Day 3 Problem solving
Day 3 Problem solving Contents Teacher’s evaluation form: Days 2 and 3
5
Handouts for Session 5 5.1 Sixes are banned 5.2 Strands in AT1 5.3 Strands in AT1: associated activities 5.4 Key Stage 2: Using and applying number 5.5 Perimeter dots 5.6 Objectives for AT1
7 9 10 11 12 13
Handouts for Session 6 6.1 Square dissection 6.2 Badminton game 6.3 Key Stage 2: Using and applying handling data
15 16 17
Handouts for Session 7 7.1 Recording sheet 7.2 Finding all possibilities 7.3 Logic puzzles 7.4 Finding rules and describing patterns 7.5 Diagram problems and visual puzzles 7.6 Attainment in using and applying mathematics 7.7 Three children’s work
19 21 22 23 24 25 27
Handout for Session 8 8.1 Two approaches to problem-solving lessons
31
Reduced copies of slides Session 5 Session 6 Session 7 Session 8
33 37 40 43
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Primary National Strategy Mathematics 3 plus 2 day course
Teacher’s evaluation form: Days 2 and 3 For completion by teachers by the end of Day 3
Day 2 Teaching division Please evaluate the usefulness of the school-based tasks for Day 2. What were the most useful aspects of Day 2?
What changes, if any, would you suggest for these tasks?
Grade: please ring 1 = Very helpful, 4 = Unhelpful Day 2, the school-based tasks
1
2
3
4
Further comment (optional)
Please turn over.
5
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Day 3
Problem solving
Please give your evaluation of Day 3, today’s sessions. What were the most successful aspects of today’s sessions?
What changes would you suggest if today’s sessions were repeated?
Please grade each session. Session
Grade: please ring 1 = Very helpful, 4 = Unhelpful
5
Using and applying mathematics
1
2
3
4
6
Working systematically
1
2
3
4
7
Types of problems and strategies for solving them
1
2
3
4
8
Teaching problem solving
1
2
3
4
1
2
3
4
Overall grade for the day Further comment (optional)
School ……………………..................................................... Name .....................……........................... Please return this form to your tutor before leaving.
6
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Handout 5.1
Sixes are banned The 6 key on your calculator is broken. Find answers to the calculations below. Work out how to do each one before trying it on your calculator. Record the calculation that you do. 1
32 + 16
2
126 58
3
48 6
4
146 ÷ 7
5
62 16
6
263 76
7
263 ÷ 62
8
36 0.6
Make up some more calculations like this, and record the calculation that you would do.
7
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8
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Handout 5.2
Problem solving
Communicating
Reasoning
Strands in AT1
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estimating
checking
scanning all possibilities
eliminating repetitions
ensuring solutions are reasonable in the context of the problem
•
•
•
•
•
working backwards
–
working systematically
–
controlling variables
trying particular cases
–
–
breaking a problem into parts
deciding on suitable problemsolving strategies, e.g.
•
–
making connections between aspects of mathematics
•
ordering, sorting or classifying information or data
•
choosing and using appropriate mathematics and resources
collecting or generating information or data
•
•
identifying information or data needed
•
Problem solving
•
–
writing presenting methods, explanations, arguments and solutions
• •
in graphs and charts
in lists and tables
– –
in diagrams
–
•
•
•
•
•
•
•
•
recording information:
•
using numbers and/or symbols
•
interpreting information
•
reading information
• •
•
talking/listening/discussing
disproving, e.g. finding counterexamples
proving
deducing
reasoning logically
justifying
explaining
testing
generalising
predicting, e.g. from patterns
recognising patterns
hypothesising
conjecturing
Reasoning
•
Communicating
Handout 5.3
Strands in AT1: associated activities
© Crown copyright 2003
Handout 5.4
Key Stage 2: Using and applying number Pupils should be taught to:
Problem solving a
make connections in mathematics and appreciate the need to use numerical skills and knowledge when solving problems in other parts of the mathematics curriculum
b
break down a more complex problem or calculation into simpler steps before attempting a solution; identify the information needed to carry out the tasks
c
select and use appropriate mathematical equipment, including ICT
d
find different ways of approaching a problem in order to overcome any difficulties
e
make mental estimates of the answers to calculations; check results
Communicating f
organise work and refine ways of recording
g
use notation, diagrams and symbols correctly within a given problem
h
present and interpret solutions in the context of the problem
i
communicate mathematically, including the use of precise mathematical language
Reasoning j
understand and investigate general statements (for example, ‘there are four prime numbers less than 10’, ‘wrist size is half neck size’)
k
search for pattern in their results; develop logical thinking and explain their reasoning
Mathematics The National Curriculum for England
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Handout 5.5
Perimeter dots Draw some polygons that have only one dot inside them. The vertices of the polygons must be on the dots. Investigate the relationship between the number of dots on the perimeter of each polygon and its area. Investigate the relationship for polygons with two dots inside them in the same way. If you have time, find a relationship between the area of a polygon with 12 dots on its perimeter and the number of dots inside it.
12
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Handout 5.6
Objectives for AT1 These objectives are drawn from the programmes of study for Key Stage 2 for using and applying number and using and applying shape, space and measures. Consider the ‘Perimeter dots’ activity. To what extent was each of these objectives met? Give each objective a rating on a five-point scale, where 5 represents ‘fulfilled objective well’ and 1 represents ‘did not fulfil objective’.
Using and applying number
Rating
a
Make connections in mathematics and appreciate the need to use numerical skills and knowledge when solving problems in other parts of the mathematics curriculum
g
Use notation, diagrams and symbols correctly within a given problem
j
Understand and investigate general statements
k
Search for pattern in their results; develop logical thinking and explain their reasoning.
Using and applying shape, space and measures a
Recognise the need for standard units of measurement
b
Select and use appropriate calculation skills to solve geometrical problems
c
Approach spatial problems flexibly, including trying alternative approaches to overcome difficulties
d
Use checking procedures to confirm that their results of geometrical problems are reasonable
e
Organise work and record or represent it in a variety of ways when presenting solutions to geometrical problems
f
Use geometrical notation and symbols correctly
g
Present and interpret solutions (in the context of the problem)
h
Use mathematical reasoning to explain features of shape and space
13
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14
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Handout 6.1
Square dissection The first square has been cut into 18 square pieces. Explore ways of cutting the other squares into square pieces.
18 square pieces
6 square pieces
10 square pieces
12 square pieces
21 square pieces
23 square pieces
Which numbers of square pieces are impossible? Explain why.
15
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Handout 6.2
Badminton game Janet, Sangita, Anne and Margaret like to play badminton together but cannot all be free to play on the same day. Janet is unable to play on Tuesdays, Wednesdays and Saturdays. Sangita is free to play on Mondays, Wednesdays and Thursdays. Anne has to stay at home on Mondays and Thursdays. Margaret can play on Mondays, Tuesdays and Fridays. None of them plays on Sunday. Can each pair find a day on which to play? Are there any days on which no games can be played? Are there any days when more than one game can be played? What if they can only get one court on any one day? How many games can they fit into a week? This problem is from the book Thinking things through by Leone Burton, published by Nash Pollock Publishing (ISBN 1 898255 06 7), and is reproduced by permission of the author.
16
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Participant’s pack 3
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Handout 6.3
Key Stage 2: Using and applying handling data Pupils should be taught to:
Problem solving a
select and use handling data skills when solving problems in other areas of the curriculum, in particular science
b
approach problems flexibly, including trying alternative approaches to overcome any difficulties
c
identify the data necessary to solve a given problem
d
select and use appropriate calculation skills to solve problems involving data
e
check results and ensure that solutions are reasonable in the context of the problem
Communicating f
decide how best to organise and present findings
g
use the precise mathematical language and vocabulary for handling data
Reasoning h
explain and justify their methods and reasoning
Mathematics The National Curriculum for England
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Handout 7.1
Recording sheet What do your colleagues do to solve the problem? Relate the problem to a similar one solved before Identify information needed to solve the problem Represent the problem in a different way (e.g. using a diagram) Decide on a system for listing possibilities or to organise recording Try particular cases or examples Check for repeats of possible solutions or answers Recognise when all possibilities have been found Look for relationships or patterns in information Fix one variable and vary the others Identify properties the answer will have Predict the next few terms in a sequence Test a term in a sequence to see if a possible rule works Describe a rule, pattern or relationship in own words Check that the answer meets all the criteria Check the solution by trying other possibilities Other:
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Handout 7.2
Finding all possibilities On the farm Jake keeps goats and ducks. He has 20 of them altogether. His animals have 54 feet between them. How many goats does Jake have?
Rounders A school’s rounders team has played five matches, and won four of them. The team’s highest score in a match was 5. Their lowest score was 2. Their median score was 4. What could the team’s five scores be?
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Handout 7.3
Logic puzzles What nationality? Amy, Bob and Cathy are three friends. One of them is English, one is Scottish and one is Welsh. They asked their teacher to guess their nationalities. The teacher said: ‘Amy is English. Bob is not English. Cathy is not Welsh.’ Only one of the teacher’s statements is correct. What nationalities are Amy, Bob and Cathy?
Ice creams Ross, Sam and Tim are brothers. Their corner shop sells three kinds of ice cream – strawberry, vanilla and banana. Each brother likes only two of the ice creams. Each kind of ice cream is liked by only two of the brothers. Sam said: ‘Ross likes strawberry and I don’t like banana.’ Which ice creams does Tim like?
22
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Handout 7.4
Finding rules and describing patterns Square areas The midpoints of the sides of a square are joined to make a smaller square in a continuing pattern. The area of the smallest white square is 3 square centimetres. What is the area of the largest white square?
Counters in a line Imagine a pattern of counters in a long line. The pattern starts like this: two grey, four white, two grey, four white, …
What colour would the 65th counter be? What position in the line would the 17th white counter be? Explain how you know.
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Handout 7.5
Diagram problems and visual puzzles Dice Each of these shapes can be folded to make a cube. For each shape, number the squares so that opposite faces of the cube add up to 7.
Dotty squares The diagram shows a 4 by 4 array of dots.
How many different squares can you draw on the array in such a way that each corner of each square lies on the dots?
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Handout 7.6
Attainment in using and applying mathematics In problem solving, pupils are increasingly able to: •
use a range of problem-solving strategies;
•
try different approaches to a problem;
•
apply mathematics in a new context;
•
check their results.
In communicating, pupils are increasingly able to: •
interpret information;
•
record information systematically;
•
use mathematical language, symbols, notation and diagrams correctly and precisely;
•
present and interpret methods, solutions and conclusions in the context of a problem.
In reasoning, pupils are increasingly able to: •
give clear explanations of their methods and reasoning;
•
investigate and make general statements;
•
recognise patterns in their results;
•
make use of a wider range of evidence to justify results through logical, reasoned argument;
•
draw their own conclusions.
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Handout 7.7
Three children’s work Nathan: Square puzzle Activity description The teacher asked pupils to find the area of a smaller square within a set of larger squares. Before starting them on the problem, the teacher reminded the class that they could use any method and materials.
Objectives The relevant Framework objectives for Year 6 are: •
explain methods and reasoning (key objective);
•
identify and use appropriate operations (including combinations of operations) to solve word problems involving numbers and quantities (key objective);
•
calculate areas of rectangles.
The problem given to the class
Each side of the large square is 10 cm. What is the area of the dark square?
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Nathan’s solutions to the square puzzle
10cm
m 7c
5c
m 5cm
3.
12.25cm2
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Rachel: Number strings Activity description The pupils had to arrange the digits 3, 1, 8 and 7 as four-digit numbers. The teacher discussed with them the need to use a systematic method. They were then encouraged to look at patterns of answers when adding and subtracting pairs of two-digit numbers made from the four digits.
Objectives The relevant Framework objectives for Year 6 are: •
use known number facts and place value to add or subtract mentally, including any pair of two-digit whole numbers (key objective);
•
explain methods and reasoning about numbers orally and in writing;
•
solve mathematical problems or puzzles, recognise and explain patterns and relationships, generalise and predict.
Rachel’s investigation into sums and differences of two-digit numbers
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Sara: Consecutive sums and products Activity description The pupils investigated the sums and products of pairs of consecutive numbers.
Objectives The relevant Framework objectives for Year 6 are: •
explain methods and reasoning about numbers orally and in writing;
•
solve mathematical problems or puzzles, recognise and explain patterns and relationships, generalise and predict.
Sara’s reasoning on consecutive sums and products
30
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Handout 8.1
Two approaches to problem-solving lessons Approach A •
Start work on a problem during an initial whole-class discussion.
•
Ask pupils to continue the activity, often in pairs or small groups, developing it to a level appropriate to their attainment.
•
Collect pupils’ responses in a plenary, and work through the solution, encouraging individual or pairs of pupils to contribute.
•
Draw attention to particular features of the solution and the strategies that pupils used.
•
Stress the stages and steps used, and how these might be applied to similar problems.
Approach B •
Work through a problem during an initial whole-class discussion, demonstrating ways of being systematic in approach and recording.
•
Follow this by providing related problems that lend themselves to similar approaches.
•
Give pupils at different levels of attainment harder or simpler but related problems, as appropriate.
•
Draw together solutions in a plenary, working from the simpler tasks to the more challenging.
•
Highlight the strategies used in the solutions, stressing the steps and stages, and how these might be applied to similar problems.
Discussion points For each approach: •
What scope does the approach offer for pupils to make their own decisions?
•
When and why should teachers intervene in what pupils are doing?
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© Crown copyright 2003
5
Handouts for Session 5 5.1 Sixes are banned 5.2 Strands in AT1 5.3 Strands in AT1: associated activities 5.4 Key Stage 2: Using and applying number 5.5 Perimeter dots 5.6 Objectives for AT1
7 9 10 11 12 13
Handouts for Session 6 6.1 Square dissection 6.2 Badminton game 6.3 Key Stage 2: Using and applying handling data
15 16 17
Handouts for Session 7 7.1 Recording sheet 7.2 Finding all possibilities 7.3 Logic puzzles 7.4 Finding rules and describing patterns 7.5 Diagram problems and visual puzzles 7.6 Attainment in using and applying mathematics 7.7 Three children’s work
19 21 22 23 24 25 27
Handout for Session 8 8.1 Two approaches to problem-solving lessons
31
Reduced copies of slides Session 5 Session 6 Session 7 Session 8
33 37 40 43
3
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Participant’s pack 3
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4
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Mathematics 3 plus 2 day course
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Participant’s pack 3
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Primary National Strategy Mathematics 3 plus 2 day course
Teacher’s evaluation form: Days 2 and 3 For completion by teachers by the end of Day 3
Day 2 Teaching division Please evaluate the usefulness of the school-based tasks for Day 2. What were the most useful aspects of Day 2?
What changes, if any, would you suggest for these tasks?
Grade: please ring 1 = Very helpful, 4 = Unhelpful Day 2, the school-based tasks
1
2
3
4
Further comment (optional)
Please turn over.
5
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Mathematics 3 plus 2 day course
|
Participant’s pack 3
© Crown copyright 2003
Day 3
Problem solving
Please give your evaluation of Day 3, today’s sessions. What were the most successful aspects of today’s sessions?
What changes would you suggest if today’s sessions were repeated?
Please grade each session. Session
Grade: please ring 1 = Very helpful, 4 = Unhelpful
5
Using and applying mathematics
1
2
3
4
6
Working systematically
1
2
3
4
7
Types of problems and strategies for solving them
1
2
3
4
8
Teaching problem solving
1
2
3
4
1
2
3
4
Overall grade for the day Further comment (optional)
School ……………………..................................................... Name .....................……........................... Please return this form to your tutor before leaving.
6
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Mathematics 3 plus 2 day course
|
Participant’s pack 3
© Crown copyright 2003
Handout 5.1
Sixes are banned The 6 key on your calculator is broken. Find answers to the calculations below. Work out how to do each one before trying it on your calculator. Record the calculation that you do. 1
32 + 16
2
126 58
3
48 6
4
146 ÷ 7
5
62 16
6
263 76
7
263 ÷ 62
8
36 0.6
Make up some more calculations like this, and record the calculation that you would do.
7
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Mathematics 3 plus 2 day course
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Participant’s pack 3
© Crown copyright 2003
8
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Mathematics 3 plus 2 day course
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Participant’s pack 3
© Crown copyright 2003
Handout 5.2
Problem solving
Communicating
Reasoning
Strands in AT1
9
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Participant’s pack 3
© Crown copyright 2003
10
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Participant’s pack 3
estimating
checking
scanning all possibilities
eliminating repetitions
ensuring solutions are reasonable in the context of the problem
•
•
•
•
•
working backwards
–
working systematically
–
controlling variables
trying particular cases
–
–
breaking a problem into parts
deciding on suitable problemsolving strategies, e.g.
•
–
making connections between aspects of mathematics
•
ordering, sorting or classifying information or data
•
choosing and using appropriate mathematics and resources
collecting or generating information or data
•
•
identifying information or data needed
•
Problem solving
•
–
writing presenting methods, explanations, arguments and solutions
• •
in graphs and charts
in lists and tables
– –
in diagrams
–
•
•
•
•
•
•
•
•
recording information:
•
using numbers and/or symbols
•
interpreting information
•
reading information
• •
•
talking/listening/discussing
disproving, e.g. finding counterexamples
proving
deducing
reasoning logically
justifying
explaining
testing
generalising
predicting, e.g. from patterns
recognising patterns
hypothesising
conjecturing
Reasoning
•
Communicating
Handout 5.3
Strands in AT1: associated activities
© Crown copyright 2003
Handout 5.4
Key Stage 2: Using and applying number Pupils should be taught to:
Problem solving a
make connections in mathematics and appreciate the need to use numerical skills and knowledge when solving problems in other parts of the mathematics curriculum
b
break down a more complex problem or calculation into simpler steps before attempting a solution; identify the information needed to carry out the tasks
c
select and use appropriate mathematical equipment, including ICT
d
find different ways of approaching a problem in order to overcome any difficulties
e
make mental estimates of the answers to calculations; check results
Communicating f
organise work and refine ways of recording
g
use notation, diagrams and symbols correctly within a given problem
h
present and interpret solutions in the context of the problem
i
communicate mathematically, including the use of precise mathematical language
Reasoning j
understand and investigate general statements (for example, ‘there are four prime numbers less than 10’, ‘wrist size is half neck size’)
k
search for pattern in their results; develop logical thinking and explain their reasoning
Mathematics The National Curriculum for England
11
|
Mathematics 3 plus 2 day course
|
Participant’s pack 3
© Crown copyright 2003
Handout 5.5
Perimeter dots Draw some polygons that have only one dot inside them. The vertices of the polygons must be on the dots. Investigate the relationship between the number of dots on the perimeter of each polygon and its area. Investigate the relationship for polygons with two dots inside them in the same way. If you have time, find a relationship between the area of a polygon with 12 dots on its perimeter and the number of dots inside it.
12
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Participant’s pack 3
© Crown copyright 2003
Handout 5.6
Objectives for AT1 These objectives are drawn from the programmes of study for Key Stage 2 for using and applying number and using and applying shape, space and measures. Consider the ‘Perimeter dots’ activity. To what extent was each of these objectives met? Give each objective a rating on a five-point scale, where 5 represents ‘fulfilled objective well’ and 1 represents ‘did not fulfil objective’.
Using and applying number
Rating
a
Make connections in mathematics and appreciate the need to use numerical skills and knowledge when solving problems in other parts of the mathematics curriculum
g
Use notation, diagrams and symbols correctly within a given problem
j
Understand and investigate general statements
k
Search for pattern in their results; develop logical thinking and explain their reasoning.
Using and applying shape, space and measures a
Recognise the need for standard units of measurement
b
Select and use appropriate calculation skills to solve geometrical problems
c
Approach spatial problems flexibly, including trying alternative approaches to overcome difficulties
d
Use checking procedures to confirm that their results of geometrical problems are reasonable
e
Organise work and record or represent it in a variety of ways when presenting solutions to geometrical problems
f
Use geometrical notation and symbols correctly
g
Present and interpret solutions (in the context of the problem)
h
Use mathematical reasoning to explain features of shape and space
13
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Mathematics 3 plus 2 day course
|
Participant’s pack 3
© Crown copyright 2003
14
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Mathematics 3 plus 2 day course
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Participant’s pack 3
© Crown copyright 2003
Handout 6.1
Square dissection The first square has been cut into 18 square pieces. Explore ways of cutting the other squares into square pieces.
18 square pieces
6 square pieces
10 square pieces
12 square pieces
21 square pieces
23 square pieces
Which numbers of square pieces are impossible? Explain why.
15
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Mathematics 3 plus 2 day course
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Participant’s pack 3
© Crown copyright 2003
Handout 6.2
Badminton game Janet, Sangita, Anne and Margaret like to play badminton together but cannot all be free to play on the same day. Janet is unable to play on Tuesdays, Wednesdays and Saturdays. Sangita is free to play on Mondays, Wednesdays and Thursdays. Anne has to stay at home on Mondays and Thursdays. Margaret can play on Mondays, Tuesdays and Fridays. None of them plays on Sunday. Can each pair find a day on which to play? Are there any days on which no games can be played? Are there any days when more than one game can be played? What if they can only get one court on any one day? How many games can they fit into a week? This problem is from the book Thinking things through by Leone Burton, published by Nash Pollock Publishing (ISBN 1 898255 06 7), and is reproduced by permission of the author.
16
|
Mathematics 3 plus 2 day course
|
Participant’s pack 3
© Crown copyright 2003
Handout 6.3
Key Stage 2: Using and applying handling data Pupils should be taught to:
Problem solving a
select and use handling data skills when solving problems in other areas of the curriculum, in particular science
b
approach problems flexibly, including trying alternative approaches to overcome any difficulties
c
identify the data necessary to solve a given problem
d
select and use appropriate calculation skills to solve problems involving data
e
check results and ensure that solutions are reasonable in the context of the problem
Communicating f
decide how best to organise and present findings
g
use the precise mathematical language and vocabulary for handling data
Reasoning h
explain and justify their methods and reasoning
Mathematics The National Curriculum for England
17
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Mathematics 3 plus 2 day course
|
Participant’s pack 3
© Crown copyright 2003
18
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Mathematics 3 plus 2 day course
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Participant’s pack 3
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Handout 7.1
Recording sheet What do your colleagues do to solve the problem? Relate the problem to a similar one solved before Identify information needed to solve the problem Represent the problem in a different way (e.g. using a diagram) Decide on a system for listing possibilities or to organise recording Try particular cases or examples Check for repeats of possible solutions or answers Recognise when all possibilities have been found Look for relationships or patterns in information Fix one variable and vary the others Identify properties the answer will have Predict the next few terms in a sequence Test a term in a sequence to see if a possible rule works Describe a rule, pattern or relationship in own words Check that the answer meets all the criteria Check the solution by trying other possibilities Other:
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Handout 7.2
Finding all possibilities On the farm Jake keeps goats and ducks. He has 20 of them altogether. His animals have 54 feet between them. How many goats does Jake have?
Rounders A school’s rounders team has played five matches, and won four of them. The team’s highest score in a match was 5. Their lowest score was 2. Their median score was 4. What could the team’s five scores be?
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Handout 7.3
Logic puzzles What nationality? Amy, Bob and Cathy are three friends. One of them is English, one is Scottish and one is Welsh. They asked their teacher to guess their nationalities. The teacher said: ‘Amy is English. Bob is not English. Cathy is not Welsh.’ Only one of the teacher’s statements is correct. What nationalities are Amy, Bob and Cathy?
Ice creams Ross, Sam and Tim are brothers. Their corner shop sells three kinds of ice cream – strawberry, vanilla and banana. Each brother likes only two of the ice creams. Each kind of ice cream is liked by only two of the brothers. Sam said: ‘Ross likes strawberry and I don’t like banana.’ Which ice creams does Tim like?
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Handout 7.4
Finding rules and describing patterns Square areas The midpoints of the sides of a square are joined to make a smaller square in a continuing pattern. The area of the smallest white square is 3 square centimetres. What is the area of the largest white square?
Counters in a line Imagine a pattern of counters in a long line. The pattern starts like this: two grey, four white, two grey, four white, …
What colour would the 65th counter be? What position in the line would the 17th white counter be? Explain how you know.
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Handout 7.5
Diagram problems and visual puzzles Dice Each of these shapes can be folded to make a cube. For each shape, number the squares so that opposite faces of the cube add up to 7.
Dotty squares The diagram shows a 4 by 4 array of dots.
How many different squares can you draw on the array in such a way that each corner of each square lies on the dots?
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Handout 7.6
Attainment in using and applying mathematics In problem solving, pupils are increasingly able to: •
use a range of problem-solving strategies;
•
try different approaches to a problem;
•
apply mathematics in a new context;
•
check their results.
In communicating, pupils are increasingly able to: •
interpret information;
•
record information systematically;
•
use mathematical language, symbols, notation and diagrams correctly and precisely;
•
present and interpret methods, solutions and conclusions in the context of a problem.
In reasoning, pupils are increasingly able to: •
give clear explanations of their methods and reasoning;
•
investigate and make general statements;
•
recognise patterns in their results;
•
make use of a wider range of evidence to justify results through logical, reasoned argument;
•
draw their own conclusions.
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Handout 7.7
Three children’s work Nathan: Square puzzle Activity description The teacher asked pupils to find the area of a smaller square within a set of larger squares. Before starting them on the problem, the teacher reminded the class that they could use any method and materials.
Objectives The relevant Framework objectives for Year 6 are: •
explain methods and reasoning (key objective);
•
identify and use appropriate operations (including combinations of operations) to solve word problems involving numbers and quantities (key objective);
•
calculate areas of rectangles.
The problem given to the class
Each side of the large square is 10 cm. What is the area of the dark square?
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Nathan’s solutions to the square puzzle
10cm
m 7c
5c
m 5cm
3.
12.25cm2
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Rachel: Number strings Activity description The pupils had to arrange the digits 3, 1, 8 and 7 as four-digit numbers. The teacher discussed with them the need to use a systematic method. They were then encouraged to look at patterns of answers when adding and subtracting pairs of two-digit numbers made from the four digits.
Objectives The relevant Framework objectives for Year 6 are: •
use known number facts and place value to add or subtract mentally, including any pair of two-digit whole numbers (key objective);
•
explain methods and reasoning about numbers orally and in writing;
•
solve mathematical problems or puzzles, recognise and explain patterns and relationships, generalise and predict.
Rachel’s investigation into sums and differences of two-digit numbers
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Sara: Consecutive sums and products Activity description The pupils investigated the sums and products of pairs of consecutive numbers.
Objectives The relevant Framework objectives for Year 6 are: •
explain methods and reasoning about numbers orally and in writing;
•
solve mathematical problems or puzzles, recognise and explain patterns and relationships, generalise and predict.
Sara’s reasoning on consecutive sums and products
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Handout 8.1
Two approaches to problem-solving lessons Approach A •
Start work on a problem during an initial whole-class discussion.
•
Ask pupils to continue the activity, often in pairs or small groups, developing it to a level appropriate to their attainment.
•
Collect pupils’ responses in a plenary, and work through the solution, encouraging individual or pairs of pupils to contribute.
•
Draw attention to particular features of the solution and the strategies that pupils used.
•
Stress the stages and steps used, and how these might be applied to similar problems.
Approach B •
Work through a problem during an initial whole-class discussion, demonstrating ways of being systematic in approach and recording.
•
Follow this by providing related problems that lend themselves to similar approaches.
•
Give pupils at different levels of attainment harder or simpler but related problems, as appropriate.
•
Draw together solutions in a plenary, working from the simpler tasks to the more challenging.
•
Highlight the strategies used in the solutions, stressing the steps and stages, and how these might be applied to similar problems.
Discussion points For each approach: •
What scope does the approach offer for pupils to make their own decisions?
•
When and why should teachers intervene in what pupils are doing?
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