Mental to Written Progression
KS2
New Mathematics Curriculum - Aims
The National Curriculum for mathematics aims to ensure that all pupils:
•
become fluent in the fundamentals of mathematics, including
through varied and frequent practice with increasingly complex
problems over time, so that pupils have conceptual understanding and
are able to recall and apply their knowledge rapidly and accurately to
problems
•
reason mathematically by following a line of enquiry, conjecturing
relationships and generalisations, and developing an argument,
justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of
routine and non-routine problems with increasing sophistication,
including breaking down problems into a series of simpler steps and
persevering in seeking solutions.
Activity
Take the cards and rank the cards in
order of importance.
Mental Strategies
How would you expect your children to solve
14 + 15 =
23+ 45 =
3x4=
25 x 14 =
6.9 + 3.4 =
16 x 4 =
4.7 + 5.6 – 0.7 =
497÷ 7 =
Reordering
Mental Strategies
Recalling facts
• What fraction is equivalent to 0.25?
• How many minutes in an hour, in six hours?
Applying facts
•
•
•
•
If 3 x 8 is 24, what is 6 x 0.8?
What is 20% of £30?
What are the factors of 42?
What is the remainder when 31 is divided by 4?
Mental Strategies
Hypothesising or predicting
• The number 6 is 1 + 2 + 3, the number 13 is 6 + 7.
Which numbers to 20 are the sum of consecutive
numbers?
• Roughly, what is 51 times 47?
• I’m 14, 069 days old today, is that possible?
Mental Strategies
Interpreting results
• Double 15 and double again; now divide your
answer by 4. What do you notice? Will this always
work?
• If 6 x 7 = 42 is 60 x 0.7 = 42?
• I know 5% of a length is 2 cm. What other
percentages can we work out quickly?
Mental Strategies
Applying reasoning
•What is the relationship between the number of
sides on a pyramid and the number of sides on the
base of the pyramid?
16 + 7 =
+ 4
16
+ 3
20
23
37 + 16 =
+ 10
37
+ 6
47
53
37 + 16 =
+ 20
37
- 4
53
57
Using the number line and bridging to count back
The baker makes 54 loaves and sells 28.
How many has he left?
54 - 28
- 4
- 4
26
- 20
30
34
26 loaves are left
54
Progression from concrete to written
Developing Fluency
Partitioning
5 6
50
6
Mental method, using partitioning
56
6 411
50 + 40 + 6
+
90
907
+
7
+
1
Subtraction – mental to written
87 - 35 =
8 7
- 3 5
80
- 30
7
5
50
2
5 2
54 + 38
Complete this question using the resources on your
table.
Consider
How you would want your teachers to model this
example using concrete resources?
What questioning and explanations would go
alongside the use of the models?
How would they ensure that this transferred into
the children’s independent work?
Subtraction – mental to written
Subtraction – mental to written
Early objectives for Multiplication
•
count in multiples of twos, fives and tens
•
recall and use multiplication and division facts for the 2, 5 and 10
multiplication tables, including recognising odd and even numbers
•
show that multiplication of two numbers can be done in any order
(commutative) and division of one number by another cannot
• calculate mathematical statements for multiplication and division
within the multiplication tables and write them using the
multiplication (×), division (÷) and equals (=) signs
• solve problems involving multiplication and division, using
materials, arrays, repeated addition, mental methods, and
multiplication and division facts, including problems in contexts
How much money in the money box?
• How would you count these coins to find the total
in the money box?
Understanding multiplication as repeated addition
5 added together 3 times is 5 + 5 + 5
or
3 lots of 5
or
5x3
Understanding multiplication as repeated addition
Multiplication
16 = 2 lots of 8 = 4 sets of 4 = 8 times 2 = 16 x 1
Multiplication The importance of arrays
The importance of arrays
5 x 2
2 x 5
Arrays Challenge
How many different arrays can you make
with 24 cubes?
Commutative law
Meaning the same
2+2+2+2+2+2+2
7 lots of 2
7 groups of 2
7 sets of 2
2 times 7
2 x 7
2 multiplied by 7
Establishing Mental Methods
Learning facts
Consolidating facts
•Reciting facts
•Using board games
•Using multiplication
charts or grids
•Auditory games e.g
‘Fizz buzz’
•Singing facts
•Applying movement
to multiples
•‘Seeing’ repeated
addition
How can you practice
multiplication facts?
Mental Strategies
How would you expect your children to solve
Double 12 =
3x4=
16 x 4 =
56 ÷ 4 =
497 ÷ 7 =
Knowing and Using Number Facts
Let’s learn to count in multiples of 17?
What facts can you use to help us?
Developing reasoning
What’s the rule?
YES
No
What’s the rule?
Multiplication interim steps…
15 x 3 =
10 x 3
0
5 x 3
30
45
Bar Model
Tom puts 5 sweets in each party bag. How many
sweets are there in 4 party bags?
5
5
5
?
5
Developing the bar model
Whilst watching the video, consider how the
teacher encourages understanding
Using models and images to support reasoning
Developing grid method
Whilst watching the video, consider how the
teacher encourages understanding
Progression in multiplication TU x U
14x7=
10
7
4
14
14
x 7
x 7
70 (10x7)
98
28 (4x7)
2
98
Progression in multiplication TU x TU
17 x 14 =
10
10
4
100
40
7
70
28
17
17
x 14
x 14
100 (10x10)
170 (17 x 10)
70 (7x10)
68 (17 x 4)
40 (10x4)
28 (7x4)
238
238
Long multiplication
Whilst watching the video, consider how the
teacher encourages understanding alongside the
more formal written algorithm.
Summary
Procedural fluency and conceptual understanding
are not mutually exclusive. The Ofsted Survey of
Good Practice in Primary Mathematics (Ofsted
2011) found that many of the successful schools
sampled teach fluency in mental and written
methods of calculation, alongside understanding of
the underlying mathematical concepts.
NCETM 2013
Summary
We need to be clear on how we develop children’s
understanding through;
• Clear use of language
• Providing opportunities to reason
• Use of appropriate visual representations
• Clear progression from mental to written