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More on calculations

More on calculations. Dr Geoff Tennant geoff.tennant@aku.edu. Vygotsky ’ s Zone of Proximal Development. I can already drive a car I cannot fly a fighter jet without a huge amount of learning which I don’t currently have.

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More on calculations

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  1. More on calculations Dr Geoff Tennant geoff.tennant@aku.edu

  2. Vygotsky’s Zone of Proximal Development I can already drive a car I cannot fly a fighter jet without a huge amount of learning which I don’t currently have. BUT: while I haven’t driven a lorry, it is reasonable to suppose that, with some lessons, I could. Driving a lorry is next to what I can already do. So, driving a lorry is in my zone of proximal development – I can’t do it now, but it is next to what I can already do. More generally, the ZPD represents what can be learnt reasonably easily given existing knowledge and understanding.

  3. Vygotsky’s ZPD: a case study (1) The task is to start from 278 and work what you need to add on to reach 500. John has successfully completed this task, although he has done very few questions in the time available. When you ask him how he went about this task, he reluctantly rescues from the bin a sheet of paper on which he has drawn tallies whilst counting 279-280-281-…-499-500, and then counted them from the beginning (tallies shown on the next slide).

  4. Vygotsky’s ZPD: a case study (2) Brief discussion: what might you say to John?

  5. Subtraction with a number line (1) 1001 – 4 998 999 1000 1001 997

  6. Subtraction with a number line (2) A train leaves London at 22.45 and arrives in Inverness at 07.20 the next day. How long was the journey? 7 hrs 15 m 1 hr 20 m 2245 0000 0700 0720 2300 So total time taken is 15 min + 1 hr + 7 hr + 20 min = 8 hrs 35 min

  7. Timestables Need as many ways as practising these as we can. Already have: • People maths activities, standing up, ‘fizz buzz’; • Can use follow on cards, Tarsia activities, bingo, and how about Countdown?

  8. Challenge Think of as many ways as possible of practising the timestable fact: 4 x 6 = 24 Include simple divisions, word problems, etc.

  9. Countdown (1) The answer is 32 You can use 4, 7, 9, 5 Use the following numbers and any of addition, subtraction, multiplication and division to get the answer. One answer: 4 x 7 + 9 – 5

  10. Countdown (2) Task Make up a Countdown activity suitable for your class. Try it out on people sitting next to you

  11. Another way of thinking about timestables….

  12. So that leaves us with…

  13. What do we know about: • The 10 timestable? • The 11 timestable? • The 5 timestable? • The 9 timestable? • The 2 timestable? • The 4 timestable? • The 8 timestable?

  14. See if you like Vedic squares:

  15. If you join the 2s:

  16. So, in summary: • Timestable facts still need to be learnt, but there are fewer to learn than may appear at first sight. Before we move on: - Do you have ideas for learning timestables we’ve not yet shared?

  17. Multiplication (1): traditional method 3 5 6 x ____1 7 9 3 2 0 4 5 5 0 9 2 2 4 3 4 3 5 6 0 0 7 4 6 3 2 1 1

  18. Multiplication (2): grid method 300 50 6 100 30000 5000 600 35600 70 21000 3500 420 24920 2700 450 54 3204 9 63724

  19. Use the grid method to do the following multiplications • 456 x 192 • 243 x 891 • 24 x 1025 If you have time, compare your answers with what they look like with the traditional algorithm. Can you see the similarities?

  20. Multiplication (3): gelosia 3 5 6 0 0 0 1 3 5 6 2 3 4 1 7 6 2 1 5 5 2 4 3 2 9 4 7 5 7 2 4 1

  21. Use the gelosia method to do the following multiplications • 456 x 192 • 243 x 891 • 24 x 1025 If you have time, compare your answers with what they look like with the traditional algorithm and with the grid method. Can you see the similarities?

  22. Happy numbers • Start eg. with 25 • 25 goes to 2x2 + 5x5 = 29 • 29 goes to 2x2 + 9x9 = 85 • 85 goes to 8x8 + 5x5 = 89 • 89 goes to 8x8 + 9x9 = 145 • 145 goes to 1x1 + 4x4 + 5x5 = 42 • 42 goes to 4x4 + 2x2 = 20 • 20 goes to 2x2 + 0x0 = 4 • 4 goes to 4x4 = 16 • 16 goes to 1x1 + 6x6 = 37 • 37 goes to 3x3 + 7x7 =58 • 58 goes to 5x5 + 8x8 = 89 Can you see what happens now? What about if you start with other numbers?

  23. Maths trails Dr Geoff Tennant geoff.tennant@aku.edu

  24. Premise of maths trails • There is maths all around us; • Excellent as an activity eg. when children change school • Gives a focus on school outings • Can be used to consolidate maths topics being learnt.

  25. Some examples Your task Devise a maths trail in your groups based around this room / this campus / nearby. Be ready to present your trail to the rest of the group, being clear what purposes it will serve in learning new material / consolidating / bringing ideas together.

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