Math has never been my favourite subject during my
school days and I’ve always asked why do we need to learn Math. It would have
changed my mind and become more interesting if students have been taught how
Math can be applied in the real world.
Take for example a simple linear equation y = mx
+ C, given two points and we can find an equation of the line.
So what’s the big deal with finding an equation
of a line and what can it practically do in the real world ??
Here’s the answer.
Suppose you would like to plot a trigonometric function of “sine”, with an amplitude of 10, over a horizontal range of 0 to 360 degrees,
we would find the graph as tabled below
FIG 1
Note that the value on the X-axis is given the
same value as the angle θ since we can assign the positive range of the X-Axis
from 0 to 360. The Y-Axis represents the amplitude of the waveform.
However, plotting similar graph with Scratch is a
different game altogether. In Scratch, the screen has a resolution of
480x360 pixels (FIG.2), such that the X-Axis range from -240 to 240 instead of
0 to 360 as seen from FIG.1
FIG. 2 Scratch screen
How
do you plot the points on the graph across the screen where the X-Axis range
from -240 to 240 with a complete cycle of 0 to 360 deg ??? In other words, we
need to translate the whole range of 0-360 degrees proportionately to a range
of -240 to 240 on the X-Axis as shown below,
FIG.3
FIG.3
denotes that if the value of the angle is 0, it is translated to -240 on the
X-Axis, 90 deg to -120, 180 deg to 0 etc.
The
translation of one range of numbers to another is often called “mapping” or
“scalling”.
Let’s
revisit the linear equation,
Note that
the x and y in eqn(1) are not the points on the X-Y coordinates of the screen.
It’s just an expression of two range of numbers where y represents a set of
number and x represents another set.
In our
case, we would like to map the angle (0-360 deg) to the full range on the
X-Axis (-240 to 240). Hence y, being the desired output, represents the values
that lies on the X-Axis and x represents the value of the angles (0 to 360 deg)
to be mapped.
Re-writing eqn(1) with its min and max value, we have,
The min
and max values of the two sets of numbers are :
Let’s solve for m and C.
Subtracting
eqn(2) from 3, we have
Re-arranging
eqn(4) yields
Replacing m from eqn(5) in eqn(3), we can now solve the constant C.
Replacing
m and C in eqn(1) :
Re-arranging
the above equation,
We had
earlier defined that
Let’s
rename the above symbols to make our life easier, i.e,
Renaming y
in eqn(7) to x (the desired points on the X-Axis) and replace with the new
symbols, we have
We can now
plot the graph as shown below.
Below is
the Scratch script to plot a “sine” function with and amplitude of 150.
A block is
also created to map the above discussed two range of numbers.
And the graph on the Stage Area is show below
I’ll cover
more on the “mapping” subject in my next post, that will involve Arduino and
external sensors. Happy Scratching………
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