Coordinates in Four Quadrants

Someone sends you a meeting place as two numbers: fifty one, and zero. Type them into a map one way round and the pin drops on London. Type the same two numbers the other way round and the pin drops into the Indian Ocean, more than four thousand miles from anywhere you meant. Nothing changed except which number went first, and that is the whole idea behind coordinates.

Maths settles that order once and for all, and it arrives with a picture worth keeping. Two number lines cross at right angles. The one running across is the x-axis, and the one running up is the y-axis. Where they cross is called the origin, and both numbers there are zero. A point is written as two numbers inside brackets, and the agreed order is x first, then y. Maps happen to put the up and down number first, and maths does the opposite. That is exactly why the order has to be a rule: two numbers mean nothing on their own until everyone follows the same one. So here are the words to keep in your head all the way to the exam hall: along the corridor, then up the stairs. You walk the corridor before you climb, and x comes before y for the same reason. Signs follow the same picture. Walk right from the origin and x is positive, walk left and x is negative. Go up and y is positive, go down and y is negative. Take the point four, three. Four along to the right, then three up. Now take three, four instead. That is three along and four up, which is a dot somewhere else entirely.

Two dots are on the grid now. One of them sits further to the right, and the other sits higher up. Which one is the point four, three? Take your pick. I'll wait. The answer is the dot further to the right. Four is the x number, so it walks you four along the corridor and only three up the stairs.

Swap those two numbers over and you have not made a small slip, because you have named a completely different point on the grid.

The two axes cut the whole grid into four regions, and each region has its own pattern of signs. Start in the top right, where x is positive and y is positive as well. That region is called the first quadrant. Now walk anticlockwise. Top left is the second quadrant, with x negative and y positive. Bottom left is the third quadrant, where both numbers are negative. Bottom right is the fourth quadrant, with x positive and y negative. The numbering always walks anticlockwise from the top right, so you never have to memorise four separate pictures. Find the first quadrant and count round from there.

Here is one to try. Four points are marked on a grid. A is at four, three. B is at negative two, five. C is at negative three, negative four. D is at five, negative one. Which point lies in the third quadrant? Pick one. I'll wait. The answer is C, at negative three, negative four. The third quadrant is the bottom left, where both coordinates are negative, and C is the only one of the four with two negative numbers.

Notice that you had to read both signs together. B and D each carry one negative number, and one negative on its own never fixes the quadrant.

Sometimes a question refuses to hand you the point at all, and gives you a condition it has to satisfy instead. A kite has one line of symmetry, and that line runs straight along the y-axis. Vertex A sits at negative three, negative two. Vertex C is the reflection of A in that line of symmetry. Write down the coordinates of C.

Have a go at this one. I'll wait. The answer is three, negative two. The y number stays exactly where it was, and only the x number changes sign.

Here is why the height cannot move. A mirror line running straight up and down sends every point across to the same level on the other side, so the up and down number is untouched and only the left to right number flips. Now the step that actually finishes the job, and it has a name of its own: the sense-check. A sits below the axis on the left, so its mirror image has to sit below the axis on the right. Three, negative two is bottom right on the drawing, the diagram agrees, and only now is the answer finished. Skip that check and a wrong sign sails through unnoticed, because a reversed point still looks like a perfectly tidy answer on the page.

There are two mistakes examiners keep writing down about coordinates, and you have already met both of them. This first line is from an examiner report, on a question where candidates had to read and plot coordinates on a grid. This was usually well done with the same reversal of coordinates sometimes seen, though not always by candidates who had reversed the coordinates in part one. Plotting the point at negative three, negative two was also seen. Read the middle of that again. The same reversal turned up twice inside one question, because once your ordering slips it goes on slipping. The second line comes from a report on a question where a point had to be found using lines of symmetry drawn on a diagram. Many students simply made the coordinates of P positive, and many others added two and six to the lines of symmetry to get the answer three, eleven. A relatively large number of students gave answers with negative coordinates. Although the graphic is not drawn accurately it should be clear that this is not a possible answer, and students should be encouraged to consider how sensible their answer is in all questions. That last sentence is the sense-check, in an examiner's own words. Ask whether your answer could genuinely sit where the drawing puts it, before you accept it. Neither of those is hard maths going wrong, and both of them are caught by the same few seconds of looking back at the grid.

So, the whole thing in one pass, starting with the words worth keeping. Along the corridor, then up the stairs. That is x first, then y, in brackets, every single time. Right and up are positive, left and down are negative, and the quadrants number anticlockwise from the top right, where both coordinates are positive. When a question describes a point instead of giving it, read the condition off the diagram, then sense-check the quadrant before you write anything down.

Two jobs sit deliberately outside this one. Working a point out from a geometric fact is one, and drawing a whole straight line through plotted points is the other, which the video on plotting straight-line graphs takes care of.

Next in the chapter: Coordinates from Geometric Information and Midpoints, where the point has to be worked out rather than read straight off the picture.

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