The Cartesian
Plane Described
The Cartesian Plane that Descartes invented was very
similar to the mathematical diagram of the two skidoo paths.
Specifically,
the Cartesian plane is just a set of two lines that cross each other at right
angles. The vertical line is called 'y' (it is often also called "the y
axis") and the horizontal line is called 'x' (it is often also called
"the x axis"). Sometimes these lines are given other names, as well,
depending on the particular mathematical problem you are working on (similar to
nick-names), but they can also always be called the x axis and the y axis.
The
point where these lines, the x axis and y axis (together called 'the axes'),
cross is given the name "The Origin".
Each axis line has
points marked out on it, and each point is labelled with a number that
represents how far away it is from the origin.
To label a point out on the middle of the plane, rather
than a point directly on one of the axes, you give the length along each of the
axes lines that a person would have to walk to get to that point in the middle
of the plane. This is like telling a person, as in section 4.1, to get to a
point out in the middle of the two skidoo tracks by saying "walk along the
first skidoo track this far, and then along beside the next skidoo track this
far".
By convention,
you give the number on the horizontal (east west) line first, and the number on
the vertical line (north south) second. You put the numbers in brackets, and
separate them by a comma. So for example if there is a point that is 5 out
along the x axis and 2 out along the y axis, the point would get the label (5,
2). This strategy gives you a label for each point in the middle of the plane.
The origin gets the special point label (0,0) because at the origin you haven't
gone any distance in any direction yet. Similarly, a point that was 4 along
the x axis but no distance along the y axis would be given the label (4, 0)
because you've gone 4 along the x axis but haven't gone any distance along the
y axis.
Now you have
variable names for the two lines, a name for the point where they cross, number
labels for each of the points on the axes and labels for each of the points on
the Cartesian plane.
Exercise 1: Draw your own Cartesian Plane and
label all of the different parts.
Exercise 2: Show where the point (1,4)
is on the Cartesian Plane
Exercise 3: Show where the point (2, 3) is on the
Cartesian Plane
Exercise 4: Show where the point (3, 2) is on the Cartesian
Plane
Exercise 5: Show where the point (0, 6) is on the Cartesian Plane
Suppose
that two kids, Okalik and Mary are playing a game by throwing small rocks
into a field. Okalik is standing 3 meters to the right of Mary. Okalik throws
a rock and it goes straight out 5 meters. Mary throws a rock and it goes
straight out 5.5 meters.
Exercise 6: Draw a picture of this
situation.
Exercise 7: Draw a mathematical diagram of the situation,
using points and lines.
Exercise 8: Draw a mathematical diagram of the
situation using the Cartesian Plane. Label the points on the plane that
represent where the rocks fell using the point labelling system described
above.
Answers
Exercise 1: It should look the same as the first picture in this part, section 4.2.
Answers for Exercise 2 – 5
Answer for Exercises 6 and 7: This will vary depending on how you drew your picture and diagram. The main thing is that these exercises will help you with Exercise 8.
Answer for Exercise 8
The answer will depend on where youÕve put Mary. Suppose you put Mary as standing at the origin, (0,0). In this case, the diagram will look like:
