Transcript Mental Computation

Common Purpose and Goals for Teaching
Mathematics at SMOT:
We aim to ensure that all students are given an equal
opportunity to achieve and to be challenged to reach
their full potential.
We aim to teach the fundamental mathematical skills
needed to function in everyday life. Learning
mathematical skills will provide our students with the
opportunity to become proficient in their understanding
of a variety of mathematical concepts, to help them
problem solve and apply strategies to life situations.
Adapted from Bern Long and Angela Rogers
presentation, 2013
K. Chiodi
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Discuss with the person next to you what you
think mental computation involves?
When should mental computation be taught?
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Mental computation is a calculation
performed entirely in the head, with only the
answer being written (McIntosh, 2005)
Reading: Mental Computation and Estimation
Read and then discuss at your table
Victoria Department of Education and Early Childhood Development, 2009
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Resource:
Mental Computation: A Strategies Approach
Alistair McIntosh, 2004
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Mental computation is based on
understanding.
Mental arithmetic is based on speed and
accuracy related to memory.
Research by Biggs (1967) revealed that:
“Allocation of time to mental arithmetic bore no relation to
attainment”
“In other words, these daily speed and accuracy tests did not
make the children noticeably more competent, but it did make
them slightly more neurotic about numbers” (McIntosh, 2004, 1)
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Warm Up activity using the Westwood
Addition and Subtraction test.
Introduce and make explicit the strategies we use
to help us complete mental computations.
 Memorisation of some basic facts required.
 It is certainly desirable for children to know the
addition facts to 20.
 Mental computation strategies must be efficient
and always allow us to arrive at the correct
answer.
 Strategies are necessary because they allow
students not only to calculate simple 1-digit
facts but also to calculate much bigger equations
e.g. 6 + 4 …….. 66 + 24
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Level
Mental Computation Link
F
Subitise small collections of objects (ACMNA003)
1
Represent and solve simple addition and subtraction problems
using a range of strategies including counting on, partitioning
and rearranging parts (ACMNA015)
2
Explore the connection between addition and subtraction
(ACMNA029)
Solve simple addition and subtraction problems using a range of
efficient mental and written strategies (ACMNA030)
3
Recognise and explain the connection between addition and
subtraction (ACMNA054)
Recall addition facts for single-digit numbers and related
subtraction
facts to develop increasingly efficient mental strategies for
computation (ACMNA055)
Level
4
Mental Computation Link
Recall multiplication facts up to 10 × 10 and related division
facts (ACMNA075)
Develop efficient mental and written strategies and use
appropriate
digital technologies for multiplication and for division where
there is no remainder (ACMNA076)
5
Use efficient mental and written strategies and apply appropriate
digital technologies to solve problems (ACMNA291)
Use equivalent number sentences involving multiplication and
division
to find unknown quantities (ACMNA121)
6
Select and apply efficient mental and written strategies and
appropriate digital technologies to solve problems involving all
four
operations with whole numbers (ACMNA123)
The Addition and Subtraction
strategies to be taught F – 6
Count on in ones
Tens Facts
Doubles (for addition)
Doubles (for subtraction)
Near Doubles (for addition)
Near Doubles (for
subtraction)
Bridging to 10
Adding 10
Communitivity (counting
on from the larger number)
Count back
Count down to
Count up from
Tens Facts (for subtraction)
Subtract 10
Bridging 10 (for
subtraction)
Inverse – Think ‘+’
You can use your fingers to count on 0, 1, 2, 3
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3+0=3
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3+1=4
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3+2=5
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3+3=6 (also a double)
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Add 1,2,3 to a multi
digit no. e.g. 3+35=
Add 10, 20, 30 to a
multiple of 10 up to
90. e.g. 80+30=
Add 10, 20, 30 to a
multi digit number
e.g. 34+30=
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Add 100, 200, 300 to a
hundreds number
E.g. 500+200=
Add 100, 200, 300 to a
multi digit number e.g.
34+300
Add 1000, 2000, 3000
to a single digit number
E.g. 3 000+9=
Add 1000, 2000, 3000
to a multi digit number
e.g. 41+2 000=
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If you spin around addition equations you get
the same answer.
2 + 4 =6
This shows children
the commutativity of
addition equations
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4+2=6
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Single digit with multi-digit
E.g. 4+64=64+4
Multi-digit with multi-digit
E.g. 97+123=123+97
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Start with real world
We need to learn these doubles.
1+1=2
2+2=4
3+3=6
4+4=8
5+5=10
6+6=12
7+7=14
8+8=16
9+9=18
10+10=20
Double multiples of 10
up to 90
e.g. 50+50
Think…
5 tens+5 tens= 10 tens=
100
 Double multiples of
100 up to 900
e.g. 600+600=
Think…
6 hundred+6 hundred=
12 hundred= 1 200
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Double multiples of 1
000 up to 9 000
e.g. 6 000+6 000
Think…
6 thousand+6
thousand=12
thousand= 12 00
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Use doubles strategy with multi-digit and
single digit numbers
e.g. 64+4
 Use doubles strategy when adding multi-digit
with multi-digit
64+24
356+36
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If we remember the doubles, we can work out
these sums.
6+7=13
5+6=11
4+3=7
9+8=17
Near doubles with multiples
of 10 up to 90 e.g. 50+60
Think…
5 tens +5tens is tens and 1
more ten is 11 tens=110
 Near doubles with multiples
of 100 to 900 e.g. 400+500
 Near doubles with multiples
of 1000 up to 9000
e.g. 7000+6000
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Use near doubles strategy
with multi-digit and single
digit numbers e.g. 64+5
 Use near doubles strategy
when adding multi-digit with
multi-digit
64+25
356+37
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These equations add to 10.
1+9=10
2+8=10
3+7=10
4+6=10
5+5=10
9+1=10
8+2=10
7+3=10
6+4=10
5+5=10
Tens facts with single
digits that add to 20
E.g. 6+14
 Tens facts with
multiples of 10 that
add to 100
E.g. 60+40
 Tens facts with
multiples of 100 that
add to 1000
E.g. 200+800
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Tens facts with
multiples of 1000 that
add to 10 000
E.g. 4000+6000
 Tens facts with single
digit that add with
multi digit numbers
E.g. 6+34
 Tens facts with multi
digit with multi digit
E.g. 64+36
 129+211
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Seven, Eight, Nine are close to Ten.
9+2=11
9+3=12
9+4=13
8+6=14
8+7=15
8+8=16
7+6=13
7+5=12
7+4=11
Bridging with multiples of 10
e.g. 40+90 think…4 tens +9 tens is 13 tens =
130
400+900
4000+9000
 Bridging single digit numbers with multi digit
numbers
 E.g. 43+9
 Bridging multi digit numbers with multi-digit
numbers
e.g. 43+59
256+349
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When we add ten, the ones number stays the
same.
2+10=12
3+10=13
4+10=14
5+10=15
6+10=16
7+10=17
8+10=18
9+10=19
Add 10 to multi-digit numbers
e.g. 10+25
10+257
 Add 100 to single digit e.g. 100+5
 Add 100 to multi-digit e.g. 100+27
 Add 1000 to single digit
e.g. 1 000+6=
 Add 1000 to multi-digit
e.g. 56+1 000
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To teach with understanding all
strategies must be taught using visual
aids and at any time when students
are experiencing difficulties teachers
must return to visual aids e.g. tens
frames, bead strings, place value
cards etc.
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2 dice are rolled
Write the number sentence on your playing
board that corresponds with the strategy you
used to work out the answer
The player to fill up their playing board first
calls ‘Bingo’ and reads out their answers
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You can count back 0, 1, 2, 3 using your
fingers or in your head
5-2=3
6-1=5
Count down to:
 Beginning at the total, count down to the number
being taken away. The answer is the number of
steps this takes.
For example, 18-13 start at 18 and count down to
13, 17, 16, 15, 14, 13= 5 numbers counted back
Count up from:
 Beginning at the number being taken away, count
up to the total. The answer is the number of
steps this takes.
For example, 21-17 start at 17 and count up to
21. 18, 19, 20, 21 = 4 numbers counted up
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Tens facts can help us work out the answer
when we subtract from ten.
E.g.. 10=5=5
10-6=4
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If you know 6+6=12, then you also know 126=6
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E.g.: 4+4=8 so 8-4=4
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10+10=20 so 20-10=10
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You can take away the ones number and it
leaves you with just the tens.
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E.g.: 13-3=10
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25-5=20
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Count back to the nearest tens number, then
its easy to take away what is left.
E.g.. 11-3=
First do 11-1=10
Then 10-2=8 (tens fact)
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If you know the doubles the near doubles can
also help.
E.g.. 12-7=
I know 12-6 (double)=6
Take one more = 5
Explain
How did you figure it out?
Justify
How did you do it like that?
Compare
Is there another way?
Which one do you like?
I really like this strategy. What other
problems will this strategy work for?
Will it always work?
Reasoning and Is your answer reasonable/ Could
reflecting
that be the answer?
How do you know your right?
Application
What would you use this for in the
real world?