Day 3 Problem solvingwsassets.s3.amazonaws.com/ws/nso/pdf/77f1a0c50d... · 8.1 Two approaches to problem-solving lessons 31 Reduced copies of slides Session 5 33 Session 6 37 Session

Day 3 Problem solving

Contents

Teacher’s evaluation form: Days 2 and 3 5

Handouts for Session 5 5.1 Sixes are banned 75.2 Strands in AT1 95.3 Strands in AT1: associated activities 105.4 Key Stage 2: Using and applying number 115.5 Perimeter dots 125.6 Objectives for AT1 13

Handouts for Session 66.1 Square dissection 156.2 Badminton game 166.3 Key Stage 2: Using and applying handling data 17

Handouts for Session 77.1 Recording sheet 197.2 Finding all possibilities 217.3 Logic puzzles 227.4 Finding rules and describing patterns 237.5 Diagram problems and visual puzzles 247.6 Attainment in using and applying mathematics 257.7 Three children’s work 27

Handout for Session 88.1 Two approaches to problem-solving lessons 31

Reduced copies of slidesSession 5 33Session 6 37Session 7 40Session 8 43

3 | Mathematics 3 plus 2 day course | Participant’s pack 3 © Crown copyright 2003


Primary National StrategyMathematics 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 divisionPlease 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?

Further comment (optional)

Please turn over.


Grade: please ring1 = Very helpful, 4 = Unhelpful

Day 2, the school-based tasks 1 2 3 4

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.

Further comment (optional)

School ……………………..................................................... Name .....................……...........................

Please return this form to your tutor before leaving.


Session Grade: please ring1 = 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 1 2 3 4for solving them

8 Teaching problem solving 1 2 3 4

Overall grade for the day 1 2 3 4

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 youwould do.


Handout 5.1


Strands in AT1


Handout 5.2C

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Strands in AT1: associated activities


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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 numericalskills and knowledge when solving problems in other parts of the mathematicscurriculum

b break down a more complex problem or calculation into simpler steps beforeattempting 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 anydifficulties

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 mathematicallanguage

Reasoning

j understand and investigate general statements (for example, ‘there are fourprime numbers less than 10’, ‘wrist size is half neck size’)

k search for pattern in their results; develop logical thinking and explain theirreasoning

MathematicsThe National Curriculum for England


Handout 5.4


Perimeter dots

Draw some polygons that have only one dot inside them. The vertices of thepolygons must be on the dots. Investigate the relationship between the number ofdots 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 onits perimeter and the number of dots inside it.

Handout 5.5


Objectives for AT1

These objectives are drawn from the programmes of study for Key Stage 2 for usingand applying number and using and applying shape, space and measures.

Consider the ‘Perimeter dots’ activity. To what extent was each of these objectivesmet? Give each objective a rating on a five-point scale, where 5 represents ‘fulfilledobjective well’ and 1 represents ‘did not fulfil objective’.

Handout 5.6

Using and applying number Rating

a Make connections in mathematics and appreciate the need to usenumerical skills and knowledge when solving problems in otherparts of the mathematics curriculum

g Use notation, diagrams and symbols correctly within a givenproblem

j Understand and investigate general statements

k Search for pattern in their results; develop logical thinking andexplain 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 geometricalproblems

c Approach spatial problems flexibly, including trying alternativeapproaches to overcome difficulties

d Use checking procedures to confirm that their results ofgeometrical problems are reasonable

e Organise work and record or represent it in a variety of ways whenpresenting 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



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



Which numbers of square pieces are impossible?

Explain why.

Handout 6.1


Badminton game

Janet, Sangita, Anne and Margaret like to play badmintontogether but cannot all be free to play on the same day. Janetis unable to play on Tuesdays, Wednesdays and Saturdays.Sangita is free to play on Mondays, Wednesdays andThursdays. Anne has to stay at home on Mondays andThursdays. Margaret can play on Mondays, Tuesdays andFridays. 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, publishedby Nash Pollock Publishing (ISBN 1 898255 06 7), and is reproduced by permission of the author.

Handout 6.2


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 thecurriculum, in particular science

b approach problems flexibly, including trying alternative approaches to overcomeany 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 theproblem

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

MathematicsThe National Curriculum for England

Handout 6.3



Recording sheet

What do your colleagues do to solve the problem?

Handout 7.1

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:



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 wonfour 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?

Handout 7.2


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 isnot 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?

Handout 7.3


Finding rules and describingpatterns

Square areas

The midpoints of the sides of a square are joined to make asmaller 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 startslike 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.

Handout 7.4


Diagram problems and visualpuzzles

Dice

Each of these shapes can be folded to make a cube.

For each shape, number the squares so that opposite faces ofthe 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 insuch a way that each corner of each square lies on the dots?

Handout 7.5


Attainment in using and applyingmathematics

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 andprecisely;

• present and interpret methods, solutions and conclusions in the context of aproblem.

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.

Handout 7.6



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 largersquares. Before starting them on the problem, the teacher reminded the class thatthey 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?

Handout 7.7


Nathan’s solutions to the square puzzle

3.5cm

7cm5cm

12.25cm2

10cm


Rachel: Number strings


The pupils had to arrange the digits 3, 1, 8 and 7 as four-digit numbers. Theteacher discussed with them the need to use a systematic method. They were thenencouraged to look at patterns of answers when adding and subtracting pairs oftwo-digit numbers made from the four digits.

Objectives


• use known number facts and place value to add or subtract mentally, includingany 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 andrelationships, generalise and predict.

Rachel’s investigation into sums and differences of two-digitnumbers


Sara: Consecutive sums and products


The pupils investigated the sums and products of pairs of consecutive numbers.

Objectives


• explain methods and reasoning about numbers orally and in writing;

• solve mathematical problems or puzzles, recognise and explain patterns andrelationships, generalise and predict.

Sara’s reasoning on consecutive sums and products


Two approaches to problem-solvinglessons

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 itto 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 thatpupils used.

• Stress the stages and steps used, and how these might be applied to similarproblems.

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 similarapproaches.

• Give pupils at different levels of attainment harder or simpler but relatedproblems, as appropriate.

• Draw together solutions in a plenary, working from the simpler tasks to themore challenging.

• Highlight the strategies used in the solutions, stressing the steps and stages,and how these might be applied to similar problems.

Discussion pointsFor 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?

Handout 8.1
















Documents

Day 3 Problem solvingwsassets.s3.amazonaws.com/ws/nso/pdf/77f1a0c50d... · 8.1 Two approaches to problem-solving lessons 31 Reduced copies of slides Session 5 33 Session 6 37 Session