Transcript Session Two

Primary
National Strategy
Mathematics
3 plus 2 day course:
Session 2
Primary National Strategy
© Crown copyright
2003
Objectives
 To consider approaches to mental and
written division calculations
 To identify ways in which a number line can
be used to teach division, including
representing the quotient as a fraction
Slide 2.1
Primary National Strategy
© Crown copyright
2003
23 people are going out,
6 people in each car.
How many cars?
Answer 4 cars
6 people sharing 23
pencils. How many each?
23 vouchers, for every 6 a
free CD.
How many CDs?
Answer 3 remainder 5
Answer 3 CDs
6 people sharing 23 cakes.
How many each?
23 ÷ 6 =
Divide 23 by 6 on a calculator.
Answer 3 5 each
6
6 people won £23 between
Answer 3.833...
them. How much each?
6 people went out for a meal which
cost £23 in total. How much each?
Answer £3.83
Answer £3.84
Slide 2.2
Primary National Strategy
© Crown copyright
2003
How do you solve these?
 363 ÷ 3
 6000 ÷ 6
 34 ÷ 7
 6 ÷ 12
 68 ÷ 17
 4÷½
 How many 30g servings can you get from a
500g packet of cereal?
Slide 2.3
Primary National Strategy
© Crown copyright
2003
Ten Problems used in the first test
Slide 2.4
Primary National Strategy
© Crown copyright
2003
Slide 2.5
Primary National Strategy
1200
© Crown copyright
2003
Attempts compared with
successful attempts in
January (Test 1)
1109
1000
800
Attempts
600
Correct
538
507
1600
400
1400
1334
273
224
199
200
176
1521200
151
135
113
108
49
5
1000
24
1
0
1(S)
2(P)
3(L)
4(H)
5(AL)
6(M)
7(WR)
8(UN)
Att
800
Co
677
600
400
Attempts compared with
successful attempts in
June (Test 2)
299
294
210
200
181
160
156
125
74
58
72
56
12
0
12
0
1(S)
2(P)
3(L)
4(H)
5(AL)
6(M)
7(WR)
8(UN)
Slide 2.6
Primary National Strategy
© Crown copyright
2003
If children are to retain confidence in their invented
strategies and see mathematical problem-solving as a
progression of procedures, it is necessary that any more
procedural algorithm is at least deferred until they have
their own reliable and efficient intuitive approaches.
Unless the algorithm complements pupils’ informal working
and builds on the understanding they display through more
intuitive approaches to division, it is likely to inhibit
performance and have a negative effect on the development
of mathematical thinking.
Julie Anghileri 2001
‘Development of Division Strategies for Year 5 Pupils in Ten
English Schools’
Slide 2.7
Primary National Strategy
© Crown copyright
2003
Informal Methods of division
 Arrays
Slide 2.8
Primary National Strategy
© Crown copyright
2003
Division as grouping
 Extract taken from Springboard 4 Unit 5
Session 1
 How are blank number lines used to help
children count in steps and keep track of how
many steps they have taken?
Grouping ITP
Slide 2.9
Number lines and grouping
2
0
2
2
2
4
2
6
8
Slide 2.10
TASK
0
What division calculation is represented if:
 the step size is 4?
 the right-hand marker represents 18?
 the middle marker represents 6?
Slide 2.11
Primary National Strategy
© Crown copyright
2003
Discussion point 1
Does this calculation have different answers in
different contexts?
23 ÷ 6
When should a remainder be expressed as
a whole number?
When should the quotient be expressed as
a fraction or decimal?
Slide 2.12
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
0
5
10
15
20
Slide 2.13
Primary National Strategy
Working with Remainders
17  5 = 3 r 2
0
© Crown copyright
2003
5
10
15
Slide 2.14
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
0
5
10
15
Slide 2.15
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
0
5
10
15
Slide 2.16
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
0
5
10
15
Slide 2.17
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
Slide 2.18
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
3
Slide 2.19
Primary National Strategy
© Crown copyright
2003
Working with Remainders
17  5 = 3 r 2
3
2
5
Slide 2.20
Remainders
-5
-5
-5
-2
0
2
7
12
17
-5
17 ÷ 5 = 3
2
5
Slide 2.21
Remainders
5
5
5
2
0
5
10
15
17
20
5
17 ÷ 5 = 3
2
5
Slide 2.22
Progression from ‘chunking’ ...
560 ÷ 24
Approximate answer:
550 ÷ 25 = 22
560
24  10 = 240
 240
320
24  10 = 240
 240
80
24  2 = 48
 48
32
Answer: 23 r 8
24  1 = 24
23
 24
8
Slide 2.23
Progression from ‘chunking’ ...
560 ÷ 24
Approximate answer:
550 ÷ 25 = 22
560
10
 240
320
10
 240
80
2
 48
32
Answer: 23 r 8
1
 24
23
8
Slide 2.24
... to efficient ‘chunking’ ...
560 ÷ 24
Approximate answer:
550 ÷ 25 = 22
Answer: 23 r 8
560
24  20
 480
80
24  3
 72
23
8
Slide 2.25
... to efficient ‘chunking’ ...
560 ÷ 24
Approximate answer:
550 ÷ 25 = 22
Answer: 23 r 8
560
20
 480
80
3
 72
23
8
Slide 2.26
... to an efficient standard method
560 ÷ 24
Approximate answer:
550 ÷ 25 = 22
24 ) 560
20 – 480
80
3 – 72
23
8
Answer: 23 r 8
Slide 2.27
Primary National Strategy
© Crown copyright
2003
Division questions at level 2 involve:
 interpretation of words such as:
how many?, half, pair, left over, divided exactly by
 simple word problems involving grouping of
two-digit numbers into 2s, 4s, 5s, 6s or 10s,
sometimes with a remainder, and sometimes
involving rounding up
 money problems involving division by 10p or 20p
(coin values), sometimes with a remainder
 recognition of multiples of 10
Slide 2.28
Primary National Strategy
© Crown copyright
2003
Division questions at level 3 involve:
 ‘missing number’ questions involving inverses, with
the ‘box’ in different positions
 word problems involving:
– division of two- and three-digit numbers by a
single digit or multiple of 10, sometimes with
remainder, sometimes involving rounding up
– mixed units of money (e.g. £4 ÷ 40p)
– in KS2 calculator paper, divisors like 20 and 25
 ‘I am thinking of a number’ problems
 recognition of multiples of 3 and 5
Slide 2.29
Primary National Strategy
© Crown copyright
2003
Division questions at level 4 involve:
 mental division of two- and three-digit numbers
 short division without calculator (e.g. 847 ÷ 7)
 more complex ‘missing number’ questions
 word problems involving:
– division of two- and three-digit numbers,
including rounding up from a decimal answer
on a calculator
– harder mixed units of money
(e.g. £12.30 ÷ 15p)
– simple direct proportion
 ‘I am thinking of a number’ problems
Slide 2.30
Primary National Strategy
© Crown copyright
2003
Division questions at level 5 involve:
 in the mental paper, division of decimals by 10,
simple direct proportion
 recognition of conventional short division layout
 ‘missing number’ problems (big numbers, decimals)
 word problems involving:
– more reading and interpretation
– division of four- and five-digit numbers, with
rounding up from decimal answer on calculator
– mixed units (e.g. 10 m ÷ 9.2 cm, 3 kg ÷ 60 g)
– direct proportion
 ‘I am thinking of a number’ problems
Slide 2.31
Primary National Strategy
© Crown copyright
2003
Criteria for calculation methods
 A calculation method for division should be:
– reliable
(the pupil gets the right answer)
– appropriate
(it is suitable for the type of calculation and
the available tools: mental, written, calculator)
– efficient
(it is not too time-consuming)
– checkable
Slide 2.32
An inefficient and inappropriate method
A tent holds 6 children.
How many tents are needed to hold 70 children?
(Paper A)
Leah achieved level 4 in the test.
Slide 3.33
An inefficient and inappropriate method
568.1 ÷  = 24.7
(Paper B)
Gemma achieved level 4 in the test. She divided by 5,
6, 8, 16 and 20 before trying 23.
Slide 3.34
Primary National Strategy
© Crown copyright
2003
Summary
 The language ‘divided by’ and images of repeated
subtraction or division on a number line help to
secure pupils’ understanding of division in KS1 and
the early years of KS2
 It is essential that these early ideas are taught well
and that pupils develop a conceptual and visual
framework linked to the language of division
 The later years of Key Stage 2 should focus on
making informal written methods for division, such
as ‘chunking’, as efficient as possible.
Slide 2.35