2. The Further Mathematics Support Programme
Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.
The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.
To find out more please visit

3. Problem solving
and proof in new GCSEs
Kevin Lord

4. Folded Rectangle
A rectangle of sides 12 and 18 is folded along its diagonal. What is the area of the new shape formed?

5. Supporting Problem Solving
Whilst working on this problem, consider,
What sort of hints, questions and prompts might help a student get started?
How can problems be adapted for different abilities?
How would you organise the classroom to encourage problem-solving?

6. Supporting Problem Solving
Things to consider
Thinking time
Banning “I can’t do it” and “I don’t know what to do”
Pairing and sharing ideas
Stand up problem solving or working on large sheets
Differentiation and support
Questioning
Particular cases
Simplifying

7. The new GCSE specifications
GCSE specifications in mathematics should enable students to:
develop fluent knowledge, skills and understanding of mathematical methods and concepts
acquire, select and apply mathematical techniques to solve problems
reason mathematically, make deductions and inferences and draw conclusions
comprehend, interpret and communicate mathematical information in a variety of forms appropriate to the information and context.

8. Use and apply standard techniques
AO1
Use and apply standard techniques
Weighting
Students should be able to:
accurately recall facts, terminology and definitions
use and interpret notation correctly
accurately carry out routine procedures or set tasks requiring multi-step solutions
(Foundation) 8

9. 30% (Higher) 25% (Foundation) AO2
Reason, interpret and communicate mathematically
Weighting
make deductions, inferences and draw conclusions from mathematical information
construct chains of reasoning to achieve a given result
interpret and communicate information accurately
present arguments and proofs
assess the validity of an argument and critically evaluate a given way of presenting information
30%
(Higher)
25%
(Foundation) 9

10. AO3
Solve problems within mathematics and in other contexts
Weighting
translate problems in mathematical or non-mathematical contexts into a process or a series of mathematical processes
make and use connections between different parts of mathematics
interpret results in the context of the given problem
evaluate methods used and results obtained
evaluate solutions to identify how they may have been affected by assumptions made.
(Foundation) 10

11. Short modelling problems
How many secondary mathematics teachers are there in England?
How many bananas do you need to make 100 banana sandwiches?
By the age of 15 what fraction of your life have you spent at school?

12. Short modelling problems
How many secondary mathematics teachers are there in England?
32,800 in November 2012 according to the DfE School Workforce in England.
Possible chain of reasoning
Each year group approximately students
Secondary school students (11-16) = students
At a school with 1000 students = 10 maths teachers
Ratio is 1 teachers to 100 students
No. of secondary maths teachers =

13. Short modelling problems
How many bananas do you need to make 100 banana sandwiches?
About 30 bananas
Possible chain of reasoning
Bananas are about 20 cm long and 3cm in diameter
Thickness of a slice = 0.5 cm giving 40 slices
Bread dimensions 10cm by 12cm
Each sandwich requires 3x4 = 12 slices
100 sandwiches = 1200 slices = 30 bananas

15. Short modelling problems
By the age of 15 what fraction of your life have you spent at school?
About an eighth
Possible chain of reasoning
School day = 8 hrs
School week = 8 x 5 = 40 hrs
School year = 40 x 40 = 1600 hrs
School time in total = 10 x 1600 = 16000hrs
No. of hours in life so far = 24x365x15 = hrs
Fraction = 12%

16. Supporting modelling
Make reasonable simplifying assumptions (keep it simple)
Articulate the assumptions made explicitly
Be imprecise and use rounded/convenient numbers
Think about upper and lower bounds
Consider validity of solution (Is it sensible?)
Consider how result depends on the assumptions made
Don’t worry about being too accurate as long as the solution is of the right order

17. Incorporating “problem solving”
Problems can be incorporated as
Starters
Extensions
Investigations
Through questioning strategies

18. Can you change one aspect of . .
a = 2 , b = 5 , c = 1 so that 2a + 4b – c is a square number?
Is it possible to change each different aspect?

19. Can you change one aspect of . .
Considering each coefficient and value independently there are 4 different equations
2a = square
4 + 4b -1 = square
4 + 5d -1 = square
c = square
Are there solutions to all 4 equations?
A spreadsheet or table is a helpful way to record results

20. Can you change one aspect of . .
2a = 2a + 19 = square (a = -9, -7.5, -5, -1.5, 3 …)
4 + 4b -1 = 4b + 3 = square (b = 0.25, 1.5, 3.25, 5.5, . . )
4 + 5d -1 = 5d +3 = square (d = 0.2,1.2, 2.6, )
c = 24 - c = square (c = 23, 20, 15, 8, -1, . . )
However if the values need to be integers then surprisingly there are no solution to 4b+3 or 5d+3.
In other words square numbers cannot be written in the form 4n+3 or 5n+3 (where n is an integer). Why not?

21. Alternative representation1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
100
The grid shows numbers arranged in 4 columns with square numbers up to 100 highlighted. It shows that square numbers are not of the form 4n+2 or 4n+3 Also odd numbers squared are in the form 4n+1 And even numbers squared are in the form 4n Similar tables can be investigated for 3n, 5n and 6n etc.

22. FMSP Resources Free resources on the FMSP website GCSE Problem Solving
GCSE Extension Materials
Extension and enrichment for KS4
Team Mathematics Competition Materials
Year 12 Problem Solving Materials

23. Other Sources nRICH (inc. STEM nRICH) UKMT challenges
Waterloo University
Brilliant
RISPS
101qs website (Dan Meyer)
Old GCSE and KS3 coursework starters