Transcript Mental Arithmetic Strategies

Mental Arithmetic Strategies
Scotts Primary School
Mental Arithmetic Scheme
.
.
Mental Strategies - Introduction
Mental strategies are just quick ways of
doing sums in your head. Sometimes it is
easier if you make notes whilst working as
this helps you check what you’re doing
and you can make sure you haven’t
missed anything. You can use these
strategies anywhere – they don’t have to
be just used in mental arithmetic tests.
Nobody minds what strategy you use as
long as you get the right answer!!!
How to use this guide
 1) Spend time going through each strategy at a
time.
 2) Do the questions at the end of each section
and check your answers carefully.
 3) If your answers are different then check
through your work carefully. Try again. If you are
still having trouble, then ask for help.
How to use this guide
4) Even when you have worked through all
of the examples don’t stop using this
guide. You can keep going over and over
the strategies, over the questions and
checking against the answers.
5) The more you practice – the better and
quicker you will be!
SUBTRACTION STRATEGIES
Difference
Take away
Subtract
Less than
Minus
1) Counting on…
Sometimes using traditional column
methods of subtraction can be confusing
especially when there are lots of zeros.
eg
3000699
_____
_____
In this case we
would have to
borrow and carry.
This can be confusing for some children and
this method would take ages to do in a
mental test.
Another method that can be used is
counting on…
Lets start with the example we used earlier
eg. 3000 – 699
 3000 – 699 =
 Set your workings out like below, with the
smaller number at the start of the number line.
699
3000
Think about what you would need to add to 699
to make 3000. HINT: go in small steps first to
round 699 to the nearest 100.
+1
699
+300
=700
+2000
=1000
=3000
So when we add 1+300+2000 we get the
answer 2301
Try these:
What is the difference between:
1) 361 and 520?
2) 410 and 280?
3) 135 and 256?
4) What is 170 less than 365?
5) What would I need to add to 153 to make
365?
Check your answers
If you are struggling with these then
please ask for help!
1) 159
2) 130
3) 121
4) 195
5) 212
2) Other subtraction strategies…
 54 – 29 is one of those sums that are tricky to do
in your head. BUT if you pretend that the sum
is…
 54 – 30 you can quickly find the answer (use
counting on if you find it tricky). 54 - 30 =24
 The answer to 54 – 30 =24 BUT REMEMBER
we added 1 to 29, so we must add 1 onto our
answer. 54 - 29 = 25
Try these…
1) 56 – 39 =
2) 66 – 29 =
3) 27 – 19 =
4) 88 – 59 =
5) 97 – 49 =
6) 77 – 29 =
7) 63 – 19 =
8) 96 – 19 =
Check your answers
 If you are struggling with these then please ask
for help!
 1) 7
 2) 37
 3) 8
 4) 29
 5) 48
 6) 48
 7) 44
 8) 77
 You won’t always be so lucky as to get a 9 on
the end. You can still do the same with 8’s and
7’s though.
 eg. 45 – 18 =
 Turn 18 into 20 (by +2) and then do 45 – 20 (do
counting on if you want)
 45 - 20 = 25 then remember to + 2 and you get
the final answer of 27
 Or what about?
 59 – 17 =
 Pretend 17 is 20 by +3 (Don’t forget you’ve done this bit!)
 59 – 20 = 39
 BUT then REMEMBER to add the 3 back on to get your
answer of 42
Try these…
1) 98 – 57 =
2) 87 – 58 =
3) 36 – 17 =
4) 67 – 48 =
5) 53 – 37 =
6) 48 – 28 =
7) 108 – 57 =
8) 156 – 48 =
Check your answers
 If you are struggling with these then please ask
for help!
 1) 41
 2) 29
 3) 19
 4) 19
 5) 16
 6) 20
 7) 51
 8) 108
STRATEGIES FOR ADDITION
Add
More than
Plus
How many…?
What quantity?
3) Finding number bonds
Sometimes you may be asked to quickly
add a list of numbers in your head.
One way of doing this quickly is by writing
the numbers in a line and finding number
bonds to add together first.
eg. 12 + 16 + 8 + 15 + 3 + 5
 First carefully look for numbers which fit together
to make whole 10’s, 100’s or 1000’s…
 12 + 16 + 8 + 15 + 3 + 5
 So,12 and 8 make 20
 15 and 5 make 20
 So far we have a total of 40
 Cross the numbers you have used out. Now see
what’s left. 16 and 3 which make 19.
 To find your answer add 40 and 19 together =
59
Try these…
Add…
1) 15, 5, 16, 4, 14, 6 =
2) 45, 8, 8, 4, 5, 17, 3 =
3) 34, 5, 6, 15, 12, 8 =
4) 56, 17, 3, 4, 20, 3 =
5) 19, 21, 1, 17, 16, 3 =
Check your answers
If you are struggling with these then
please ask for help!
1) 60
2) 90
3) 80
4) 103
5) 78
4) Partitioning
 A way of adding numbers that are not in lists is by using the partition
method.
 108 + 57 =
 This sum can be broken down into 100 + 8 + 50 + 7 = ?
 The easiest way of dealing with this is to add the bigger numbers
first and then add on the smaller ones (if you are lucky you may find
number bonds too).
 100 + 50 = 150 and 8 + 7 = 15
 150 + 15 = 165
 165 is the final answer.
Try these…
1) 167 + 47 =
2) 256 + 14 =
3) 163 + 17 =
4) 187 + 35 =
5) 234 + 26 =
Check your answers
If you are struggling with these then
please ask for help!
1) 214
2) 270
3) 180
4) 222
5) 260
5) Counting on …
 Counting on can also be used for addition (this
is just another way of partitioning really).
 365 + 65
 Start with 365
+5
+30
+30
 365_____370_____400_____430
And add little bits of 65 at a time…
eg. 365 + 5 = 370, 370 + 30 = 400, 400 + 30 = 430
Try these…
1) 365 + 56 =
2) 456 + 38 =
3) 247 + 45 =
4) 189 + 98 =
5) 276 + 78 =
Check your answers
If you are struggling with these then
please ask for help!
1) 421
2) 494
3) 292
4) 287
5) 354
STRATEGIES FOR MULTIPLICATION
Product
Multiply
3x greater than
10x greater than
100x greater than etc etc
Times
Lots of
STRATEGIES FOR DIVISION
Quotient
Divide
How many into
Shared
10x, 100x etc smaller than
Groups of
6) Multiplying by 10, 100, 1000
When multiplying by 10, all of the numbers
you are multiplying move one space left
to the next column.
eg. 5 x 10 = 50
1000 100 10 1
.
1/10 1/100
5

5
0
We put a 0 in the space where the 5 was.
Some people will tell you that when you
multiply by 10 all you need to do is add a 0
on the end. This does work, although not
when we are looking at decimals. This is
why we teach the method of moving
numbers at school as it is easy to use and
will help you get the right answer every
time!
The same happens with decimals…
1000
100 10
1
1
.
.
1/10 1/100
5
X 10 (Numbers move 1 place to the left)
=
1000 100 10 1
.
1/10 1/100
1
.
5

1 5
Try these…
1) 6 x 10 =
2) 0.5 x 10 =
3) 5.6 x 10 =
4) 7.8 x 10 =
5) 9.3 x 10 =
6) 1.23 x 10 =
7) 3.25 x 10 =
8) 1.75 x 10 =
Check your answers
 If you are struggling with these ask for help!
 1) 60
 2) 5
 3) 56
 4) 78
 5) 93
 6) 12.3
 7) 32.5
 8) 17.5
6) Multiplying by 10, 100, 1000
When multiplying by 100, all of the
numbers you are multiplying move two
spaces left.
eg. 5 x 100 = 500
1000 100 10 1
.
1/10 1/100
5

5
0 0
We put a 0 in the spaces.
The same happens with decimals…
1000
100 10
1
5
.
.

5
7 0
We put a 0 in the spaces.
1/10 1/100
7
Try these…
1) 13 x 100 =
2) 3.2 x 100 =
3) 4.6 x 100 =
4) 4.25 x 100 =
5) 0.2 x 100 =
6) 2.355 x 100 =
7) 0.525 x 100 =
8) 65.543 x 100 =
Check your answers
 Remember, IF YOU ARE STUCK – ASK!
 1) 1300
 2) 320
 3) 460
 4) 425
 5) 20
 6) 235.5
 7) 52.5
 8) 6554.3
6) Multiplying by 10, 100, 1000
When x by 1000 the numbers move 3
places to the left…
Try these…
1) 4.5 x 1000 =
2) 3.25 x 1000 =
3) 1.5 x 1000 =
4) 4.75 x 1000 =
5) 45.25 x 1000 =
6) 0.32 x 1000 =
7) 0.175 x 1000 =
Check your answers
ASK IF YOU NEED HELP
1) 4500
2) 3250
3) 1500
4) 4750
5) 45250
6) 320
7) 175
7) Dividing by 10, 100 and 1000
When dividing by 10, all of the numbers
you are dividing move one space right to
the next column.
eg. 5 divided by 10 = 0.5
1000 100 10 1
.
1/10 1/100
5

0
.
5
We put a 0 in the space where the 5 was.
The same happens with decimals…
1000
100 10
1
1
.
.
1/10 1/100
5
Divide by 10 (Numbers move 1 place to
the right)
1000 100 10
1 .
1/10 1/100
1
.
5
0
.
1
5
Try these…
1) 15 divided by 10 =
2) 1.3 divided by 10 =
3) 25 divided by 10 =
4) 0.25 divided by 10 =
5) 13.25 divided by 10 =
6) 16.25 divided by 10 =
7) 123.5 divided by 10 =
8) 234.24 divided by 10 =
Check your answers
 ASK IF YOU NEED HELP
 1) 1.5
 2) 0.13
 3) 2.5
 4) 0.025
 5) 1.325
 6) 1.625
 7) 12.35
 8) 23.424
7) Dividing by 10, 100 and 1000
When dividing by 100, all of the numbers
you are dividing move two spaces right.
eg. 5 divided by 100 = 0.05
1000 100 10 1
.
1/10 1/100
5

0
.
We put a 0 in the spaces.
0
5
The same happens with decimals…
1000 100
10
1
.
1/10
1
.
5
1/100 1/1000
Divide by 100 (Numbers move 2 places
to the right)
1000
100
10
1
.
1/10
1
.
5
0
.
0
1/100 1/1000
1
5
Try these…
 1) 1.5 divided by 10 =
 2) 2.35 divided by 100 =
 3) 67 divided by 100 =
 4) 34.5 divided by 1000 = (Think about how many spaces
you will have to move the numbers)
 5) 56.7/10 (/ means divide)
 6) 123.67/100 =
 7) What number is 100 x smaller than 54.75?
 8) What number is 10 x smaller than 185?
And these…
 9) 567/100 =
 10)123 = (This means 123 divided by 10 – this

10
is a fraction!)
 11) 5 =

10
 12) Find the quotient of 560 and 100. (This just
means divide 560 by 100).
More!!!
13) Find the quotient of 43 and 100
14) Find the quotient of 456 and 10
15) Find the quotient of 1.6 and 10
16) What number is 100 x smaller than
67?
17) What number is 100 x smaller than
3.2?
18) Find the difference between the
quotient of 60 and 10, and 10 x 0.5
Check your answers











ASK IF YOU NEED HELP!
1) 0.15
2) 0.0235
3) 0.67
4) 0.0345
5) 5.67
6) 1.2367
7) 0.5475
8) 18.5
9) 5.67
10) 12.3
Check your answers
11) 0.5
12) 5.6
13) 0.43
14) 45.6
15) 0.16
16) 0.67
17) 0.032
18) 1
8) Multiplying by multiples of 10…
When we multiply by multiples of 10 the
best thing to do is to partition the sum eg.
30 x 15 could be written as 10 x 15

10 x 15

10 x 15
 (10 x 15 = 150, 150 x3 = 450
You could also partition this way if you
prefer…
30 x 8 =
You could do (3 x 8) x 10(because it is 30
not 3). 3x 8 = 24 and 24 x 10 = 240
Answer = 240
Another example…
20 x 17 =
10 x 17 = 170
10 x 17 = 170
170+170 = 340
Try these…
1) 30 x 14 =
2) 20 x 18 =
3) 40 x 15 =
4) 10 x 16 =
5) 60 x 14 =
Check your answers
IF STUCK – ASK
1) 320
2) 360
3) 600
4) 160
5) 840
9) Multiplying 2 digit numbers by a single
digit …
A quick way of working out sums with 2
digits x by 1 digit is to partition…
eg. 14 x Could be done mentally like this..

7
10 x 7 = 70 (remember the 1 is actually a 10)
 4 x 7 = 28 Then just add the two numbers
together = 98
Try these…
1) 18 x 5 =
2) 15 x 8 =
3) 19 x 8 =
4) 17 x 9 =
5) 19 x 9 =
6) 18 x 6 =
7) 13 x 7 =
8) 16 x 8 =
Check your answers
 IF STUCK - ASK!
 1) 80
 2) 120
 3) 152
 4) 153
 5) 171
 6) 108
 7) 91
 8) 128
10) Multiplying by 20
When multiplying by 20 simply multiply by
10 and double the answer!
eg. 36 x 20 =
36 x 10 = 360, now double 360 to get 720
Try these…
1) 45 x 20 =
2) 34 x 20 =
3) 18 x 20 =
4) 56 x 20 =
5) 43 x 20 =
6) 23 x 20 =
7) 20 x 20 =
8) 30 x 20 =
Check your answers
 IF STUCK – ASK
 1) 900
 2) 680
 3) 360
 4) 1120
 5) 860
 6) 460
 7) 400
 8) 600
11) Multiplying by 50
When multiplying by 50, just x the number
by 100 and half it!
eg. 24 x 50 =
24 x 100 = 2400, half of 2400 = 1200
So 24 x 50 = 1200
Try these…
1) 26 x 50 =
2) 18 x 50 =
3) 12 x 50 =
4) 22 x 50 =
5) 30 x 50 =
6) 45 x 50 =
7) 36 x 50 =
8) 65 x 50 =
Check your answers
 IF STUCK ASK FOR HELP
 1) 1300
 2) 900
 3) 600
 4) 1100
 5) 1500
 6) 2250
 7) 1800
 8) 3250
12) Multiplying by 25
When a number is multiplied by 25, simply
multiply it by 100, half it and half it again!
eg. 24 x 25 =
24 x 100 = 2400
Half 2400 = 1200
Half 1200 = 600
24 x 25 = 600
Try these…
1) 24 x 25 =
2) 12 x 25 =
3) 16 x 25 =
4) 18 x 25 =
5) 25 x 25 =
6) 60 x 25 =
7) 40 x 25 =
8) 20 x 25 =
Check your answers
 REMEMBER TO ASK FOR HELP IF NEEDED!
 1) 600
 2) 300
 3) 400
 4) 450
 5) 625
 6) 1500
 7) 1000
 8) 500
13) Doubling and halving
When doubling numbers, simply multiply
by 2!
When halving numbers, simply divide by 2!
Doubling…
Double 356 (use the partition method if it helps!)
 eg. 300 x 2 = 600

50 x 2 = 100

6 x 2 = 12
 Add these together to get 712
Try these…
 Double…
 1) 712
 2) 452
 3) 321
 4) 754
 5) 435
 6) 1345
 7) 2345
 8) 3543
Check your answers
 ASK FOR HELP IF NEEDED!!!
 1) 1424
 2) 904
 3) 642
 4) 1508
 5) 870
 6) 2690
 7) 4690
 8) 7086
Halving
 Partitioning can also be done when halving large
numbers in your head.
 eg. 1232 can be halved by halving in bits.
 1000 halved = 500
 200 halved = 100
 30 halved = 15

2 halved = 1
 When you add these up you get 616
Try these…
 Halve these…
 1) 456
 2) 986
 3) 1344
 4) 2468
 5) 2464
 6) 4322
 7) 2453
 8) 2545
Check your answers
 REMEMBER…(do I need to say it again?)
 1) 228
 2) 493
 3) 672
 4) 1234
 5) 1232
 6) 2161
 7) 1226.5
 8) 1272.5
14) The factor method
 This is another quick mental way of doing multiplication…
 eg. 16 x 15










First find the factors of 16 in factor pairs.
The factor pairs are (1,16), (2,8), (4,4)
Now times 15 by one of the pairs of numbers.
15 x 2 = 30 and then times your answer by the other number in the factor pair.
30 x 8 = (remember the strategy for this one?)
Either do…
Or do…
10 x 8 = 80
(3 x 8) x 10
10 x 8 = 80
Which is simply 3 x 8 = 24
10 x 8 = 80
and then 24 x 10 = 240
Add them up and you get 240
 Another way of solving this would have been to choose the factor pair (4,4)
 15 x 4 = 60
 And 60 x 4 = 240
 The factor method is a good one to use when multiplying
2 digit even numbers with decimal numbers like 4.25 or
3.5 for instance.
 eg.
 3.25 x 24 could be done by finding the factors of 24 first.
 The factors of 24 are (1,24), (3,8), (4,6), (2,12)
 Choose a factor pair which will help you get rid of the .25
when you multiply. For example if we take the factor pair
(4,6) we can x 3.25 by 4 and we get 13.
 (3 x 4 = 12 and 0.25 x 4 = 1, add them to get 13)
 We then take our answer 13 and multiply it by 6 (the
other number in the factor pair) to get 78
Try these…
1) 15 x 16 =
2) 3.25 x 16 =
3) 4.5 x 14 =
4) 1.25 x 32 =
5) 2.25 x 24 =
Check your answers
IF STUCK - ASK!
1) 240
2) 52
3) 63
4) 40
5) 54
15) Partitioning for long multiplication
 When multiplying 2 digit numbers by 2 digit
numbers we can break the numbers up to help
us work out the sum.
 eg. 23 x 31 can be done like this…
 Find the answer to 23 x 30 and then add the
answer to 23 x 1.
 23 x 10 = 230; 230 x 3 = 690, then add 23 x 1 =
23
 690 + 21 = 711
Try these
1) 43 x 23 =
2) 35 x 27 =
3) 56 x 31 =
4) 41 x 29 =
5) 32 x 38 =
Check your answers
ASK IF YOU NEED HELP!!!
1) 43 x 23 = 989
2) 35 x 27 = 945
3) 56 x 31 = 1636
4) 41 x 29 = 1189
5) 32 x 38 = 1116
16) Multiplying numbers with zeros to
other numbers with zeros
 When multiplying whole 10’s, 100’s or 1000’s by
other whole 10’s, 100’s or 1000’s, multiply the
first digit of each number…
 eg. 70 x 40 =
 (7 x 4 = 28)
 Now count how many zeros in the sum (2)
 Next, add the 2 zeros to the answer 28.
 The answer to 70 x 40 = 2800
Another example…
500 x 800 =
First, 5 x 8 = 40
Then count the number of zeros in the
sum 500 x 800 (4 zeros)
Answer = 400000
Try these
1) 60 x 70 =
2) 80 x 90 =
3) 200 x 300 =
4) 40 x 80 =
5) 700 x 50 =
6) 400 x 200 =
7) 2000 x 60 =
8) 500 x 900 =
Check your answers
 REMEMBER…REMEMBER…
 1) 4200
 2) 7200
 3) 60000
 4) 3200
 5) 35000
 6) 80000
 7) 120000
 8) 450000
17) Multiplying by 99
 When multiplying by 99, pretend you are multiplying by 100 and then
take away 1 for every 99 you have to multiply by…
 eg. 4 x 99 =
 Pretend this is 4 x 100 = 400; then take away 4 because…




99 + 1 = 100
99 + 1 = 100
99 + 1 = 100
99 + 1 = 100
 400 – 4 = 396
 Answer = 396
Here is another example…
7 x 99 = (pretend the 7 lots of 99 are 100)
7 x 100 = 700 (then take away 7)
Answer = 693
Try these
1) 5 x 99 =
2) 3 x 99 =
3) 8 x 99 =
4) 6 x 99 =
5) 2 x 99 =
Check your answers
IF YOU GET STUCK – ASK!
1) 495
2) 297
3) 792
4) 594
5) 198
18)The Italian Method
 This isn’t strictly a
mental method of
calculation but if you
can learn to do it
quickly it is a really
easy way of doing
long multiplication!!!
 eg. 13.25 x 12 =
 Answer = 159
1
3
0
.
0
2
5
0
X
0
1
1
0
0
3
0
2
0
5
1
2
2
1
5
6
9
.
4
0
0
0
Draw a grid to start. There should be
enough columns for the digits in one
number and enough rows for the digits of
the number you are multiplying by.
eg. 57 x 46 =
• 4
You try…
6
5
7
Try these using the Italian method – you
will practice this in class.
FINALLY!
 REMEMBER there are lots of strategies – you use which
suits you (which ones you can use quickly and
accurately!)
 There are lots of other strategies too! Whatever strategy
you learn, you must practice. Keep working over and
over the strategies in this guide. The more you do the
better!
 Also REMEMBER to ask if you need help!!!
 Good Luck!
Mr Abeledo