Stretch and Challenge 3 A problem-solving, cross-curricular programme for children working above end-of-year expectations
Peter Clarke
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Contents Quick reference guide to Stretch and Challenge 3
4
Introduction
5
The features of Stretch and Challenge
7
A possible Stretch and Challenge teaching and learning sequence
10
Links to the Year 3 Mathematics National Curriculum Programme of Study and Attainment Targets
11
Cross-curricular links to the National Curriculum Programme of Study
19
Resources used in Stretch and Challenge 3
21
The Issues
23
Teacher’s notes
95
Resource sheets Record of completion
209
My notes
211
Pupil self assessment booklet
213
Other Resource sheets
215
3
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Quick reference guide to Stretch and Challenge 3 Domain(s)
Topic
Number:
Number
1
96
– Number and place value
Number
2
99
Number
3
103
Number
4
106
Number:
Addition
5
109
– Addition and subtraction
Addition
6
112
Subtraction
7
115
Subtraction
8
118
Number:
Multiplication
9
121
– Multiplication and division
Multiplication
10
124
Division
11
127
Division
12
130
Number:
Mixed operations
13
133
– Addition and subtraction
Mixed operations
14
136
Mixed operations
15
139
Mixed operations
16
142
Number:
Fractions
17
145
– Fractions
Fractions
18
148
Fractions
19
151
Fractions
20
154
Length
21
157
Mass
22
160
Capacity and volume
23
163
Time
24
166
Measurement
25
169
Measurement
26
173
Geometry:
2-D shapes
27
177
– Properties of shapes
3-D shapes
28
180
Symmetry
29
183
Position and direction
30
187
Movement and angle
31
190
Geometry
32
193
Geometry
33
196
Statistics
34
199
Statistics
35
202
Statistics
36
205
– Multiplication and division
Measurement
Statistics
Issue number
Teacher’s notes page number
4
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Introduction
Introduction The National Curriculum emphasises the importance of all children mastering the programme of study taught each year and discourages the acceleration of children into content from subsequent years. The National Curriculum states: ‘The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.’ 1 However, the National Curriculum also goes on to say that: ‘Within each key stage, schools [therefore] have the flexibility to introduce content earlier or later than set out in the programme of study. In addition, schools can introduce key stage content during an earlier key stage, if appropriate.‘ 2 Stretch and Challenge aims to provide support in meeting the needs of those children who are exceeding age-related expectations by providing a range of problem-solving and cross-curricular activities designed to enrich and deepen children’s mathematical knowledge, skills and understanding. The series provides opportunities for children to reason mathematically and to solve increasingly complex problems, doing so with fluency, as described in the aims of the National Curriculum: ‘The National Curriculum for mathematics aims to ensure that all pupils: • become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately • reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language • can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.’ 3 Stretch and Challenge has been designed to provide: • a flexible ‘dip-in’ resource that can easily be adapted to meet the needs of individual children, and different classroom and school organisational arrangements • enrichment activities that require children to use and apply their mathematical knowledge, skills and understanding to reason mathematically and to solve increasingly complex problems • mathematical activities linked to the entire primary curriculum, thereby ensuring a range of cross-curricular contexts • an easy-to-use bank of activities to save teachers time in thinking up new enrichment activities • an interesting, unique and consistent approach to presenting enrichment activities to children. The Stretch and Challenge series consists of six packs, also available digitally on Collins Connect, one for each year group from Year 1 to Year 6.
1 Mathematics programmes of study: key stages 1 and 2 National Curriculum in England, September 2013, page 3 2 Mathematics programmes of study: key stages 1 and 2 National Curriculum in England, September 2013, page 4 3 Mathematics programmes of study: key stages 1 and 2 National Curriculum in England, September 2013, page 3
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Introduction Printed resources Containing: • Pupil activity booklets (Issues)
Stretch an d Ch al
len ge
3
• Teacher’s notes • Resource sheets Enriching at Assessment mathematical the heart and understanding from the start
Online resources at connect.collins.co.uk
PETE R CLARK E SERIE S EDITO R
Containing editable: • Pupil activity booklets (Issues) • Teacher’s notes • Resource sheets It is envisaged that the activities in Stretch and Challenge will be used by either individuals or pairs of children. However, given the flexible nature of the resource, if appropriate, children can work in groups. The activities are intended to be used: • as additional work to be done once children have finished other set work • by those children who grasp concepts rapidly and need to be challenged through rich and sophisticated problems • as in-depth work that is to be undertaken over a prolonged period of time, such as during the course of several lessons, a week or a particular unit of work • as a resource for promoting mathematical reasoning and problem solving and developing independent thinking and learning • as a springboard for further investigations into mathematics based on the children’s suggestions.
6
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Introduction
The features of Stretch and Challenge Pupil activity booklets (Issues) • Each of the 36 Issues in Stretch and Challenge 3 consists of a four-page A5 pupil activity booklet (to be printed double sided onto one sheet of A4 paper). • The 36 Issues cover the different domains and attainment targets of the Mathematics National Curriculum Programme of Study (see pages 11–18). • The Issues have been designed to resemble a newspaper, with each of the Issues consisting of between five and eight different activities, all related to the same mathematical topic. • It is important to note that children are not expected to complete all the activities in an Issue nor work their way through an Issue from beginning to end. For many children not all of the activities offered in an Issue will be appropriate. When choosing which activities a child is to complete, teachers should ensure that the activities do not accelerate the child into mathematical content they may not be familiar with, or are unable to reason more deeply in order to develop a conceptual understanding. Rather, activities should be chosen on the basis that they engage the child in reasoning and the development of mathematical thinking, as well as enriching and deepening the child’s mathematical knowledge, skills and understanding. • The terms ‘Issue’ and ‘Volume’ have been used rather than ‘Unit’ and ‘Year group’ because they are in keeping with the newspaper theme. Types of activities • Each of the 36 Issues in Stretch and Challenge 3 are designed to deepen children’s mathematical knowledge, skills and understandings, and enhance their use and application of mathematics. There are four different types of ‘using and applying’ activities in the series: What’s the Problem?
The Puzzler
Looking for Patterns
Let’s Investigate
• Alongside developing children’s problem-solving skills, the series also provides activities with cross-curricular links to other subjects in the primary curriculum. The following shows the Stretch and Challenge features and its corresponding primary curriculum subject. Curriculum subject
Stretch and Challenge feature
English
The Language of Maths
Science
Focus on Science
Computing
Technology Today
Geography
Around the World
History Mathematics
In the Past Famous Mathematicians
Art and design / Music
The Arts Roundup
Design and Technology
Construct
Physical Education
Sports Update
• As well as the features mentioned above, other regular features in Stretch and Challenge include: Money Matters
At Home (home–school link activities).
• A chart showing the link between the Issues, the Stretch and Challenge features and cross-curricular links can be found on pages 19 and 20. • Inquisitive ant is a recurring feature of the series. In each Issue there is an ant holding a mathematical word or symbol. Children locate the ant and write about the meaning of the word or symbol.
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Issue 11 - Division
Issue 11 Division Prerequisites for learning
Resources
• Identify patterns and relationships involving numbers • Recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables • Recognise multiples of 2, 3, 4, 5, 8 and 10, and begin to recognise multiples of 6 and 7 • Write and calculate mathematical statements for division, including for two-digit numbers by one-digit numbers, using mental and progressing to formal written methods • Begin to understand that in a division situation a remainder (abbreviated as ‘r’) is the amount left over after an equal sharing or grouping has been completed
pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) 30 counters (optional) calculator (optional)
Teaching support Page 1 Looking for Patterns • If necessary work through the first set of numbers with the children, for example: 9 and 81 81 ÷ 9 = 9 18 – 9 = 9 • Can the children find other pairs of one-digit and two-digit numbers that have the same relationship? Looking for Patterns • Provide the children with 30 counters to assist them with working out the solution to the problem.
Page 2 The Arts Roundup • This activity involves the children calculating with numbers that they may not yet be familiar with. However the aim of this activity is to encourage children to use known multiplication and division facts for the 8 multiplication table in order to solve the problem. • To cover the total cost of production, each advert costs £120 000 (£960 000 ÷ 8). • You may wish to allow the children to use a calculator for this activity. The Puzzler • Ensure children understand the description: “The sum of the two digits is 16.” For example, the sum of the digits of the number 54 is 9, i.e. 5 + 4 = 9. • Children write a similar puzzle for a friend to solve. Looking for Patterns • You may wish to introduce the children to the concept of common multiples as a way of helping them work out the answer to this problem. (84 is the only common multiple of 3, 4 and 7 less than 100.)
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Issue 11 - Division Page 3 Let’s Investigate • Ask the children to investigate what happens if you start with a four-digit number. (No similar pattern occurs.) What’s the Problem? • If all the heads belonged to men, there would only be 120 legs. However, there are 40 more legs than this. Each horse has two more legs than a man, so there must be 40 ÷ 2 horses = 20 horses for the number of heads and legs to be correct.
Page 4 What’s the Problem? • One method of solving this problem is to find the lowest common multiple of 2, 3, 4, 5 and 6 and add one. • What is the smallest three-digit number that has a remainder of 1 when divided by 3, 5 and 8? (121) • An important aspect of this activity is the explanation children give for discovering the number. Therefore, when children have finished, arrange them into pairs or groups to compare and discuss their methods. Let’s Investigate • Ask children to comment on the strengths and limitations of each of their methods. Which one do they prefer to use? Why? • When the children have used a range of different methods to work out the answer to the calculation, arrange them into pairs. Children discuss and compare the different methods used. Sports Update • The first flag is at the Start line and the eleventh at the Finish, so the sixth flag is half-way along the course. • Suggest children draw a diagram to help them, for example:
4UBSU
LN
'JOJTI
LN
AfL • What patterns did you notice? How did this help you? • How did you work out the answer to this problem? Did you use a diagram to help you? How did that help? • Explain the different methods you used to work out the answer to the calculation. Which of these methods do you prefer? Why?
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Issue 11 - Division
Answers
Page 3 Let’s Investigate The final answer is always 11.
Page 1 Looking for Patterns 9 and 81 3 and 72 2 and 94 81 ÷ 9 = 9 72 ÷ 3 = 24 94 ÷ 2 = 47 18 – 9 = 9 27 – 3 = 24 49 – 2 = 47 The answers to both calculations are the same. Looking for Patterns Emily picked up 21 counters.
What’s the Problem? There are 20 horses. Page 4 What’s the Problem? 61 Explanations will vary.
Page 2
Let’s Investigate Calculating methods will vary.
The Arts Roundup Each advert costs £120 000.
Sports Update When Omar passes the sixth flag he has run 6 km.
The Puzzler The number in the box is 97. Looking for Patterns There are 84 apples in the crate.
Inquisitive ant
factor A whole number that will divide exactly into another whole number.
129
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Issue 12 - Division
Issue 12 Division Prerequisites for learning
Resources
• Identify patterns and relationships involving numbers • Recall and use multiplication and division facts for the 2, 3, 4, 5, 8 and 10 multiplication tables • Recognise multiples of 2, 3, 4, 5, 8 and 10, and begin to recognise multiples of 6 and 7 • Identify the doubles of two-digit numbers and use these to calculate doubles of multiples of 10 • Multiply and divide numbers to 1000 by 10 (wholenumber answers) • Write and calculate mathematical statements for division, including for two-digit numbers by one-digit numbers, using mental and progressing to formal written methods • Begin to understand that in a division situation a remainder (abbreviated as ‘r’) is the amount left over after an equal sharing or grouping has been completed
pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) counters and matchsticks, or similar (optional) set of 0–9 digit cards
Teaching support Page 1 What’s the Problem? • If all the heads belonged to people, there would be only 26 × 2 = 52 legs. However, there are 76 – 52 = 24 extra legs present. Each dog has two more legs than a person, so there are 24 = 12 dogs present. 2 • Use counters and matchsticks (or similar) to represent the heads and legs, and discuss the above explanation graphically. The Arts Roundup • Suggest children draw a table to find out the number of members of BADS, for example: Size of groups Number of members
2
3
4
5
6
50
✓
✗
✗
✓
✗
51
✗
✓
✗
✗
✗
52
✓
✗
✓
✗
✗
53
✗
✗
✗
✗
✗
54
✓
✓
✗
✗
✓
55
✗
✗
✗
✓
✗
56
✓
✗
✓
✗
✗
• What patterns do they identify? Can they use these patterns to work out the answer to the problem? Focus on Science • Ensure children know that one tonne is equivalent to 1000 kg.
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Issue 12 - Division Page 2 What’s the Problem? • Although this problem involves no actual mathematical calculation, it does require the children to problem solve, reason and think logically. • The second sock that she chooses may be a different colour to the first, but the third is certain to match either the first or the second. Let’s Investigate • Remind the children that a square is a type of rectangle and therefore should be included in this investigation. • Do not give the children counters. Can they still make all the multiplication calculations possible with products from 10 to 24? • What do the children notice about the numbers that make squares? Looking for Patterns • Suggest the children try each of the multiples of 8 in ascending size.
Page 3 Let’s Investigate • Ensure children realise that the numbers 2, 4, 6, 8, 10, 12, 14, 16, … are referred to as multiples of 2; and that the numbers 4, 8, 12, 16, 20, 24, 28, 32, … are referred to as multiples of 4. Looking for Patterns • The number must be even, so it must be an even multiple of 7, i.e. a multiple of 14. The smallest number that is divisible by both 14 and 6 is 42, so the number is a multiple of 42. 42 is not divisible by 8, nor is 84, nor 126; however, 168 is. Looking for Patterns • The example in the issue uses nine out of the ten cards to show different multiples of 6. Encourage the children to use as many of the digit cards as possible when making multiples of any number. Can they use all ten cards?
Page 4 At Home • Once the children have completed the investigation, ensure that there is an opportunity in class for pairs or groups of children to discuss their results. Let’s Investigate • Ask the children to write about the limitations of this method.
AfL • Explain to me how you worked out the answer to this problem. • Tell me some of your calculations. What do you notice about them? How else could you describe this array? What does this mean? • What patterns did you notice in the units digits? • Is there another way that you could arrange these cards to show me some other multiples of …? Is there a way you could use even more of the cards? • What did you find out about the way different food and other items are packaged? Why do you think this is? • What do you think of this method for dividing a number by 5? When it is not useful?
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Issue 12 - Division
Answers
Page 3
Page 1 What’s the Problem? There are 14 people and 12 dogs in the dog training class. The Arts Roundup There are 60 members of BADS.
Let’s Investigate The patterns of the 2 units digit for the multiples of 2 and 4 are: ②, 4, 6, 8, 10, 1②, 14, 16, 18, 20, 2②, 24, 26, 28, … 4, 8, 1②, 16, 20, 24, 28, 3②, 36, 40, 44, 48, 5②, 56, … 2 as the units digit appears in every fifth number in both sequences. Similar patterns occur for the other units digits. Patterns for the multiples of 3 and 6, and 4 and 8 will vary.
Focus on Science Peter’s supply of logs will last for 75 days.
Looking for Patterns 168
Page 2 What’s the Problem? The minimum number of socks that Kylie needs to take out of the drawer in order to be certain of getting a matching pair is three socks. Let’s Investigate 10: 1 × 10, 2 × 5 11: 1 × 11 13: 1 × 13 14: 1 × 14, 2 × 7 15: 1 × 15, 3 × 5 16: 1 × 16, 2 × 8, 4 × 4 17: 1 × 17 18: 1 × 18, 2 × 9, 3 × 6 19: 1 × 19 20: 1 × 20, 2 × 10, 4 × 5 21: 1 × 21, 3 × 7 22: 1 × 22, 2 × 11 23: 1 × 23 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6
Looking for Patterns Results of the investigation will vary. Page 4 At Home Results of the investigation will vary. Let’s Investigate 95 ÷ 5 = 19; 180 ÷ 5 = 36; 425 ÷ 5 = 85; 715 ÷ 5 = 143. The method works for all multiples of 5.
Looking for Patterns 34
Inquisitive ant
multiple A number that can be divided exactly by another smaller number without a remainder.
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