I provided my students with a laptop, which had Geogabra on it, and the following picture:
I then showed, on my iPad, how the rockets travels around the point. When the rope is cut, the rocket will fly tangent to the curve.
I used this in my calculus class and asked "Determine the exact location the rocket should be when the rope is cut"
Students determined the equation of the circle, found the derivative and made it equal to the slope between any point (x,y) on the circle and the co-ordinates of the mouth of the monster (or goblin).
**WARNING, this is a hard question**
After, I was thinking this could be used in lower classes, by changing to one the questions below:
allow students to use the tangent button on geogabra, and ask for the reference angle.
Determine the amount of time which should pass before cutting the rope.
There are always two tangent lines on a circle through an exterior point, are there two times you could cut the rope in the picture?
There are more I could think of.
My students enjoyed the activity, and feel free to change and adapt as you see necessary.
First provide the students with a laptop with Geogebra, and the following photo.
Introduce the meaning of Piecewise functions, and how the yellow bird, when clicked, shoots off at a tangent to the curve.
Ask the students to describe what are the two functions that create this curve (Parabola and a line).
Using prior knowledge have students graph the maximum point on the parabola, and use the dot where the yellow bird took off at a tangent (B), to create the equation of the parabola.Have students graph the parabola in geogabra, overtop the picture, to ensure all calculations have been done correctly.
Next, ask students how to determine the equation of line.
We will need either a) two points or b) a slope and a point.Both of which is impossible, without the use of the tangent button in Geogabra.I explained, we can calculate this without that button!
Have students pick another “point on” the parabola (c), and to calculate the slope between B and C.Ask how do we make this slope more accurate to the slope of the tangent…
Move C closer to B..
Have students move C closer to B, while still staying on the parabola and calculate the new slope.Eventually move C as close as possible to B.The slope should be -0.5Below is a picture of one of the students’ work
Next using the point B and the slope you can create the equation of line.
From here your choice for extensions: I had students graph the piecewise functions on their calculator and got the following image
Lastly, here is a video on how to determine the slope at a point: