Issuu

Stretch and Challenge 6 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 6

Introduction

The features of Stretch and Challenge

A possible Stretch and Challenge teaching and learning sequence

Links to the Year 6 Mathematics National Curriculum Programme of Study and Attainment Targets

Cross-curricular links to the National Curriculum Programme of Study

Resources used in Stretch and Challenge 6

The Issues

Teacherâ€™s notes

Resource sheets Record of completion

215

My notes

217

Pupil self assessment booklet

219

Other Resource sheets

221

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Quick reference guide to Stretch and Challenge 6 Domain(s)

Topic

Issue number

Teacher’s notes page number

Number: – Number and place value

Number

100

Negative numbers

103

Number:

Addition

106

– Addition, subtraction, multiplication and division

Subtraction

110

Multiplication

113

Division

117

Mixed operations

120

Mixed operations and algebra

123

Number:

Fractions

126

– Fractions (including decimals and percentages)

Fractions

130

Decimals

134

Percentages

137

Percentages

140

Ratio and proportion

143

Ratio and proportion

146

Ratio and proportion

149

Algebra

152

Algebra – The Financial Issue

155

Length

158

Mass

161

Volume and capacity

164

Time

167

Area

171

Perimeter

174

Measurement

178

Geometry:

2-D shapes

181

– Properties of shapes

3-D shapes

185

Movement and angle

188

Geometry:

Reflections

191

– Position and direction

Translations

194

Geometry:

Geometry

197

– Properties of shapes

Geometry

200

Geometry

203

Statistics

206

Statistics

209

Statistics

212

Ratio and proportion

Algebra

Measurement

– Position and direction Statistics

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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.

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Introduction

The features of Stretch and Challenge Pupil activity booklets (Issues) • Each of the 36 Issues in Stretch and Challenge 6 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 11–20). • 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 6 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

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 22 and 23. • 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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Introduction Teacher’s notes Each of the 36 Issues includes a set of teacher’s notes, including answers. Issue 3 - Addition

Issue 3

Prerequisites for learning Lists the prerequisites for learning that children need to have acquired prior to this Issue.

Mathematics topic

Addition

Issue number

Resources

• • • •

pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) ruler dominoes calculator (optional)

Identify prime numbers to at least 100 Use tests of divisibility Calculate mentally with integers and decimals Use the formal written method to add whole numbers and decimals • Use approximations to estimate and check results

To aid preparation, the resources needed for the Issue are listed.

Teaching support

Page 1

=6 =5 =3 =8 =4 =7

=7 =5 =4 =3 =2

Looking for Patterns • Tell the children the value of one (or more) of the shapes in each of the grids, i.e. The Puzzler • Tell the children the value of one of the numbers (see Answers).

Lists the associated knowledge and skills that contribute to understanding the Issue topic.

Provides teaching points for each of the activities in the Issue. These may be helpful when introducing the Issue to the children, or when children experience difficulty whilst working on a particular activity.

The Puzzler • Ensure that the children are familiar with the term ‘consecutive’ and can identify consecutive odd numbers, e.g. 1, 3, 5, 7, … • Ask the children to show all their working as this will help them when writing about how they discovered what the numbers are. 2 3 5 7 11 The Puzzler • Ensure that the children can identify prime numbers to at least 100. If not, provide them with a list:

Page 2

97 The Language of Maths • Before the children work independently on this activity, ensure that they understand how to work out a numerical tautonym. • Children can spend as much or as little time on this activity as is appropriate. You may want some children to ﬁnd only one or two four-, six- or eight-letter tautonyms, while other children might be expected to ﬁnd more.

Simplifications Where appropriate, offers suggestions for supporting children who may be experiencing difficulties understanding the main mathematical ideas.

Almost all of the activities in Stretch and Challenge can be undertaken either individually or in pairs (or sometimes in small groups).

Let’s Investigate • Expect the children to use the given method to make several whole number palindromes before working with decimal numbers. • Children should discover that the method is the same for decimal numbers as it is for whole numbers, but that the position of the decimal point plays an important role in creating perfect decimal palindromes such as 19·91 and 18·81, as opposed to imperfect decimal palindromes such as 111·1 and 66·6.

Where an activity is particularly suitable for pairs to work on, this is denoted by .

Page 3 Famous Mathematicians • Most children will probably be familiar with Pascal’s Triangle. If not, you may need to spend some time discussing it with the children. • The important part of this activity is not identifying the rule, rather it is in investigating other rules for producing triangles of numbers.

Extensions Where appropriate, offers suggestions for extending children’s understanding if you feel they are developing a good understanding of the main mathematical ideas.

Resources

Prerequisites for learning

When children are likely to need access to a computer, this is denoted by .

106

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AfL • How did you work out the answer to this puzzle? What strategies did you use? How else could you have gone about it? • Did you use trial and improvement to help you work out the answer to this puzzle? How did you know where to start? Then what did you do? • How did you organise your thoughts? Did this help? What else could you have done? Why might this have been more effective?

When children are likely to need access to a computer with Internet access, this is denoted by .

Assessment for Learning Each Issue includes a list of questions specifically designed to assist in assessing pupils’ understanding of the Issue topic. Issue 3 - Addition

Answers These are provided where appropriate.

Answers

A perfect palindrome can be made by altering the position of the decimal point, e.g. 8·6 8·6 + 6·8 = 15·4 15·4 + 4·51 = 19·91

Page 1 Looking for Patterns 7 7 2 5 21

5 7 3 5 20

4 4 5 2 15

3 4 4 3 14

19 22 14 15

6 3 6 5 20

5 4 7 8 24

3 6 3 4 16

8 5 8 7 28

22 18 24 24

The Puzzler 67356_P099-133.indd 108

6 9

Other solutions are possible.

The Puzzler 17, 19, 21, 23, 25, 27, 29, 31, 33 The Puzzler 43

Page 2 The Language of Maths Many words are possible.

13·5 13·5 + 5·31 = 18·81

Page 3 Famous Mathematicians In Pascal’s Triangle, each number is produced by adding the two numbers above:

2 3

Let’s Investigate 10 = 7 + 2 + 1 16 = 8 + 7 + 1 22/07/16 20 = 8 + 7 + 5 The numbers 4 and 19 cannot be made using the given set. The set of numbers {1, 2, 3, 7, 8} can make all the numbers 1 to 21 and none above.

12:44 PM

The Arts Roundup The gates opened at 5 p.m. Page 4 Looking for Patterns The numbers are consecutive. The sums of two consecutive numbers are all odd numbers. The sums of three consecutive numbers are all divisible by 3. The sums of four consecutive numbers start at 10 and increase by 4 each time. The sums of ﬁve consecutive numbers start at 15 and increase by 5 each time. Let’s Investigate Other solutions are possible.

Four-letter numerical tautonyms: BUDS, BULK, CHEF, COME, DIAL, DONE, EXIT, FOIL, GRIP, HEAL, HUMP, LAID, LOVE, MEND, MOTH, NEAR, OKAY, PALE, PINK, RUST, SELL, SPOT, SPUN, TABS, TAIL, TAPE, THIS, TIDY, TOPS, UNTO, VAIN Six-letter numerical tautonyms: ASSENT, ASSUME, BOTTLE, BUTTER, CREATE, HUNTER, REPENT, SAVERS, SHADES, SMACKS, TIMERS, VACATE, WASTER

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Introduction Class/Teacher:

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Stretch and Challenge 6 Record of completion

Resource sheet 1

THE

Maths

Date of starting issue:

Number: – Number and place value

Addition

Fractions

Decimals

Percentages

Ratio and proportion

Multiplication

Division

Mixed operations

Ratio and proportion

15 16

Algebra

Algebra – The Financial Issue

Resource sheet 2

Ratio and proportion

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Algebra

Subtraction

Mixed operations and algebra

Ratio and proportion

Number:

Negative numbers

– Addition, subtraction, multiplication and division

– Fractions (including decimals and percentages)

Number

Date of finishing issue:

My notes

Stretch and Challenge Issue

Topic

Name:

Date: Date:

Name:

Domain(s)

S&CCNPM Volume 6 Volume 1

TheHerald Maths Herald

Names

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Record of completion

To assist in keeping a record of which Issues children have completed. Once a child has completed an Issue you could either put a tick or write the date in the corresponding box.

THE

Maths

Name:

The pupil activity booklets have been designed to resemble a newspaper. This means that quite often there is insufficient space in the booklets for children to show their working and answers. You may decide to simply provide children with pencil and paper to record their work or an exercise book that they use as their ‘Stretch and Challenge Journal’. Alternatively, you could provide them with a copy of the A5 booklet: ‘My notes’ (to be printed double sided onto one sheet of A4 paper). This can then be kept, together with the child’s copy of the Issue and, if appropriate, their ‘Pupil self assessment booklet.’

S&CCNPM Volume 6 Volume 1

TheHerald Maths Herald

Date of starting issue:

My notes

Name:

Date: Date:

Date of finishing issue:

What have you learned?

Whichever method you choose for the children to record their working and answers, i.e. on sheets of paper, using a ‘My notes’ booklet, in an exercise book, or any other method, children need to be clear and systematic in their recording. What did you use to help you?

Isometric paper

Resource sheet 10

Resource sheet 3

Pupil self assessment booklet Each Resource Pack in the Stretch and Challenge series includes an age-appropriate pupil self assessment A5 booklet (to be printed double sided onto one sheet of A4 paper). This booklet is a generic sheet that can be used for any, or all, of the 36 Issues in the Resource Pack. The booklet is designed to provide children with an opportunity to undertake some form of self assessment once they have completed the Issue. After the children have completed the booklet, discuss with them what they have written. This can then be kept, together with the child’s copy of the Issue and their working out and answers, including, if appropriate, ‘My notes’.

Busy Ant Maths Stretch and Challenge 6

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Other Resource sheets

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For some of the activities, children are required to use a specific Resource sheet. These are included both in the back of this Resource Pack and the online resources.

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Issue 11 - Decimals

Issue 11 Decimals Prerequisites for learning

Resources

• Calculate mentally with integers and decimals • Use efﬁcient written methods to add and subtract decimals with up to two places • Use understanding of place value to multiply and divide decimals by 10, 100 or 1000 • Use a calculator to solve problems, including those involving decimals • Select and use standard metric units of measure and convert between units using decimals to two places • Use the order of operations, including brackets

pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) Resource sheet 5: 1 cm squared paper (optional) ruler section from a newspaper showing exchange rates calculator cheques from a range of different banks material for making a poster computer with Internet access

Teaching support Page 1 Money Matters • Either provide the children with the section from a newspaper that shows exchange rates (ﬁnancial and / or travel section) or access exchange rates from the Internet. • Ensure that children understand the difference between imports and exports. • The most important aspect of this activity is the last section – drawing conclusions about the effects of exchange rates on the prices of imported and exported goods. • Ask the children to draw a conversion graph showing the relationship between the pound and another major currency such as the Euro or US dollar. Let’s Investigate • Using two, three or four digits it is possible to make the following combinations of decimals with one, two or three places: – decimals with one place: O·t, TO·t, HTO·t – decimals with two places: O·th, TO·th – decimals with three places: O·thth Let’s Investigate • Ensure that children understand the difference between a terminating decimal and a recurring decimal. • Children also need to be conﬁdent with using a calculator and understand the error display ‘e’.

Page 2 Money Matters • The main purpose of this activity is for the children to investigate for themselves how cheques work, what are their advantages and disadvantages, and how they are designed to ensure security. • Ensure that there is a range of cheques from different banks available for the children to refer to when designing their school’s cheque. At Home • After children have completed the activity at home, ensure that there is an opportunity in class for pairs or groups of children to compare and discuss their lists, and to draw conclusions about the usage of decimals with different decimal places. The Language of Maths • Encourage the children to be as detailed as possible when deﬁning the word ‘decimal’.

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Issue 11 - Decimals Page 3 Let’s Investigate • Ensure that children are confident in using the order of operations, including brackets. Looking for Patterns • Remind the children that they are designing a poster with the purpose of explaining to others how to effectively and efficiently convert between different metric units for length, mass and capacity. Therefore, the poster needs to be easy to read and follow, and illustrated with examples. The Language of Maths • Children should be able to infer the meaning of the prefixes ‘centi’, ‘milli’ and ‘deci’ from their association with words such as centimetre, millimetre and decimal. Likewise, they should be able deduce the meaning of the words ‘decimetre’, ‘centigram’, ‘centilitre’ and ‘decilitre’.

Page 4 The Puzzler • The amount of money in each envelope can be found by subtracting the total of the other two pairs of envelopes from £10. For example, Envelope E = £10 – (£5.20 + £3.40) = £1.40. The Puzzler • Tell the children that the common total is 3·4.

AfL • • • •

What did you ﬁnd out? How did you ﬁnd this out? What does it mean? How do you know that you have found all the different possibilities? How did you organise your work? Explain to me how you worked that out. What strategies did you use? How did they help you?

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Issue 11 - Decimals

Answers

Looking for Patterns Results will vary. The Language of Maths centi – one hundredth milli – one thousandth deci – one tenth decimetre – metric unit of length that is equal to one tenth of a metre centigram – metric unit of mass that is equal to one hundredth of a gram decilitre – metric unit of capacity that is equal to one tenth of a litre centilitre – metric unit of capacity that is equal to one hundredth of a litre

Page 1 Money Matters Results of the investigation will vary. Let’s Investigate Results of the investigation will vary. Let’s Investigate 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , 2 4 5 8 10 16 20 25 32 40 50 64 80

Page 4

Page 2

The Puzzler A = £2.70 B = £2.50 C = £1.80 D = £1.60 E = £1.40

Money Matters Results of the investigation will vary. At Home Results of the investigation will vary. The Language of Maths ‘Decimal’ means using the number ten as a base. Numbers are expressed in a counting system that uses units of ten.

The Puzzler 1·6

0·3

0·2

1·3

0·5

1·1

0·8

0·9

0·6

0·7

1·2

0·4

1·5

1·4

0·1

Page 3 Let’s Investigate Results of the investigation will vary.

Inquisitive ant

Each row, column and diagonal totals 3·4. Other solutions are possible.

10–3 = 0·001 (milli)

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Issue 12 - Percentages

Issue 12 Percentages Prerequisites for learning

Resources

• Recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per hundred’ • Express tenths and hundredths as percentages • Express one quantity as a percentage of another • Recognise approximate proportions of a whole and use percentages to describe and compare them

pencil and paper Resource sheet 2: My notes (optional) Resource sheet 3: Pupil self assessment booklet (optional) ruler pair of compasses protractor newspapers, magazines and catalogues data-handling software calculator (optional) computer with Internet access

Teaching support Page 1 Let’s Investigate • Discuss with the children all the different regular activities they do in a weekday. Encourage the children to organise all the activities into six to ten different categories. This will make it easier for the children to draw their pie chart. • It is advised that children create their pie chart using ICT. • Children draw their pie chart using a pair of compasses, a protractor and a ruler. The Puzzler • Ensure children are familiar with magic squares. Ask the children to create a 4 × 4 magic percent square. Focus on Science • Children will need to use the Internet for this activity.

Page 2 The Language of Maths • Briefly introduce the children to the terms ‘deposit’, ‘down-payment’ and ‘balance’. Ensure that they are secure in their understanding of these terms before setting them to work on the activity. Let’s Investigate • At this stage, children should be able to recognise approximate proportions of a whole and use percentages to describe and compare them. Encourage the children to make comparisons between three, four or more groups. Money Matters • Most children should be able to calculate successfully the VAT on goods and services and the new price once VAT has been added. • It will be easier to find prices in magazines and catalogues that are VAT inclusive. Services, as opposed to goods, are often quoted without VAT. To find prices of goods and services that do not include VAT refer to appropriate websites. • If the price of an item already includes VAT, some children may need assistance in calculating the cost of the item before VAT (see Answers).

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Issue 12 - Percentages Page 3 Money Matters • Below are the VAT rates for some European countries: 18% – Malta 19% – Germany 20% – Austria and France 21% – Belgium and Spain 22% – Italy 23% – Ireland, Poland and Portugal 24% – Finland and Greece 25% – Denmark and Sweden Note: Rates current as of 1 June 2016 The Language of Maths • This activity requires children to be able to recognise approximate proportions of a whole and use percentages to describe and compare them. • Explain to the children that each sector of the pie chart is a multiple of 5%. Money Matters • You may wish to provide children with appropriate clippings from newspapers rather than have them look for them themselves.

Page 4 Money Matters • Draw children’s attention to the illustrations displaying different percentages and talk about what they all have in common (they are multiples of 10% or 25%). Technology Today • To be able to answer this problem, children need to be able to express successfully one quantity as a percentage of another. • Work through one or more examples with the children. If appropriate, provide the children with a calculator. e.g. 58 out of 80. 58 × 100 = 580 = 72·5% 80 1 8

AfL • • • •

What did you ﬁnd out? What does it mean? How did you go about this? How did you go about working out the percentage? How would you do this on a calculator? What calculation did you do? What does the answer tell you? Approximately what percentage of the pie chart is this sector? Why do you think it is that percent?

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Issue 12 - Percentages

Answers

To calculate the price of an item including VAT: (price of item before VAT x percentage rate of VAT) + price of item e.g. price of item before VAT costs £24 VAT rate is 20% (£24 × 0·2) + £24 = £28.80

Page 1 Let’s Investigate Results of the investigation will vary.

To ﬁnd the price of an item before VAT:

The Puzzler 60% 10% 80%

The magic percent is 150%. Other solutions are possible.

70% 50% 30%

Page 3

20% 90% 40%

Money Matters Results of the investigation will vary.

Focus on Science A humidity meter measures the amount of moisture in the air. RH% refers to the percentage of relative humidity in the air. Relative humidity is the amount of water vapour in the air expressed as a percentage of the maximum amount of water vapour that can exist in the air at that temperature. Page 2

The Language of Maths Answers will vary. However, they should be similar to the following: Sector 1: 25% Sector 2: 10% Sector 3: 5% Sector 4: 5% Sector 5: 5% Sector 6: 5% Sector 7: 15% Sector 8: 10% Sector 9: 20% Money Matters Results of the investigation will vary.

The Language of Maths Results of the investigation will vary.

Page 4

Let’s Investigate Results of the investigation will vary. Money Matters VAT stands for Value-Added Tax. It is a tax added to the value of a product at its production or distribution and is paid by the consumer when they purchase the product. Every quarter, the producer (or distributor) pays to the government the VAT they receive from consumers. The rate of VAT in the UK as of 1 January 2011 is 20%.

Inquisitive ant

price of item including VAT × 100 100 + percentage rate of VAT e.g. price of item including VAT costs £28.80 VAT rate is 20% £28·80 × 100 2880 = = £24 100 + 20 120

Money Matters Results of the investigation will vary. However, most sales have discounts of multiples of 10% (or 25%). This is because such percentage discounts are easy to calculate, which means consumers can quickly work out how much money they are saving. Technology Today Hector did best on Friday. He did worst on Wednesday. Gomez did best on Tuesday and Friday. He did worst on Saturday.

‰ ‘Per mil’ means parts per thousand. It is a tenth of a percent.

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