Cartesian Coordinates — Plotting on a 2D Plane

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates measuring distance from an origin (0,0) first horizontally (the x-axis) and then vertically (the y-axis). This is an extension of last week’s lesson on algebraic notation where we are identifying a location with the use of two coordinates (GPS; longitude-latitude; Ng3 chess notation; games like Bingo and Battleship; streets and avenues; north, east, south, west).

We attribute the first link from algebra to geometry to Rene Descartes, the French mathematician and philosopher after which the Cartesian Coordinates are named. Although the concept of negative integers is very new for them, many of the students are familiar with negative numbers because the Mathlete cheer at the end of each class is to pick a negative number and count from a multiple up towards zero (the first non-negative even number) after which they say “2, 4, 6, 8, who do we appreciate, Mathletes.” For example, we could start with -91 and count by 13 so “-91, -78, -65, -52, -39, -26, -13, ‘and we never say negative’ 0, 2, 4, 6, 8, who do we appreciate, Mathletes.”

I do not expect them to master negative integers this week but here is the way I introduced the use of the four quadrants of the Cartesian Plane: 

• x comes before y in the alphabet so we alway plot x first.
• the x axis is horizontal and the y-axis is vertical; H comes before V in the alphabet so we do horizontal before vertical
• now my favorite: we enter a building in the lobby at the origin (0,0). We want to go to point (3,10). This is the 3rd elevator bank and the 10th floor. Would we ever go to the 10th floor before we picked our elevator? NO. So, we first chose our elevator, in this case 3; then we push the 10th floor button and go up 10 floors.
• if our elevator is negative, we go to the left from the origin and if our elevator is positive we go to the right
• after we are in our elevator on the x-axis, we choose our floor: if we have a positive floor, we go up, if we have a negative floor, we go down.

The beauty of this analogy is that there is direct evidence for this in most buildings in Europe where the lobby is considered the zeroth floor and one floor up is called floor 1; the basement is called -1 and the second basement is called -2 and so on. This is fun to see on cruise ships as you can have as many as 7 basement levels and 7 positive floor levels.

This coordinate system is also used in many programming languages.

The children should practice writing coordinates from the pictures I provided. The first is a cube with an oblique pyramid. The second is the word MATH in the 1-4th quadrants. They should also try to draw a picture from the list of coordinates as well. 

The most fun is for them to draw pictures on their own and then list the coordinates in order of connecting the points. To go to the next level, they should then copy down those coordinates on to a blank graph page and copy these so their friends, parents, teachers, etc. can draw their picture from just the coordinates.

As extra credit, I challenged the older children to draw a circle on the graph. You will see one of the documents is a picture of 11 concentric-like circles for which they can find the coordinates.

My holiday present to the children will be a book of these coordinates that they can draw over the holidays.