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KS3 Mathematics N9 Mental methods

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1 KS3 Mathematics N9 Mental methods
The aim of this unit is to teach pupils to: Consolidate understanding of the operations of multiplication and division, their relationship to each other and to addition and subtraction; know how to use the laws of arithmetic. Know and use the order of operations, including brackets. Consolidate the rapid recall of number facts and use known facts to derive unknown facts. Consolidate and extend mental methods of calculation, accompanied where appropriate by suitable jottings. Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp N9 Mental methods

2 N9 Mental methods Contents N9.1 Order of operations
N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

3 Using the correct order of operations
What is 7 – 3 – 2? When a calculation contains more than one operation it is important that we use the correct order of operations. The first rule is we work from left to right so, Stress that this rule is most important when repeatedly subtracting or dividing because when we subtract or divide numbers the order is important. When we repeatedly add or multiply the order is not important. 7 – 3 – 2 = 4 – 2 NOT 7 – 3 – 2 = 7 – 1 = 2 = 6

4 Using the correct order of operations
What is × 4? The second rule is that we multiply or divide before we add or subtract. 8 + 2 × 4 = 8 + 8 NOT 8 + 2 × 4 = 10 × 4 = 16 = 40

5 Brackets What is (15 – 9) ÷ 3? When a calculation contains brackets we always work out the contents of any brackets first. (15 – 9) ÷ 3 = 6 ÷ 3 = 2

6 Nested brackets Sometimes we have to use brackets within brackets.
For example, 10 ÷ {5 – (6 – 3)} These are called nested brackets. We evaluate the innermost brackets first and then work outwards. Point out that it is common when using nested brackets to use a different style of bracket, such as the ‘curly’ brackets shown, to distinguish between the two pairs. Square brackets can also be used. 10 ÷ {5 – (6 – 3)} = 10 ÷ {5 – 3} = 10 ÷ 2 = 5

7 Using a division line 13 + 8 What is ? 7
When we use a horizontal line for division the dividing line acts as a bracket. 13 + 8 7 = (13 + 8) ÷ 7 = 21 ÷ 7 = 3

8 Using a division line 24 + 8 What is ? 24 – 8
Again, the dividing line acts as a bracket. = (24 + 8) ÷ (24 – 8) 24 + 8 24 – 8 = 32 ÷ 16 = 2

9 Multiplying by a bracket
When we multiply by a bracket it is not always necessary to use the symbol for multiplication, ×. For example, 8 + 3(7 – 3) is equivalent to × (7 – 3) = × 4 = = 20 Compare this to the use of brackets in algebraic expressions such as 3(a + 2).

10 Indices What is 100 – 2(3 + 4)2 When indices appear in a calculation, these are worked out after brackets, but before multiplication and division. 100 – 2(3 + 4)2 Brackets first, = 100 – 2 × 72 then Indices, Introduce BIDMAS to remember the correct order of operations. = 100 – 2 × 49 then Division and Multiplication, = 100 – 98 and then Addition and Subtraction =

11 B I D M A S BIDMAS Remember BIDMAS: RACKETS NDICES (OR POWERS) IVISION
ULTIPLICATION Remind pupils of the correct order of operations using the mnemonic: BIDMAS. A DDITION S UBTRACTION

12 Using BIDMAS (3.4 + 4.6)2 What is – 8 × 0.5 ? (6 + 5 × 2)
Brackets first, 82 16 – 8 × 0.5 = then Indices, 64 4 – 8 × 0.5 = Demonstrate the order of operations with this example. Explain that a square root is classed as an index. (We can write 16 as 16½). then Division and Multiplication, = – 4 and then Addition and Subtraction =

13 Using a calculator We can use a calculator to evaluate more difficult calculations. ( ) 7 - 5 For example, This can be entered as: ( 5 x2 + 7 ) ÷ - Note that the dividing line acts as a bracket and so it is necessary to use brackets when entering 7 – 5 on a calculator. Talk about how can check this answer using an approximation. = = 74. Since 74 is between 64 and 81, the square root or 74 must be between 8 and 9. We then divide this by 2 and so we can estimate the answer as approximately 4. = 4.3 (to 1 d.p.) Always use an approximation to check answers given by a calculator.

14 Positioning brackets Explain to pupils that the positioning of brackets is important. Demonstrate this by positioning a single pair of bracket in the expression × 5 – 4 ÷ 2 many different answers are possible. For example, (1 + 3) × 5 – 4 ÷ 2 = 18 (1 + 3 × 5 – 4) ÷ 2 = 6 1 + (3 × 5 – 4) ÷ 2 = 6.5 1 + 3 × (5 – 4) ÷ 2 = 2.5 1 + 3 × (5 – 4 ÷ 2) = 10 Ask pupils to suggest possible positions for the brackets. Alternatively, ask pupils to position the brackets to make a given answer.

15 Target numbers Give pupils 4 digits, for example, 2, 4, 5 and 8 and challenge them to make a target number (for these digits we could have 70) using all four digits and any of the given operations. Allow pupils a few moments to jot down their answers and then ask a volunteer to demonstrate their solution on the board by dragging the required cards into the large box. For example, 8 × (4 + 5) – 2. The solution 70 will appear in the small box. Some target numbers may have several solutions using the required digits. If other pupils have an alternative solution invite them to demonstrate it on the board. Other examples (with the target number shown in bold): (7 + 8) × 2 – 5 = 25 ( )2 + 4 = 40 (3 + 7) ÷ (1 + 4) = 2 ( – 5) = 4 2 × – 3 = 100

16 N9.2 Addition and subtraction
Contents N9 Mental methods N9.1 Order of operations N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

17 Complements match Explain to pupils that we we are adding it is often useful to look for complements in 1 (for adding decimals) and 10, 50, 100 and 1000 for adding larger whole numbers. Numbers can be broken apart mentally to make complements. For example, = = = 9.1 This activity practises identifying complements. Challenge pupils to match pairs of cards as quickly as possible. When the first card is revealed ask pupils to tell you what its complement is before trying to find it. By clicking on the blue bar you can choose to match complements in 1, 10, 50, 100 or 1000.

18 Counting on and back Use this counter to practise counting on and back in whole number steps with different start numbers. The start number can be selected by changing the step size to the required start number and moving the counter one step clockwise. The whole class could count together or you may like to select individuals. Extend the activity by selecting more difficult start numbers and counting back to produce negative numbers. Link: A4 Sequences – linear sequences Adapt the activity to play the following game: Left hand, right hand Use the counter with start number 1 and step size 1. Tell the class that when they see a multiple of 3 they must put their left hand up and when they see a multiple of 5 they must put their right hand up. If a number is a multiple of both 3 and 5 they should put both hands up. (You can start with different multiples if you wish.) Call out the numbers as you go through them and increase speed as necessary. You could start this game with everyone standing up initially. If anyone makes a mistake or doesn’t put their hand up when they should, they must sit down. The winners are the last people to remain standing. N3 Multiples, factors and primes – multiples and factors

19 Using partitioning to add
What is ? = = = = 344 What is ? The following sequence of slides demonstrates a variety of mental methods used to add and subtract whole numbers and decimals. The same examples are used to highlight the differences between each method. These may be edited to give different examples as required. Explain that using mental methods means that we are not using a standard written procedure – that does not necessarily mean that we are not allowed to write anything down. On the contrary, many mental methods require the use of ‘jottings’. These examples show every step in the calculation. However, it should be pointed out to pupils that most of these steps would be calculated mentally. The method chosen depend on the numbers, and on individuals’ preference. For example, some people may prefer counting up when subtracting, whereas others may find that using compensation comes more naturally. Often a combination of methods is required. This slide demonstrates the use of partitioning to add. This could be considered as a mental version of the standard column procedure. = = = = 68.5

20 Adding by counting up What is 276 + 68? 276 + 60 + 8 = 344 276 336 344
Many mental methods use a sketch of a number line (this doesn’t have to be drawn to scale). With practice, the number line can be imagined. To add by counting up, start with the larger number and add on the smaller number one part at a time. Ask pupils how we could check each answer using, for example, inverse operations. 63.8 + 4 + 0.7 = 68.5 + 4 + 0.7 63.8 67.8 68.5

21 Using compensation to add
What is ? 276 + 70 – 2 = 344 + 70 – 2 276 344 346 What is ? Tell pupils that this method involves rounding up one of the numbers, adding it on and then subtracting the amount that we added on during rounding. 63.8 + 5 – 0.3 = 68.5 + 5 – 0.3 63.8 68.5 68.8

22 Using partitioning to subtract
What is 564 – 437? 564 – 400 – 30 – 7 = 127 – 7 – 30 – 400 127 564 134 164 What is 22.5 – 6.4? This method involves partitioning the number we are subtracting and subtracting these one part at a time. In the second example the 6 is partitioned into two further parts to aid subtracting 6 from In this example we could also take the 0.4 away first and then the 2 and then the 4. Ask pupils how we could check each answer using, for example, inverse operations, for example. 22.5 – 2 – 4 – 0.4 = 16.1 – 0.4 – 4 – 2 16.1 22.5 16.5 20.5

23 Subtracting by counting up
What is 564 – 437? 100 + 3 + 20 + 4 = 127 + 100 + 3 + 20 + 4 437 537 540 560 564 What is 22.5 – 6.4? To subtract by counting up we start at the smaller number and count up to the larger number in parts. These parts are added together to give the answer. 10 + 6 + 0.1 = 16.1 + 10 + 6 + 0.1 6.4 16.4 22.4 22.5

24 Using compensation to subtract
What is 564 – 437? 564 – 500 + 63 = 127 – 500 + 63 64 564 127 What is 22.5 – 6.4? This method involves rounding up the smaller number (to a number that’s easy to subtract), subtracting this number and then adding the required amount back on to give us the answer. 22.5 – 6.5 + 0.1 = 16.1 – 6.5 + 0.1 16 22.5 16.1

25 Addition pyramid Use mental methods to find the hidden values in the pyramid. Reveal the numbers on the bottom row to practise addition or reveal one or two numbers from each row to practise addition and subtraction.

26 N9.3 Multiplication and division
Contents N9 Mental methods N9.1 Order of operations N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

27 Using partitioning to multiply whole numbers
What is 7 × 43? We can work out 7 × 43 mentally using partitioning. 43 = So, 7 × 43 = (7 × 40) + (7 × 3) Remind pupils that partitioning a number means to break a number down into smaller parts (usually hundreds, tens, units, tenths, hundredths etc.). An alternative to the method shown on this slide would be to think of 43 as 50 – 7 and use the distributive law. 7 × (50 – 7) = 350 – 49 = 301 Ask pupils to look at the completed calculation and ask a sequence of questions whose answers can be derived from this solution using inverse operations and place value, for example: What is 0.7 × 43? (30.1) What is 7 × 4.3? (30.1) What is 301 ÷ 43? (7) What is 301 ÷ 7? (43) and so on. A more challenging question would be: What is 43 ÷ 301? (1/7) Compare this with cancelling fractions. = = 301

28 Using partitioning to multiply decimals
What is 3.2 × 40? We can work out 3.2 × 40 by partitioning 3.2 3.2 = So, 3.2 × 40 = (3 × 40) + (0.2 × 40) In this example, we could also work out 3.2 × 10 and then × 4. 3.2 × 10 is 32. 32 × 4 is equal to 32 × 2 × 2, and 64 × 2 equals 128. This method is called using factors. 10 × 4 is equal to 40 and 2 × 2 is equal to 4. A further method would be to multiply 3.2 by 10 and divide 40 by 10, to give the equivalent calculation 32 × 4. Ask pupils to look at the completed calculation and ask a sequence of questions whose answers can be derived from this solution using inverse operations and place value, for example, What is 128 ÷ 32? (4) What is 128 ÷ 0.32? (400) What is 32 × 40? (1280) What is 3.2 × 4? (12.8) What is 128 ÷ 40? (3.2) What is 128 ÷ 3.2? (40) and so on. = = 128

29 Using the distributive law to multiply
What is 0.6 × 29? We can work out 0.6 × 29 using the distributive law. 29 = 30 – 1 So, 0.6 × 29 = (0.6 × 30) – (0.6 × 1) This example demonstrate the use of the distributive law to multiply. Explain that this method is useful when multiplying by a number that is close to a multiple of 10. Ask pupils to look at the completed calculation and ask a sequence of questions whose answers can be derived from this solution using inverse operations and place value, for example, What is 6 × 29? (174) What is 17.4 ÷ 29? (0.6) What is 17.4 ÷ 0.6? (29) What is 17.4 ÷ 6? (2.9) What is 17.4 ÷ 2.9? (6) What is 174 ÷ 29? (6) What is 128 ÷ 0.32? (400) and so on. = 18 – 0.6 = 17.4

30 Using a grid to multiply
Use this grid to demonstrate the use of the distributive law to multiply single-digit whole numbers by two-digit whole numbers. Stress to pupils that these calculations should be done mentally (using jottings if required). The grid is used here for practice and to aid visualisation. There are two ways to solve each problem. Either the two-digit number can be divided into tens and units which are added together or it can be rounded to the nearest ten with the amount that has been added on subtracted. Notice that clicking on the plus sign changes it to a minus sign. Discuss which of these methods is best for each problem. Point out that because it is often usually easier to add than to subtract we usually only use the second of these methods for two-digit numbers that end in 8 or 9. This avoids having to subtract larger numbers to obtain the solution.

31 Using a grid to multiply
Use this grid to demonstrate the use of the distributive law to multiply single-digit whole numbers by two-digit numbers written to one decimal place. Once the solution is revealed ask pupils to answer related questions whose answers can be derived from the given solution using inverse operations and place value. For example, if the board shows that 7 × 4.3 = 30.1, ask pupils, What is 7 × 43? (301) What is 70 × 4.3? (301) What is 0.7 × 4.3? (3.01) What is 7 × 0.43? (3.01) What is 30.1 ÷ 7? (4.3) What is 30.1 ÷ 4.3? (7) What is 301 ÷ 4.3? (70) and so on.

32 Using factors to multiply whole numbers
What is 26 × 12? We can work out 26 × 12 by dividing 12 into factors. 12 = 4 × 3 = 2 × 2 × 3 So we can multiply 26 by 2, by 2 again and then by 3: An alternative method could also be to think of 12 as We can then work out (10 × 26) + (2 × 26). = 312. Also, 25 × 12 = 25 × 4 × 3 = 100 × 3 = 300. So, 26 × 12 = 312 Ask pupils to look at the completed calculation and ask a sequence of questions whose answers can be derived from this solution using inverse operations and place value, for example: What is 2.6 × 12? (31.2) What is 26 × 0.12? (3.12) What is 312 ÷ 26? (12) What is 312 ÷ 12? (26) What is 312 ÷ 1.2? (260) What is 312 ÷ 120? (2.6) and so on. 26 × 2 × 2 × 3 = 52 × 2 × 3 = 104 × 3 = 312

33 Using factors to multiply decimals
What is 0.7 × 18? We can work out 0.7 × 18 by dividing 18 into factors. 18 = 9 × 2 So we can multiply 0.7 by 9 and then by 2: Stress that we multiply by 9 and then by 2 because as the numbers get larger it is easier to double than it is to multiply by 9. An alternative method could also be to think of 18 as 20 – 2. We can then work out (20 × 0.7) – (2 × 0.7). This uses the distributive law. 14 – 1.4 = 12.6. Also, point out to pupils that if they find multiplying by a decimal difficult they could work out 7 × 18 and then divide the answer by 10. Ask pupils to look at the completed calculation and ask a sequence of questions whose answers can be derived from this solution using inverse operations and place value, for example: What is 0.7 × 180? (126) What is 7 × 1.8? (12.6) What is 0.7 × 1.8? (1.26) What is 12.6 ÷ 0.7? (18) What is 12.6 ÷ 7? (1.8) What is 12.6 ÷ 18? (0.7) What is 126 ÷ 18? (7) and so on. 0.7 × 18 = = 0.7 × 9 × 2 = 6.3 × 2 = 12.6

34 Using doubling and halving
What is 7.5 × 8? Two numbers can be multiplied together mentally by doubling one number and halving the other. We can repeat this until the numbers are easy to work out mentally. An alternative would be to recognise that because 8 = 10 – 2, 7.5 × 8 = (7.5 × 10) – (7.5 × 2) = 75 – 15 = 60. We could also use factors to double 7.5 three times (because 8 = 2 × 2 × 2). Reveal the solution 7.5 × 8 = 60. Ask pupils to look at this calculation and ask: What is 60 ÷ 8? and What is 60 ÷ 7.5? Ask pupils to look at the completed calculation and ask a sequence of questions whose answers can be derived from this solution using inverse operations and place value, for example: What is 75 × 8? (600) What is 7.5 × 0.8? (6) What is 75 × 0.8? (60) What is 12.6 ÷ 0.7? (18) What is 60 ÷ 8? (7.5) What is 60 ÷ 7.5? (8) What is 6 ÷ 8? (0.75) and so on. 7.5 × 8 = 15 × 4 = 30 × 2 = 60

35 Using factors to divide whole numbers
What is 68 ÷ 20? We can work out 68 ÷ 20 by dividing 20 into factors. 20 = 2 × 10 So we can divide 68 by 2 and then by 10: 68 ÷ 20 = 68 ÷ 2 ÷ 10 An alternative could be to partition 68 into 60 and 8. 60 ÷ 20 is 3 and 8 ÷ 20 is 8/20 or 4/10. 4/10 is 0.4 so 68 ÷ 20 is 3.4 Discourage the use of remainders at this stage. Remainders should be expressed as fractions or decimals (unless a problem asked in context demands otherwise). Reveal the solution, ask pupils to look at this calculation and ask, What is 3.4 × 20? What is 68 ÷ 3.4? = 34 ÷ 10 = 3.4

36 Using factors to divide decimals
What is 12.4 ÷ 8? We can work out 12.4 ÷ 8 by dividing 8 into factors. 8 = 2 × 2 × 2 So we can divide 12.4 by 2, by 2 again and then by 2 a third time: 12.4 ÷ 8 = 12.4 ÷ 2 ÷ 2 ÷ 2 31 ÷ 2 = 15.5 so 3.1 ÷ 2 =1.55 This works for any number divided by 8. Halve, halve and halve again. Reveal the solution, ask pupils to look at this calculation and ask, What is 15.5 × 8? What is 124 ÷ 15.5? = 6.2 ÷ 2 ÷ 2 = 3.1 ÷ 2 = 1.55

37 Using partitioning to divide
What is 486 ÷ 6? We can work out 486 ÷ 6 by partitioning 486. 486 = So, An alternative method could be to use factors. To divide by 6 we can divide by 2 and then divide by divided by 2 is ÷ 3 = 81 (from (240 ÷ 3) + (3 ÷ 3)). The last part of this calculation still uses partitioning but, unless you know your six times table very well, it is easier to mentally divide by 3 than to divide by 6. Reveal the solution, ask pupils to look at this calculation and ask, What is 81 × 6? What is 486 ÷ 81? 486 ÷ 6 = (480 ÷ 6) + (6 ÷ 6) = = 81

38 Using fractions to divide whole numbers
What is 420 ÷ 40? We can simplify 420 ÷ 40 by writing the division as a fraction and then cancelling. 420 40 420 ÷ 40 = 21 Remind pupils that any division of two whole numbers can be written as a fraction. If the numbers share a common factor then we can simplify the calculation by cancelling. Ask pupils to give you the highest number they can think of that will divide into both 420 and 40. Dividing by 20 gives us 21/2. 21 divided by 2 is 10½ or Point out that if the fraction cannot be written as a terminating decimal, it is often better to leave it as a fraction. Reveal the solution, ask pupils to look at this calculation and ask, What is 10½ × 40? What is 420 ÷ 10½? 420 40 21 2 = 2 = 101/2 = 10.5

39 Using fractions to divide decimals
What is 2.6 ÷ 0.8? We can simplify 2.6 ÷ 0.8 by writing the division as a fraction. × 10 2.6 0.8 26 8 2.6 ÷ 0.8 = = 13 Reveal the solution, ask pupils to look at this calculation and ask, What is 3.25 × 0.8? What is 2.6 ÷ 3.25? 26 8 13 4 = 4 = 31/4 = 3.25

40 Multiplying by multiples of 10, 100 and 1000
We can use our knowledge of place value to multiply by multiples of 10, 100 and 1000. What is 7 × 600? What is 2.3 × 4000? 2.3 × 4000 = 2.3 × 4 × 1000 7 × 600 = 7 × 6 × 100 Point out that we can use factors to write 7 × 600 as 7 × 6 × 100. Similarly, 2.3 × 4000 is equivalent to 2.3 × 4 × 1000. Link: N1 Place value, ordering and rounding – multiplying and dividing by powers of ten = 42 × 100 = 9.2 × 1000 = 4200 = 9200

41 Dividing by multiples of 10, 100 and 1000
We can use our knowledge of place value to divide by multiples of 10, 100 and 1000. What is 24 ÷ 80? What is 4.5 ÷ 500? 24 ÷ 80 = 24 ÷ 8 ÷ 10 4.5 ÷ 500 = 4.5 ÷ 5 ÷ 100 Stress that 24 ÷ 80 = 24 ÷ (8 × 10) is equivalent to 24 ÷ 8 ÷ 10. Similarly, 4.5 ÷ 500 = 4.5 ÷ (5 × 100) is equivalent to 4.5 ÷ 5 ÷ 100 Link: N1 Place value, ordering and rounding – multiplying and dividing by powers of ten = 3 ÷ 10 = 0.9 ÷ 100 = 0.3 = 0.009

42 Noughts and crosses 1 Make sure that activity is set to multiply and divide by whole numbers. Divide the class into two teams, noughts and crosses. Decide which team will start . The starting team chooses a number from the board. Click on the number to highlight it. The box will randomly show an operation to be preformed on that number. Select a pupil from the team to give you their answer. Check their answer by clicking on the show answer button. If the answer is correct, select that team’s symbol (a nought or a cross). If the answer is incorrect select the opposing team’s symbol. It is then the turn of the opposing team to choose a number from the grid. The game is over when one of the teams gets three of their symbols in a row, horizontally, vertically, or diagonally. (Or when the board is full, in which case, the game ends in a draw).

43 N9 Mental methods Contents N9.1 Order of operations
N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

44 Multiplying by multiples of 0.1 and 0.01
Dividing by 10 is the same as Multiplying by 0.01 Dividing by 100 is the same as What is 4 × 0.8? What is 15 × 0.03? 4 × 0.8 = Remind pupils of the rules for multiplying by 0.1 and 0.01. Use these facts to multiply 4 by 0.8 and 15 by 0.03. Stress that multiplying by number between 0 and 1 makes the number smaller (we are finding a fraction of the number). Link: N1 Place value, ordering and rounding – multiplying by 0.1 and 0.01 4 × 8 × 0.1 15 × 0.03 = 15 × 3 × 0.01 = 32 × 0.1 = 45 × 0.01 = 32 ÷ 10 = 45 ÷ 100 = 3.2 = 0.45

45 Dividing by multiples of 0.1 and 0.01
Multiplying by 10 is the same as Dividing by 0.01 Multiplying by 100 is the same as What is 36 ÷ 0.4? What is 3 ÷ 0.02? 36 ÷ 0.4 = Remind pupils of the rules for dividing by 0.1 and 0.01. Use these facts to divide 36 by 0.4 and 3 by 0.02. Stress that dividing by number between 0 and 1 makes the number bigger. Link: N1 Place value, ordering and rounding – dividing by 0.1 and 0.01 36 ÷ 4 ÷ 0.1 3 ÷ 0.02 = 3 ÷ 2 ÷ 0.01 = 9 ÷ 0.1 = 1.5 ÷ 0.01 = 9 × 10 = 1.5 × 100 = 90 = 150

46 Multiplying by small multiples of 0.1
This activity show a diagrammatic representation of multiplying by small multiples of 0.1. Start by explaining to pupils that the large square represents one whole. It is one unit across and one unit down. The square is divided into ten equal parts across the top and so each division across the length must be equal to 0.1. The same is true down the side. Establish that the unit square is divided into 100 equal parts and so each small part is one hundredth of the large square or 0.01. Compare this with the fact that there are 100 mm2 in 1 cm2. Demonstrate various calculations by changing the size of the shaded area. For example, start by showing that 0.1 × 0.1 = One tenth of the width multiplied by one tenth of the length equals one hundredth of the whole unit. Ask pupils what 0.2 × 0.4 is equal to. It is a common error to give the answer as 0.8. Show, using the grid that the answer is 0.08. Change the number of unit squares to extend to more difficult problems. For example, 0.4 × 3.7 is equal to = 1.48 Click on the yellow rectangles to hide the answers. An ask pupils to calculate the solutions mentally using the diagram. Link: N1 Place value, ordering and rounding – multiplying by 0.1 and 0.01

47 Multiplying by decimals between 1 and 0
When we multiply a number n by a number greater than 1 the answer will be bigger than n. When we multiply a number n by a number between 0 and 1 the answer will be smaller than n. When we divide a number n by a number greater than 1 the answer will be smaller than n. This slide summarizes the rules for multiplying and dividing by decimals between 0 and 1. When we divide a number n by a number between 0 and 1 the answer will be bigger than n.

48 Noughts and crosses 2 Make sure that activity is set to multiply and divide by decimals. Divide the class into two teams, noughts and crosses. Decide which team will start . The starting team chooses a number from the board. Click on the number to highlight it. The box will randomly show an operation to be preformed on that number. Select a pupil from the team to give you their answer. Check their answer by clicking on the show answer button. If the answer is correct, select that team’s symbol (a nought or a cross). If the answer is incorrect select the opposing team’s symbol. It is then the turn of the opposing team to choose a number from the grid. The game is over when one of the teams gets three of their symbols in a row, horizontally, vertically, or diagonally. (Or when the board is full, in which case, the game ends in a draw).

49 N9 Mental methods Contents N9.1 Order of operations
N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

50 Chequered sums Explain that the numbers in the white squares are equal to the sum of the numbers in the four coloured cells touching them. Explain clearly how to work out the answer in one of the squares. Now ask pupils for the number which goes into each square. Pupils may give their answers using individual white boards or by putting hands up. Make the activity more difficult by hiding integers in the coloured cells.

51 Arithmagons – whole numbers
Explain to pupils that in this puzzle the numbers inside the squares are equal to the sum of the two circles on either side. Use the arithmagon on this slide to practise adding and subtracting whole numbers. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left. To practise addition, only reveal the numbers inside the circles. To practise subtraction, reveal some numbers inside the squares and some of the numbers inside the circles.

52 Arithmagons - decimals
Use the arithmagon on this slide to practise adding and subtracting decimals. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left.

53 Arithmagons –integers
Use the arithmagon on this slide to practise adding and subtracting positive and negative integers. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left.

54 Arithmagons – two significant figures
Use the arithmagon on this slide to practise adding and subtracting numbers of the same magnitude written to two significant figures (including negative numbers). Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left.

55 Circle sums – whole numbers
Explain to pupils that in this puzzle, the numbers on the right have to be dragged into each intersection so that the sum of the three numbers in each circle is same. The ‘circle sum’ can be hidden or revealed to make the puzzle more or less difficult. The circle sum puzzle on this slide uses whole numbers.

56 Circle sums - integers The circle sum puzzle on this slide uses positive and negative integers.

57 Circle sums – one decimal place
The circle sum puzzle on this slide uses numbers written to one decimal place.

58 Circle sums –two decimal places
The circle sum puzzle on this slide uses numbers written to two decimal places.

59 Productagons – using times tables
Explain to pupils that in this puzzle the numbers inside the squares are equal to the product of the two circles on either side. Use the productagon on this slide to practise multiplying and dividing by whole numbers between 2 and 12. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left. To practise multiplying, only reveal the numbers inside the circles. To practise dividing, reveal some numbers inside the squares and some of the numbers inside the circles.

60 Productagons – using factors
Use the productagon on this slide to practise multiplying and dividing using factors. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left.

61 Productagons – using partitioning
Use the productagon on this slide to practise multiplying and dividing using partitioning. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left. Notice that the numbers that appear in the circles alternate between one-digit and and two-digit numbers. This means the the hexagonal and square puzzles are easier than the triangular and pentagonal puzzles.

62 Productagons – using place value
Use the productagon on this slide to practise multiplying and dividing using place value. Press reset for a new puzzle or change the shape of the puzzle by selecting one of the shapes on the left.

63 Product triangle Arrange the numbers on the right so that the product of the three numbers along each side of the triangle is the same. The product can be hidden or revealed to make the puzzle more or less difficult.


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