Stretch and Challenge 4 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 4
4
Introduction
5
The features of Stretch and Challenge
7
A possible Stretch and Challenge teaching and learning sequence
10
Links to the Year 4 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 4
21
The Issues
23
Teacher’s notes
95
Resource sheets Record of completion
205
My notes
207
Pupil self assessment booklet
209
Other Resource sheets
211
3
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Quick reference guide to Stretch and Challenge 4 Domain(s)
Topic
Issue number
Teacher’s notes page number
Number: – Number and place value
Number
1
97
Number
2
100
Number
3
103
Number
4
106
Number
5
109
Negative numbers
6
112
Number:
Addition
7
115
– Addition and subtraction
Subtraction
8
118
Number:
Multiplication
9
121
– Multiplication and division
Division
10
124
Number:
Mixed operations
11
127
– Addition and subtraction
Mixed operations
12
130
Mixed operations
13
133
Mixed operations
14
136
Mixed operations
15
139
Number:
Fractions
16
142
– Fractions (including decimals)
Fractions
17
145
Decimals
18
148
Decimals
19
151
Length
20
154
Mass
21
157
Volume and capacity
22
161
Time
23
164
Area
24
167
Perimeter
25
170
Measurement
26
173
Measurement
27
176
Geometry:
2-D shapes
28
179
– Properties of shapes
3-D shapes
29
182
Symmetry
30
185
Movement and angle
31
188
Position and direction
32
191
Geometry
33
194
– Position and direction
Geometry
34
197
Statistics
Statistics
35
200
Statistics
36
203
– Multiplication and division
Measurement
Geometry: – Position and direction Geometry: – Properties of shapes
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
• 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 printable: • 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 4 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 4 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 crosscurricular 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 - Mixed operations
Issue 11
Mixed operations Prerequisites for learning
Resources
• Identify and use patterns, relationships and properties of numbers • Recall multiplication and division facts for the multiplication tables up to 12 × 12 • Understand decimal notation for tenths. • Derive and recall multiples of numbers to 10 up to the 12th multiple • Calculate mentally with whole numbers • Use formal written methods to add, subtract, multiply and divide whole numbers • Understand and use the order of operations, and begin to use brackets
pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) Resource sheet 7: Function machines selection of travel / holiday brochures calculator data handling software computer with Internet access
Teaching support Page 1 Money Matters • Ask the children to create their spreadsheet on the computer using an appropriate data handling software program. • Encourage the children to be as detailed as possible with their expenses sheet, accounting for all possible expenses. Around the World • This is a very open-ended investigation. You may want to leave it open for the children, or you may wish to narrow the investigation to focus on one or more of the following: – Difference in prices between different airlines – Difference in prices between different classes – Difference in prices at different times of the day – Difference in prices at different times of the year
Page 2 Looking for Patterns • You may need to spend some time before the children start on this activity discussing with them the concept of algebra. Use known rules and formulae, such as calculating the perimeter of a rectangle, to illustrate the concept. • Ensure the children have an understanding of the order of operations, including the use of brackets. ) can be substituted with a letter such • You may also wish to discuss with the children how the cloud ( as a, n or x.
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Issue 11 - Mixed operations Page 3 The Puzzler • Children use the function machines at the top half of Resource sheet 7 to write their own, two-step puzzles.
• Children use the function machines at the bottom half of Resource sheet 7 to write their own, three-step puzzles.
Technology Today • When pairs of children have completed the activity, ask them to compare the different calculations they performed to get each of the answers without using the ‘4’ key. Which is the most efficient method? Can the children work together to find an even more efficient method? • What if the ‘2’ and ‘6’ keys on the calculator were broken?
Page 4 Looking for Patterns • Once children have completed the activity, discuss and compare the strategies that the children used to solve the different grid problems. Were some easier to solve than others? Why? • What would happen if the part of each of the grids in the issue were not the top of the grid, but taken from the middle of the grid? • What happens if you shade two different sets of multiples on the same grid? Let’s Investigate • Ensure the children have an understanding of the order of operations, including the use of brackets. • Can the children find more than one way to make some of the numbers? • Can the children find three digits that will make all the odd numbers from 1 to 29 using any of the four operations?
AfL • • • • • •
How did you spend your £2000? How did you work out the answer to this problem? How did you begin? What patterns did you notice? Can you explain this in words / using symbols? How can you explain this using letters and symbols? Without using the number four, what calculations did you have to do to work out the answer? What is the correct answer? Why is it correct?
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Issue 11 - Mixed operations
Answers
Page 3 The Puzzler 3 6 27
Page 1
Technology Today Calculations will vary.
Money Matters Expenses sheets will vary.
Page 4
Around the World Results of the investigation will vary. Page 2 Looking for Patterns Expressions will vary. However, may be similar to the following: Think of a number. Multiply it by 2.
× 2 =
Add 4.
+ 4 =
Halve it.
÷ 2 =
Subtract 2.
− 2 =
Inquisitive ant
24
?
Looking for Patterns Large complete grid: Multiples of 4 are shaded. 1st partial grid: Multiples of 5 are shaded. The width of the grid could be 8, 13, 18, … 2nd partial grid: Multiples of 7 are shaded. The width of the grid could be 10, 17, 24, … 3rd partial grid: Multiples of 6 are shaded. The width of the grid could be 11, 17, 23, … 4th partial grid: Multiples of 3 are shaded. The width of the grid could be 8, 11, 14 … Let’s Investigate 6=1+9–4 8=9–1 10 = 9 + 1 12 = 9 + 4 – 1 14 = 9 + 4 + 1 16 = 4 × 4 18 = 14 + 4
20 = (9 – 4) × 4 22 = 41 – 19 24 = 19 + 4 + 1 26 = (9 × 4) – (9 + 1) 28 = (9 × 4) – 9 + 1 30 = 49 – 19 Other answers are possible.
formula A rule or principle represented in symbols, numbers, or letters, often written in the form of an equation.
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Issue 12 - Mixed operations
Issue 12 Mixed operations Prerequisites for learning
Resources
• Identify and use patterns, relationships and properties of numbers • Recall multiplication and division facts for the multiplication tables up to 12 × 12 and use them to multiply pairs of multiples of 10 and 100 • Derive squares of numbers to at least 12 × 12 • Calculate mentally with whole numbers • Use formal written methods to add, subtract, multiply and divide whole numbers • Understand and use the order of operations, and begin to use brackets • Understand simple expressions and formulae in words and symbols • Make estimations and approximations
pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) set of dominoes calculator school’s electricity bill (optional) data handling software computer with Internet access
Teaching support Page 1 What’s the Problem? • You may need to spend some time before the children start on this activity discussing with them the concept of algebra. Use known rules and formulae, such as calculating the perimeter of a rectangle, to illustrate the concept. • Ensure children realise that the values of a and b are the same for both the addition and the multiplication calculation. Let’s Investigate • Ask the children to use a calculator to check that the short-cut method for finding the square of a number ending in 5 works.
Page 2 Technology Today • Ask the children to write a calculation that illustrates the problem, i.e. (7 × 10) – (5 × 2) = 60. The Puzzler • Ensure children are familiar with how a magic square works. • Do not provide the children with a set of dominoes.
Page 3 Money Matters • This activity will probably take the children some time to complete. Not only are they required to find out about the cost of growing their own fruit and vegetables, they also need to find out about the cost of similar fruit and vegetables from a supermarket, and compare the difference. • Another important aspect of this activity is the justifications that children offer to support their conclusions.
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Issue 12 - Mixed operations Focus on Science • A watt is a unit for measuring electrical power. 1 kW (kilowatt) = 1000 watts 1 kWh (kilowatt hour) = 1000 watts of electricity used for one hour • Approximately 20% of all electrical consumption is from lighting. • If possible, show the children the school’s electricity bill and talk about it with them. Discuss how there are many different appliances and devices in the school that use electricity, but that lighting accounts for almost one-fifth of the total electricity bill.
Page 4 Money Matters • Ask the children to create a spreadsheet of the expenses on the computer using an appropriate data handling software. The Puzzler • Ensure the children have an understanding of the order of operations, including the use of brackets. +3– = 0 (see Answers). • Provide the children with one number in each calculation, e.g. What’s the Problem? • Once the children have completed the activity, provide an opportunity for children to talk about, and compare, their methods for working out the answer to the problem. Which were the most effective strategies?
AfL • • • •
What are the values of a and b? How did you work this out? What can you tell me about square numbers? How did you work out the answer to this problem / puzzle? Where did you start? What did you discover about ‘growing your own’ / the amount of electricity used by the lights in your classroom in a year / the cost of a wedding?
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Issue 12 - Mixed operations
Answers
Page 3 Money Matters Results of the investigation will vary.
Page 1 What’s the Problem? The values of a and b (or b and a) are 7 and 16. Let’s Investigate The method works for any number ending in 5. Page 2
Page 4 Money Matters Results of the investigation will vary. The Puzzler 5+3–8=0 16 ÷ 8 + 3 = 5 16 – 3 × 5 = 1 3 × 16 ÷ 8 = 6 16 ÷ (12 – 8) = 4 (8 + 12) ÷ 5 = 4 Other answers are possible.
Technology Today Lakshmi hit 7 flying saucers. The Puzzler Many solutions are possible. Here is one: The magic number is 6.
Inquisitive ant
Focus on Science Results of the investigation will vary.
What’s the Problem? Pascal has baked 18 loaves of bread by 5:00 a.m.
average The degree to which something is typical of a group or class of people or things. A number that can be regarded as typical of a group of numbers, calculated by adding the numbers together, then dividing the total by the amount of numbers.
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