Developing
Mental Strategies
Mathematics CPD course 04/05
Nigel Davies
Frequently asked questions
How can I teach 30 children some
Mathematics that they didn’t know before so
that they can understand it?
How can I do this so that they feel involved &
interested?
What strategies & techniques can I use?
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Effective Maths Teaching
“Good teachers put
themselves in the position of
learners”
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Effective teaching involves …
Directing – explaining what has to be done & when
Demonstrating & modelling – showing children how
to do something or providing an image to help them
understand something
Instructing – running through a procedure or process
to be followed
Explaining & illustrating – providing reasons & giving
examples
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Effective teaching (contd) …
Questioning & discussing – encouraging children to
interact with the Mathematics, with the teacher &
each other
Developing & consolidating – reinforcing what has
been taught, providing opportunities for repetition,
practice & problem solving
Evaluating children’s responses – giving them
feedback & dealing with misconceptions & errors
Summarising – stressing key teaching points, ideas &
vocabulary & making links to other work
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Watching the video, do you see
teachers …
… demonstrate or model a piece of Mathematics?
… explain or illustrate something?
… question & discuss?
… develop & consolidate?
… evaluate children’s responses?
… summarise?
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Context in which good teaching takes
place :
To teach effectively, the teacher needs to
have :
High expectations of what the children can do
Clear objectives, outlining what is to be taught &
learned
A careful plan for each of the three parts of the
lesson
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Context (contd) :
An understanding of the appropriate & correct
vocabulary & notation for this topic
A suitably organised class for each lesson or partlesson
Well-organised resources
Any support staff used effectively
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Different types of questions
Recalling facts
What is … 1 more than 6 … 7 times 8 …
How many metres in 1 kilometre?
Using facts
7 x 6 = 42, so what is 7 x 12?
Double 250
Hypothesising or predicting
What is the next number : 1, 3, 7, 15, …
Estimate the number of words on this page
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4 squared?
Different types of questions (contd)
Designing or comparing procedures
How could we count this pile of coins?
How could you subtract 97 from 210?
Are there other ways of doing it?
Interpreting results
What do you notice about multiples of 5?
What does this bar chart tell you?
Applying reasoning
Why is the sum of any two odd numbers an even number?
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Ways of answering
Recalling facts
Children show a number of fingers
Children hold up number cards
Children say an answer in unison
Using facts
Give 1 minute time limit, then children show an answer using
whiteboards
Give a “thinking time”, then ask children to answer aloud
Hypothesising or predicting
Children work in pairs. Give a longish thinking time.
Children write or say their answers
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Ways of answering (contd)
Designing or comparing procedures
Children work in groups & then report back as a group
Interpreting results
Children work in pairs. Give a longish thinking time.
Choose several pairs to explain their answers
Applying reasoning
Children work in pairs. Give a longish thinking time.
They can present their answers on a board or flipchart.
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Four elements
The “success” of the lesson
What its objectives are & what you will do in the plenary to check
what has been learned
The “story” of the lesson
The sequence of events
The “stuff” of the lesson
What ideas, methods or skills are being taught & the models or
images used to teach them
The “speed” of the lesson
How long each activity should last & how to keep children who can
work fast occupied at the same time as allowing for children who
work more slowly
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The “story” of the lesson
Problem :
Planning too much for each part of the
lesson so that the mental/oral starter runs
over, or activities in the main part are
rushed, & there is little or no plenary.
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The “story” of the lesson
Possible solutions :
Have only one or, at most, two objectives for the mental/oral work.
Don’t have too many objectives for main activity.
Link objectives, specifically in relation to how you teach them.
If the main activity involves a high percentage of whole-class direct
teaching, keep the mental/oral part short.
If the main activity involves mainly group work, the mental/oral
can be longer.
Tell the children how long they have to complete a task. Give a 5
minute warning before the finish time.
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The “stuff” of the lesson
Problem :
Teaching a topic, & realising that despite
your best efforts, only half the class have
understood it. You either teach it all over
again or repeat yourself by re-teaching
small groups or, worse, individual
children.
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The “stuff” of the lesson
Possible solutions :
Plan a clear model or image so that you know how you are
teaching the topic & how it will be represented.
Plan a second model or image so that it is not repetitious to teach
this a second time.
Have resources for small groups or individuals as well as for
demonstration to the whole class.
Practice any prerequisite skills or facts during the mental/oral so
that these are uppermost in children’s minds.
Display any supporting information if it will help the topic being
taught.
Plan carefully what any support staff will do.
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The “speed” of the lesson
Problem :
Some children regularly work so fast that
you run out of things for them to do;
others work so slowly that they never
finish their work.
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The “speed” of the lesson
Possible solutions :
Have some targeted work – two types : extension & support - for
individuals or groups who are likely to need something different
from the ‘norm’.
Make it very clear to children how much work you expect. Be
realistic, but don’t be satisfied with less than they can do.
Help children who work slowly by encouraging them to do ‘just one
more’ each day.
Set limited, but achievable, goals for improvement.
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Approaches to calculation
Mental first
Moving to written procedures too fast can
mean :
Children following the procedure but
not getting a ‘feel’ for the numbers
Children missing the obvious
Children looking for a calculator if the
written procedure looks too difficult
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In your heads
10000 – 10
25 x 19
5% of 86
248 – 99
84 – 77
Half of 378
1+2+3+4+5+6+7+8+9
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Using paper & pencil …
… show your working
10000
25
10
X 19
-
5% of 86
1
2
3
4
5
6
248
84
99
- 77
-
7
2 378
8
+9
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Differences
What are the differences between mental
& written methods?
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Mental
We may break the calculation into manageable parts,
e.g. we do
248 - 100 + 1 instead of 248 - 99.
We say the calculation to ourselves, and therefore are aware of
what numbers are involved,
e.g. 2000 - 10 is not much less than 2000.
We choose a strategy to fit the numbers, e.g. 148 - 99
may be done differently from 84 - 77, although they are both
subtractions.
We draw upon specific mathematical knowledge, an
understanding of the number system, learned number facts and
so on.
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Written
We never change the calculation to an equivalent one,
e.g. 148 - 99 is done as it is.
We don't say the numbers to ourselves, but start a procedure
such as 148 - 99
by saying,
'8 take away 9, you can't, so make it 18 take away 9....'
We always use the same method.
We draw upon a memory of a procedure, and possibly, though
not necessarily, an understanding of how it works.
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Structured approach for + & Mental counting, Years 1 to 6
Counting in ones from and back to zero
Counting in tens: 46, 56, 66, 76, 86, 96, 106 . . . then 100s
Counting in decimals . . . in fractions . . . through zero to negative numbers
Early stages of mental calculation, learning facts, Years 1, 2 and 3
Knowing all addition and subtraction facts to 10, then 20
Knowing that 6 + 4 = 10, so 10 - 6 = 4
Knowing that 8 + 8 = 16, so 8 + 9 = 17, and 7 + 8 = 15
Knowing that 67 - 10 = 57, so 67 - 9 = 58
Working with larger numbers and informal jottings, Years 2 and 3
Adding and subtracting multiples of 10 and 100
Finding complements in 100, e.g. 66 +
= 100
56 + 37 = 56 + 30 + 7
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Structured approach for + & (contd)
Non-standard expanded written methods, beginning in late Year 3
Standard written methods, Years 4 and 5
Column addition and subtraction, with 'carrying' and 'adjustment'
Money and decimals
Use of calculators, beginning in Year 5
Addition and subtraction problems using realistic data
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Structured approach for x & 
Mental counting and counting objects, Years 2 and 3
Counting in twos or fives . . .
Breaking off 'sticks' of cubes: How many twos make 20?
Counting on from and back to zero in threes, fours . . .
Early stages of mental calculation, learning facts, Years 2 and 3
Knowing doubles of small numbers, and corresponding halves
Knowing that 10 + 10 + 10 = 30
Knowing that 10 x 3 = 30, and 30 ÷ 3 = 10
Working with larger numbers and informal jottings, Years 2, 3 and 4
Double 17 is double 10 plus double 7, or 20 + 14
37÷ 5 = 7 R 2
Multiplying, then dividing, by 10, 100, 1000
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Structured approach for x & 
(contd)
Non-standard expanded written methods, beginning in Year 4
Standard written methods, Years 4, 5 and 6
Short multiplication, long multiplication, short division
Multiplication and division of decimals by whole numbers
Use of calculators, beginning in Year 5
Find
Find
Find
Find
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12.5% of £15.36
the area of a lawn measuring 21.3m x 7.65m
two consecutive numbers with a product of 2352
two integers, each less than 20, with a quotient of 1.1818181
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Written Mathematics
Children can begin to make written
records at any stage BUT:
the introduction of standard written methods
of addition and subtraction should be delayed
until children can add and subtract two-digit
numbers mentally;
before beginning standard written methods of
multiplication and division, children should be
confident with multiplication tables for at least
2, 3, 4, 5, and 10.
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Modelling strategies
Direct teaching of mental calculation strategies
involves :
adopting a structured approach so that skills and
strategies are developed systematically;
modelling a strategy using an image, a model, a 'real
life' scenario, or some structured apparatus;
explaining to the children how to do something;
using a child's error or a less efficient strategy as a
starting point for a demonstration of a better
strategy;
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Modelling strategies (contd)
encouraging children to compare strategies
and improve on their strategies;
talking about choosing a strategy;
showing the children how to use a known fact
in developing a strategy;
providing the children with a 'prompt' to help
them recall and then use an appropriate
number fact.
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Teaching Mental Calculations
Foundations of mental calculation
understanding of the counting system, the order of
numbers, where they 'fit in', whether they are close
together or far apart, and so on;
understanding of place value, and in particular thinking of
48 as 40 + 8 rather than 4 tens and 8 units, and 325 as
300 + 20 + 5 rather than 3 hundreds, 2 tens and 5 ones;
knowing familiar number facts by heart;
understanding and use of the relationships between the
four operations, and the principles (not the names!) of
the commutative, associative and distributive laws.
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Commutative Law
This applies when the order of the
numbers in the calculation doesn’t
matter :
2+3=3+2
3x4=4x3
Note : 7 – 5 does not equal 5 – 7
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Associative Law
This applies when numbers can be
regrouped without affecting the result :
2 + (3 + 4) = (2 + 3) + 4
(2 x 3) x 4 = 2 x (3 x 4)
Note : 60  (12  3) does not equal
(60  12)  3
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Distributive Law
This applies when partitioned numbers
are multiplied :
(2 + 3) x 4 = (2 x 4) + (3 x 4)
27 x 5 = (20 x 5) + (7 x 5)
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+ & - strategies?
45 – 37
45 + 8
262 – 95
307 + 95
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+ & - strategies
45 – 37
start at 37, add 3, add 5
45 + 8
add 5, then add 3 more
262 – 95
subtract 100 then add 5
307 + 95
add 95 to 300 then add 7 more
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+ & - strategies (contd)
3468 – 3288
2460 + 3540
3.07 + 1.2
3.07 - 1.2
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+ & - strategies (contd)
3468 – 3288
2460 + 3540
288 + 200 = 488, which is 20
too much so 200 less 20 is 180
the hundreds and tens add up to
1000, so there are 2 + 3 + 1
thousands
3.07 + 1.2
altogether there are 4 units, 2 tenths
and 7 hundredths, which is 4.27
3.07 - 1.2
from 1.2 to 3 is 1.8, and 7
hundredths more is 1.87
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Skills in early addition
Counting all - a child doing 2 + 3 counts out two
bricks and then three bricks and then finds the total
by counting all the bricks.
Counting on from the first number - a child finding 3
+ 5 counts on from the first number 'four, five, six,
seven, eight'.
Counting on from the larger number - a child chooses
the larger number, even when it is not the first
number and counts on from there.
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Skills in early addition (contd)
Using a known addition fact - where a child gives an
immediate response to facts known by heart, such as
6 + 4 or 3 + 3 or 10 + 8.
Using a known fact to derive a new fact - where a
child uses a number bond that they know by heart to
calculate one that she or he does not know, e.g.
using knowledge that 5 + 5 = 10 to work out 5 + 6
= 11 and 5 + 7 = 12.
Using knowledge of place value - where a child uses
knowledge that 4 + 3 = 7 to work out 40 + 30 = 70,
or knowledge that 46 + 10 is 56 to work out 46 + 11
= 57.
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Skills in early subtraction
Counting out - a child finding 9 - 3 holds up nine
fingers and folds down three.
Counting back from - a child finding 9 - 3 counts back
three numbers from 9: 'eight, seven, six'.
Counting back to - a child doing 11 - 7 counts back
from the first number to the second, keeping a tally
using fingers of the number of numbers that have
been said, 'ten, nine, eight, seven', holding up four
fingers.
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Skills in early subtraction
(contd)
Counting up - a child doing 11 - 7 counts up from 7 to
11, 'eight, nine, ten, eleven', sometimes keeping a
count of the spoken numbers using fingers (not a
'natural' strategy for many children because of the
widely held perception of subtraction as 'taking away').
Using a known fact - a child gives a rapid response
based on facts known by heart, such as 8 - 3 or 20 - 9.
Using a derived fact - a child uses a known fact to
work out a new one, e.g. 20 - 5 is 15, so 20 - 6 must
be 14 (more unusual in subtraction than in addition).
Using knowledge of place value - a child finding 25 - 9
knows that 25 - 10 is 15, and uses this to give an
answer of 16.
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x &  strategies?
30 x 70
1000 ÷ 200
37 x 4
472 ÷ 4
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x &  strategies
30 x 70
30 x 7 = 210 and
210 x 10 = 2100
1000 ÷ 200
10 x 100 is 1000, so 5 x 200 is
1000
or 10 ÷ 2 = 5, so 1000 ÷ 200 is 5
37 x 4
double 37 = 74 and double 74 = 148
or 30 x 4 = 120, 7 x 4 = 28,
and 120 + 28 = 148
472 ÷ 4
half 472 is 236, and half 236 is 118,
or 400 ÷ 4 = 100 and 72 ÷ 4 = 18
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x &  strategies (contd)
36 x 50
2000 ÷ 5
4.2 x 1.1
9.3 ÷ 0.3
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x &  strategies (contd)
36 x 50
36 x 100 = 3600, so 36 x 50 is
half of 3600,which is 1800
2000 ÷ 5
20 ÷ 5 = 4, so 2000 ÷ 5 = 400
4.2 x 1.1
4.2 x 1 = 4.2 and 4.2 x 0.1 = 0.42,
So 4.2 x 1.1 = 4.62
9.3 ÷ 0.3
is equivalent to 93 ÷ 3 which is 31
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Multiplication : Years 2, 3 & 4
Repeated addition - working out 3 x 4 by counting
out four groups of three.
Multiplication as counting in equal jumps along the
number line - 'five, ten, fifteen, twenty', or in twos or
tens or other multiples.
Starting with tables for 2 and 10, knowing by heart
facts such as 'seven twos' or 'four tens', progressing
to facts in the 5 times-table, then others.
Recognising that multiplication can be done in any
order - e.g. realising that 5 x 2 is the same as 2 x 5.
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Multiplication : Years 2, 3 & 4
(contd)
Doubling two-digit numbers - double 10, 20, 30, 40
... then, for example, double 23 as double 20 plus
double 3.
Using doubling - e.g. working out the 4 times-table
by doubling the 2 times-table.
Using related facts - e.g. using 5 x 4 = 20 to find 6 x
4 = 24.
Knowing by heart the 3 and 4 times-tables.
Recognising the effect of multiplying a number by 10.
Approximating, e.g. working out 19 x 17 by doing 20
x 17 and then subtracting 17.
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Multiplication : Years 4, 5 & 6
Knowing all multiplication facts up to 10 x 10.
Partitioning, e.g. working out 23 x 4 as 20 x 4 and 3
x 4.
Understanding the effect of multiplying numbers by
10, or 100 or 1000.
Multiplying multiples of 10, e.g. 30, 50,700 x 20.
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Multiplication : Years 4, 5
& 6 (contd)
Using factors, e.g. 51 x 12 = 51 x 4 x 3, or 12 x 25 =
12 x 100 ÷ 4.
Using related facts, e.g. using 20 x 17 to work out 19
x 17.
Approximating, e.g. recognising that 43 x 18 is close
to 40 x 20 = 800, or that 372 ÷ 5 is about 350 ÷ 5 =
70.
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Difficulties
Have you worked with children or adults, who:
do almost any addition or subtraction by counting on
or back in ones?
subtract numbers that are close together, such as 42
- 38 by trying to 'take away' or 'count back' 38 from
42, rather than counting up from 38?
don't recognise that to add 10 or 100 is no more
difficult than to add 1?
don't recognise numbers which are 10, or a multiple
of 10, apart?
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More difficulties
…
who:
never spot a double or a number bond?
never change a calculation to make it easier, e.g. in
242 - 99 they subtract 99 rather than subtracting
100?
to calculate mentally, turn to a standard written
method and try to visualise it?
don't see facts which would make the calculation
easier, such as the fact that 50 is half of 100 in
calculating 36 x 50
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Teaching & learning number
facts & tables
What kind of learner?
Take different learning styles into
account :
Aural
Visual
Written
Kinaesthetic
Pattern
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Strategies for memorising
Kinaesthetic - this involves memorising through movement,
learning by matching facts to specific ways of moving, such as
finger counting or action sequences.
Visual - some children have a good visual memory, and can 'see'
facts on the page or on the board. Flash cards can be helpful, as
can visual representations of the facts in question.
Aural - some children remember things by 'hearing' them
repeated. Chanting the sequence of numbers, or the number
facts in order can be useful strategies for such children, as can
matching facts to rhymes, songs or music.
Written - writing something can help the facts to travel from the
pen to the brain! Children can also see how the facts connect
together when they are presented in written form.
Pattern - some children find it much easier to recall facts when
they understand the structure of patterns in which they are
embedded.
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Kinaesthetic strategies
Number bonds to ten using fingers
Doubles using fingers
Counting in fives or twos
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Kinaesthetic strategies (contd)
Nine times-table
For three times, fold down the third
finger
Actions to go with tables facts
'Eight eights are sixty four,
clap, clap, knock on the floor.'
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Aural strategies
Chant tables
Make up a tune
Make up a ‘rap’
Call out together
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Chant as whole number sentences:
“One six is six, two sixes are twelve . .”
Chant as lists of multiples:
“Six, twelve, eighteen, twenty-four . .”
Chant them forwards and backwards
Sing a table.
Accompany it with instruments . . .
Make a rap for a particular timestable.
Perform it for the rest of the school.
e.g. making ten:
teacher says, “Four and . . .”
children say, “. . . six make ten”.
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Aural strategies (contd)
Make up rhymes
Make up rhyming words for 'difficult' facts -
“six sevens are forty-two, boo, hoo, hoo”
“eight eights are sixty-four, what a bore”
Fact of the day
Choose a 'fact of the day' when school starts, then ask different
children what it is at frequent intervals through the day.
Use funny voices
Say a number sentence or 'difficult' fact in different voices:
Low, high, squeaky, loud, soft, tired, excited . . .
In the voice of a frog, elephant, mouse, lion . . .
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Visual strategies
Number bonds with pegs/key
rings on a hanger
Matching card games such as Pelmanism or Snap,
with questions on one set of cards and answers on another
Flash cards
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Visual strategies (contd)
Bead strings
Sets and dividers
Coloured pairs on a number line
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43 - 26
You can find the first 43 beads and then …
…count out ( or take away) 26
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=17
66
Calligrams
Para
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cute
e
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Written strategies
Writing out number facts as 'sums with holes'
Chains
Posters
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Written strategies (contd)
Lists or notices of key facts
Index cards
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Patterning strategies
Number lines
Number grid 1 - 100
with multiples coloured
Multiplication grid with 1x, 2x, 5x, 9x, 10x shaded
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Patterning strategies (contd)
Number grids of different sizes
Snowflakes
Squares
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Ways to help children remember
Practise with just one fact a day, or try a 'fact for the
week'.
Practise 'fact families'. e.g.
6 + 8 = 14, 8 + 6 = 14, 14 - 6 = 7, 14 - 8 = 6
4 x 5 =20, 5 x 4 = 20, 20 ÷ 4 = 5, 20 ÷ 5 = 4
Work from answers back to facts - how many facts
do you know with an answer of 12?
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Ways to help children remember
(contd)
Make an addition or multiplication table and cross out
all those facts you already know. Now focus on those
you need to learn.
Encourage children to work out their own ways to
remember facts.
Draw pictures to accompany particular facts.
Repeat it and repeat it!
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Two principles
Once children have learned some facts, bear in mind:
known facts can always be used to derive
new facts, e.g. if you know that 3 x 7 = 21,
you can work out other 'facts', such as:
21 ÷ 7 = 3
by using the relationship between
3 x 8 = 24
6 x 7 = 42
30 x 7 = 210
13 x 7 = 91
multiplication and division
by adding one more 3
by doubling 3 x 7 = 21
by using understanding of multiplying by 10
by partitioning 13 into 10 + 3
facts will be forgotten unless there are
frequent, varied opportunities to practise
recalling them and using them in the oral and mental
starter of your lessons, in games, puzzles, the everyday life of
the classroom, other subjects . . .
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Teaching strategies that assist
learning of number facts
Significantly increasing the time given to regular
mental work to keep skills sharp.
Providing varied opportunities for regular mental
practice and recall, in the oral and mental starter, in
work out-of-class and at home.
Enlisting the support of adults to help children learn
and remember number facts in interesting and varied
ways.
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Teaching strategies that assist
learning of number facts (contd)
Using resources such as number lines
and squares, counting sticks, and the
abacus, to give children a visual model
for their calculation methods.
Emphasising strategies which help children to learn new
facts and recover forgotten ones.
Linking the number operations: + and - facts taught
together, x &  facts taught together.
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Number lines
+10
76
+10
86
+10
96
+10
106
+7
116
+ 40
+7
116
76
76 + 47 = 123
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123
77
123
Some activities for learning
number facts and tables
Counting and chanting in different ways – whole class,
taking turns, as a rap, using rhyming slang, using
different voices, etc.
Playing ‘Beat the calculator’ games.
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Some more activities for
learning number facts and tables
Playing number games with dice,
cards, dominoes, target boards, etc.
Having a ‘fact of the day’ or ‘fact of the week’.
Setting children personal time challenges.
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Yet more activities for learning
number facts and tables
Giving complements of numbers (progressively to
complements in 100).
Using open question variations of the type
‘The answer is 12, what is the question?’
Giving a fact and asking for related facts,
e.g. to ‘4 x 5 = 20’, say ‘20  4 = 5’.
Using x2s to get x4s, x4s to get x8s, etc.
Exploring patterns in tables, etc.
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Developing Mental Strategies
in KS1
Where do mental
strategies come from?
Skills
What can
you
DO?
Developing
skills such as:
• counting
• partitioning
and
recombining
numbers
Ideas
What do you
REALISE?
What can you
REMEMBER?
Building
awareness of:
• the number system
• the relative value of
numbers
• number relationships
Recalling facts
such as:
• halves and
doubles
• number
bonds
• multiplication
facts
Construct a
strategy
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Facts
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Approaches to mental
calculations
AS 1.2 & 1.3
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Anisha in November
As you watch the video, consider :
What skills & number facts does Anisha
draw upon?
What ideas about the number system
does she use?
How does Ian develop Anisha’s
understanding?
Does she learn any new strategies?
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Developing Mental Strategies
in KS2
Approaches to mental
calculations
AS 3.2 & 3.3
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Cheryl in November
As you watch the video, consider :
What skills & number facts does Anisha
draw upon?
What ideas about the number system
does she use?
How does Ian develop Anisha’s
understanding?
Does she learn any new strategies?
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Developing Mental Strategies
How do we help children to learn new strategies?
Modelling numbers using equipment
Modelling number operations
using number lines or number grids
Teaching
Providing short exercises in which
all the examples follow
the same structure
Explaining how to use a model
Sorting and classifying problems
to help to identify those
with particular features
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Building confidence in understanding
situations through structured
number games and activities
88
Developing Mental Strategies
Could you double this number?
What facts could you use?
What 10s number do you
pass on your way?
Questioning
Can you see a pair of numbers
that make 100?
Which number did
you start with?
Do you know what 10
more than that is?
How would you read these decimal
numbers if they were money?
What ideas? What skills? What facts?
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