Where Does a Graph Go?

Where Does a Graph Go? — page 3 — THE LINES MADE BY ROADS ON A MAP FORM A MATHEMATI CAL-LOOKING GRID 6

Where Does a Graph Go? — page 4 — COMMON LOCATORS MUCH OF THE MATHEMATICS THAT YOU DO INVOLVES WORK ING WITH A ONE-DIMENSIONAL SET OF NUMBERS Even when you add two numbers together the final sum is just one number The same holds true for the other operations that you do The result will always be another number You are essentially just moving forward and backward on a giant number line The numbers that you calculate with can be considered points on a number line into a two-dimensional space One of the main principles of ge ometry is that two points make up a line You may also know that two points can be used for graphing in This idea expands as you get algebra but it wasn’t always that way Mathematicians were able to argue ideas in geometry and solve problems with algebra but the two branches didn’t come together until the early 17th century Think of those two points in space as being two cities The road connecting those two cities can be thought of as a line If you travel between the cities you are traveling on that line If you stay on that road to go to another city you are still just traveling in one di mension Essentially you’re mov ing along a number line by doing calculations But what happens if you want to go to a city that isn’t on the same road In real life you can use a map 7

Where Does a Graph Go? — page 5 — to represent all the different cities as points Mathematicians would call those points loci or groups of points that follow a specific rule Always looking for ways to apply their knowledge in daily life mathematicians re alized they needed a more general way of discussing things such as loci Loci are a way to talk about geometric figures with more algebraic rules How would you describe a circle to a friend Would you say it is a shape that doesn’t have any corners That’s a great description and your friend would probably be able to picture a circle However a mathematician would describe a circle as the set of all points that are the same distance from a given point Describing it like that is talking about a circle as a locus Prior to the development of the coordinate plane though that was difficult having a system for naming or identifying their positions Think about it It was also difficult to pinpoint specific points within loci without PTOLEMY’S GEOGRAPHY MAPPED THE PLACES KNOWN TO FIRST CENTURY ROMANS this way If someone wanted to leave a note on your desk at school but had never been inside your classroom how would you explain the location to them You would need to use terms that have mean ing to both of you You couldn’t just say two in from the right in the first row Depending on which direction you’re facing that could mean different things You need an origin point to reference like the teacher’s desk or the door From that origin point it becomes easier to describe the locations of other objects in the room You also need common axes or straight fixed 8

Where Does a Graph Go? — page 6 — lines to talk about such as rows and columns of desks French philosopher and mathematician Rene Des cartes tackled this topic of loci origin points and axes When Descartes was a little boy he had some health issues that kept him in bed until 11 00 each morning He ended up making a habit of late rising throughout his life It is said that while lying in bed one day Descartes noticed a fly on his ceiling He be gan to wonder how he might be able to tell a friend about this fly’s posi tion as well as track the fly’s movements Legend has it that from this experience Descartes developed the system that now bears his name the Cartesian coordinate plane The coordinate plane is now a common system for explaining locations in two-dimensional space Ancient civilizations such as the Greeks had used a rectangular grid system for dividing and mapping particular areas For example around the year 100 Claudius Ptolemy proposed using such a system for mapping specific regions Other civilizations used similar methods as they constructed maps for navigation exploration and trade However these civilizations did not have a method common to each other which is what the Cartesian plane gave mathematicians 1,500 years later Descartes generated his coordinate plane by unknowingly following in the footsteps of fellow French mathematician Pierre de Fermat Fermat whose everyday job was practicing law is most commonly associated with number theory He postulated or proposed an idea about the rela tionship of powers of numbers that remained unproven for approximately ORIGIN POINT You need an to reference 9

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