A little room to wonder

Coordinates & Graphs

Use labelled axes and ordered pairs to locate points and explain changes without trusting appearance alone.

What does a point mean before you draw a line through it?

Robert Hoddle’s 1837 survey map of Melbourne, showing streets, blocks and the river.
The approved historical map illustrates a designed spatial representation. It is not the fictional coordinate grid and does not show the A-to-B route.Robert Hoddle / Government of New South Wales, 1837, via Wikimedia Commons. Public domain. · Source · Public domain

Build a coordinate agreement

A rectangular coordinate system uses two perpendicular axes with an agreed origin and scale. In the pair (x, y), the first number gives the horizontal coordinate and the second gives the vertical coordinate. Negative values indicate the opposite direction from the positive axis. The origin has coordinates (0, 0).

The ordered pair (−2, 1) differs from (1, −2). Reading a pair consistently matters more than remembering the rough appearance of a picture. Label the axes and state what one grid interval represents before plotting any points. Here both axes measure fictional map units, rather than a real location.

Worked example: describe a route

On a fictional grid, A is (−2, 1) and B is (3, 4). From A to B, the horizontal change is 3 − (−2) = 5 units and the vertical change is 4 − 1 = 3 units. A route travelling only horizontally and vertically therefore needs at least eight units: five plus three.

That eight-unit result is not the length of the straight diagonal between the points. It answers a particular route rule. Nor do the coordinates describe a speed: there is no elapsed time in this drawing. State the question before deciding which difference or distance to calculate.

Read the scale, not just the steepness

A graph can stretch a drawing without changing its data. If one horizontal grid interval represents two units while a vertical interval represents one, the same numerical changes look different from a graph with equal scales. A visually steep line is not an explanation until the axis quantities and units are known.

Connecting two points also adds an assumption about what happens between them. A route map, a sequence of observations and a continuous relationship do not automatically justify the same connecting line. Missing points can remain missing rather than being filled with a confident-looking curve.

Make a graph auditable

Write the plotted pairs next to the drawing so someone can reconstruct it. Check a point by projecting to each labelled axis. If the plotted point and written pair disagree, correct the representation before interpreting a trend or route.

Coordinates can support game worlds, engineering views and data displays. They are a language for position, not proof that the represented place or process exists. Use imaginary maps here; there is no need to share personal locations or track anyone’s movements.

Only if you feel like trying

Draw and explain a fictional grid route

  1. Label the origin and one-unit scale, then plot the supplied A and B.
  2. Draw two different horizontal-and-vertical routes of eight units and explain why their order can differ.
  3. Change the horizontal drawing scale and state which coordinates and route quantities stay the same.

Optional. Use paper or talk through the supplied fictional example. No purchase, personal information, account, public sharing or practical experiment is needed.