Idly wondering about the tangents from a point to a circle I constructed the figure below. The point A is on a circle and can be moved round the circle. The chord CD then moves around. The interesting thing is to see the envelope of this chord as the point A is moved. ..
Circles and hyperbolas galore, but when the big circle passes through the centre of the smaller circle the envelope is the surprise parabola. The question is “How do I find the equation of such a parabola? Sensible choice of origin and axes is the first thing. the y-axis is best as the line joining the centres of the two circles. Then you need an equation for the line CD, with a suitable parameter. Then a little bit of calculus………..Nice one for calculus students
Constructions done with my web based program. Try it yourself: GEOSTRUCT
And now for January in Puerto Rico. I don’t know the name of the pink and white flowering tree, but I do know that without my machete the whole garden would be infested with babies from the first one.
This one below is a yellow eleconia, also spreading madly.
And this is a cute little tree with probably poisonous berries.
And what the hell have they done to the post editor. It only works in BOLD
Glaring omissions to me, that is.
The obsession with Al Gebra and manipulations has used up loads of time which could have been spent on
The sudden appearance of the word “parameter” in High School :
“Interpret expressions for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context.”
The idea of a parameter is basic to the study of functions and relationships. At the start the equation y = mx + b has four letters in it. x and y are variables. What on earth are m and b? Numbers? Fixed numbers? Variable numbers, but not as variable as variables? No, they are parameters for the line. For a given line they are fixed, but for different lines one or both are different.
(When I was at school we, that is the kids, used to call them “variable constants”)
2. Parametric representation of curves and relationships.
For example a circle. With parameter θ a point (x,y) on the unit circle is described by x = cos(θ), y = sin(θ)
and a parabola, parameter a, point on curve given by x = a, y = a2
and for a lot of curves the only neat way.
It also allows for ease in programming graphics of curves.
3. Polar coordinates. The ONLY mention of the word “polar” is with regard to representation of complex numbers. With no way of simple plotting them ?????
How about the function representation of a circle as r = 2 ??
There are others!
It was admitted at the time of development of the CCSSM that too much time was spent on K-8, and HS math was a rough job – so why can it not be modified ???????