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