Video time

These video links will help you understand the topics we're discussing:

• https://www.youtube.com/watch?v=0T0z8d0_aY4
• https://www.youtube.com/watch?v=B22VtuU3koQ

Let's Explore!

An exploration is a detailed research paper where you discuss a certain problem related to a mathematics topic and try to solve it using technology, usually Geogebra.

We will show you 3 different explorations about modeling mentioned above but more detailed.

1.Height of a tunnel 

2. The Mosque Dome

My goal is to find a function that would help me model a moderate dome, designed for increasing the sound in a mosque, and also find the point where the sound is maximum and where the sound source should be placed, and then find the two pointes where the dome meets the supporting wall.                                                                 

 The parameter a of the function should be negative because the dome has a maximum value, not a minimum value.

 3. Flashlight Parabola

Linear Regression

Here's a new type of regression called the "Linear Regression"

What does linear regression mean?

In statistics, it is an advance or an approach that models the relationship between two variables by forming a linear equation. One variable is called an explanatory variable, and the other is considered to be a dependent variable. Before trying to form a linear equation we must first know whether there’s a relationship between the two variables.

What is the formula in which we use to check if there is a relation?

A linear regression line has an equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable. The slope of the line is b, and a is the intercept. In the case of which there’s one explanatory variable it is then called simple linear regression. For more than one explanatory variable, the process is called multiple linear regressions. 

Example 1: 

In simple linear regression, we predict scores on one variable from the scores on a second variable. The variable we are predicting is called the criterion variable and is referred to as Y. The variable we are basing our predictions on is called the predictor variable and is referred to as X. When there is only one predictor variable, the prediction method is called simple regression. In simple linear regression, the topic of this section, the predictions of Y when plotted as a function of X form a straight line.

Here's the table of data that we're going to plot.

Here are the results we got.

You can see that there is a positive relationship between X and Y. If you were going to predict Y from X, the higher the value of X, the higher your prediction of Y.

Linear regression consists of finding the best-fitting straight line through the points. The best-fitting line is called a regression line. The black diagonal line in the following picture is the regression line and consists of the predicted score on Y for each possible value of X. The vertical lines from the points to the regression line represent the errors of prediction. 

As you can see, the red point is very near the regression line; its error of prediction is small. By contrast, the yellow point is much higher than the regression line and therefore its error of prediction is large.

→ therefore, the more the point is higher the more its error of prediction is large, while if it's close to the regression line, its error of prediction is small.

Source:http://onlinestatbook.com/2/regression/intro.html

Another Type of Modeling

Here's a problem you can face; if you want to find an equation or a formula (like in the previous examples) that models certain values or numbers? What can you do? You won't be able to use Geogebra because you're trying to model the relation between some values, not a picture or a logo. 

Here comes the "REGRESSION"

So regression is the statistics that determine whether there's a relation between two variables. And if there's a relationship between them, regression helps us identify the strength of this relation and whether it's positive or negative. 

For example, you can use regression to know the relation between the height of a point from which you threw a ball and the time the ball took to hit the ground. You can find the relation by collecting data from the experiment. Then you plot this data and here you go! you get the relationship between the two variables and how strong or weak it is! 

This is how a regression looks like:

Example of a strong relationship between two variables.

Remember, sometimes you might not find the plotted points forming a line or even close to forming one, which means there's no relationship between the two variables you're investigating about. 

Some Real-Life Modeled Examples :D

If you're still confused about modeling and its uses , we're going to show you some real-life situations that have been solved by modeling.
1. If an architect wants to build a tunnel, he needs to know about         several points. First he needs to know the average height of               tunnels and the average height of trains that pass by the tunnel.         And finally he will need to form an equation for this tunnel.

    Here's the picture of the model using technology. (Geogebra)

Here's the function the architect detected:

f(x) =  - 0.06 x^2 + 1.3x +2.43

Using this function he can know the dimensions he needs in order to accomplish the bridge!

2. If a man wants to build a mosque, he surely has to build a dome       for it. Domes have lots of uses especially in raise the voice               produced under it. So the man needs to know the exact point on       the dome where the sound will be maximum (the center of the         dome.) 

Here's the equation found:

f(x) = - 0.03 x^2 + 0.81x - 0.06

Using this function and the model he created, now he can easily detect the center of the dome and the dimensions needed.

3. Now, we will show you a simpler example. If you want to                 calculate the equation of the parabola made by a flashlight.               Here's what you have to do: 

And you've just turned a flashlight into mathematics!

4. Let's have some fun and model a roller-coaster! Roller-coasters       make parabolas out of their rails. And you can model this                 parabola using Geogebra. Here's the modeling of the roller-               coaster.

5. Here comes our favorite part, McDonald's! Did you know that         even the McDonald's can be mathematically modeled? Here's its       model and the functions.  

Hope after these examples of different applications of modeling, you understand it and its uses clearly!

Let's start modeling!

• What is modeling?

Have you ever thought about how bridges, tunnels, buildings and even dog houses are built? Well the answer is modeling. Modeling is the "simplification" of reality. It helps in so many things, like:

1. Describing real world phenomena
2. Investigating important questions about the world
3. Testing ideas
4. Making predictions about the real world
5. Visualizing something we want to build in the future. 

This diagram can help you understand.

For example, an architect can use modeling in order to help him first visualize how his building can look like and second help him define the steps he needs to follow in order to accomplish his building. 

Even teenagers can use modeling! for example, if you want to design a T-Shirt with a logo on it that has a certain shape (ex: a parabola-shaped logo) you can model your logo first on the computer and calculate the dimensions needed and then, print it out! Enjoy!