# Coordinate Grid Location: Descartes, Random Monsters, Shapes, and Pictures in 1 and 4 Quadrants

After learning the wonders of chess and recording games with algebraic notation, the natural progression was to explore the origins of coordinate grid location. The coordinate grid of an horizontal x-axis and vertical y-axis in two dimensions was proposed by the French mathematician René Descartes (1596 - 1650). (Pronounced "day CART"). He proposed further that curves and lines could be described by equations using this technique, thus being the first to link algebra and geometry. In honor of his work, the coordinates of a point are often referred to as its Cartesian coordinates, and the coordinate plane as the Cartesian Coordinate Plane.

Using Cartesian Coordinates you mark a point on a graph by how far along and how far up it is. So for example, to find the location (3,5) [we simply say “3  5”], you first locate the position of 3 on the horizontal x-axis, then you move from the zero position on the vertical line through x=3 until you reach 5 units up (if you look to the left, you will see 5 on the vertical y-axis).

For time immemorial, students have been mixing up the location (3,5) with the location (5,3) with little rhyme or reason. Sometimes they get it right and sometimes not. Guessing a location can be very disruptive in so many ways.

Wielding little more than a pencil, a slide rule and one of the finest mathematical minds in the country, Katherine Johnson calculated the precise trajectories that would let Apollo 11 land on the moon in 1969 and, after Neil Armstrong’s history-making moonwalk, let it return to Earth. A single error, she well knew, could have dire consequences for craft and crew. Her impeccable calculations had already helped plot the successful flight of Alan B. Shepard Jr., who became the first American in space when his Mercury spacecraft went aloft in 1961. The next year, she likewise helped make it possible for John Glenn, in the Mercury vessel Friendship 7, to become the first American to orbit the Earth. Ms. Johnson died at 101 on February 20, 2020 at a retirement home in Newport News, Va. She was one of a group of black women mathematicians at NASA and its predecessor who were celebrated in the 2016 movie “Hidden Figures.”

I know that your children are not responsible for sending rockets to the moon just yet, but maybe in a few years …. So, I devised a method for making location easy and memorable. I call it the (elevator, floor) method of plotting location on a graph. Think of walking into a building on the zeroth floor (incidentally, in Europe, most buildings number the lobby floor as zero) and choosing your elevator by number. Then you get in the elevator and choose your floor location. So for location (12,5), you get into the 12th elevator in the horizontal zeroth floor lobby and hit the 5 button to go up 5 floors. For location (0,6), you get into elevator zero and push the 6th floor; for location (-4,2), you walk a little to the left of the zeroth elevator to find elevator number -4 and push floor 2; likewise, to find location (8,-5), you get into elevator 8 and hit the -5 button sending you down 5 floors into the basement. Again, in Europe, floors are numbered zero for the lobby, positive 1 for the first floor up (floor 2 in the United States), and negative for basement floors (you will literally see -1 for the first basement floor, -2 for the second and so forth).

On the x-axis, values to the right are positive and those to the left are negative. On the y-axis, values above the origin are positive and those below are negative. A point's location on the plane is given by two numbers, the first tells where it is on the x-axis and the second which tells where it is on the y-axis. Together, they define a single, unique position on the plane. Note that the order is important; the x coordinate is always the first one of the pair.

I took the children through the lobby of One World Trade Center, the tallest building in North America and the 7th tallest building in the world. It has 71 elevators. We explored going to several locations in this building such as (1,100), (17,0), (3,-8), and (-1,5). They especially liked location (17,0) since you got into elevator 17 and just hit the hold button and went nowhere. Just remember, you had to first GO to elevator 17 in the lobby and that IS a specific location.

I then showed them a location on a 1st Quadrant map where treasure is buried. If they make the common mistake of switching the x and y values, they dig up dirt. With the older students, we used a 4 Quadrant Coordinate Plane with positive and negative numbers and found that there are potentially 7 possible mistakes that can be made in finding the treasure. So, bottom line, be careful to use my (elevator, floor) metaphor method.

We then explored pictures of squares, triangles, kites, rhombuses (or rhombi), trapezoids, octagons, hexagons, pentagons, and dodecahedrons and the children were tasked to plot the points that would complete the pictures. They noticed that they had to plot one more point then the number of vertices (points) to complete the diagram. For example, to plot a square, they would need to show 5 coordinate points to complete the square.

For the older students, the answer key also shows how to find the area of these figures using the coordinate plane, breaking the figures up into rectangles and triangles. Remember, all area is in square units not linear units.

NOW FOR THE FUN PART (for me at least): what would happen if you plotted random points and connected them? With the first quadrant children, we used dice and the site https://www.piliapp.com/random/dice/?num=2

to generate random coordinates with rolling two dice. Of course, it is even more fun for the children to roll dice on their own. How often I hear children say, “I don’t have dice.” I ask them if they have Monopoly, Yahtzee, Backgammon, etc.

Rolling ten times and then recording the first roll at the end, the children and I generated a figure that looked to me like a dog so I called it “Dice Dog.” Then we created our own and it looked something completely different.

For the older kids (3rd grade and up) I used a site called https://www.random.org/sequences/?min=-10&max=10&col=2&format=html&rnd=new that I set up to create two columns of 10 coordinate points with negative numbers from -10 to 10. Our first try, we created Random Pelican, but of course, every subsequent design will be different.

We even looked at the probabilities of making a roll. I had the children write down their dice roll and then rolled the dice on the random number site. From time to time, one child would have picked that roll. We looked at an image of all of the 36 possible rolls so the probability is 1/36 to generate any combination (1/6 times 1/6 = 1/36). With the negative and positive numbers the probability of choosing the random coordinate is 1/441 since there are 21 digits possible and 1/21 times 1/21 is 1/441.

I hope they create dozens of random monsters as I call them (as Forest Gump said, “life is like a box of box of chocolates, you never know what youre going to get”).

Finally, I shared with them my favorite non-random monster from Monsters Inc., Mike. I created the picture on 4 Quadrants and then recorded the 10 different pictures with coordinate points writing STOP after each picture (so the Head was a septagon with 7 vertices requiring 8 coordinate points, then I wrote STOP).

I challenged the children to look at the ear on the left and determine how many coordinate points were needed to make it. Most said 4 since a triangle needs three vertices and then back to the first one; however, other students pointed out that it only required three coordinates, since the Head already provided the third side.

Finally, we looked at the triangles created by the empty space between Mike’s arm and his head and I had them physically do the with their own arm. With the older children we analyzed those triangle to see if they were isosceles (triangles with at least two equal sides). Finding the length of segments on a coordinate plane that are horizontal or vertical are easy; otherwise, you need the power of Pythagorus to find the length (MORE IN A FUTURE LESSON).

If your children create Random Monsters or other pictures, I will distribute their coordinate points to the other Mathletes to create. Send me good images if you can.

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