modelling tasks lessonsassessment reflecting. what is modelling? modelling tasks lessonsassessment...

ModellingModelling

Modelling Tasks Lessons Assessment Reflecting

What is modelling?

ModellingModelling

Modelling Tasks Lessons Assessment Reflecting

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Session 1Insight into modellingInsight into modelling

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What is modelling? Why modelling?

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You will:

• Work on different reality-based tasks.

• Reflect on the features of the tasks.

• Think about criteria to identify modelling tasks from other reality-based tasks.

Objectives

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• Criteria to identify modelling tasks

• Overview of the modelling process

Outcomes

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Activity 2

Reflection on

the features of

the given

situations

Activity 3

Sharing

reflections

[Small groups] [Small groups] [Whole group]

Activity 1

Working on

the given

situations

Activity 4

Developing

criteria

[Whole group]

Session structure

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Solving some tasks.

Activity 1

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Task 1: “Signing against a new law”Recently, the 25th of April of 2006, the Spanish party in the opposition presented in the

congress 4.000.000 signatures against a new law promoted by the government.

All Spanish newspapers published pictures with the big boxes and the 10 vans needed to

transport the sheets of paper to the congress. Do you think there was a political intention

behind this staging or all these boxes and vans were really necessary to carry the 4000000

signatures?

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For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency.

For years the relationship between a person’s recommended maximum heart rate and the person’s age was described by the following formula:

Recommended maximum heart rate = 220 – age

Recent research showed that this formula should be modified slightly. The new formula is as follows:

Recommended maximum heart rate = 208 – (0.7 x age)

A newspaper article stated: “A result of using the new formula instead of the old one is that the recommended maximum number of heartbeats per minute for young people decreases slightly and for old people it increases slightly.”

From which age onwards does the recommended maximum heart rate increase as a result of the introduction of the new formula? Show your work.

Task 2: “Heartbeat”

Retrieved from www.pisa.oecd.org/dataoecd/46/14/33694881.pdf

http://www.pisa.oecd.org/dataoecd/46/14/33694881.pdf



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Task 3: Music festival

The Glastonbury Festival of

Contemporary Performing Arts

is the largest greenfield music and

performing arts festival in the world. For

2005, the enclosed area of the festival was

over 900 acres (3.6 km²), and had over 385

live performances. Many of the festival

goers carry their own tents to sleep inside

the festival area.

Organisers needs to limit the number of tickets and the number of tents allowed

in order to guarantee the security. What advice would you offer?

Thanks to Logan1138, published at Wikimedia Commons

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Task 4: Natural gasIn 1993 the worldwide reserves of natural gas were estimated to be 141.8 billion cubic metres. Since then 2.5 billion cubic metres have been used every year on average.

Calculate when the reserves of natural gas will be exhausted. Use different assumptions and models. Explain all your steps.

Picture: Thanks to Stan Shebs, published at Wikimedia CommonsTasks: © 2007 Cornelsen Verlag Scriptor – Mathematisches Modellieren

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Task 5: Easter eggs

Danielle found 23 eggs.

She smiled broadly because she had found nine more eggs than Chris.

Jennie smiled even more. She had found exactly as many eggs as Chris and Danielle together.

How many eggs did Jennie find?

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Task 6: Neighbours

In your opinion, how many people live in

this block of flats?

© Cornelsen Verlag Scriptor - Mathematisches Modellieren

Bell signs in the

entrance area:

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In groups, compare your solutions:

What are the differences?

What are similarities?

Record your ideas on the given charts(use a different chart for each task)

Activity 2

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Guidelines for reflection

ContextContext of of

the taskthe task

ContextContext of of

the taskthe task

MathematicalMathematical

knowledgeknowledge

involvedinvolved

MathematicalMathematical

knowledgeknowledge

involvedinvolved

ExpectedExpected

solutionssolutions

ExpectedExpected

solutionssolutions

Main features ofMain features of

the solver’sthe solver’s

activityactivity

Main features ofMain features of

the solver’sthe solver’s

activityactivity

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What similarities/differences can you establish among these tasks?

Context Mathematical knowledge

Expected

solutions

Solver’s activity

Activity 3: Discussion

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Concerning the context of the task

Concerning the mathematical knowledge involved

Concerning the expected solutions

Concerning the solver’s activity

Some conclusions

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What features should a task have to be considered as a

modelling task?

In relation to:

Context?Context? Mathematical Mathematical knowledge?knowledge?

Expected Expected

solutions?solutions?Solver’s Solver’s

activity?activity?

Activity 4: Developing criteria

Session 2Describing the modelling processDescribing the modelling process

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You will:

• Reflect on the problem solving processes you used in Session 1.

• Summarise these processes in a common schema.

• Discuss a possible schema you could use to describe the modelling

process

• Learn about the modelling process

Objectives

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• Description of the modelling process.

Outcomes

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Activity 2

Sharing our

reflections

[Small groups] [Whole group]

Activity 1

Reflection on

your problem

solving

processes

Session structure

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Working in groups:

• Look on the tasks solved so far again

• How did you proceed to find a solution? Reflect on your problem solving processes on a general level

• Sketch a only diagram synthesising these processes

[ Task 1 – Task 2 – Task 3 – Task 4 – Task 5 – Task 6 ]

Activity 1

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Introducing a description of the modelling process

“Real world” “Mathematical world”

1 2

3

4

5

5

Real-world

problem

Real-world

problem

Mathematical

Problem

Mathematical

Problem

Mathematical

solution

Mathematical

solutionReal solutionReal solution

The modelling cycle (from the PISA study, 2003)

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Share diagrams

What similarities/differences can you establish among these?

Activity 2: Sharing reflections

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1 2

3

4

5

5

Real-world

problem

Real-world

problem Mathematical

Problem

Mathematical

Problem

Mathematicalsolution

MathematicalsolutionReal solutionReal solution

The modelling cycle (from the PISA study, 2003)

1. Starting with a problem situated in reality

2. Organising it according to mathematical concepts and identifying the relevant

mathematics

3. Gradually trimming away the reality through processes such as making

assumptions, generalising and formalising, which promote the mathematical

features of the situation and transform the real-world problem into a

mathematical problem that faithfully represents the situation.

4. Solving the mathematical problem

5. Making sense of the mathematical solution in terms of the real situation

1

2

3

4

5

Extended description… Examples…

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Important remarks

• The modelling cycle is not an algorithm

• On many occasions it is necessary to think ahead to the next step and

backward to a previous step

• You may need to go round the cycle several times to arrive at a

solution

• More than one solution is possible

• Many times the solution depends on the person working on the tasks


1 2

3

4

5

5

Real-world

problem

Real-world

problem Mathematical

Problem

Mathematical

Problem

Mathematicalsolution

MathematicalsolutionReal solutionReal solution

Extra SlidesSession 1: Insight into modellingSession 1: Insight into modelling

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Concerning the context of the task

Real and authentic? Interesting for students? Relevant for student’s

Task 1 Yes It could be Yes

Task 2 Not sure It could be It could be

Task 3 Yes It could be It could be

Task 4 Yes It could be Yes

Task 5 No Probably not Definitely not

Task 6 Yes It could be It could be

Back to conclusions

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Concerning the mathematical knowledge involved

Back to conclusions

Unique and completely determined in advance?

Promotes the use of different pieces of math knowledge?

Task 1 NoEstimation, arithmetic calculations,

measures, geometry

Task 2 Yes Linear functions


measures, geometry


measures, algebra, functions

Task 5 Yes Arithmetic

Task 6 No Estimation, arithmetic calculations

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Concerning the expected solutions

Back to conclusions

One or several?Nature of the expected

solution?Relation between the solution

and the initial context?

Task 1 SeveralA number, an interval, a

statementRelevant

Task 2 One A number Relevant

Task 3 Several Measures, intervals Relevant

Task 4 SeveralNumbers, intervals,

statements, functions, patterns

Relevant

Task 5 One A number Not relevant at all

Task 6 Several Numbers, intervals Relevant

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Concerning solver’s activity

Back to conclusions

To perform an “optimal and only” procedure?

To explore, make hypothesis, look for different ways of working, interpret and validate his/her

solutions,…?

Task 1 No Yes

Task 2 Yes No

Task 3 No Yes

Task 4 No Yes

Task 5 Yes No

Task 6 No yes

Extra SlidesSession 2: Describing the modelling process

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From the “problem in the real world” to the “mathematical problem” (1, 2, 3)

(horizontal mathematization, De Lange, 1987)

identifying the relevant mathematics with respect to a problem situated in

reality;

representing the problem in a different way, including organising it according to

mathematical concepts and making appropriate assumptions;

understanding the relations between the language of the problem, and symbolic

and formal language needed to understand it mathematically;

finding regularities, relations and patterns;

recognising aspects that are isomorphic with known problems;

translating the problem into mathematics; i.e., to a mathematical model

Modelling (mathematization) process – PISA framework 2003 – p. 39Modelling (mathematization) process – PISA framework 2003 – p. 39

The modelling cycle (PISA, 2003)

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Working in the “mathematical world” (4)

(vertical mathematization, De Lange, 1987)

using and switching between different representations;

using symbolic, formal and technical language and operations;

refining and adjusting mathematical models;

combining and interpreting models;

argumentation;

generalisation.



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Going back to the real world situation (5)

(interpreting and validating both the solution and the model)

understanding the extent and limits of mathematical concepts;

reflecting on mathematical arguments, and explaining and justifying results;

communicating the process and solution;

critiquing the model and its limits.

Back to presentation



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Examples

Task 1: Signing against a new law

Task 2: Heartbeat

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Example 1: Signing against…

Back to presentationText of the task

Tasks 1Tasks 1Modelling task (all the cycle and steps have to Modelling task (all the cycle and steps have to

be considered)be considered)


Tasks 1Tasks 1Modelling task (all the cycle and steps have to Modelling task (all the cycle and steps have to

be considered)be considered)


1 2

3

1 2

3

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55

55

Real-world

problem

Collecting signatures

Carrying them to the Congress

Are 11 vans really needed?

Real-world

problem

Collecting signatures

Carrying them to the Congress

Are 11 vans really needed?

Mathematical

Problem

How may sheets of paper?

What is the volume occupied

by n sheets of paper?

Mathematical

Problem

How may sheets of paper?

What is the volume occupied

by n sheets of paper?

Mathematical

solution

Arithmetic calculations

Calculating a volume

Mathematical

solution

Arithmetic calculations

Calculating a volume

Real solutionComparing volumes (n sheets

of paper vs. 11 vans)

Arguing about the situation

Real solutionComparing volumes (n sheets

of paper vs. 11 vans)

Arguing about the situation

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Example 2: Heartbeat

Back to presentationText of the task

Tasks 2 Tasks 2 Application task (steps 2 and 3 do not have to be Application task (steps 2 and 3 do not have to be considered: the mathematical model is provided)considered: the mathematical model is provided)


Tasks 2 Tasks 2 Application task (steps 2 and 3 do not have to be Application task (steps 2 and 3 do not have to be considered: the mathematical model is provided)considered: the mathematical model is provided)


1 2

3

1 2

3

44

55

55

Real-world

problem

Two math. models (linear) and a qualitative statement are given.

Which age onwards does the new model increase the recommended frequency given by the old one?

Real-world

problem

Two math. models (linear) and a qualitative statement are given.

Which age onwards does the new model increase the recommended frequency given by the old one?

Mathematical

Problem

Comparison of two functions:

x / 220 – x < 208 – 0,7x?

Mathematical

Problem

Comparison of two functions:

x / 220 – x < 208 – 0,7x?

Mathematical

solutionSolving a linear inequality:

x > 40

Mathematical

solutionSolving a linear inequality:

x > 40

Real solutionInterpreting this inequality in

terms of age and

recommended max. heart rate.

Real solutionInterpreting this inequality in

terms of age and

recommended max. heart rate.

Extra SlidesTasks

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LessonsModelling


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Task 1: “Signing against a new law”Recently, the 25th of April of 2006, the Spanish party in the opposition presented in the

congress 4.000.000 signatures against a new law promoted by the government.

All Spanish newspapers published pictures with the big boxes and the 10 vans needed to

transport the sheets of paper to the congress. Do you think there was a political intention

behind this staging or all these boxes and vans were really necessary to carry the 4000000

signatures?

LessonsModelling


42

For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency.

For years the relationship between a person’s recommended maximum heart rate and the person’s age was described by the following formula:

Recommended maximum heart rate = 220 – age

Recent research showed that this formula should be modified slightly. The new formula is as follows:

Recommended maximum heart rate = 208 – (0.7 x age)

A newspaper article stated: “A result of using the new formula instead of the old one is that the recommended maximum number of heartbeats per minute for young people decreases slightly and for old people it increases slightly.”

From which age onwards does the recommended maximum heart rate increase as a result of the introduction of the new formula? Show your work.

Task 2: “Heartbeat”

Retrieved from www.pisa.oecd.org/dataoecd/46/14/33694881.pdf

LessonsModelling


43

Task 3: Music festival

The Glastonbury Festival of

Contemporary Performing Arts

is the largest greenfield music and

performing arts festival in the world. For

2005, the enclosed area of the festival was

over 900 acres (3.6 km²), and had over 385

live performances. Many of the festival

goers carry their own tents to sleep inside

the festival area.

Organisers needs to limit the number of tickets and the number of tents allowed

in order to guarantee the security. What advice would you offer?

Thanks to Logan1138, published at Wikimedia Commons

LessonsModelling


44

Task 4: Natural gasIn 1993 the worldwide reserves of natural gas were estimated to be 141.8 billion cubic metres. Since then 2.5 billion cubic metres have been used every year on average.

Calculate when the reserves of natural gas will be exhausted. Use different assumptions and models. Explain all your steps.

Picture: Thanks to Stan Shebs, published at Wikimedia CommonsTasks: © 2007 Cornelsen Verlag Scriptor – Mathematisches Modellieren

LessonsModelling


45

Task 6: Neighbours

In your opinion, how many people live in

this block of flats?

© Maaß, Katja (2009): Mathematisches Modellieren im Grundschulunterricht. Cornelsen Verlag, Berlin© Maaß, Katja (2009): Mathematisches Modellieren im Grundschulunterricht. Cornelsen Verlag, Berlin

Bell signs in the

entrance area:

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