Mental Arithmetic Strategies Rush Green 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 3000- In this case we 699 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.

So when we add 1+300+2000 we get the answer 2301 699 =700 =1000 =3000 So when we add 1+300+2000 we get the answer 2301 +1 +300 +2000

Try these: What is the difference between: 1) 361 and 520? 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 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 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 365 365_____370_____400_____430 And add little bits of 65 at a time… eg. 365 + 5 = 370, 370 + 30 = 400, 400 + 30 = 430 +5 +30 +30

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 . 1/10 1/100 1 . 5 X 10 (Numbers move 1 place to the left) = 1 5

Try these… 1) 6 x 10 = 2) 0.5 x 10 = 3) 5.6 x 10 = 4) 7.8 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 . 1/10 1/100 5 . 7 5 7 0 We put a 0 in the spaces.

Try these… 1) 13 x 100 = 2) 3.2 x 100 = 3) 4.6 x 100 = 4) 4.25 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 . 1/10 1/100 1 . 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 =

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 . 0 5 We put a 0 in the spaces.

The same happens with decimals… 1000 100 10 1 . 1/10 1/100 1/1000 1 . 5 Divide by 100 (Numbers move 2 places to the right) 1000 100 10 1 . 1/10 1/100 1/1000 1 . 5 0 . 0 1 5

Try these… 1) 1.5 divided by 10 = 2) 2.35 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 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 = 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 =

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 =

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 =

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 =

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 =

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 =

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 =

Check your answers ASK IF YOU NEED HELP!!! 1) 43 x 23 = 989

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 =

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 = 100 400 – 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 . 2 5 X 1 1 3 2 5 6 4 1 2 5 9 . 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 6 You try… 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