maths-4-looking beyond the uk

8/8/2019 MATHS-4-Looking Beyond the UK

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MathematicsKS3 Cycle Curriculum PackProject four Looking Beyond the UK

Page 4.1

Summary

It is curious that with the advent of the automobile and the airplane, thebicycle is still with us. Perhaps people like the world they can see from abike, or the air they breathe when theyre out on a bike. Or they like thebicycles simplicity and the precision with which it is made. Or becausethey like the feeling of being able to hurtle through air one minute, andsaunter through a park the next, without leaving behind clouds of chokingexhaust, without leaving behind so much as a footstep.Gurdon S. Leete

This is not Disneyland, or Hollywood. Ill give you an example: Ive readthat I flew up the hills and mountains of France. But you dont fly up a hill.You struggle slowly and painfully up a hill, and maybe, if you work veryhard, you get to the top ahead of everybody else.Lance Armstrong

Looking Beyond the UK is a project that combines a number of highlyrelevant educational issues and Key Stage 3 (KS3) mathematical content:

Mathematical conclusions about the urban transport issues facing citiestoday, particularly focused on London in comparison with similar citiesor developing countries

A wide range of mathematical skills and KS3 coverage of investigativework and processes, as well as use of communication andinterpretative skills

A practical and real use of mathematics within the context of todayssociety and international and environmental issues

A wide range of whole-school and cross-curricular coverage, shown bythe teaching delivery map, School Travel Plan (STP) and all five objectivesof Every Child Matters (ECM)

Looking Beyond the UKsummary

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Page 4.2

Summary

Looking Beyond the UK challenges students to think and act within acycling context. The project is based on a comprehensive set of dataconcerning cycling in London and other parts of the world. The data areflexible enough for students to be able to follow their own ideas and testtheir own hypotheses. It also gives students a grounding in manipulatingstatistics to prove their arguments in todays communication-rich society.There is enough data supplied for students to immediately investigate, butthis can be enhanced by students collecting their own data or accessingother sources via the internet.

The Looking Beyond the UK project is flexible and consists of many parts,which are composed of a number of lessons. Each section can be taughtwithin one day, or over consecutive days, or over a number of weeks. Theapproximate time for this to be delivered is three to five hours, dependingon the route taken through the project.

This activitys other focus is cycling. Given that cycling has so manypositive impacts, for example on health, on the environment and oneconomics, it is perhaps a concern that cycling is not a more significantpart of urban life.

Looking Beyond the UKsummary

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Page 4.3

Teaching delivery map

Mathematical contextThe following sections have been taken from the National Curriculum for Mathematics onthe Qualifications and Curriculum Authoritys (QCAs) website.

New focus on aims and skills; the curriculum should enable all young people to become:

Successful learners who enjoy learning, make progress and achieve

Confident individuals who are able to live safe, healthy and fulfilling lives

Responsible citizens who make a positive contribution to society

These aims, which incorporate the five outcomes of Every Child Matters (ECM), have been the

starting point for all the changes to the secondary curriculum. The new curriculum continuesto recognise the importance of subjects while at the same time placing emphasis on thedevelopment of skills for life and work.

1. Key concepts of mathematicsThere are a number of key concepts that underpin the study of mathematics. Students needto understand these concepts in order to deepen and broaden their knowledge, skillsand understanding.

Unit no. Key concepts Project summary

1.1 Competence Applying suitable mathematics accurately

within the classroom and beyond Communicating mathematics effectively Selecting appropriate mathematical tools

and methods, including information andcommunication technology (ICT)

1.2 Creativity Combining understanding, experiences,imagination and reasoning to constructnew knowledge

Using existing mathematical knowledge tocreate solutions to unfamiliar problems

Posing questions and developingconvincing arguments

Looking Beyond the UKteaching delivery map

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Page 4.4


Key concepts of mathematics (continued)

Unit no. Key concepts Project summary

1.3 Applications and implications Knowing that mathematics is a rigorous,of mathematics coherent discipline

Understanding that mathematics is used asa tool in a wide range of contexts

Recognising the rich historical and culturalroots of mathematics

Engaging in mathematics as an interestingand worthwhile activity

1.4 Critical understanding Knowing that mathematics is essentiallyabstract and can be used to model,interpret or represent situations

Recognising the limitations and scope ofa model or representation

2. Key processesThese are the essential skills and processes in mathematics that students need to learnto make progress.

Unit no. Key processes Project summary2.1 Representing Students should be able to:

Identify the mathematical aspects of asituation or problem

Choose between representations Simplify the situation or problem in order to

represent it mathematically, using appropriatevariables, symbols, diagrams and models

Select mathematical information, methodsand tools to use


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Page 4.5


Key processes (continued)

Unit no. Key processes Project summary

2.2 Analysing Students should be able to: Make connections within mathematics Use knowledge of related problems Visualise and work with dynamic images Identify and classify patterns Make and begin to justify conjectures and

generalisations, considering special casesand counter-examples

Explore the effects of varying values andlook for invariance and co-variance Take account of feedback and learn

from mistakes Work logically towards results and

solutions, recognising the impact ofconstraints and assumptions

Appreciate that there are a number ofdifferent techniques that can be used toanalyse a situation

Reason inductively and deduce Make accurate mathematical diagrams,

graphs and constructions on paper andon screen

Calculate accurately, selecting mentalmethods or calculating devices,as appropriate

Manipulate numbers, algebraic expressionsand equations and apply routine algorithms

Use accurate notation, including correctsyntax, when using ICT

Record methods, solutions and conclusions Estimate, approximate and check working


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Page 4.6


Key processes (continued)

Unit no. Key processes Project summary

2.3 Interpreting and evaluating Students should be able to: Form convincing arguments based on

findings and make general statements Consider the assumptions made and the

appropriateness and accuracy of resultsand conclusions

Be aware of the strength of empiricalevidence and appreciate the difference

between evidence and proof Look at data to find patterns and exceptions Relate findings to the original context,

identifying whether they support orrefute conjectures

Engage with someone elses mathematicalreasoning in the context of a problem orparticular situation

Consider the effectiveness ofalternative strategies

2.4 Communicating and reflecting Students should be able to: Communicate findings effectively Engage in mathematical discussion of results Consider the elegance and efficiency of

alternative solutions Look for equivalence in relation to both the

different approaches to the problem anddifferent problems with similar structures

Make connections between the currentsituation and outcomes, and situations andoutcomes they have already encountered


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Page 4.7


3. Range and contentThis section outlines the breadth of the subject on which teachers should draw when teachingthe key concepts and key processes. The study of mathematics should enable students toapply their knowledge, skills and understanding to relevant real-world situations.

The study of mathematics should include:

Unit no. Range and content Project summary

3.1 Number and algebra (not relevant in this section)

3.2 Geometry and measures (not relevant in this section)

3.3 Statistics The data-handling cycle Presentation and analysis of grouped and

ungrouped data, including time series andlines of best fit

Measures of central tendency and spread

4. Curriculum opportunitiesDuring the key stage students should be offered the following opportunities that are integralto their learning and enhance their engagement with the concepts, processes and content ofthe subject.

The curriculum should provide opportunities for students to: Develop confidence in an increasing range of methods and techniques Work on sequences of tasks that involve using the same mathematics in increasingly

difficult or unfamiliar contexts, or increasingly demanding mathematics in similar contexts Work on open and closed tasks in a variety of real and abstract contexts that allow them

to select the mathematics to use Work on problems that arise in other subjects and in contexts beyond the school Work on tasks that bring together different aspects of concepts, processes and

mathematical content Work collaboratively as well as independently in a range of contexts Become familiar with a range of resources, including ICT, so that they can

select appropriately


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Page 4.8


Assessment criteria

Attainment target 4: Handling data

Level 4 Students collect discrete data and record them using a frequencytable. They understand and use the mode and range to describesets of data. They group data in equal class intervals whereappropriate, represent collected data in frequency diagrams andinterpret such diagrams. They construct and interpret simpleline graphs.

Level 5 Students understand and use the mean of discrete data. Theycompare two simple distributions using the range and oneof the mode, median or mean. They interpret graphs anddiagrams, including pie charts, and draw conclusions.

Level 6 Students collect and record continuous data, choosingappropriate equal class intervals over a sensible range to createfrequency tables. They construct and interpret frequencydiagrams. They construct pie charts. They draw conclusions fromscatter diagrams, and have a basic understanding of correlation.

Level 7 Students specify hypotheses and test them by designing andusing appropriate methods that take account of variability or bias.

They determine the modal class and estimate the mean, medianand range of sets of grouped data, selecting the statistic mostappropriate to their line of enquiry. They use measures of averageand range, with associated frequency polygons, as appropriate,to compare distributions and make inferences.

Level 8 Students interpret and construct cumulative frequency tablesand diagrams. They estimate the median and interquartile rangeand use these to compare distributions and make inferences.

Exceptional performance Students interpret and construct histograms. They understandhow different methods of sampling and different sample sizes

may affect the reliability of conclusions drawn. They select and justify a sample and method to investigate a population.


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Source: www.dcsf.gov.uk/ Crown copyright 2007


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Page 4.9


Areas within a School Travel Plan (STP)

This project addresses all five aims of the STP

Significantly reduce the number of car trips on journeys to and from school Remove the barriers, both perceived and actual, to walking, cycling and using public

transport for school journeys Increase the number of young people and adults choosing active travel options over that

of the car Increase understanding among whole-school communities of the travel options that are

open to them Provide information to allow school communities to understand the benefits of active,

sustainable transport and to use this information to inform how they choose to travel

Areas within Every Child Matters

The aims of ECM

Be healthy Stay safe Enjoy and achieve Make a positive contribution Achieve economic well-being

It could easily be argued that this project covers all aims of the ECM agenda but specificallyaddresses the elements of a healthy lifestyle and making a positive contribution to societyas well as the economics of sustainable travel.

Areas within other central government initiatives/programmes

Initiative/programme Explain the projects link to this initiative/programme

Social and Emotional Although SEAL is aimed at KS1 and 2, the Looking Beyond theAspects of Learning (SEAL) UK project links in well through its emphasis on group work

and self-awareness as well as increased motivation towards

learning in mathematics.


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Page 4.10

Teaching notes andlesson plans

Handling data is illustrated by the cycle shown in the diagram. 1

Data handling is best taught in a coherent way in the context of realstatistical enquiries so that teaching objectives arise naturally from thewhole cycle. As an enquiry develops, you will need to reinforce and developcertain skills by direct teaching of particular objectives. This project enablesyou to address both the statistical enquiry and the specific skill objectives. 2

Young Dragons, Local Community, Local Discovery, The Big Debate andLooking Beyond the UK all address the four key elements in the cycleabove and outlined below:

Specify the problem and plan Students need to be specific about howthey intend to address the data to substantiate their arguments to a wideraudience. This is best learnt when students have choices and control overtheir choice of data. All these projects need discussion either in controlledsmall-group work or in general class discussion led by the teacher toformulate the purpose and process of data collection.

Collect data from a variety of sources Enclosed is a substantial set ofelectronic data as well as hard copy, but also listed are up-to-date websites

to support the data-collection process. These data sets are designed tobe easy enough to be accessible but substantial enough to have to useelements of sampling, which is important to students.

Looking Beyond the UKlesson plans introduction

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Specify theproblemand plan

Process andrepresent data

Collect datafrom a variety

of sources

Interpret anddiscuss data

evaluate results

1, 2 Source: www.dcsf.gov.uk/ Crown copyright 2007


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Page 4.11


Process and represent data Students need to consider carefully thebest way to present their data rather than simply answer a question in atextbook. To this end, a series of student handouts are available eitherfor teaching purposes or as reminder prompts when completing thebigger projects.

Interpret and discuss data This is a key element of handling data andconclusions must be drawn in response to the initial specification ofthe problem. If teachers are using up-to-date relevant data, then theseconclusions should have a wider audience than the students in theclassroom or the teacher. This could be in the form of a letter orelectronic presentation to a specific group (governors, press, Sustrans,etc), which will make the whole process real for the students.As students move through Key Stage 3, the cross-curricular aspects of datahandling become more important. It is usually best for a cross-curricularenquiry to be defined in the other subject, but good preparation is neededto check that the mathematical skills, techniques and representationsthat students need to learn next are likely to arise. In Year 7, much of thework may take place in mathematics lessons, with small sets of data thatstudents can generate readily from simple experiments and easily accessiblesecondary sources. In Year 9, students should engage with large sets of realdata from a much wider range of sources and contexts. After all, theirGCSE coursework in Key Stage 4 may require them to undertake a majorstatistical investigation, with supporting information and communicationtechnology (ICT). The experience of working with real data in Key Stage 3is an important preparation.

Primary and secondary sources

Give students experience of collecting and using primary data from, forexample, questionnaires or results of an experiment, and secondary datafrom published sources, including reference materials, ICT databases andthe internet. Plan carefully how to balance and use the various sourcesacross the key stage.


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Page 4.12


Real data present problems that textbook or contrived data can skirtaround, such as the accuracy of recording, or how to deal with data that areambiguous. The sizes of numbers can be problematic, either because theyare large or, in the case of a pie chart, because they are not factors of 360.The time needed to process and represent real data is likely to be greaterthan with textbook examples, but by using it students will have gaineduseful skills that can be transferred to other investigations.

Features of handling data in Key Stage 3

To summarise, the distinctive features of handling data in Key Stage 3 are:

Basing work on purposeful enquiry , using situations of interest andrelevance to students and making appropriate links to other subjects

Placing an emphasis on making inferences from data, drawing on a rangeof secondary sources to ensure that samples are sufficiently large

Using ICT as a powerful source of data, and as a means of processingdata and simulating situations

Lesson plans

Enclosed are a variety of lesson plans, but the best way of using thisresource is for students to use the data for their own enquiry in the form

best suited to you (the teacher), them (the students) and the currenteducational situation you are in.


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Page 4.13


Collecting dataLesson 1/2 of 6 (40 minutes. This timing allows the teacher to completea starter and plenary in a standard lesson of 60 minutes.)

Starter : Use additional resource: Cycling survey of USA. Display the sourceon the board or give out the resource sheet for the class to discuss:

Discuss which countries you think cycle most. Why is that? (China/India/Asia: cars too expensive; Europe, Holland, France, Germany, Italy: goodweather; Africa: too expensive to buy cars and roads in poor condition;Canada: too cold; USA: petrol is cheap and people are too used to cars)

What do you think Middleweight/Cruiser means? (It is an American termfor cycles that are built for roads but are similar to mountain bikes.)

What word do the British use instead of sidewalk? (pavement)

What is an unpaved road? (a dirt track)

Main task: Decide which of the tasks would be best for the class ordifferentiate by group or individual):

1. Draw a bar chart of the types of bicycle that Americans use. Thendraw a second bar chart with the types of bicycle that British peopleuse (use data from the supplied cycling database). Write one or twosentences explaining the differences. If possible, draw both bar chartsusing the same axes.

2. As above, but represent your information in pie charts. In this case,say whether bar charts or pie charts would be best to comparethe data.

3. Calculate the percentage of people in the UK that have a bell or hornusing a sample from the cycling database, and compare this with theUSA data. Do the same for a helmet.

4. Collate all the data together and write conclusions.

Looking Beyond the UKlesson 1/2 structure

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Collecting data (continued)Plenary: Get students to say whether it was better to use bar chartsor pie charts to compare the data. Also, what were the differences inthe data between the two countries? Can they suggest a reason forthe differences?

Resources:

Worksheet 1 Data collection sheet

Additional resource Cycling survey of USA

Additional resource Cycling database


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Page 4.15


Looking Beyond the UKlesson1/2 worksheet 1

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These questions are designed for use withwhole classes. Doing a show of hands is quick but some students might copy friends

or be too embarrassed to give an honest answer. You might want to givea sheet to each student

On the last question you might want to list the other reasons

You might consider boys and girls separately, but you would have tocollect this data individually

When you ask the group, give them all the options before you ask fortheir responses


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Looking

Beyondthe UK...Handouts and worksheets for photocopyingLesson 1/2


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Page 4.19

Lesson 1/2Worksheet 1

Page 1 of 1

Class

Name Date

Data collection sheet

Date Year group Total number in class

Q1. How did you travel to school this morning?

Car Cycle Walk Bus Train/Underground

Other:

Q2. Do you think the school should do more to support walking andcycling to school?

Yes No Dont know

Q3. What stops you walking or cycling to school?

I dont have a bike Too far

Too dangerous My bike is broken

Too lazy Nowhere safe to leave bike

Too dark and wet Other:


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Page 4.21



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Process and represent dataLesson 3/4 of 6 (40 minutes)

Starter : Use additional resource: Cycling in Holland. Display the source onthe board or give out the resource sheet for the class to discuss:

How big is 14bn kilometres? (Approximately 350,000 times round theequator; 15,000 return trips to the moon; everybody in London going ona mass cycle ride for 30km.)

Do you think that in the UK we cycle an average of 2.5km per person perday? (The average daily journey distance for the UK is 2.4km per day butthis is for cyclists, not for the whole population.)

Why do people in Holland cycle a lot? (Flat countryside; people live nearto towns and cities; good facilities; more environmentally aware, etc)

Main task:

1. In the UK, do men cycle more than women and by how much? Use asample from the cycling database for this analysis. Compare the resultsfor the UK with those for Holland by writing some sentences indicatingthe differences. (Try to include averages and percentages.)

2. Take a sample of the 12-18 age group from the cycling database andcalculate how far they travel on average. Compare this with the datafor Holland.

3. By choosing suitable age groups from the cycling database, find outwhich age range cycles the most in the UK. (Represent your findingsin tables and graphs, etc.)


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Page 4.22



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Process and represent data (continued)Plenary: Class discussion on what they have found out about the differencesbetween Holland and the UK. Can they suggest reasons for this or makesuggestions for encouraging more use of cycling in the UK?

Resources:

Handout 1 Bar charts

Handout 2 Pictograms

Handout 3 Line graphs

Handout 4 Two-way tables

Handout 5 Pie charts

Additional resource Cycling in Holland


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Looking



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Page 4.25

Lesson 3/4Handout 1

Page 1 of 1

Bar charts

Data collection for use in producing bar charts

In the example below you are reminded of some of the keyfeatures that make a good bar chart.

Age of bike (in years) Tally Frequency

0-2

3-5

6-8

9-11

12-14

15-19

20+

15

26

13

9

3

1

7

Keep talliesneat and aligned

0

10

20

0-2 3-5 6-8 9-11 12-14 15-19 20+

Age

Age of bicycles

5

15

25

30

F r e q u e n c y

Even scale forthe frequency

Lables forthe axes

Simple shadingor colouring

Equallyspaced bars

Title forthe graph

Total amountof bikes

per category


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Page 4.26

Lesson 3/4Handout 2

Page 1 of 1

Pictograms

Pictograms are a special type of bar chart where, instead of simple bars,a series of basic pictures is used.

There are some differences, which are noted below in the example.

Pictogram showing peoples favourite fruit

Pictograms are used for simple data showing a quick overview.They should not be overcomplicated.

No scale onvertical axis

= 2 people

Parts of picturesare allowed

Labels are not needed if the pictures are obvious

A key is vital. The picturecan represent any number

All picturesmust be

same sizeand aligned


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Page 4.27

Lesson 3/4Handout 3

Page 1 of 1

Line graphs

Line graphs can be used when the data you are using are a measure.This means they have two characteristics:

They are numbers

They are continuous. (They can take any value between a rangeof numbers.)

Examples age, height, weight, time, price, etc.

Examples you cannot use favourite colour, type of transport, etc.

To draw a line chart, set it out like a bar chart but put a small point insteadof a bar and then join up the points.

Example

These data show the age that people first learnt to ride a bicycle:

Age 5

3

6

9

7

12

8

20

9

34

10

17

>10

4Frequency

5 >10109876

Age

0

10

20

30

40

F r e q u e n c y

Line graph plotted from the above data


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Page 4.28

Lesson 3/4Handout 4

Page 1 of 1

This is a good way to compare the relationship between two sets of data.

Example 1

In this case you are looking at whether left- or right-handed people kickwith their left or right foot:

Two-way tables

Right-handed

Left-handed

Right-footed

12

2

Left-footed

1

3

Both

1

0

BoyGirl

Tennis

54

Not tennis

55

Year 7

Cricket

14

26

3419

21

114

Athletics

35

26

2218

15

116

Tennis

52

48

2519

34

178

Swimming

18

21

3224

17

112

Total

119

121

11380

87

520

Year 8

Year 9Year 10

Year 11

Total

Gender

Tennis or not

B

N

B

N

G

Y

B

Y

G

N

G

N

G

N

B

Y

G

N

B

Y

G

Y

B

N

B

Y

G

Y

G

N

B

Y

B

N

B

N

G

Y

L/R handed

L/R footed

R

R

R

R

R

Both

L

L

R

R

L

R

R

R

R

R

L

R

R

L

R

R

R

R

R

R

R

R

L

L

R

R

R

R

L

L

R

R

Example 2

This example looks at whether more boys or girls play tennis:

Example 3

This shows sports played in different years of school:

The two-way table looks like this:

The two-way table looks like this:


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Page 4.29

Lesson 3/4Handout 5

Page 1 of 3

Pie charts

Before drawing your circle for the pie chart you must draw out the frequencychart. The key elements here are the total frequency and the multiplier column.

Type of cycle Tally FrequencyRacing

Multiplier Angle(o)

Mountain

BMX

Touring

Childs

Folding

Total

17

13

11

17

7

2

67 360

Type of cycle Tally Frequency

Racing

Angle(o)

Mountain

BMX

TouringChilds

Folding

Total

17 5.4

13 5.4

11 5.4

17 5.47 5.4

2 5.4

67 360

To calculate the multiplier you divide 360 by the total frequency.

In this case, 360 67 = 5.373134 = 5.4 (to one decimal place).

Multiplier


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Page 4.30

Lesson 3/4Handout 5

Page 2 of 3

Pie charts

17 x 5.4 = 91.8 = approx 92 (rounded to nearest whole number).

Do the same for all other values.

Note that the total comes to 362 (not 360) due to rounding, but it iscustomary to shave off 1 from each of the two biggest sectors.

Type of cycle Tally Frequency

Racing

Mountain

BMX

Touring

Childs

Folding

Total

17 5.4 92

13 5.4 70

11 5.4 59

17 5.4 92

7 5.4 38

2 5.4 11

67 362

Multiplier Angle(o)

Draw your circle and start with a straight line from the centre to the edge.This is where you start measuring from.


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Lesson 3/4Handout 5

Page 3 of 3

Pie charts

Start measuring the next sector from the end of the previous one.

Fully labelled pie chart

Folding cycle

Touring cycleMountain cycleBMX cycle

Racing cycle

Childs cycle2

171311

17

7

91

Racing cycle 17


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Teachersonly

Interpret and discuss data (handling data cycle)Lesson 5/6 of 6 (40 minutes)

Starter : Use additional resource: World bicycle production. Display thesource on the board or give out the resource sheet for the class to discuss:

How many cycles did India produce in 1991?

Who produces the most each year? Why do you think this is?Who produced the least in the table in 1998?

What does N/A mean?

Main task:

1. Choose the UK, China and one other country to draw a line graph ofthe number of million cycles produced each year. Write one or twosentences saying what your graph shows.

2. Draw two pie charts for 1990 and 2000 showing the distribution ofcycle production, and say what the differences are.

Plenary: Discuss what the class found out about production trends. Ask ifproduction influences trends in the use of bicycles, ie more cycles = moreuse. Do they think different countries produce different types of cycle?

Resources:

Handout 1 Mode, median and mean

Handout 2 Scatter diagrams

Handout 3 Cumulative frequency

Handout 4 Box and whisker plot

Handout 5 Random sampling

Additional resource World bicycle production


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Looking



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Page 4.37

Lesson 5/6Handout 1

Page 1 of 5

Mode, median and mean

These are three types of average. An average is a way of giving a singlevalue that represents a whole set of data.

The mode is the value that occurs most often in a set of data.

The median is the middle number when the data are in order.

The mean is all the data added together and divided by the numberof pieces of data.

Example 1

A survey of the number of pieces of homework completed last night gave

this data:

There are more 1s than anything else, so mode = 1

The middle number in the list is 2, so median = 2

Mean = 0 + 0 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 4 = 26

26 15 = 1.73

Example 2

The number of vehicles in each family is surveyed, giving this data:

Here there are equal number of 2s and 3s so we say the mode = 2 and 3

For the median there is no exact middle as it lies between 2 and 3

We say the median = 21 2 or 2.5

Mean = 1 + 2 + 2 + 2 + 3 + 3 + 3 + 4 = 20

20 8 = 2.5

0 0 1 1 1 1 1 2 2 2 2 3 3 3 4

1 2 2 2 3 3 3 4


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Lesson 5/6Handout 1

Page 2 of 5

The mean uses good mathematics but sometimes gives a value thatdoes not make real sense (in the example above, what does 1.73 piecesof homework look like?). It can also be influenced by one extreme pieceof data.

The mode is very simplistic and just gives you the number written downthe most. It can be useful for non-numerical data.

The median can be good if you have extreme values at the top or bottomof your data list.

The best average to use will always depend on your data and what you aretrying to show. If you calculate all three you should say which is the bestaverage and why.

Which average do you use?


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Lesson 5/6Handout 1

Page 3 of 5

Mean, median and mode fromgrouped data

This frequency table looks at how much pocket money children geteach week.

Mode = 3 to 4 (this is often described as the modal group) as it has thehighest frequency.

There are 4 + 3 + 5 + 7 + 4 + 2 = 25 pieces of data so the middle piece ofdata is the 13th. If you count along the median then the 13th piece of datais the first one in the 3 < 4 group.

Median group = 3 < 4

The mean is difficult as there are no exact values to add up. Here we haveto assume that each amount will be, on average, near the central value ofthe group. We now need a new table.

Total = 2 + 4.5 + 12.5 + 24.5 + 18 + 12 = 73.50

Number of pieces of data = 25

Mean = 73.50 25 = 2.94

Note The central value in the last group (over 5) is a best guess as thereis no way of knowing what the values are. We could easily have taken5.50 or higher.

Money ()

Money ()

1

1

Frequency 4

1


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Page 4.40

Lesson 5/6Handout 1

Page 4 of 5

Range

The range is a measure of how spread out the data are. It is often usedalongside one of the three averages to describe a set of data.

To calculate the range , subtract the lowest value from the highest.

Example 1

These data are about the number of millimetres of rain falling in aperiod of 10 days:

Range= 5 0 = 5

Example 2

These data are about the price of second-hand cars:

Highest possible price = 4000

Lowest possible price = 500

Range = 4000 500 = 3500

131 0 0 1 5 0 0 1

Price () 500


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Page 4.41

Lesson 5/6Handout 1

Page 5 of 5

Range

Why range is usefulMaths Group 1 results:

Mean = 70

Range = 74 65 = 9

Maths Group 2 results:

Mean = 70

Range = 90 50 = 40

Both maths groups have the same mean, so it could be said they areequal. But Group 2 has a much bigger range, so is spread out more, as thedata show. There are also some very high and very low marks. Group 1 ismuch more grouped together, as can be seen from its low range.

Therefore, we can say that the students in maths Group 1 are all of asimilar standard, whereas maths Group 2 has some higher and lower abilitystudents, but on average they are the same.

67 72 71 65 74 72 69

50 53 90 85 63 79


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Page 4.42

Lesson 5/6Handout 2

Page 1 of 3

Scatter diagrams

For a scatter diagram you need two sets of numerical data.

For example:

Choose the scales for your axes so that the values are spread out evenly.Choose Height from 0 to 20 and Weight from 40 to 80 Note: you do not have to start at 0,if you go from 0 to 80 then the values will be too bunched up.

Scatter diagram plotted from the above data

Mark each pair of values with a small x using a sharp pencil.

Height (cm)

70

60

50

50 10 15 2040

80

W e i g h

t ( g )

Height (cm)

Weight (g)

11

64

10

65

11

64

12

60

13

63

16

68

17

72

15

63

12

63

14

67

13

68

11

62

9

59

17

68

13

69

12

62

15

65

12

67


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Page 4.43

Lesson 5/6Handout 2

Page 2 of 3

Outcomes

Scatter diagrams

Diagram 1. Shows strong positive correlation Diagram 2. Shows strong negative correlation

Diagram 3. Shows weak positive correlation Diagram 4. Shows weak negative correlation

Diagram 5. Shows no correlation


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Page 4.44

Lesson 5/6Handout 2

Page 3 of 3

Scatter diagrams

Line of best fit

If your scatter diagram shows correlation then draw a straight line throughyour data which best fits the data.

You can now calculate the gradient (m) and Y-intercept (c) to calculate aformula linking weight with height.

In the case above the Y-intercept is approximately 45 (where the line of

best fit crosses the Y axis).The gradient is found from the triangle on the line of best fit. This couldbe any size but the bigger the better. The gradient is the vertical dividedby the horizontal distances.

In this case Gradient = (68-52) (15-5) = 16 10 = 1.6

So (using Y = mx + c):

Weight = 1.6 x Height + 45

This formula can now calculate the weight if you know the height

(approximately, as the line of best fit is only approximate).For example, if height is 18, we can substitute this in the formula:

W = (1.6 x 18) + 45 = 28.8 + 45 = 73.8

Height (cm)

70

60

50

50 10 15 2040

80

W e i g h

t ( g )


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Lesson 5/6Handout 3

Page 1 of 1

Cumulative frequency

This is used to calculate the median and interquartile range for a set ofgrouped data.

A new table needs to be drawn. The groups of continuous data are calledclass intervals.

Points are plotted at the end of each class interval.

Cumulative frequency graph plotted from the above data

Total frequency = 55 so median value is at 27.5 ( 1 2 of 55)

Lower quartile = 13.75 (1 4 of 55). Upper quartile = 41.25 ( 3 4 of 55)

Length (l) 0l


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Lesson 5/6Handout 4

Page 1 of 1

To show this you need to calculate the median and upper and lowerquartiles (see separate sheet).

If your values are:

Median = 32

Lower quartile = 26

Upper quartile = 42

Lowest value = 0

Highest value = 60

The box and whisker plot would look like this:

This is a pictorial way of showing the median as well as the range andinterquartile range. It does not matter how high the middle rectangle is.The interquartile range contains half of the values.

A box and whisker plot is a good way of comparing two differentdistributions using the same single-number scale.

0 10 20 30 40 50 60

Interquartile range

LQ M UQ

Box and whisker plot20

5030

10

40


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Page 4.47

Lesson 5/6Handout 5

Page 1 of 2

When working with a large amount of data you need a balance betweennot using everything (as this would take too long) and using enough datato represent the whole group (or population).

This is called a sample .

Examples:

If you have 100 people your sample size should be about 20, but youcould use all 100

If your group size is 1,000 then your sample size should be at least 50

If your group size is 1,000 from two different types then you wouldhave to take at least 25 from each to avoid bias

If your group size is 1,000 (700 from one group and 300 from another)then your sample of 50 would have to be 35 from Group 1 and 15 fromGroup 2 keeping the sample in the same proportion as the group

Remember: the bigger your sample, the more accurate your results.

You need to pick this sample carefully to avoid bias, and we try to usea random sample.

Closing your eyes and waving your pencil around before selecting is not

a good mathematical method.Scientific calculators normally carry a random button (see calculatorinstructions), which gives a decimal number to three decimal placesbetween 0 and 1.

Multiply this decimal by the number of pieces of data and round to thenearest whole number to get the piece of data required.

Random sampling


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Lesson 5/6Handout 5

Page 2 of 2

Random sampling

Example You have 1,156 pieces of data

Discard the last selection, as it has already been used. Continue until youhave the required sample size.

Random (RND) number

0.638

0.123

0.003

0.943

0.124

x 1156

737.528

142.818

3.468

1090.109

143.344

Data number

738

143

3

1090

143

maths-4-looking beyond the uk

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