1. South Yorkshire Maths Hub
Year 9 Problem Solving Research Pilot

2. Intended outcomes of this work
To produce written guidance for to support the teaching of problem solving strategies to Y9 pupils

3. A children’s nursery uses one room for babies and one room for toddlers.
Each baby needs at least 3.5 m2 of floor space.
Each toddler needs at least 2.5 m2 of floor space.

4. A children’s nursery uses one room for babies and one room for toddlers.
Each baby needs at least 3.5 m2 of floor space.
Each toddler needs at least 2.5 m2 of floor space.

5. Room A
Area = 40 m2
Room B
Area = 37 m2
Show that the total number of children allowed is larger if the toddlers are in Room A and the babies are in Room B.
[4 marks]

6. The perimeter of an isosceles triangle
is 25 cm.
The length of each side, in cm, is a
prime number.
Work out the lengths of the sides of the
two possible isosceles triangles.
[4 marks]

7. The reciprocal of any prime number p (where p is neither 2 nor 5) when written as a decimal, is always a recurring decimal.
A theorem in mathematics states
The period of a recurring decimal is the least value of n for which p is a factor of 10n – 1
Hugo tests this theorem.
He uses his calculator to show that 37 is a factor of
103 – 1
Hugo then makes this statement,
“The period of the recurring decimal equal to the reciprocal of 37 is 3 because 37 is a
factor of 103 – 1. This shows the theorem to be true in this case.”
Explain why Hugo’s statement is incomplete.

8. Programme
0930 – 1100 Session – 1130 Coffee 1130 – 1230 Session – 1315 Lunch 1315 – 1500 Session – Next Steps
There have been statements in the past that have caused similar challenges based on systematic evaluation at the time.
Whilst the statement is of some concern we have had previous reports through history that are located in a system of continuous improvement
We should welcome the challenge and continue to raise our expectations

9. Session 1
PD1: Can we classify the range of problem solving strategies that should be taught?

10. 1. Generating Data and Listing
Pupils should be taught to:
Generate data from a given rule or a set of conditions
Derive a set of numbers or shapes that meet a list of criteria
Find the largest and/or the smallest cases or values for given circumstances and conditions
Systematically list and record all the possibilities in a set given a number of conditions

11. 2. Sorting and Classifying
Pupils should be taught to:
Sort objects, numbers or shapes by deciding whether they meet a given criteria
Classify a set of objects numbers, or shapes using a number of criteria or properties
Identify criteria to describe sets of numbers, objects or shapes that have been sorted or classified
Use sorts and/or classifications to complete sets with missing items

12. 3. Explaining and reasoning
Pupils should be taught to:
Organise data to generate and complete patterns
Use symmetric properties in shapes, sets of numbers and calculations to establish relationships and enumerate lists
Organise information into tables, charts and diagrams in order to recognise and discover patterns and relationships
Describe relationships and patterns. Manipulate these to generate new ones

13. 4. Explaining and reasoning
Pupils should be taught to:
Use calculations to support explanation and argument
Look for a counter example to define the conditions and limits of a rule
Use a relationship or pattern to justify or confirm others.
Use properties and relationships to reason and deduce.
Use a diagram to support an explanation

14. Playing Cards and Square Numbers

15. Prisoner Cell Problem

16. Teaching a problem solving strategy
The full time score in a game of
hockey was 3 – 4. What could the
score have been at half time?
Work with a friend and write down as any
different half-time scores as you can on the
blank cards provided.

17. 0 - 1
0 - 3
0 - 4
2 - 1
3 - 2
1 - 0
1 - 1
2 - 2
2 - 4
1 - 3
2 - 0
0 - 2
3 - 4
1 - 2
0 - 0
3 - 1
3 - 3
1 - 4
2 – 3
3 - 0

18. Sum of the scores 0 - 0 0 - 1 1 - 0 0 - 2 1 - 1 2 - 0 0 - 3 1 - 2
2 - 1
3 - 0
0 - 4
1 - 3
2 - 2
3 - 1
3 - 2
1 - 4
2 - 3
2 - 4
3 - 3
3 - 4

19. 0 diff
1 diff
2 diff
3 diff
4 diff
0 - 0
1 - 1
2 - 2
3 - 3
0 - 1
1 - 0
1 - 2
2 - 1
2 – 3
3 - 4
3 - 2
0 - 2
1 - 3
2 - 0
2 - 4
3 - 1
0 - 3
1 - 4
3 - 0
0 - 4

20. 0 - 0
0 - 1
0 - 4
0 - 3
0 - 2
1 - 0
1 - 1
1 - 4
1 - 3
1 - 2
2 - 0
2 - 1
2 - 4
2 – 3
2 - 2
3 - 0
3 - 1
3 - 4
3 - 3
3 - 2

21. DRAWS
2 - ?
1 - ?
0 - ?
3 - ?
0 - 0
1 - 1
2 - 2
3 - 3
0 - 2
0 - 1
1 - 0
1 - 4
1 - 3
2 - 0
3 - 0
3 - 1
3 - 4
3 - 2
2 - 1
2 – 3
2 - 4

22. DRAWS
2 - ?
1 - ?
0 - ?
3 - ?
0 - 0
1 - 1
2 - 2
3 - 3
0 - 2
0 - 1
1 - 0
1 - 4
1 - 3
2 - 0
3 - 0
3 - 1
3 - 4
3 - 2
1 - 2
2 - 1
0 - 3
2 – 3
0 - 4
2 - 4

23. DRAWS
2 - ?
1 - ?
0 - ?
3 - ?
0 - 0
1 - 1
2 - 2
3 - 3
0 - 2
0 - 1
1 - 0
1 - 4
1 - 3
2 - 0
3 - 0
3 - 1
3 - 4
3 - 2
1 - 2
2 - 1
0 - 3
2 – 3
0 - 4
2 - 4

24. DRAWS
2 - ?
1 - ?
0 - ?
3 - ?
0 - 0
1 - 1
2 - 2
3 - 3
0 - 2
0 - 1
1 - 0
1 - 4
1 - 3
2 - 0
3 - 0
3 - 1
3 - 4
3 - 2
1 - 2
2 - 1
0 - 3
2 – 3
0 - 4
2 - 4

25. 0 - 0
0 - 1
0 - 4
0 - 3
0 - 2
1 - 0
1 - 1
1 - 4
1 - 3
1 - 2
2 - 0
2 - 1
2 - 4
2 – 3
2 - 2
3 - 0
3 - 1
3 - 4
3 - 3
3 - 2

26. 0 - 0
0 - 1
0 - 4
0 - 3
0 - 2
1 - 0
1 - 1
1 - 4
1 - 3
1 - 2
2 - 0
2 - 1
2 - 4
2 – 3
2 - 2
3 - 0
3 - 1
3 - 4
3 - 3
3 - 2

27. Number of half time scores = (n+1)(m+1)
0 - 0
0 - 1
0 - m
1 -m
1 - 0
n - 0
n - 1
n - m
Number of half time scores = (n+1)(m+1)

29. 1

30. 1 5

31. 1 5 9

32. 1 3 5 9

33. 1 3 5 7 9

34. 1 3 5 7 9 2

35. 1 3 5 7 4 9 2

36. 1 6 3 5 7 4 9 2

37. 8 1 6 3 5 7 4 9 2

39. 1st
2nd
3rd
4th
5th
6th
7th
8th
9th T

40. Starting Numbers
Magic Total

41. Starting Numbers
Magic Total 36 45 51 66 60

42. b a + c

43. b a
a+b
a+2b
a+c
a+b+c
a+2b+c
a+2c
a+2c+b
a+2c+2b + c

44. a+2c+b a
a+2b+c
a+2b
a+b+c
a+2c
a+c
a+2c+2b
a+b

45. Session 2
PD2: Would it be helpful to classify these strategies against different type of problems as set out below?

47. Session 3
PD3: Would it be useful to describe specific teaching approaches using particular activities in relation to the four strands?

48. Opportunities Higher order thinking, reasoning and proof
Intensifying learning
Enjoy more the richness and beauty of mathematics
New examination questions present opportunities for learning

49. Mathematical thought can have no success where it cannot generalise
Algebra
Mathematical thought can have no success where it cannot generalise
Charles Pierce
The essence of mathematics

50. Algebra - Important Principles
Grammar
Symbolism
Generalisations
Connections

51. Developing algebraic imagery
4, 7, 10, …….
nth term = 3n + 1

52. See, say, record 4, 4+3, 4+3+3, leads to 4+3(n-1)
SEE: 4 and adding 3 more each time
4, 4+3, 4+3+3, leads to 4+3(n-1)

53. See, say, record 1+3, 1+3+3, 1+3+3+3, leads to 1+3n
SEE 1 and 3 more, and 3 more each time
1+3, 1+3+3, , leads to 1+3n

54. 4, 8-1, 12-2, leads to 4n-(n-1) See, say, record
SEE groups of 4, with some lines counted twice
4, 8-1, 12-2, leads to 4n-(n-1)

55. See, say, record Horizontal lines: 2n Vertical lines: n + 1
SEE horizontal and vertical lines
Horizontal lines: n
Vertical lines: n + 1
Total = 2n + (n+1)

56. See, say, record
The sequence is 4, 12, 24, 40, 60,….

57. Find the rule...
4, 12, 24, 40, 60,….
2n2 + 2n
2n(n + 1)

58. See, say, record: a route to algebra
Find a way of seeing
this pattern of matches
and use it to arrive at a
statement of generality.

59. See, say, record: a route to algebra
Look at the matches
on the left hand side
and across the bottom

60. See, say, record
There are 2n
matches

61. See, say, record
Now look for these
L shaped arrangements
of matches

62. See, say, record
Can you see n2 sets
of 2?

63. Developing Teacher Craft5 6 11 17 28 45

64. Developing Teacher Craft3 ? ? 93

65. Developing Teacher Craft

66. Developing Teacher Craft

67. Developing Teacher Craft+

68. Developing Teacher Craft+ =

69. Developing Teacher Craft+ = 2 =

70. Developing Teacher Craft3 ? ? 93

71. Developing Teacher Craft? ? 93 3

72. Developing Teacher Craft? ? 93 3 +

73. Developing Teacher Craft? ? 3 93 +

74. Developing Teacher Craft? ? 3 93 + =

75. Developing Teacher Craft? ? 3 93 96 + =

76. Developing Teacher Craft? ? 3 93 96 2 + = =

77. Developing Teacher Craft? 3 93 96 2 ? + = =

78. Developing Teacher Craft? 3 93 96 2 48 + = =

79. Developing Teacher Craft3 ? 48 93

80. Developing Teacher Craft3 45 48 93

81. Developing Teacher Craftb
a+b
a+2b
3a+2b

82. Developing Teacher Craft26 69

83. Developing Teacher Craft
a+b
2a+3b
a + b =26
2a + 3b = 69

84. Questions pertaining to the Points of Departure
Is the list of problem solving strategies as defined complete?
How useful is it to classify the different types of problems?
Are the strategies specific to the teaching of problem solving taught or are they over-arching teaching approaches that teachers should use in all of their teaching?
Would it be useful to link particular problems to age related year groups in order to exemplify the particular problem solving strategy being taught?

85. Next Steps Modification of framework and template
Begin to capture teaching activities and teaching approaches
Sharing and dissemination