Workshop Notes

Embedding problem solving in
teaching and learning at
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
• 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
• Learners’ progress is assessed regularly (various methods)
• Teachers have high expectations
• Teachers create a settled and purposeful atmosphere for
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
• 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.
• 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
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
• 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
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?
• 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?
• Can you find a fraction between ½ and ¾?
• Can you find a fraction between any two
• Can you devise a rule that will always do this?
• How can you show why it works (not algebraic
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?
• 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!)
• 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
• How does the pattern continue?

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