South Yorkshire Maths Hub Year 9 Problem Solving Research Pilot

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

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.

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.

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]

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]

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.

Programme 0930 – 1100 Session 1 1100 – 1130 Coffee 1130 – 1230 Session 2 1230 – 1315 Lunch 1315 – 1500 Session 3 1500 – 15300 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

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

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

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

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

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

Playing Cards and Square Numbers

Prisoner Cell Problem http://www.transum.org/software/SW/prison/prison.asp

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.

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

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

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

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

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

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

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

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

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

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

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)

1

1 5

1 5 9

1 3 5 9

1 3 5 7 9

1 3 5 7 9 2

1 3 5 7 4 9 2

1 6 3 5 7 4 9 2

8 1 6 3 5 7 4 9 2

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

Starting Numbers Magic Total

Starting Numbers Magic Total 4 3 5 36 2 6 7 45 8 3 6 51 9 9 4 66 5 7 8 60

+ b a + c

+ b a a+b a+2b a+c a+b+c a+2b+c a+2c a+2c+b a+2c+2b + c

a+2c+b a a+2b+c a+2b a+b+c a+2c a+c a+2c+2b a+b

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

https://www.youtube.com/watch?v=xkbQDEXJy2k

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

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

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

Algebra - Important Principles Grammar Symbolism Generalisations Connections

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

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)

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, 1+3+3+3, leads to 1+3n

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)

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

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

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

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.

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

See, say, record There are 2n matches

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

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

Developing Teacher Craft 5 6 11 17 28 45

Developing Teacher Craft 3 ? ? 93

Developing Teacher Craft

Developing Teacher Craft

Developing Teacher Craft +

Developing Teacher Craft + =

Developing Teacher Craft + = 2 =

Developing Teacher Craft 3 ? ? 93

Developing Teacher Craft ? ? 93 3

Developing Teacher Craft ? ? 93 3 +

Developing Teacher Craft ? ? 3 93 +

Developing Teacher Craft ? ? 3 93 + =

Developing Teacher Craft ? ? 3 93 96 + =

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

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

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

Developing Teacher Craft 3 ? 48 93

Developing Teacher Craft 3 45 48 93

Developing Teacher Craft b a+b a+2b 3a+2b

Developing Teacher Craft 26 69

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

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?

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