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
3
LessonsModelling
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
LessonsModelling
What is modelling? Why modelling?
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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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What is modelling? Why modelling?
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Solving some tasks.
Activity 1
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What is modelling? Why modelling?
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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
What is modelling? Why modelling?
9
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
What is modelling? Why modelling?
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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
LessonsModelling
What is modelling? Why modelling?
11
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
What is modelling? Why modelling?
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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?
LessonsModelling
What is modelling? Why modelling?
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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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What is modelling? Why modelling?
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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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What is modelling? Why modelling?
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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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LessonsModelling
What is modelling? Why modelling?
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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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What is modelling? Why modelling?
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“Real world” “Mathematical world”
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
“Real world” “Mathematical world”
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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LessonsModelling
What is modelling? Why 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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What is modelling? Why modelling?
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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
Task 3 NoEstimation, arithmetic calculations,
measures, geometry
Task 4 NoEstimation, arithmetic calculations,
measures, algebra, functions
Task 5 Yes Arithmetic
Task 6 No Estimation, arithmetic calculations
LessonsModelling
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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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What is modelling? Why modelling?
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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
33
LessonsModelling
What is modelling? Why modelling?
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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.
Modelling (mathematization) process – PISA framework 2003 – p. 39Modelling (mathematization) process – PISA framework 2003 – p. 39
The modelling cycle (PISA, 2003)
LessonsModelling
What is modelling? Why modelling?
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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
Modelling (mathematization) process – PISA framework 2003 – p. 39Modelling (mathematization) process – PISA framework 2003 – p. 39
The modelling cycle (PISA, 2003)
LessonsModelling
What is modelling? Why modelling?
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Examples
Task 1: Signing against a new law
Task 2: Heartbeat
LessonsModelling
What is modelling? Why modelling?
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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)
“Real world” “Mathematical world”
Tasks 1Tasks 1Modelling task (all the cycle and steps have to Modelling task (all the cycle and steps have to
be considered)be considered)
“Real world” “Mathematical world”
1 2
3
1 2
3
44
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
LessonsModelling
What is modelling? Why modelling?
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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)
“Real world” “Mathematical world”
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)
“Real world” “Mathematical world”
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
40
LessonsModelling
What is modelling? Why modelling?
41
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
What is modelling? Why modelling?
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
What is modelling? Why modelling?
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
What is modelling? Why modelling?
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
What is modelling? Why modelling?
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: