2. What do you see? (1) Watch what happens.

12. Start again

13. What do you see? (1) Describe what happens. How many dots are added each time?

15. What do you see? (2) Watch what happens; it starts with a square of dots.

19. Start again

20. What do you see?(2) Can you describe what happens? Would the same thing happen if you started with a larger square? Or a smaller square? Can you write this algebraically?

21. a a Start with any square: a²

22. Take one away: a²-1 a a

23. Move the top row to become a column… a a

24. a a

25. a a

26. a-1 a Move the top row to become a column…

27. a a-1 Move the top row to become a column…

28. a+1 a-1 Move the top row to become a column…

29. a+1 a-1 Move the top row to become a column…

30. (a-1)(a+1) a+1 a-1

31. a²-1 = (a-1)(a+1)

33. What do you see? (3) Watch what happens; the initial shape is a square.

35. The shape removed is a square.

38. Start again

39. What do you see? (3) Can you describe what happens? Would the same thing happen if you with a larger square? Or a smaller square? Can you write this algebraically?

41. What do you see? (4) Watch what happens.

43. Start again

44. What do you see? (4) What would happen if it kept on going? Describe what happens in words. Can you describe what happens using numbers?

46. Start again What should these be?

47. What do you see? (4) Does this help you to find the sum of the following:

49. What do you see? (5) Watch what happens.

51. Start again

52. What do you see? (5) Describe what happens in words. What would happen if it kept on going? Can you describe what happens using numbers?

54. What do you see? (5) Does this help you to find the sum of the following:

55. What do you see? (5) Can you think of a similar geometrical proof to find the sum of the infinite series:

57. Teacher notes: What do you see? This edition looks at a range of visual stimuli and asks ‘What do you see?’ Many of the activities are visual representations of proof, including sums of infinite series and algebraic equivalence. The pedagogical focus for these activities is on generating discussion amongst small groups of students in order that they arrive at an understanding. At various times, the teacher might circulate and listen in unobtrusively to discussions, ask probing questions, or draw out conflicting or differing responses from groups to contribute to a whole class plenary. The activities could be used as a series of starter activities. These are some of the approaches highlighted within the AfL session run by Simon Clay and Debbie Barker at the MEI 2014 Conference. Additionally, these are strategies which many girls find particularly supportive – but that’s not to suggest that boys don’t find them supportive too!

58. Teacher notes: What do you see? (1) Make it ‘girl friendly’: Ask students to discuss it with a friend before giving an answer

59. Teacher notes: What do you see? (2) This activity shows that a²-1 = (a-1)(a+1) Show the sequence to students and ask them to describe what they see. A specific size of square is shown, ask students to consider what would happen with larger or smaller starting squares. They might like to sketch them out to check, so some squared paper might be useful. Make it ‘girl friendly’: Ask students to “think, pair, share”: give students silent thinking time before a short discussion with a partner and then others.

60. Teacher notes: What do you see? (3) This activity shows that a²-b 2 = (a-b)(a+b) Show the sequence to students and ask them to describe what they see. A generic square is shown and a smaller generic square is removed. Can students convince themselves that the rectangle ‘on top’ will always fit ‘at the side’? Encourage students to write this algebraically. Make it ‘girl friendly’: Ask students to work with a partner; you circulate, listening in and asking questions quietly to pairs.

61. Teacher notes: What do you see? (4) This activity shows the sum of the reciprocals of powers of 2. Show the sequence to students and ask them to describe what they see. Transitions are timed – no mouse click needed. Students in KS3 and 4 may describe the series in words. Ask how much of the white square is used each time. The second part of the sequence also shows the fractional values. Make it ‘girl friendly’: Show the sequence two or three times through. Accept and validate all responses.

62. Teacher notes: What do you see? (5) This activity shows the sum of the reciprocals of powers of 3. Show the sequence to students and ask them to describe what they see. Transitions are timed – no mouse click needed. Students in KS3 and 4 may describe the series in words. Ask how much of the white square is used each time. The second part of the sequence also shows the fractional values. The third part of the activity asks if students can think of something similar for reciprocals of powers of 4. An example shown on the following slides shows that the sum is 1 / 3. Make it ‘girl friendly’: Ask students to Convince themselves Convince a friend Convince the class

63. Teacher notes: What do you see? (5) In this case, divide the square into quarters, colour one quarter dark and two light.

64. Teacher notes: What do you see? (5) With the remaining space, divide it into quarters, colour one quarter dark and two light.

65. Teacher notes: What do you see? (5) With the remaining space, divide it into quarters, colour one quarter dark and two light.

66. Teacher notes: What do you see? (5) For every one dark section there will be two light sections.