The following instructions lead through steps that build linked numeric, graphic, algebraic, and geometric models for the flight of the ball in the video above.
The instructions are also available as a PDF document.
Note: A journal article paralleling the ideas presented in this workshop appeared in the June 2013 issue of the Ontario Mathematics Gazette and is available by clicking the following link.
Roulet, G., & Lazarus, J. (2013). Linking representations using GeoGebra. Ontario Mathematics Gazette, 51(4), 2833.
A Numeric Representation of the Ball’s Flight
Click: ball flight start.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_start.html if GeoGebra is not installed on your computer.
The Spreadsheet View in GeoGebra provides an Excellike spreadsheet. The numbers in columns A and B record the distances in feet, right and up from the point on the floor occupied by the ball in the first image (left end of the folded bleachers).
Graph Representation
Move the cursor to the graph pane and adjust the axes by:
Rightclick > Graphics View > Axes > x and set x axis 1 to 15
> y and set y axis 1 to 15 and set xAxis : yAxis = 1:1
Highlight the A and B columns of the spreadsheet:
Hold shiftkey and click on each column
rightclick > Create List of Points
Problems? Do the following to see the results of the above:
Click: ball flight graph.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_graph.html if GeoGebra is not installed on your computer.
The spreadsheet values and points are linked. Add 1.0 to any of the A column values and note the effect on the related point. Restore the A value by subtracting 1.
What type of curve might pass through the points?
Algebraic Representation: Equation in Standard Form: y=ax^{2}+bx+c
Click: View > Algebra View (to show Algebra window)
Click: View > Input Bar
In the Input Bar type y=3x^2 and Enter
This is obviously not the correct equation. We will adjust and attempt to make the curve fit the points.
In the Free Objects list doubleclick: y=3x^{2}
Modify the equation and check fit by Enter. Repeat to improve the fit.
A simpler approach to modifying the equation
In the Input Bar type:
a=1 Enter
b=1 Enter
c=1 Enter
y=a*x^2+b*x+c Enter
This puts the parameters a, b, and c in the Free Objects list and the equation y=x^{2}+x+1 or y=ax^{2}+bx+c in the Dependent Objects list.
Click (highlight): a=1 and use the up and down cursor arrow keys to adjust the parameter, a
Using this approach with all three parameters adjust the curve to pass through the points.
Alternatively:
Rightclick a > click Show Object
Use the slider for a to adjust the value of the a parameter
Repeat for b and c
Use the sliders to adjust the values of all three parameters and adjust the curve to pass through the points. If the range of values is not sufficient:
Rightclick on the slider > click Object Properties…
Adjust the minimum and/or maximum values
We now have linked numeric (spreadsheet) and algebraic (equation & graph) representations.
Algebraic Representation: Function in Vertex Form: f(x)=a(xh)^{2}+k
If you would like to retain an unexpanded form for the expression in x then function notation is required.
Create two new sliders for parameters h and k.
Click on the slider tool and select Slider
Click on the graph window where you want the slider. Change the label to ‘h’, and Apply.
Repeat to get a slider for parameter ‘k’.
We now have two more Free Objects h and k to use in a function
In the Input Bar type f(x)=a*(xh)^2+k and Enter
Use the sliders for a, h, and k to adjust this new curve to pass through the points.
Problems? Do the following to see the results of the above:
Click: ball flight alg curve.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_alg_curve.html if GeoGebra is not installed on your computer.
Reversing the Algebraic and Numeric Linking
Put a point on the curve for the function f(x)
Click: New Point
Click on the curve
Click the Move tool
Rightclick on the point and select Trace to Spreadsheet. Click and drag the point.
Problems? Do the following to see the results of the above:
Click: ball flight numeric plus.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_numeric_plus.html if GeoGebra is not installed on your computer.
The algebraic (equation & graph) and numeric (spreadsheet) representations are linked.
A Statistical Modelling Approach to Constructing an Algebraic Representation
For some courses a statistical modelling process may be appropriate for developing a function and curve that best models the data. GeoGebra 4.0 provides statistics tools including modelling by regression. Return to the initial version of GeoGebra by:
Click: ball flight start.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_start.html if GeoGebra is not installed on your computer.
Highlight the numeric data in the A and B columns of the spreadsheet. You do this by the usual method  leftclick on the top left cell, hold down the left mouse button, and drag down to the bottom right cell.
Click on the Analysis icon and in the drop down menu select Two Variable Regression Analysis
In the popup analysis window click on the Regression Model dropdown list and select Polynomial. We do this because the scatter plot appears to take the shape of of a parabola, the curve of a second degree polynomial. Degree 2 will automatically be selected since it is the smallest available. If you wish to select a higher degree use the dropdown list.
GeoGebra will use regression analysis to generate the best fitting model using a second degree polynomial (quadratic) function and display the function and its graph. In this case, since the points were selected to generate perfect data the model is a perfect fit.
In the analysis window click Options > Show Statistics. This gives us statistical information on the x (distance) and y (height) data and the R^{2} value of 1 tells us the model fit is perfect.
Additional information on using the model building feature of GeoGebra can be found on the Modelling Gas Prices page.
Building a Geometric (Conic) Model
Click: View > Spreadsheet (to hide Spreadsheet window and create space)
We will use the Euclidean focusdirectrix construction for a parabola. A parabola is the curve traced out by a point moving so that it is equidistant from a fixed point (focus) and a fixed line (directrix). (see the definition at: http://www.mathwords.com/d/directrix_parabola.htm)
Directrix:
Put a point on yaxis at approximately y=13: Click: New Point > Click on yaxis
Construct a line through this new point perpendicular to yaxis. Note the steps are opposite to GSP – select the type of construction then the objects.
Select: Perpendicular Line and click the new point and the yaxis
Focus:
Place another new point below the perpendicular (horizontal) line
Construct the Parabola:
Click the Conic option and select Parabola
Click: the directrix (horizontal line) and focus (new point below the directrix)
Using the Move tool, adjust the locations of the focus and directrix to create a parabola that passes through the points.
Select the parabola and rightclick. Select Equation y=ax^{2}+bx+c
Problems? Do the following to see the results of the above:
Click: ball flight geometric.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_geometric.html if GeoGebra is not installed on your computer.
The geometric (parabola) and algebraic (equation) representations are now linked.
Using the Algebraic Representation
At this point all the parabolas should coincide and pass through the points. If you are having problems:
Click: ball flight all.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_all.html if GeoGebra is not installed on your computer.
We could use any of the models to determine the point where the ball would have hit the floor a second time if there was no net, but the function representation, f(x), provides a direct route to calculating the xcoordinate or distance to the right.
Click Command…
Scroll down the list of commands and select Root
Adjust the contents of the Input Bar to look read Root[f(x)] and Enter
In the Dependent Objects, read the xcoordinate of the new root point.
Adding Images to Connect to the Problem Context
You can add images (gif, jpeg, jpg, tif, png, bmp) to GeoGebra constructions using the Insert Image tool. The image may be set as background, pegged to the axes system, or may be linked to any point on the graphics plane. With this we can connect the ‘quadratic function – parabola’ activity to the original basketball context. In the GeoGebra constructions linked below the picture of a basketball can be moved along the parabola to score a basket.
Click: ball flight bg parabola plus ball.ggb and Open if you have GeoGebra on your computer
or
Click: ball_flight_bg_parabola_plus_ball.html if GeoGebra is not installed on your computer.


Comments (0)
You don't have permission to comment on this page.