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Modelling with GeoGebra

Gustavo Aguilar, Jul 19, 2017

Modeling work

Introduction

1. What is a maths exploration project?

What is a maths exploration project?

[justify]An essential part of the IB are the internal assessments, in this exploration students must put in practice what the have learned thought the IB courses adding their own thinking and creativity. Creativity is an essential part of the Maths internal assessment.[br][br]First of all we MUST take into account how are we going to be evaluated. A maths internal is not only about doing the maths right; it also has to do with communication, critical thinking and showing your own approach to problem solving. [br][br]These are the criteria by which you are going to be evaluated in the IB maths Standard internal. [/justify]

[justify]As you may have noticed there is a great impact of the communication, personal engagement and reflection in the Maths internal final grade. Remember that this internal assessment is 20% of your final IB maths standard grade.[br][br]The file that you will have to deliver to the teacher will be a [b]word document [/b]so you should include in the word document all the importan information gathered in this project. That document must be well organised and concise and concrete to achieve a good mark. Also you must do a well use of the equation editor and include pictures of what you are doing. Also every step and every calculation must be [b]relevant, well explained and justified. [/b][/justify][b][br][/b][center][b]So lets get it on ![/b][/center][justify][/justify]

2.

Model from an image

1. Model from an image
2. Using integrals

2.1.

Model from an image

[justify]One of the ways in which you can do the Maths internal assessment is looking for a Maths model of something in real life, in this case we will use a mixture of Maths and architecture because we are going to look for a model that fits the roof of aeropuerto internacional de Carrasco. [br][br]So first create a [b]word document[/b] with the title Maths Project "Aeropuerto de Carrasco" and create an introduction explaining what are you going to do. [br][br]As we said a Maths internal is not only doing calculations so you must create an introduction explaining that you are going to find a model for the roof of the airport maybe because you want to see if it was done following a maths model or maybe because you are fascinated about architecture but you must find a reason to do so in other to give coherence to your internal assessment. [br][br]Create the introduction explaining what you are going to do, find an aim for that to be done.[br]Introduce a picture of the aeropuerto, give some information about it.[br][br]Then continue below...[/justify]

[justify]In the following applet:[br]Insert 7 points in the roof of the aeropuerto. You must use the tool [icon]/images/ggb/toolbar/mode_point.png[/icon] (firs click on the picture, one example is done)[br]Then looking at the x and y coordinates in the algebra view copy them in the spreadsheet alongside.[br][br]The x-coordinate in column A and the y-coordinate in column B. (an example has been done)[/justify][br]

Then select the two culumns (A and B) and press the tool [icon]/images/ggb/toolbar/mode_twovarstats.png[/icon](two variable analysis)[br]Select the regression model that you think will be the best fit for this roof.[br][br]Now introduce the algebraic model that GeoGebra found to check if the model really fits. Be careful to input the model as a function: use [math]f\left(x\right)=[/math] instead of y=.

You should get something similar to this.

[justify]Obviously there is a way of getting the model that does not requires the software. Once we have some coordinates of the points we can find the model by some algebraic means. [br][br]For example if we think of a third degree function we could have used simultaneous equations to obtain the coefficients. [br][br]Also we can use the vertex form of the equation. Estimate the vertex, fin any other point and we one simple equation we have to model. Remember the vertex form is [math]f\left(x\right)=a\left(x-h\right)^2+k[/math] being [math]h[/math] and [math]k[/math] the coordinates of the vertex. [br][br]An example of this:[br][br]Estimated vertex: (7.15,1.67)[br][br]Another point in the graph: (1.72, 1.16)[br][br][math]f\left(x\right)=a\left(x-h\right)^2+k[/math][br][br]Substitute vertex coordinates: [br][br] [math]f\left(x\right)=a\left(x-7.15\right)^2+1.67[/math][br][br]Substitute coordinates of point C:[br][br][math]1.16=a\left(1.72-7.15\right)^2+1.67[/math][br][br][br]Solve equation to get [math]a[/math]:[br][math]1.16-1.67=a\left(1.72-7.15\right)^2[/math][br][math]-\frac{0.51}{\left(1.72-7.15\right)^2}=a[/math][br][br][math]-0.0173\approx a[/math][br][br][br][math]Model:f\left(x\right)=-0.0173\left(x-7.15\right)^2+1.67[/math][/justify]

This model looks like this.

[justify][b]Continue the WORD document:[/b][br][br]Explain the equations or the process you did to find the model, put every equation in maths format using the equation editor. Put a picture to show how you model fitted the roof. [br][br]Be sure to include all details about the process.[/justify]

More maths in the internal.

[justify]To include and use in a very good way some more of the maths seen in this course we are going to use integrals to find the area under this curve. This obviously must be justified in the aim of the project. For example we are going to calculate the area to see how many square meters of glass will be needed to close the front of the airport. [br][br]To calculate the integral we must find the roots of the function because in this case, the picture was uploaded in a clever way fitting it in the x-axis so now calculations are a little bit more easy. [br][br]In the input bar write: 'roots' or 'raíces' depending on the language. Then introduce the required data and you will have the roots. (In the real internal assessment you will have to do this algebraically.)[br][br]Then in the input bar write: 'integral' and complete the data required. You will now notice that GeoGebra has calculated the area below the curve. So we can use that data in our project. (Latter in this course you will be able to calculate this algebraically with no need of this software.[br][br][b]Continue the WORD doc:[br][br][/b]Complete the word with this picture, the new calculations, why was this done? what it means? how this helps you to do the project, etc.[br][br]Now think about the scale of this project. Are you using the real scale? how can you arrange it to fit reality?[br][br]Create a final [b]conclusion [/b]of what you did so far, does this model present some limitations? which limitations? can it be improved? how? did you applied maths? how? [/justify]

2.2.

3.

Model from a set of data

1. Model a set of data
2. Logistic model

3.1.

Model a set of data

There are different ways to model a set of data. In this task we will see how to create a model with the tools that GeoGebra has. [br][br]Remember that in the internal assessment you will have to perform algebraically most of this processes.

Comenzamos trabajado...

In the applet below you will find attached the US. Dollar/Uruguayan peso Exchange rate for some months.[br][br]Select columns B and C and then right click > create > list of points. the coordinates [math]\left(B_i,C_i\right);i=1\cdots14[/math] will appear.[br][br]What type of function can model this data?[br][br]Select columns B and C again and with the tool [icon]/images/ggb/toolbar/mode_twovarstats.png[/icon] Find the model that you think will fit the most.[br][br]Write your model in the input bar and check if it really fits your model.

Predic the exchange rate for Dicember 2016.

State some limitation that you see of the model.

Now look for a model using the sine function ([math]Asen\left(Bx-C\right)+D[/math] ) [br][br]Use the coordinates that GeoGebra provides but do it yourself. Do not use any GeoGebra has to analyze the data,[br][br]Input your model in the input bar.

In July 2016 the exchange rate is increasing or decreasing?

3.2.

4.

Hands on

1. Ponte Vecchio
2. Fictitious Sales

4.1.

Ponte Vecchio

Busca una imagen del puente Vecchio en Florencia, Italia[br][br]Inserta la imagen en GeoGebra y ubícala de manera que sea conveniente para trabajar con ella. Herramienta: [icon]/images/ggb/toolbar/mode_image.png[/icon] [br][br]Ponla como imagen de fondo. (propiedades de la imagen)[br][br]Busca un modelo para el arco central y encuentra el área que debe tener una bandera para cubrir ese arco.[br][br]Sabiendo que en la realidad ese arco central mide 30 metros en su parte más ancha calcula el área de la bandera a escala real.