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Mathematical Modeling

Learning Targets

1.05B - I can describe and sketch linear, quadratic, and inverse patterns and identify similarities and differences. (more about hand-drawing/identifying with real data on graphs)
1.06A - I can interpret linear mathematical models, including the meaning of slope, y-intercept, and individual points.
1.06B - I can interpret inverse or quadratic mathematical models, including the meaning of slope, y-intercept, and individual points.
1.06C - I can interpret any mathematical model, including the meaning of slope, y-intercept, and individual points.
1.07C - I can take a set of data and represent on a graph with the correct model.  I can follow the conventions for creating a graph and include error bars.

Can you...

  • Use Logger Pro software to perform graphical analysis of data.
  • Make test plots of data to find linear relationships
  • Write mathematical models, in standard y = mx + b form, for linearized data. Replace m and b with constants including units and replace y and x with dependent and independent variable names.
  • Provide interpretations for the physical significance of the slope and y-intercept.
  • Relate mathematical and graphical expressions
  • Use proportional reasoning in problem solving

Interpreting Graphs

  • Points:(x,y) => (I.V., D.V.) A point will ALWAYS give us a prediction of our dependent variable for a specific independent variable value
  • Slope: Slope ALWAYS tells us specifically how the dependent variable changes (increases/decreases) as we increase the independent variable,  but this relationship might not always be constant
  • Y-intercept: The y-intercept ALWAYS tells us about the dependent variable when our independent variable is ZERO
  • ​Conclusion: SLOPE is the difference​
Desmos - Exploring Relationships

Graphical Models

Linear
  • Points:(x,y) => (I.V., D.V.) A point will give us a prediction of our dependent variable for a specific independent variable value
  • Slope: Slope tells us specifically how the dependent variable changes (increases/decreases) as we increase the independent variable. BUT it looks like the slope is different at different places…
  • Y-intercept: The y-intercept tells us about the dependent variable when our independent variable is ZERO
Quadratic
  • Points:(x,y) => (I.V., D.V.) A point will give us a prediction of our dependent variable for a specific independent variable value
  • Slope: Slope tells us specifically how the dependent variable changes (increases/decreases) as we increase the independent variable. BUT it looks like the slope is different at different places…
  • Y-intercept: The y-intercept tells us about the dependent variable when our independent variable is ZERO​
Inverse
  • Points:(x,y) => (I.V., D.V.) A point will give us a prediction of our dependent variable for a specific independent variable value
  • Slope: Slope tells us specifically how the dependent variable changes (increases/decreases) as we increase the independent variable. BUT it looks like the slope is different at different places…
  • Y-intercept: The y-intercept tells us about the dependent variable when our independent variable is ZERO. BUT this looks like it might not cross the y-axis…
Flat-line
  • Points:(x,y) => (I.V., D.V.) A point will give us a prediction of our dependent variable for a specific independent variable value
  • Slope: Slope tells us specifically how the dependent variable changes (increases/decreases) as we increase the independent variable. BUT it looks like the slope is zero…
  • Y-intercept: The y-intercept tells us about the dependent variable when our independent variable is ZERO

Determining Fit

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"There is no science in this world like physics. Nothing comes close to the precision with which physics enables you to understand the world around you. It's the laws of physics that allow us to say exactly what time the sun is going to rise. What time the eclipse is going to begin. What time the eclipse is going to end."
​
-Neil deGrasse Tyson
  • Home
  • General Physics
    • Class Policies >
      • Grading Policy
    • Unit 1 - Patterns and Measurements >
      • Experimental Design
      • Data Collection
      • Mathematical Modeling
    • Unit 2 - Motion >
      • Definitions and Terms
      • Position Time Graphs
      • Velocity Time Graphs
      • Relating Different Representations
      • Problem Solving
    • Unit 3 - Forces >
      • Nature of Forces
      • Newton's Laws
      • Problem Solving
    • Leader Board >
      • Quarter 1
      • Quarter 2
  • AP Physics
    • Lab Portfolios >
      • Sample Lab
      • Hall of Fame
    • Patterns and Measurements
    • Kinematics
    • Forces
    • 2D Motion
    • Energy
    • Momentum
    • Rotation
    • Electrostatics & Circuits >
      • Electric Charge
      • Electric Potential
      • Circuits
      • Ohm's Law
      • Kirchoff's Rules
  • Explore Something New
  • About
  • Contact
    • Digital Portfolios