While Physical and Conceptual models are interesting to look at,

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While Physical and Conceptual models are
interesting to look at,

• They wont represent any numerical data that can
  be used to make predictions about the future
  behavior of the object.

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Physical and Conceptual models are useful, but
they are only the first step towards making a
model that can make predictions.

• The next step is most often a graph.
• A Graphical Model shows how two variables are
  related with a picture that is easy to
  understand.
• Because this type of graph uses numbers it is
  also called a Mathematical Model.
• There are many different types of graphs.
• They fall into four general forms.

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Line Graphs
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Bar Graphs
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Pie Graphs
Student and faculty response to the poll 'Should
Avenue High School adopt student uniforms?'
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Pictographs
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In this class we will be using line graphs more
than any other type so we are going to look at
them more closely then the other types.

• In the line graphs we use a series of lines to
  represent the two variables that are related in
  any given experiment.
• These lines show where the information from the
  experiment intersects.
• This information is plotted according to whether
  the variable is dependant or independent.
• Because the line graphs uses a coordinate system
  of lines we are mapping the numbers out
  according to where they are found on each
  opposing AXIS.
• There are two axis on any line graph, and they
  are known as the X and the Y axis.
• The X axis is found along the bottommost line
  of the graph or horizontal line, and is known as
  the Independent Variable.
• The Y axis is found along the far left side of
  the graph and it is the vertical line also known
  as the Dependent Variable.

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What does that mean Dependent and Independent?

• When something is Independent it means that it
  doesnt rely on anything else in order to achieve
  its results.
• Time is probably the best example of an
  Independent Variable.
• No matter what you do, Time Marches On!
• And you cant change units of measurement.
• For example you cant change the order of a
  distance measurement, that is to say 3 feet
  follows 2 feet not 4!
• And when something is Dependent it means that it
  does rely on something else in order to achieve
  its results.
• Actions are usually found on this axis.
• Such as Speed and Temperature.

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So why use a graph such as the types that we have
been discussing?

• One purpose of making a graph is to organize the
  data into a model that you can use and make sense
  of in order to make predictions.
• Pictures are easier to understand than data
  tables with just numbers in them.
• This is where a graphical model comes into play.
• It shows the exact relationship between the
  variables.

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Graph vs. Data Table
Temperature (deg C) Temperature (deg C) Mean Impact Energy (joules) Mean Impact Energy (joules)

20   70.4  
100   77.3  
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In many experiments we are looking for a cause
and effect relationship.

• That is to say, How does changing one variable
  change the other?
• Graphs are good ways to see whether there is a
  connection between two variables or not.
• You cant always see the connection with a data
  table, but with a graph the connection is always
  clear!
• When there is a relationship between the
  variables the graph will show a clear pattern.

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Does there have to be a relationship between the
variables?

• The answer to this is NO!
• This will show up as a random, scattered bunch of
  data points.
• This information should never be thrown out!
• It is meaningful!
• It could mean that you messed up something and
  have to try again, or it could mean that the two
  variables really dont have any relationship to
  one another at all!
• For example if you plotted the number of homes in
  Sylvania with 3 bedrooms vs. the number of
  students in this class that have blue eyes.
• At any rate it will tell you something!

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Now, when there is a pattern between the two
variables the pattern will be very self-evident.

• These relationships can be strong or weak
  depending upon how one variable effects the
  other.
• In a Strong relationship, a small change in one
  variable will cause a big change in the other.
• In a Weak relationship, even a big change in one
  or the other variable wont have much of an
  effect on the other!
• What do these look like?

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Strong Relationship

• The following two charts show the strong
  relationship between the available energy and the
  time prior Tornadic development.
• The graph below shows the strong relationship
  between the value of the U.S. dollar and the
  price of gold.

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Weak Relationship

• In the following graph we see very little change
  in amplitude when frequency is changed.
• And then we can see how little the temperature
  will change in a given time frame.

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Some Relationships are Direct.

• This means that as one variable increases the
  other increases as well.
• For example if we plotted the amount of time
  spent on studying vs. the grades earned you would
  see a direct effect on the relationship between
  the two.
• You would start out with little time and low
  grades, and as the time spent studying increased,
  so would the grade!

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The graph below shows the direct relationship
between age, weight, and the amount of exercise.
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Some relationships are Inverse.

• This means that as one variable increases, the
  other variable decreases.
• For example if you graphed the amount of money
  that you started out with vs. the amount of money
  that you end up with, you will see an inverse
  relationship.
• You would start out with a lot and end up with a
  little! And the graph would be slanted downwards.

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A cars value goes down as the mileage goes up!
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And then you can have random scatterings of points

• In these situations you want to choose the best
  fitting line that represents as many of the
  points as possible.