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  1. “They didn’t do it like that in my day!”

  2. Do your children ask for help with their maths homework and start talking in another language, using words like ‘partitioning’, ‘chunking’, ‘grid multiplication’…..? • If so, you may feel the need for some translation. This booklet is designed to explain some of the methods used to teach calculation in schools following the Framework Review in 2006.

  3. mental calculation ? written ? Which is more important: or

  4. This will depend on the numbers involved and the individual child.

  5. When faced with a calculation, no matter how large or difficult the numbers may appear to be, all children should ask themselves:

  6. Can I do this in my head? Do I know the approximate size of the answer? If I can’t do it wholly in my head, what do I need to write down in order to help me calculate the answer? Will the written method I know be helpful?

  7. When do children need to start recording?

  8. When do children need to start recording? • The following table shows how some sort of recording is relevant throughout the primary years with mental strategies playing an important role throughout.

  9. It is important to encourage children to look first at the problem and then get them to decide which is the best method to choose – pictures, mental calculation with or without jottings, structured recording or calculator.

  10. ? • Children attempting to use formal written methods without a secure understanding will try to remember rules, which may result in unnecessary and mistaken applications of a standard method.

  11. Can anyone explain to me why the answer to this calculation is incorrect?

  12. Some of the methods explained in this presentation involve ‘partitioning’ and a set of place value cards. These are easy to find online.

  13. Equipment that we use

  14. ADDITION How would you complete this calculation? • If you went shopping and bought a bunch of bananas for £1.25 and a box of tea bags for £2.38, how much would you have spent altogether?

  15. These are the methods that we use to teach the children addition.

  16. Where do they begin?

  17. Problems will start off being verbal questions and will become more formal as they progress through Key Stage 1.

  18. Children are encouraged to develop a mental picture of the number system in their heads to use for calculation. • They develop ways of recording calculations using pictures, etc.

  19. Bead strings or bead bars can be used to illustrate addition 8+2=10

  20. 4 5 6 7 • Children then begin to use numbered lines to support their own calculations using a numbered line to count on in ones. E.g. 4+3=7

  21. ADDITION • Using an informal method by counting on in multiples of 10 with a number line Why use a number line? TU + TU 86 + 57 It helps me to show on paper what is going on in my head

  22. +50 86 Stage 1: Partitioning through a 10 means counting on in a multiple of 10 and then adding the remaining amount. Start at 86 (the larger number) on the number line. Partition the smaller number 57 into tens and units and count on the multiples of 10 first and then the units. TU + TU 86 + 57 +4 +3 136 140 143 86 + 57 = 143

  23. Using a number line to add too much and then subtract (compensate) I noticed that 96 is close to 100. 100 is easier to add than 96 but that means I’ve added 4 too many. I need to subtract 4 from the number I reach. Why are you subtracting when you should be adding? HTU + TU 754 + 96

  24. +100 -4 754 854 850 Start with the larger number 754. Add on 100 and then subtract 4. HTU + TU 754 + 96 754 + 96 = 850

  25. Partition into tens and ones and recombine 36 + 53 = 53 + 30 + 6 = 83 + 6 = 89

  26. Pencil and paper procedures 83 + 42 = 125 80 + 3 + 40 + 2 120 + 5 = 125

  27. I know that I can add numbers in any order and the total will be the same. My teacher has told me that I need to practise adding the units first. The next method I will learn works this way. I must remember to line the numbers up in the correct columns. Expanded method: moving on from adding the most significant digits first to adding leastsignificant digits first. Why switch to adding the units (least significant digits) first? HTU + HTU 625 + 148

  28. Add most significantdigits first: Add least significant digits first: (in this example, units) (in this example, hundreds) HTU + HTU625 + 148 625 + 148 13 60 700 773 625 + 148 700 60 13 773 5 + 8 20 + 40 600 + 100 600 + 100 20 + 40 5 + 8 Mentally add 700 + 60 + 13 = 773 625 + 148 = 773

  29. Using a standard method HTU + HTU 587 + 475 Why do you say 80 + 70 instead of 8 + 7? I need to remember the value of each digit, so I know the size of the numbers I am adding and whether my answer makes sense.

  30. HTU + HTU587 + 475 7+5= 12 Place the2in the units column and carry the 10 forward to the tens column. 587 + 475 1062 1 1 80 + 70 =150then + 10 (carried forward) which totals160. Place 60 in the tens column and carry the100forward to the hundreds column. 500+ 400=900 then + 100which totals1000. Place this in the thousands column. 587 + 475 = 1062

  31. Subtraction • How would you complete this calculation? • If you were shopping and wanted to find out how much change you have from £5 when you spend £2.70, how would you work it out in your head?

  32. These are the methods that we use to teach the children subtraction.

  33. Where do they begin?

  34. They develop ways of recording calculations using pictures.

  35. Pictures / marks Sam spent 4p. What was his change from 10p?

  36. Bead strings or bead bars can be used to illustrate subtraction including bridging through ten by counting back 3 then counting back 2. 6-2=4

  37. The numberline should also be used to show that 6 - 3 means the ‘difference between 6 and 3’ or ‘the difference between 3 and 6’ and how many jumps they are apart. +1 +1 +1 3 6

  38. Counting on or counting back? A B SUBTRACTION How do you decide whether to count on or count back? TU - TU 84 - 56 If the numbers are close together like 203 – 198 it’s quicker to count on. If they’re a long way apart like 203 – 5 it’s quicker to take away. Sometimes I count on because that’s easier than taking away.

  39. +20 +4 +4 56 60 84 0 80 A Start Here B 0 -50 -4 -2 30 84 28 TU - TU 84 - 56 Find the difference between the two numbers. Count on from 56 to 84. 20 + 4 + 4 = 28 Start by ‘taking away’ (crossing out) the 56. 34 Partition 56 and count back (subtract) 50 and then 6. Start HERE

  40. Complementary addition A Number lineB Written method HTU - HTU 954 - 586 The number line method is very clear. Why do you use method B and write the numbers vertically? I could make mistakes. Method B helps me line the numbers up and see what I need to add.

  41. +300 +300 +50 +50 +4 +4 590 590 900 900 0 600 600 586 586 Find the difference between the two numbers. Count on from 586 to 954. 300 + 50 + 10 + 4 + 4 = 368 Find the difference between the two numbers. Count on from 586 to 954. 300 + 50 + 10 + 4 + 4 = 368 START HERE START HERE ‘Take away’ the 586. ‘Take away’ the 586. START HERE START HERE 954 - 586 4 10 300 50 4 368 954 - 586 4 10 300 50 4 368 Count on to the next multiple of 10. Count on to the next multiple of 10. Count on to the next multiple of 100. Count on to the next multiple of 100. To make 590 To make 600 To make 900 To make 950 To make 954 To make 590 To make 600 To make 900 To make 950 To make 954 Count on in 100s. Count on in 100s. Count on to the larger number in the calculation which is 954. Count on to the larger number in the calculation which is 954. 954 - 586 = 368 954 - 586 = 368 HTU - HTU 954 - 586 +10 +4 950 954

  42. Standard method (decomposition) HTU - HTU 754 - 286 Because all the stages I have learnt before have really helped me understand exactly what I’m doing. Why didn’t you use the standard method straight away?

  43. 40 1 700+50+4 -200+80+6 600 54 is the same value as 40104 . Now 6 can be subtracted from14. 1 700 +40+14 -200+80+6 740 is the same value as 600 + 100 + 40 . Now 80 can be subtracted from140. 600+140+14 -200+80+6 400+60+8 = 468 1 6 4 1 754 - 286 468 HTU - HTU 754 - 286 Or, more efficiently the standard method. 754 – 286 = 468

  44. multiplication • How would you complete this calculation? • If I buy 5 bunches of bananas at £1.35 per bunch, how much would they cost me altogether?

  45. These are the methods that we use to teach the children multiplication.

  46. Where do they begin?

  47. Children will experience equal groups of objects. • They will count in 2s and 10s and begin to count in 5s. • They will work on practical problem solving activities involving equal sets or groups.

  48. Pictures and symbols There are 3 sweets in one bag. How many sweets are there in 5 bags?

  49. Repeated addition3 times 5 is 5 + 5 + 5 = 15 or 3 lots of 5 or 5 x3 Repeated addition can be shown easily on a number line:

  50. and on a bead bar: