Transcript Progression In Calculations.

Progression In Calculations at
Lyndhurst First School.
Addition and
Subtraction
Mathematical Calculations in School Today.
The phases indicate progression in the development of children’s
jottings to support their calculations. The phases do not
represent any specific Year Group or National Curriculum level.
Remember each child progresses at their own pace.
Phase 1 - Understanding Addition and Subtraction.
Understanding addition as combining two
groups
Children need to experience counting out a
set of objects and combining them with
another set of objects to make a total
amount.
Eg. 4 plus 2 makes 6
Count out a set of 4 and another set of 2.
Then count them altogether to reach a
total of 6.
Vocabulary/Language
More, more than, one more
Less, less than, one less
Add, plus, count
Take away, subtract,
Leaves, equals, makes
Understanding subtraction as taking away
Children need to experience counting out a set
number of objects and then removing/taking
away a certain number from that group.
Eg.
7 take away 3 leaves 4
Count out 7 objects. Then remove 4 from that
set and count the objects that are left over.
Explanation
The physical act of counting out
a set number of objects, and combining two
groups or taking some away, is an important
step for children to explore. This is best
done in a ‘real life/play’ context as much as
possible.
Phase 1 - Using a Number track.
1
2
3
4
5
6
7
8
9
10
7
8
9
10
One more than four is five
1
2
3
4
5
6
One less than nine is eight
Vocabulary/Language
Find the number that is one more than
one less than
Count on from/count back from
How many…?
Explanation
Number tracks can be used for children
to locate a number, learn the order of
numbers, and to begin to find out one
more and one less.
Phase 1 – Simple jottings/mark making for addition
3
+
Vocabulary/Language
Count, count on
More, add, plus
Sum, altogether, total, equals
How many altogether?
2
=
5
Explanation
Simple mark making is the first stage of
children’s independent jottings to help them
solve additions. They draw or make the
appropriate number of marks under each
number then count them up to reach the
total.
Phase 1 – Simple jottings/mark making for subtraction
5
-
Vocabulary/Language
Count, count back
Subtract, take away, cross off
Difference between
Leaves, equals
How many are left over?
2
=
3
Explanation
Simple mark making is the first stage of
children’s independent jottings to help them
solve subtractions. They draw the initial
number of objects and then cross
off the number it says to take away and
count the ones left over.
Phase 1 – Using a blank number line for addition.
At phase 1, children learn to use ones jumps, adding single digit numbers and working
within a range up to about 20. It is only necessary to record where they start and where
they end up after adding on. They can then progress to using this method of single jumps
when adding ‘teen’ numbers and working up to about 30.
5+3=8
5
4 + 7 = 11
8
Vocabulary/Language
Count on, count on in ones
Add, plus, more than, total
Equals, makes
7
11
Explanation
Blank number lines are used to enable
children to count on and back with more
than one jump. Children are taught to
draw their own number line and start with
the biggest number. There is no need to
write +1 in each jump.
Phase 1 – Using a blank number line for subtraction.
At phase 1, children learn to use ones jumps, subtracting single digit numbers and working
within a range up to about 20. Their recording methods should be the same as for
addition, except that with subtraction they start at the right hand end of the line and
jump back. They can progress to using larger numbers when they are secure in this
method.
9–4= 5
5
14 – 6 = 8
9
Vocabulary/Language
Count back, count back in ones
Less than, take away, subtract
Leaves, equals
8
14
Explanation
Blank number lines are used to enable
children to count on and back with more
than one jump. Children are taught to
draw their own blank number lines,
enabling them to do calculations within
any range of numbers. There is no need to
write -1 in each jump.
Phase 2 – Using a hundred square for addition and subtraction
Count on and
back in tens
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2
3
4
5
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7
8
9
10
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25
26
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30
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35
36
37
38
39
40
41
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43
44
45
46
47
48
49
50
51
52
53
54
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57
58
59
60
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68
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70
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74
75
76
77
78
79
80
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83
84
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99
100
Count on and back in ones
Vocabulary/Language
One more, one less, ten more, ten less
Count on in ones/tens
Count back in ones/tens
Add, plus, more than
Subtract, take away, minus, less than
Explanation
Working with a hundred square helps
develop a childs’ understanding of the
number system. Looking at the patterns in
numbers along the rows, and up and down
the columns, will support their counting in
ones and tens.
Phase 2 – Developing use of number lines, adding tens.
Children need to be able to work out ten more than any 2 digit number in order to
progress to this stage.
+10
26 + 10 = 36
26
36
33p + 20p = 53p
+10
33p
Vocabulary/Language
Addition, add, plus, more, more than
Subtract, take away, minus, less than
Ten more, ten less, count in tens
Difference between
Leaves, equals, altogether
+10
43p
53p
Explanation
Once children can count on in ones jumps
on a number line, they learn to do tens
jumps. To begin with they add 10, then
multiples of ten, to 2-digit numbers. We
write +10 inside the jumps. Begin to use
these strategies in context, such as
money and measures.
Phase 2 – Developing use of number lines, subtracting tens.
Children need to be able to work out ten less than any 2 digit number in order to
progress to this stage.
43 – 10 = 33
-10
33
43
61cm– 20cm = 41cm
-10
41cm
Vocabulary/Language
Addition, add, plus, more, more than
Subtract, take away, minus, less than
Ten more, ten less, count in tens
Difference between
Leaves, equals, altogether
-10
51cm
61cm
Explanation
Once children can count back in ones
accurately on a number line, they learn to
jump back in tens. To begin with they take
away 10, then multiples of ten, to
2-digit numbers. We write -10 inside the
jumps.
Phase 2 – Developing use of number lines, adding tens and ones.
Children need to understand place value in order to partition 2-digit numbers
into tens and ones at this phase.
+10
26p + 12p = 38p
26p
36p
38p
33 + 21 = 54
+10
33
Vocabulary/Language
Addition, add, plus, more, more than
Subtract, take away, minus, less than
Ten more, ten less, count in tens
One more, one less, count in ones
Difference between
Leaves, equals, altogether
+10
43
53
54
Explanation
When children understand that 15 is made
up of one ten and five ones, they can learn
a more efficient method of using a
number line than just doing 15 single
jumps. We do the number of tens jumps
first, followed by the number of ones
jumps.
Phase 2 – Developing use of number lines, subtracting tens and
ones.
Children need to understand place value in order to partition 2-digit numbers
into tens and ones at this phase.
48g – 13g = 35g
-10
35g
38g
48g
116 – 24 = 92
-10
92
Vocabulary/Language
Addition, add, plus, more, more than
Subtract, take away, minus, less than
Ten more, ten less, count in tens
One more, one less, count in ones
Difference between
Leaves, equals, altogether
96
-10
106
116
Explanation
Subtracting tens and ones is the same as
for addition. Jottings are set out as
shown, with a record of where you have
reached kept under the line and the jumps
done recorded over the line. There is no
need to write +1 or -1 in the small jumps,
this would be inefficient.
Phase 2 – Partition and recombine
To add 23 and 35,
First add the number of tens,
so 20 + 30 = 50
Then add the number of ones,
23
+
35
+
8
So 3 + 5 = 8
Finally combine the answers to give the
total,
so 50 + 8 = 58
Vocabulary/Language
Tens digit, ones digit
Partition, split, recombine
How many tens? How many ones?
How many altogether?
50
= 58
Explanation
Some children find this strategy a quick
and easy method for addition, that they
soon are able to do mentally. Initially it is
important to give calculations where the
ones digits do not total more than 10, (we
say they don’t cross the tens boundary).
Phase 2 – Add or subtract 9, by compensation.
To add 9, + 10 then -1
25 + 9 = 34
To take away 9, -10 and then +1
+10
25
34 35
46 – 9 = 37
-10
36 37
Vocabulary/Language
Add, plus, more than, sum
Subtract, take away, minus, less than
Equals, leaves, totals
adjust
46
Explanation
When adding or taking away 9, children
are taught that it as quicker to
add/subtract ten and then adjust by one
accordingly.
Phase 3 – Develop efficient use of number lines.
74 + 43 = 117
+40
74
+3
114
117
158 – 62 = 96
-60
-2
96
Vocabulary/Language
Addition, add, plus, more, more than
Subtract, take away, minus, less than
Ten more, ten less, count in tens
One more, one less, count in ones
Difference between
Equals, leaves, altogether
98
158
Explanation
Once children are confident and accurate
in the use of tens and ones jumps, they
can progress to using multiple of tens
jumps. This obviously requires them to be
able to add/subtract multiples of tens in
their head.
Phase 3 – Partition and recombine
46
+
70
38
+
80
125
14
+
4 = 84
Vocabulary/Language
Hundreds digit, tens digit, ones digit
Partition, split, recombine
How many hundreds? tens? ones?
How many altogether?
+
400 + 60
349
+
14
400 + 70 +
4
= 474
Explanation
At this phase. Children can partition and
recombine numbers that may cross the
tens boundary. They will also be able to
use this method with 3-digit numbers.
Phase 3 - Compensation
25 + 19 = 44 (+20, then -1)
754 + 96 = 850
+20
(+100 then -4)
+100
25
44
45
754
850
854
63 – 28 = 35 (-30, then + 2)
-30
33
34 35
Vocabulary/Language
Add, plus, more than, sum
Subtract, take away, minus, less than
Equals, leaves, totals
Multiples of tens, adjust
63
Explanation
To add, select the next multiple of 10 (or
100), add that number and then subtract
the difference to compensate/adjust.
To subtract, select the next multiple of 10 (or
100), subtract that number and then add
to compensate.
Phase 3 – Subtraction as complementary addition
Complementary addition is often used in the context of money and finding change. It is also
useful when finding a small difference between two numbers.
£1.00 – 63p = 37p
63
+7
+10
+10
70
80
+10
90
100
106 – 98 = 8
+6
+2
98
100
Vocabulary/Language
106
Subtract, take away, minus, less than
Difference between
Equals, leaves
Count on, count up
Explanation
Start at the number being subtracted and
then count on to the biggest number.
When using this method, it is important
to remember that the answer is now
found above the line in the total that was
added on.