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Embedding problem solving in teaching and learning at KS2 Dr Ray Huntley Brunel University, London September 2, 2013 How to be good • Most learners make good progress because of the good teaching they receive • Behaviour overall is good and learners are well motivated • They work in a safe, secure and friendly environment • Teaching is based on secure subject knowledge with a wellstructured range of stimulating tasks that engage the learners • The work is well matched to the full range of learners’ needs, so that most are suitably challenged • Teaching methods are effectively related to the lesson objectives and the needs of the learners Assessment for Learning • Ensure that every learner succeeds: set high expectations • Build on what learners already know: structure and pace teaching so that they can understand what is to be learned, how and why • Make learning of subjects and the curriculum real and vivid • Make learning enjoyable and challenging: stimulate learning through matching teaching techniques and strategies to a range of learning needs • Develop learning skills, thinking skills and personal qualities across the curriculum, inside and outside of the classroom • Use assessment for learning to make individuals partners in their learning Personalisation • Teaching is focused and structured • Teaching concentrates on the misconceptions, gaps or weaknesses that learners have had with earlier work • Lessons or sessions are designed around a structure emphasising what needs to be learnt • Learners are motivated with pace, dialogue and stimulating activities • Learners’ progress is assessed regularly (various methods) • Teachers have high expectations • Teachers create a settled and purposeful atmosphere for learning Main part of a lesson • • • • • • • Introduce a new topic, consolidate previous work or develop it Develop vocabulary, use correct notation and terms and learn new ones Use and apply concepts and skills Assess and review pupils’ progress This part of the lesson is more effective if you… Make clear to the class what they will learn Make links to previous lessons, or to work in other subjects • • • • • Give pupils deadlines for completing Maintain pace, making sure that this part of the lesson does not over-run and that there is enough time for the plenary When you are teaching the whole class, it helps if you: Demonstrate and explain using a board, flipchart, computer or OHP Highlight the meaning of any new vocabulary, notation or terms, and encourage pupils to repeat these and use them in their discussions and written work Involve pupils interactively through carefully planned and challenging questioning • • • • • • • • • • • activities, tasks or exercises methods and solutions Ask pupils to offer their to the whole class for discussion Identify and correct any misunderstandings or forgotten ideas, using mistakes as positive teaching points Ensure that pupils with particular needs are supported effectively When pupils are working on tasks in pairs, groups or individuals it helps if you… Keep the whole class busy working actively on problems, exercises or activities related to the theme of the lesson Encourage discussion and cooperation between pupils Where you want to differentiate, manage this by providing work at no more than three or four levels of difficulty across the class Target a small number of pairs, groups or individuals for particular questioning and support, rather than monitoring them all Make sure that pupils working independently know where to find resources, what to do before asking for help and what to do if they finish early Brief any supporting adults about their role, making sure that they have plenty to do with the pupils they are assisting Fishy Problem • A fish has a head that is 9cm long. • Its tail is the same length as its head plus half its body. • Its body is the same length as its head and tail together. • How long is the fish? A Fishy Solution Head = 9cm Tail = Head + ½ Body, so ½ Body = Tail – 9 Body = Head + Tail, so Body = Tail + 9 ½ Body = Tail – 9, so Body = 2 Tails – 18 So Tail + 9 = 2 Tails – 18, so Tail = 27 And Body = 36, so fish is 72cm long Equal Sets • Take a set of digits, say 1 to 8. Can you split them into 2 sets with the same total? • What about other sets of digits, say 1 to 5? 1 to 7? 1 to 20? • How can you decide whether it can be done? Equal Sets Solution Need total of the set to be an even number. Each set totals to a triangle number. So it can be done for any set that totals to an even triangle number. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, … These are the 3rd, 4th, 7th, 8th, 11th, 12th, etc.. So it can be done if the number in the set is a multiple of 4, or 1 less than a multiple of 4. Quadrilaterals • Start with a circle marked with 8 points evenly spaced on the perimeter. • What different quadrilaterals can you draw by joining 4 points? • How many are there? How do you know you have them all? • Can you sort the quadrilaterals by some criteria? Quads solution There are 8 different shapes that can be made. Denoting each shape by the number of spaces between the points round the circle, they are: 1,1,1,5 - trapezium 1,1,2,4 – trapezium (different one) 1,1,3,3 - kite 1,2,1,4 – another trapezium 1,2,2,3 - irregular 1,2,3,2 – another trapezium 2,2,2,2 - square 1,3,3,1 - rectangle Houses • There are some houses arranged around a square green. • There is the same number of houses on each side of the square. • Number 9 is opposite number 47. • How many houses are there around the square? Houses solution • With 9 houses in each row, 9 is opposite 19 • With 10 in each row, 9 is opposite 22 • With 11 in each row, 9 is opposite 25 • • • • Continuing this pattern, With 18 in a row, 9 is opposite 46 With 19 in a row, 9 is opposite 49 So 9 can never be opposite 47!! (Sorry!) Thank you! Please try these activities with your children and your teaching colleagues! Any feedback is always welcome! [email protected] Coordinate shapes • Start with 2 points on a rectangular grid, marked by coordinates, say (2,1) and (4,3). • Can you find 2 more points to make a square (rhombus, trapezium, etc) ? • Can you do it in different ways? How many can you find? Halftimes • If a sporting match (football, hockey) has a final score of 3-1, what are the possible half time scores? • If the fulltime score is a-b, what is the connection between a and b and the number of halftime scores possible? Fractions • Can you find a fraction between ½ and ¾? • Can you find a fraction between any two fractions? • Can you devise a rule that will always do this? • How can you show why it works (not algebraic proof!) 9s to make 1000 • Use a dice to generate 9 digits, each in the range 1-6. • Arrange them into three 3-digit numbers. • Add them. • Largest/Smallest/Closest to 1000 wins! Magic square patterns • Draw a 3x3 magic square, where each row, column and diagonal adds to 15 using 1-9. • Find pairs of numbers in the square that have the same totals. Record this on a blank square colouring the positions of one pair in one colour and the other pair in another. • How many different coloured relationships can you find? 5 presents • • • • • • • There are 5 presents, labeled A, B, C, D, E. A and B together cost £6 B and C cost £10 C and D cost £7 D and E cost £9. All 5 presents together cost £21. How much is each present? Digit Sums • Without 0, write down as many 2-digit numbers as you can where the digits add to 6. • Now do the same for 3-digit numbers. • How many 4-digit numbers do you think there might be? Try it. • Now 5-digits, and finally 6-digits. 15 cards • From a set of cards numbered 1 to 15, put down 7 cards in a row, face down. • Cards 1&2 add to 15, 2&3 add to 20, 3&4 add to 23, 4&5 add to 16, 5&6 add to 18 and cards 6&7 add to 21. • From this information can you work out what numbers are on the cards? • How many solutions are there? Numbers of triangles • Take an integer number for the perimeter of a triangle, say 12. • What integer sides are possible? • Find all permutations. • How many possible permutations are there? • Is it always a triangle number? IRATs • Start with an isosceles right-angled triangle (IRAT). Fold along the line of symmetry. • Open and cut along the fold line, what do you get? (Predict first!) • Now start with a new IRAT, fold along the line of symmetry and then again. • Open just the last fold and cut along the fold line. What do you get? (Predict first!) IRATs • Now do 3 folds along lines of symmetry and cut along the 2nd fold line… what do you get? • What about 4 folds and cut along the 3rd fold line? • How does the pattern continue?