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 www.furthermaths.org.uk

Problem solving and proof in new GCSEs   Kevin Lord

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

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?

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

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.

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

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

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

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?

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 500 000 students Secondary school students (11-16) = 2 500 000 students At a school with 1000 students = 10 maths teachers Ratio is 1 teachers to 100 students No. of secondary maths teachers = 25 000

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

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 = 131400 hrs Fraction = 12%

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

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

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?

Can you change one aspect of . . Considering each coefficient and value independently there are 4 different equations 2a + 20 -1 = square 4 + 4b -1 = square 4 + 5d -1 = square 4 + 20 - c = square Are there solutions to all 4 equations? A spreadsheet or table is a helpful way to record results

Can you change one aspect of . . 2a + 20 -1 = 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, 4.4 . . . ) 4 + 20 - 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?

Alternative representation 1 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.

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

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