
know?
Developing strategies
2. Multiply and divide by multiples of 10 with whole
and decimal numbers (link to place value)
3. Doubling and Halving
Have a go
• 509 x 3
• 5 x 23 x 20
• 112 ÷ 8
• 750 ÷ 25
Which strategies did
you use?
Issues: Mental Calculations
• methods are all valuable if they are quick and
accurate
• emphasis on place value and usually work from
left to right
• May involve adjusting numbers
• Need a distinction drawn between special cases
and general strategies – a repertoire from which
you select according to the particular numbers
• Need to be explained in words and described
using correct equations (partly to check
understanding)
• May be accompanied by structured jottings
Structured Jottings
•
•
•
•
Reflect and support mental strategies
May need to be tidied up for an audience
Should lead to shortened forms
Are flexible
Examples of children’s jottings
Children’s recording in Reception
Yr 1. My shepherd looks after 8 sheep but has lost 5 and
he has 3 left..
Key Stage 1 examples: Children’s
recording in Year 2
1. Write the
answer: 51- 27 = 24
Taken from Standards at Key Stage 1, QCA 2001 p.37
2. Write the number that is half of 38
30 + 8
15 + 4
As above, p 39
3 + 20 + 1
Open number lines
25 + 38
5
25
33
2
30
38
63
23
40
63
Answer 63
246 – 78
20
2
78
80
using complementary
addition and looking at
the difference
146
100
Answer 146 + 20 + 2 = 168
246
Ofsted 2011
Stresses the importance of children demonstrating a
fluency in calculating, solving problems and reasoning
about number.
Key findings:
Practical, hands-on experiences are crucial in EYFS and KS1
couple with opportunities to develop mathematical language.
Understanding place value, fluency in mental methods and
good recall of number facts.
Subtraction should be taught with its inverse addition and
division taught alongside its inverse, multiplication.
Ofsted 2011 (cont.)
• Children need to increasingly develop more sophisticated
mental and written methods
• Children need to be taught to be flexible in their thinking
and approaches
• Needs to be a strong emphasis on problem solving
• Teachers need to recognise and quickly intervene when
misconceptions occur so that progress is not impeded
• Teachers need good subject knowledge and subject
specific teaching skills.
Ofsted (2011) Good practice in primary mathematics Manchester: Ofsted
(Full report available from: www.ofsted.gov.uk/resources/110140)
Children who experience problems
• do not look for alternative methods
• overlook number properties
• try to replicate standard written methods in
their heads
• depend on counting strategies
• have limited strategic methods
• do not treat number holistically
Useful resources
• NCETM calculations microsite and videos to
support teaching of calculations
• Teaching Mental Calculation booklet in your
Arithmetic self study work book
• Resources on Moodle linked to mental
calculation workshops.