Transcript Skills in Early Division

Division
Leicestershire Numeracy Team 2003
division 1
The problems with division
Try these:
6 18
24 202
Leicestershire Numeracy Team 2003
division 2
What is division?
How would you illustrate this division to a
child? What would you draw and what
language would you use?
12  3 = 4
Leicestershire Numeracy Team 2003
division 3
Skills in Early Division
12  3 = 4
Sharing
There are three children and 12 cakes. How many
can they each have, if I share them out equally?
(Sharing 12 things equally into 3 piles. How many in
each)
Leicestershire Numeracy Team 2003
division 4
Skills in Early Division
12  3 = 4
Grouping
There are 12 cakes. How many children can have
three each?
(How many threes are there is 12?)
Leicestershire Numeracy Team 2003
division 5
Language and division
Since the  sign represents both the
sharing and grouping aspects of
division, encourage the children to
read this as ‘divided by’ rather than
‘shared by’.
Leicestershire Numeracy Team 2003
division 6
6000  1000 =
Would you group or share for this calculation?
Leicestershire Numeracy Team 2003
division 7
Introducing division
In Year 2 children are encouraged to understand the
operation of division as:
• sharing equally
• grouping or repeated subtraction e.g. How many
tens are in 60?
Leicestershire Numeracy Team 2003
division 8
18  3 =
Leicestershire Numeracy Team 2003
division 9
Sharing
• Supports an understanding of halving and the 1
to 1 correspondence between objects.
• Requires little knowledge or skill beyond
counting.
• Becomes more difficult to visualise as the divisor
increases.
• Is inefficient.
Leicestershire Numeracy Team 2003
division 10
Division and number lines
18  3 =
0
3
6
9
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12
15
division 11
18
Modelling division on beadstrings
20  4 =
Leicestershire Numeracy Team 2003
division 12
20  4 =
Leicestershire Numeracy Team 2003
division 13
20  4 =
Leicestershire Numeracy Team 2003
division 14
20  4 =
Leicestershire Numeracy Team 2003
division 15
20  4 =
Leicestershire Numeracy Team 2003
division 16
Key Stage 1 - Calculations
• Encourage children to use jottings, as well, to check answers to
calculations that have been reached by mental methods
Q29
1
2c
4%
4%
2b
2a
3
All
12% 27% 61% 31%
Leicestershire Numeracy Team 2003
division 17
Grouping
• Links to counting in equal steps on a number line.
• Requires knowledge of subtraction facts (repeated
subtraction) and addition facts (counting up).
• Is more efficient than sharing as the divisor
increases.
• Provides a firmer basis on which to build children’s
understanding of division.
Leicestershire Numeracy Team 2003
division 18
Introducing division
In Year 3 and 4 children also need to know that:
• dividing a whole number by 1 leaves the number
unchanged: e.g. 12  1 =12
• 16  2 does not equal 2  16
• division reverses multiplication (the inverse) – this allows
them to solve division calculations by using
multiplication strategies (18  3 by counting the hops
of 3 to 18)
•there will be remainders for some division calculations (to
be expressed as whole-number remainders).
division 19
Leicestershire Numeracy Team 2003
How many eights in 48?






Leicestershire Numeracy Team 2003
division 20
Continuing division
In Year 4 children need to begin to :
• relate division and fractions
• use a written method for division (chunking).
Leicestershire Numeracy Team 2003
division 21
the number to be divided
2

3
the divisor
Leicestershire Numeracy Team 2003
division 22
the number to be divided
2
3
the divisor
Leicestershire Numeracy Team 2003
division 23
the number to be divided
23
the divisor
Leicestershire Numeracy Team 2003
division 24
Teaching chunking - partitioning
72  5
Partition 72 in to a convenient multiple of 5 + the rest
72 = 50 + 22
Divide each part
50 ÷ 5 = 10
22 ÷ 5 = 4 rem 2
Recombine the parts
Answer: 14 remainder 2
Leicestershire Numeracy Team 2003
division 25
Teaching chunking - number line
72 ÷ 5 =
Grouping - How many 5’s are there in 72?
Adding groups of 5
5 x 10 or
5 x 4 or
10 groups of 5
0
5
10
15
20
25
30
4 groups of 5
35
40
45
Leicestershire Numeracy Team 2003
50
55
60
65
division 26
70 72
Teaching chunking - vertical
5x1 =5
5 x 2 = 10
72  5 =
72
50
5 x 5 = 25
(5 x 10)
22
5 x 10 = 50
20
(5 x 4)
2
Answer: 14 remainder 2
Leicestershire Numeracy Team 2003
division 27
Using calculators for repeated
subtraction
The constant function
To calculate 72  5 using repeated subtraction
Press 5 - - =
then press 72
Leicestershire Numeracy Team 2003
division 28
Teaching chunking - larger numbers
256  7
256 = 210 + 46
7x1=7
7 x 2 = 14
210 ÷ 7 = 30
46 ÷ 7 = 6 remainder 4
256  7 =
7 x 5 = 35
7 x 10 = 70
or
256
210
46
42
4
(7 x 30)
(7 x 6)
Answer: 36 remainder 4
Leicestershire Numeracy Team 2003
division 29
Continuing division
In Year 5 and 6 children also need to understand:
• that a number cannot be divided by zero
• how a quotient can be expressed as a fraction and as a
decimal fraction
• how to interpret the display when dividing with a calculator.
Leicestershire Numeracy Team 2003
division 30
185 people go to the school concert.
They pay £1.35 each.
How much ticket money is collected?
£
Programmes cost 15p each.
Selling programmes raises £12.30
How many programmes are sold?
Show your
method you
may get a
mark.
Leicestershire Numeracy Team 2003
division 31
Leicestershire Numeracy Team 2003
division 32
Solve these word problems
To make a box pieces of wood 135mm long have to be
cut from a 2.5m length. How many lengths of wood can
be cut?
Train fares cost £14.50. I have £52. How many people
can I take on the journey?
Leicestershire Numeracy Team 2003
division 33