MENTAL CALCULATION 1. RE-ARRANGING When trying to add a row of numbers, we should look for pairs that add up to make a multiple of 10 or 100 e.e. 13 + 8 + 7 + 6 + 2 13 + 8 + 7 + 6 + 2 20 10 2. UNITS, 20 TENS + 10 AND + 6 SO = 36 ON When adding two numbers the units, tns etc can be taken separately. i.e we are using our knowledge of addition number bonds (0-9) and an understanding of place value. 4 7 + 2 6 47 + 26 Adding 6 + 7 = 53 + 20 Then adding 20 = 73 OR 47 + 26 Adding 40 + 20 Adding 7 + 6 23 = 67 + 6 = 73 1
3. COMPENSATION We can sometimes add or subtract more than we should and then compensate. We usually round the number to the nearest 10. e.e. 37 + 19 We can round up the 19 to 20 and then compensate by subtracting 1, because we have added 1 too much = (37 + 20) - 1 = 57-1 = 56 e.e. 6.7 + 3.9 We can round the 3.9 up to 4.0 and then compensate by subtracting 0.1, because we have added 0.1 too much. = (6.7 + 4.0 ) 0.1 = 10.7-0.1 = 10.6 A similar method can be used in subtraction: e.e. 137-28 We can round the 28 up to 30 and compensate by adding 2, because we have subtracted 2 too many = (137-30) + 2 = 107 + 2 = 109 4. NEAR DOUBLES If we are adding two numbers that are near to each other, we can double one number and then compensate. We can double the smaller number and add or double the larger number and subtract. e.e. 13+14 This can be considered as double 13 add one or double 14 subtract one 13 + 13 + 1 = 26 + 1 = 27 neu 14 + 14-1 = 28-1 = 27 2
MULTIPLICATION AND DIVISION Most mental strategies for multiplication and division depend on a knowledge of tables. This must be extended to the multiplication and division of larger numbers: x 2 double 2 x 56 = (2 x 50) + (2 x 6) = 100 + 12 = 112 x 3 double, then add the number 3 x 125 = (2 x 125) + 125 = 250 + 125 = 375 x 4 double and double again 4 x 34 = (2 x 34) x 2 = 68 x 2 = 136 x 5 multiply by 10 and halve 5 x 240 = (240 x 10) / 2 = 2400 / 2 = 1200 x 6 multiply by 5 and add the number x 7 double, double and double again and subtract the number x 8 double, double again and double again 8 x 24 = (24 x 2) x 2 x 2 = (48 x 2) x 2 = 96 x 2 = 192 x 9 multiply by 10 and subtract the number 9 x 57 = (10 x 57) 57 = 570-57 = 513 x 10 move the numbers to the left 10 x 12 = 120 10 x 3.75 = 37.5 3
1. ADDITION (a) Carrying from one column to the next, starting with the units. (a) This is the usual method (i) 7 6 4 8 + 1 4 8 6 9 1 3 4 1 1 1 We can use the methods above with decimals but we must remember to place the decimal points underneath each other and to fill every gap with 0 (zero) as required carrying under the line 124.90 + 73.25 198.15 2. SUBTRACTION (a) Counting on method In this method we gradually add to the lower number The bill at a market stall is 6.34. How much change should be given from 20? 6.34 + 6 6.40 0.06 + 10 0.10 + 6.50 0.50 adding 50 3.00 these 7.00 + 10.00 + 3.00 13.66 10.00 + 10.00 20.00 4
(b) Written calculation: decomposition method In this method we take from the next column. 5 6 1 4 7 8 1 4-2 6 6 7 3 8 1 7 (c) We can use the above methods with decimals but we must remember to place the decimal points underneath each other and to fill every gap with 0 (zero) as required. 3 1 2 1 4. 8 9 1 0-7. 2 3 3 1 7. 6 7 decomposition 5
3. MULTIPLICATION (a) Area Method In this method we will break down the numbers as the sides of a rectangle. The area of the rectangle will be the answer to the multiplication (i) 56 x 34 50 and 6 30 and 4 50 x 30 30 50 6 1500 180 30 x 6 4 200 24 4 x 50 4 x 6 The answer is the total area therefore:- 1500 180 300 + 24 1904 (ii) 236 x 27 200 & 30 & 6 20 & 7 20 x 30 20 x 200 20 200 30 6 4000 600 120 20 x 6 7 x 200 7 1400 210 42 7 x 6 7 x 30 6
236 x 27 = 4000 600 120 1400 210 + 42 6372 (b) Multiplying Decimals We can adapt the previous methods of multiplication to multiply decimals. To simplify the multiplication process we will eliminate the decimal point and then put it back in the right place at the end, after multiplying. 1 (linked to the preceding example, above) 236 x 27 = 6372 3.27 is 100 times smaller than 327, so tye answer will be 100 times smaller, that is, 150.42 4.6 is 10 times smaller than 46, so the answer will be 10 times smaller, that is, 15.042 2 3.8 x 2.5 2 x 3 3 0.8 2 6 1.6 2 x 0.8 0.5 1.5 0.40 0.5 x 0.8 0.5 x 3 6 1.6 1.5 + 0.4 9.5 7
(b) Partition Method In this method the smaller number is partitioned. (i) 352 x 27 352 x 7 2464 352 x 20 + 7040 352 x 27 = 9504 it is broken down into tens and units 4. DIVISION (a) Method 4 1 8 7 2 9 1 2 5 6 If the number does not divide exactly we can show the answer with a remainder or as a mixed number. 4997 6 = 832 g 5 neu 832 5/ 6 8
This is te traditional method (b) Long Division method In this method it is important that you set out work with the tens and units columns correctly underneath each other (i) 782 34 23 (answer line) 34 782-680 (34 x 20 = 680, put 2 in the tens column on the answer line) 102-102 (34 x 3 = 102, put 3 in the units column on the answer line) 000 So, 782 34 = 23 (i) 977 36 27 (answer ;ine) 36 977-720 (36 x 20 = 720, put 2 in the tens column on the answer line) 257-252 (36 x 7 = 252, put 7 in the units column on the answer line) 5 So, 782 36 = 27 remainder 5 or 782 36 = 27 5/ 36 (ch) Dividing Decimals 9.72 3.6 The traditional method can be used to calculate 972 36 = 27 9.72 is 100 times smaller than 972, so the answer will be 100 times smaller, that is, 0.27. 3.6 is 10 times smaller than 36 but the answer will be 10 times more, that is, 2.7 9
3. FRACTION, PRCENTAGES AND DECIMALS 3.1 FRACTIONS A fraction is one whole number divided by another whole number. this fraction is ¾, that is three shaded parts and four equal parts in total. Here are other examples Equal/Equivalent Fractions The shaded parts are the same size in the three diagrams so ½ = 2 / 4 = 4 / 8. 10
We can create equal fractions by multiplying or dividing. We must treat the top number [the numerator] and the bottom number [the denominator] in exactly the same way. 3.1 PERCENTAGES A percentage means OUT of 100. % is the percentag symbol. We say that one whole is 100%. 3.2 CHANGING A FRACTION TO A PERCENTAGE Convenient numbers e.g 7 20 Try to change the number on the bottom (dnominator) to 100 and changing the top number in excatly the sam way. In this example we can change 20 to 100 by multiplying by 5. So, we also multiply the 7 by 5 to make 35. So 7 changes into 35 which is 35% 20 100 11
Less convenient numbers e.g. 23 36 Here we use a calculator : 23 36 = 0.638888. = 0.638 neu 0.64 i ddau le degol So, 23 is equivalent to 0.64 which is worth 64 which is 64% 36 100 PLACE VALUE: understanding place value thoroughly is a cornerstone when working with fractions, decimals and percentages. Place Value M HTh TTh Th H T U. t h th Million Thousand thousandth Hundred Thousand Hundred hundredth Ten Thousand Ten tenth Units Decimal Point 12
3.3 FINDING A PERCENTAGE OF A NUMBER (a) What is 10% of 40? 10% is 10 or 0.1 100 So 10% makes a sum 0.1 smaller which is 10 times smaller. 10 times smaller than 40 is 4 So 10% of 40.00 is 4 (b) What is 5% of 50kg? 10% is 50kg 10 = 5kg 5% = 2.5kg (5% is half of 10%) (c) What is 20% of 80? 10% is 8 20% is 16 (ch) What is 8% of 250kg 10% is 25kg 1% is 2.5kg 2% is 5kg 8% is 4 x 2% sef 4 x 5kg = 20kg 3.4 FRACTIONS AND DECIMALS To change a fraction to a decimal we must divide the top (the numerator) by the bottom (the denominator) using a calculator if necessary. (a) (b) Expressing 3 / 8 as a decimal 3 8 = 0.375 Expressing 4 2 / 5 as a decimal 2 5 = 0.4 so 4 2 / 5 = 4 + 0.4 = 4.4 13
To cahnge a decimal to a fraction we must create a fraction over 10, 100, 1000 and so on, and then cancel if necessary. (c) M Expressing 0.54 as al fraction HTh TTh Th H T U. t h th 0. 5 4 0.54 = 54 = 27 100 50 (ch) Expressing 3.6 as a fraction 3.6 = 3 + 0.6 0.6 = 6 = 3 10 5 3.6 = 3 + 3 / 5 = 3 3 / 5 3.5 DECIMALS AND PERCENTAGES (a) To understand changing a decimal to a percentage one must understand place value. M HTh TTh Th H T U. t h th Million Thousand thousandth Hundred Thousand Hundred hundredth Ten Thousand Ten tenth Unit Decimal Point 14
Expressing 0.35 as a percentage M HTh TTh Th H T U. t h th 0. 3 5 That is 35 per cent = 35 which is 35% 100 Expressingi 1.275 as a percentage M HTh TTh Th H T U. t h th 1. 2 7 5 That is 1 whole and 27.5 hundredth = 127.5% (b) The same method is usedto change a percenmtyage to a decimal. Expressing 45% as a decimal 45% means 45 which is 45 hundredth 100 M HTh TTh Th H T U. t h th 0. 4 5 So, 45 % = 0.45 Expressing 17½% as a decimal 17½% = 17.5% which is 17.5 100 M HTh TTh Th H T U. t h th 0. 1 7 5 So 17½% = 0.175 15
3.6 Exchanging common fractions, decimals and percentages A table of common fractions, decimals and percentages. FRACTION DECIMAL PERCENTAGE ½ 0.5 50% ¼ 0.25 25% ¾ 0.75 75% 1 / 8 0.125 12.5% 1 / 10 0.1 10% 1 / 5 0.2 20% 3 / 10 0.3 30% 3 / 5 0.6 60% 16