Transcript Progression In Calculations.

Progression In Calculations.
Subtraction
Mathematical Calculations in East Worthing Schools.
This document has been discussed and agreed by all East Worthing Maths
Leaders and is designed to help you to understand the calculation
methods your child will be taught in school. When supporting your child
at home with Maths work it would be helpful if you could reinforce these
methods rather than teach them the way that you were taught. Your
child’s teacher will be able to direct you to the appropriate method
within this document to use at home.
Remember each child progresses at their own pace.
Understanding Subtraction.
Explanation
The physical act of counting out
a set number of objects and taking some away
is an important step for children to explore.
This is best done in a practical play based
context as much as possible.
Count out a given number of objects and take
away 1 of them... How many do you have now?
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. Initially this
needs to be taking away 1 from a set.
Eg.
Explore 1 less through simple songs and rhymes, for
example ’10 green bottles’ or ‘5 little speckled frogs’.
7 take away 1 leaves 6
Count out 7 objects. Then remove 1 from that set
and count the objects that are left over.
Key Questions/Vocabulary
Less, less than, one less, before, take away, subtract, leaves,
equals
What is the number before 5?
How many different ways can you show me that 8 subtract 1 is
7?
Can you show me how you worked out 1 less than.....?
Success criteria
•I can use objects to take away 1 from
any number up to 10.
• I can work out how much I have left
from an amount up to 10p when I buy a
sweet costing 1p (using 1p coins to work
practically).
Simple Jottings/Mark Making for Subtraction.
12
-
3
=
9
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. Children can also use
objects, such as counters, sweets, beads, to create
the initial group and then physically take away the
right number to find the answer to the subtraction.
Key Question/Vocabulary
Count, count back, subtract, take away, cross off
Difference between, leaves, equals
How many are left over?
Find the number that is 6 less than...?
Count back 5 from 16, what number do you get to?
At this level,
children need to
see that when
doing subtraction
the biggest number
needs to be first
and you take away
the smaller
number.
Success criteria
• I can use simple jottings
to support the subtraction
of two numbers starting
from numbers up to 10, then
20.
• I can begin to select
appropriate apparatus to
support subtraction.
Using a Number Track for Subtraction.
1
2
3
4
5
Explanation
Number tracks can be used for
children to locate a number, learn the
order of numbers, and to begin to
find out one less and then a few less
than a given number.
Key Questions/Vocabulary
Count back, take away, subtract, less than, difference
between, distance between, leaves, equals
Find the number that is one less than ....? Five less
than....?
Count back 4 places from 8, where do you land?
How many is 5 less than 9?
How many between?
How far apart?
6
7
8
9
10
Three less than nine is six
Children need to be able to understand the order of
numbers remains the same and that when counting
back the numbers get smaller by 1. They need lots of
practice in counting backwards... Count back as you go
down the stairs.... Do a countdown from 10 or 20
before you leave the house....Write the numbers to 10
on separate pieces of paper and get your child to
order them in reverse, then try from 20.
Order Numicon
1 – 10 set, note
the difference
between two
shapes next to
each other is 1.
Success criteria
• I can use a number track to
count back and find the answer
to a given question starting from
numbers up to 10, then up to 20.
Using a Blank Number Line for Subtraction.
Explanation
Blank number lines are used to enable children to count 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. Children learn to use ones jumps, subtracting single
digit numbers and working within a range up to about 20, then 30. The start and end numbers are
recorded below the line and there is no need to write -1 in the jumps. With subtraction they start at
the right hand end of the line and jump back.
9–4= 5
5
14 – 6 = 8
9
Key Questions/Vocabulary
Count back, count back in ones, less than, take away,
Subtract, leaves, equals
Where are you going to start your number line?
Which number are you starting with?
How many jumps back do you need to do?
What number have you reached?
8
Children can use
Deines (one blocks)
to make the start
number and then
place one in each
jump to support the
visual image of how
many they need to
count back.
14
Success criteria
• I can use number lines to
support the subtraction of
two numbers starting from
numbers up to 20, then 30.
• I can use a number line to
help me solve subtraction
problems involving money ,
up to 30p, and measures, up
to a similar amount.
Understanding Subtraction as the Difference Between.
Explanation
When children have a secure understanding of subtraction as take away, they can learn how to solve
subtraction calculations by counting on to find the difference between the numbers. Counting on is much
easier than counting backwards and generally far fewer mistakes are made, however it is important that
children know the link between addition and subtraction, that they undo each other, we call this
INVERSE. It is also important for children to begin to get an idea of when counting on or counting back
are the best methods to use we find the difference between when the numbers are close together and
it is easier and quicker to count on than back. (eg for 87 – 12 counting back would be quicker and more
efficient but for 76 – 59 finding the difference would be preferable)
37 – 32 (What is the difference between 32 and 37?) Children start with the smaller number under the
32
37
number line and jump on until they reach the
larger number. They need to understand that
when finding the difference between two
numbers, the answer to the question is found by
counting how many jumps they did.
So the difference between 32 and 37 is 5.
Key Questions/Vocabulary
Count on, difference between, distance between, how many
more than… , How many between? How far apart?
Equals, inverse
Where are you going to start your number line?
Which number are you starting with?
Which number are you going to stop at?
How many jumps on did you do?
What was the difference between the numbers?
Children can use
Numicon to create
both the numbers
they are finding
the difference
between , laying
them on top of
each other to see
the visual
difference.
Success criteria
• I can use a number line to
find the difference
between two numbers by
counting on.
• I can use a number line to
help me solve subtraction
problems involving money
and measures, using the
method of counting on.
Developing use of Number Lines, Subtracting Tens and Ones.
Children need to understand the place value of each digit in order to partition
2-digit numbers into tens and ones so they know how many tens and ones
jumps to do.
-10
48g – 13g = 35g
(as subtraction by
counting back)
35g
38g
48g
60p – 35p = 25p
(as the difference
between so counting on)
+10
35p
Key Questions/Vocabulary
Subtract, take away, minus, less than,
ten less, count back in tens, one less,
count back in ones, leaves, equals,
inverse
Difference between (Eg for 53 – 47, it
is easier to count on from 47 to find the
difference between the two numbers
than count back 47 places.)
+10
45p
55p
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 in the
small jumps, this would be inefficient. When
confident subtracting ‘teen’ numbers,
progress to subtracting numbers with more
than one ten.
60p
Success criteria
• I can use
number lines more
efficiently to
subtract a 2-digit
number by
counting back or
counting on the
tens jumps and
then the ones
jumps.
Develop Efficient Use of Number Lines.
112cm – 94cm = 18cm
+6cm
(as the difference
between so counting on)
94cm
100cm
152 – 68 = 84
(as subtraction by
counting back)
+2cm
110cm
112cm
-60
-6
84
+10cm
90
-2
150
152
Explanation
Once children are confident and accurate in the use of tens and ones jumps, they can progress to using
multiple of tens jumps. Encourage children to use their knowledge of number bonds to bridge through the nearest
multiple of 10 to make counting easier (as in the second eg). Make sure they keep a record inside their jumps of
what they are doing so that they can check they have subtracted the correct number.
Key Questions/Vocabulary
Subtract, take away, minus, less than
Ten less, count in tens
One less, count in ones
Difference between, inverse
Equals, leaves, altogether
Use mental recall of + and –
facts to 20 and apply to
problems.
Know all the + and –facts for
multiples of 5 to 100 (eg 35 +
45, 80 – 55).
Derive rapidly all number pairs
that total 100 (eg 61 + 39,
22 + 78)
Success Criteria
• I can subtract chunks of
tens and ones to make my
calculations more efficient.
• I can bridge through the
nearest multiple of ten.
Partition for Subtraction.
To subtract 24 from 68, first subtract the tens and record
where you have reached. Then you need to continue and
subtract the units. It can be recorded initially as a number
line so that the children can see the link between the two
recording methods given here.
TU
TU
68
-
24
68
-
20
48
-
4 =
= 48 (subtract the tens first)
44
(subtract the units second)
SO 68 – 24 = 44
- 10
44
48
-10
68
Explanation
Some children will find this a quick and easy method for subtraction, when
the digits being subtracted are smaller than the starting number, and
they will be able to do it mentally. Initially it is important to give
calculations where the ones and tens digits do not cross the tens boundary.
Key Questions/Vocabulary
Subtract, minus, take away, less
than, tens, units, partition, inverse
How many units are you
subtracting?
How many tens are you subtracting?
Confidently recall + and – facts to 10
then 20 (eg, 9 + 6, 13 – 7, 15 + 4)
Know all + and – facts for multiples of
10 to 100 (eg, 30 + 50, 90 – 20)
Children can make the
start number from Deines
or Numicon and physically
remove the tens and then
the units. This will act as a
strong visual image to
support the subtraction
they are doing.
Success Criteria
• I can partition the
number being
subtracted into tens
and units to support
subtraction.
• I can subtract the
units and then the tens
for 2-digit subtraction
calculations.
Expanded Column and Formal Written Subtraction.
87 – 52 = 35
Key Questions/Vocabulary
Subtract, minus, take away, less
than, hundreds, tens, units, partition,
inverse
How many units are you
subtracting?
How many tens are you subtracting?
How many hundreds are you
subtracting?
Explanation
This method moves the
children on from number lines
and horizontal partitioning to
vertical methods including the
partitioning still.
They are able to see the
number split up then
recombine at the end.
**
If you wish to insert an + sign
between the partitioned
numbers then this is an
individual school decision.
e.g. 100 + 50 + 4
Success Criteria
• I can partition the
number being
subtracted into
hundreds, tens and units
to support subtraction.
• I can subtract the
units and then the tens
for 2-digit subtraction
calculations and
continue for 3-digit
calculations
Formal Written Subtraction.
Key Questions/Vocabulary
Subtract, minus, take away, less
than, tens, units, partition, inverse
Decompostition
Estimation
Operation
Explanation
This formal written method will
only work once the children are
secure in their understanding of
place value and partitioning of
numbers. Applying the process of
units, tens, hundreds will ensure
that they are subtracting in the
correct manner.
Some calculations will be
straightforward whilst others
will require further thought as
you ‘take’ a ten or hundred from
the next column
Success Criteria
•I can subtract the
units, then the tens,
then the hundreds for
3-digit or more
subtraction calculations.