Students and Teacher discuss the notion of setting up shooting area and collecting data:
How might we measure height and distance of shots from release point with accuracy?
How might we record the shots for future analysis?
How many shots may be needed to accumulate sufficient data?
How might we determine the functional curve of the shots?
Students collaborate for 5 to 10 minutes and we return to group discussion:
Firm decision is made on graduating a board both horizontally and vertically
Board prep is begun by a selection of students (or done overnight by teacher)
Video recorder (phone, iPad, etc...) is set up and students ensure visual parameters
A set of 6 uniformly made paper balls are created (suggest 3 shots per student)
PART TWO -
Students are chosen to shoot 3 shots at the trash can and their efforts are video recorded:
Each student shoots from the designated location their three shots
Teacher gathers paper balls and calls next student
This process is repeated until ALL students have taken three shots
NOTE: Do Not Allow Criticizing or Negativity while students are shooting
View selected successful shots from video as a class and analyze the data frame by frame:
Distribute (or display DATA TABLE TEMPLATE for student generation) of data table blank form
Confer and concur to build student confidence in their analysis skills
EACH STUDENT creates their own data table for this exercise
It is recommended that you chose only 10 successful shots for analysis
If you DO NOT have the resources to video and/or display... here is a video you can view and/or assign to be viewed at home. The dilemma is you do not have the control over the data collection methods when students attempt on their own. Click HERE for video sample.
PART THREE -
Students plot the data points on provided graph paper or LAB REPORT TEMPLATE:
Each output point must be converted to decimal feet from feet and inches
Each plot must be VERY accurate to represent true trajectory as best as possible
Draw a SMOOTH curve through the points (Best Fit)
Inspect curves...What do you notice? What type of functions MIGHT these curves represent?
Students explore mathematical implications of the shape:
Consider the shape of the curves... is there a Maximum value it achieves?
Consider the shape of the curves... can you determine the extremes (starting and ending points)?
Is it possible to determine the "coordinate points" of these previous features we just located?
Determine what those coordinate values might be for each shooter's curve (trajectory).
Students explore mathematics and manipulate equations: (Use calculation template as necessary)
Might this curve be part of a larger circle? . . . How might we check this hypothesis?
~ What might I say about the relationship between tangent lines and circles?
~ What might I say about the relationship between tangent lines and the radii of circles?
~ How might we create lines tangent to our curves to use radii to find the center of our proposed circle?
~ How might we use a compass (or a piece of string) to test or assumptions on our curves?
Could the FOUR QUESTIONS method of predicting functional shapes be of use here? (inspection and analysis)
~ How might we find the highest value (Local Maxima) by inspecting the curve? ~ How might we assign a "coordinate" value to this point (Local Maxima)? ~ How might we assign "coordinate" values to the beginning and end points (Extrema)? ~ Write the Master Equation for a function we learned in the FOUR QUESTIONS exercises.
Could the FOUR QUESTIONS method of predicting functional shapes be of use here? (equation justification)
~ Is the function even or odd? <Is this function based upon a line or a curve? ...check Extrema... ~ Is the function positive or negative? <y is what? what sign is the lead coefficient? ~ How many curves? <how does the number of curves relate to the exponent? ~ ( h ,k ) = ( ?, ? ) <what point do you think represents that "center of attention" on these curves?
How might I calculate for the exact "a" value for all of these various curves? (equation refinement)
~ Begin with the Master Equation for a function we learned in the FOUR QUESTIONS exercises. ~ Looking at the "a" in the equation... Is it alone? What is happening to it? How do I undo it? ~ Using inverse relationships to undo... What undoes addition? What undoes an exponent? ~ Solving for "a" should take around 3 steps to complete... Now you try...
These new equations are now written in Vertex Form ... Just like our "Master Equation".
Extension Activity One:
PART FOUR -
Students consider release angle as a function of curve shape:
Begin with the Master Equation for a function we learned in the FOUR QUESTIONS exercises.
How might we expand the (x - h) binomial and rewrite the equation?
How might we determine which are like terms based upon our knowing input the variable?
Organize your new expression into Polynomial Form. y(x) = . . .
Given y(x) = ax^2 + bx + c consider the following questions:
~ Which variable (or cluster) in your equations might you predict to correlate to the "a" term? ~ Which variable (or cluster) in your equations might you predict to correlate to the "b" term? ~ Which variable (or cluster) in your equations might you predict to correlate to the "c" term?
Now use this insight to convert all of your vertex form equations into polynomial form equations.
Write the polynomial form equation under the graph of each curve...right below your vertex forms.
PART FIVE - A Calculus Interlude
APPLY the power rule to all polynomial forms (I give brief overview of Power Rule here) USE these derivatives (dy/dx) and input ZERO in for X ... then simplify and rewrite. THINKING ABOUT IT... WHY might I be using an input of ZERO? Which angle am I trying to find? Where will I find an input value of x = 0? NOW you have the simplified statement dy/dx = b
SINCE dy is a vertical change in distance... let's call it RISE SINCE dx is a horizontal change in distance... let's call it RUN
THEREFORE we might call dy/dx what?
Looking further at dy and dx... in a UNIT CIRCLE, a quadrant 1 reference triangle, the side opposite Reference AngleQ is vertical. SINCE vertical is a y direction attribute then one might say dy = opposite ALSO since horizontal is an x direction attribute one might say dx = adjacent
Looking at this relationship what might we conclude dy/dx is equivalent to?
Which trigonometric function could dy/dx relate to then?
Let's discuss how this information could be used to help us calculate release angle...
- CLASS DISCUSSION -
PART SIX -
Students calculate release angle using inverse trig function:
Since dy/dx = opposite/adjacent... it therefore follows: tan Q = dy/dx
If I want to solve for the angle Q ... I must use: ArcTan (dy/dx) = Q
REMEMBER: Since I used the x input of ZERO when evaluating the derivative this is the release angle.
Write the Release Angle for each curve within the provided area on the LAB REPORT TEMPLATE
DRAW in a release angle tangent line arrow to represent the release angle. Label the arrow.
Let's discuss if these are the only things that matter in our shots... Release angle and trajectory? - CLASS DISCUSSION -