Coordinates & maps
Coordinates: describe/plot points in the first quadrant; translations (units left/right/up/down); locate/move on plans & maps using cardinal points and scale-based distances.
A coordinate is an address for a point: a pair of numbers that says exactly where it sits on a grid. Maps use the same idea — a way to name a place and measure how far and in which direction to travel.
What it means
Draw two number lines that cross at right angles. The horizontal one is the x-axis, the vertical one is the y-axis, and the point where they meet — where both are 0 — is the origin.
Any point on the grid is named by two numbers written in a pair, like (3, 2). The first number is the x-coordinate: how far to go across (right) from the origin. The second is the y-coordinate: how far to go up. Order matters — (3, 2) and (2, 3) are different points. A handy phrase: “along the corridor, then up the stairs.”
The region where both numbers are positive — right of the origin and above it — is called the first quadrant. That is where we work at this stage. (Later you’ll meet the other three quadrants, using the negative numbers you already know from the number line.)
A translation slides a point or a shape without turning or flipping it: so many units left/right, so many up/down. Every point moves the same way, so the shape keeps its size and looks. Moving (3, 2) right 2 and up 1 lands you at (5, 3).
Maps apply the same thinking to real places. Instead of “right” and “up” we use the cardinal points — North, South, East, West. A scale tells you how map distance turns into real distance (e.g. 1 cm on the map = 1 km on the ground), so you can say not just which way but how far.
Worked examples
Plot a point. Where is (4, 1)? Start at the origin, go 4 right along the x-axis, then 1 up. Mark it.
Translate a shape. A triangle has corners at (1, 1), (3, 1), (2, 3). Translate it right 2, up 1 — add 2 to every x and 1 to every y:
(1, 1) -> (3, 2)
(3, 1) -> (5, 2)
(2, 3) -> (4, 4)The new triangle is the same shape, just shifted.
Read a map with scale. Scale: 1 cm = 2 km. The park is 3 cm due East of the school on the map. Real distance = 3 x 2 = 6 km, heading East.
The generative-art connection
Coordinates are the native language of the screen: every pixel and every dot an artwork draws is placed by an (x, y) pair. So this concept is not a metaphor for generative art — it is how the art gets made. In Desmos you can type a handful of points, drag them, and watch the coordinates update live; in the Geoboard you stretch a shape across a peg grid.
The nico.art tool dot-multiplier shows the payoff: it places dots at computed coordinates and multiplies them into a symmetric starburst. Change the numbers feeding the positions and the whole picture moves — a translation you can see. A child who can plot (x, y) can start steering the picture: shift every dot by the same amount and the pattern slides across the canvas, exactly like translating the triangle above.
Common misconceptions
- Swapping x and y.
(2, 5)is not(5, 2). Always across first, then up. - Counting the squares instead of the lines. Coordinates count grid lines from the origin, not the boxes between them; the origin itself is
(0, 0). - Thinking a translation can rotate or resize. A slide keeps orientation and size — every point moves by the same left/right and up/down amount, nothing else changes.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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On a grid, plot these four points and join them in order: `(1, 1)`, `(5, 1)`, `(5, 4)`, `(1, 4)`. What shape do you get, and what are the lengths of its two different sides?
Answer
A rectangle. Reading across then up: the bottom side runs from `(1,1)` to `(5,1)`, so it is `5-1 = 4` units long; the left side runs from `(1,1)` to `(1,4)`, so it is `4-1 = 3` units. So it is a `4` by `3` rectangle.
Art hook Canvas grid where the child clicks 4 corners; the app rubber-bands the rectangle live and prints each side length. Add a slider that translates the finished rectangle, redrawing it as every corner's coordinates update together.
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A shape has corners at `(2, 1)`, `(4, 1)` and `(3, 4)`. Translate it **right 3 and up 2**. Write the three new corner coordinates.
Answer
Add `3` to every x and `2` to every y: `(2,1)->(5,3)`, `(4,1)->(7,3)`, `(3,4)->(6,6)`. It is the same triangle, just slid across.
Art hook Show the original triangle in one colour and the translated triangle in another; a dashed arrow from each old corner to its matching new corner makes the identical slide visible.
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A treasure map has a scale of `1 cm = 5 m`. On the map the chest is `4 cm` due North of the rock and `3 cm` due East of the tree. How many real metres North of the rock, and how many East of the tree, is the chest?
Answer
`4 x 5 = 20 m` North of the rock, and `3 x 5 = 15 m` East of the tree.
Art hook A small map canvas: place rock, tree and chest as dots; label each map distance in cm and its converted real distance in m, with N/S/E/W compass arrows.
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Point `A` is at `(2, 3)`. It is translated to `(6, 5)`. Describe the translation in words (how many left/right and how many up/down).
Answer
Right `6-2 = 4` and up `5-3 = 2`. (Find the change in x and the change in y.)
Art hook Interactive: drag a dot from a start to an end spot; the app reads out the translation ('right 4, up 2') and draws the arrow between the two positions.
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Sam says the point `(3, 0)` is not on the grid because 'one of the numbers is zero, so it isn't really a point.' Is Sam right? Explain where `(3, 0)` is.
Answer
Sam is wrong. `(3, 0)` means go `3` right and `0` up, so it sits right on the x-axis, `3` units from the origin. Zero is a perfectly good coordinate. The origin itself is `(0, 0)`.
Art hook Highlight the axes as special 'roads': plotting any point with a `0` lights up whichever axis it lands on, showing the x-axis and y-axis as lines of points.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Starting at the origin `(0, 0)`, which way do you go for the **first** number in a coordinate pair: across (right) or up? And which way for the **second** number?
Answer
The first number is across (right); the second number is up. A helper phrase: 'along the corridor, then up the stairs.'
Art hook A dot walks the path for any pair you type: first it slides right along the x-axis, then climbs up, tracing the two-step route to the point.
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Where is the point `(3, 2)`? Say in words how you get there from the origin.
Answer
Go `3` right along the x-axis, then `2` up. Mark the dot there.
Art hook Place a single dot at `(3,2)` on a labelled grid; a faint L-shaped path shows the 3-right-then-2-up journey from the origin.
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Three points are plotted: a red dot at `(1, 4)`, a blue dot at `(4, 1)`, and a green dot at `(4, 4)`. Which dot is furthest to the right? Which is highest up?
Answer
Furthest right: the biggest x-coordinate is `4`, shared by the blue and green dots. Highest up: the biggest y-coordinate is `4`, shared by the red and green dots. So the green dot `(4,4)` is both furthest right and highest.
Art hook Colour each dot by its coordinates: let the x-value drive the red channel and the y-value drive the blue channel, so a dot's position maps directly onto its colour.
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Plot and join in order to make a closed shape: `(2, 2)`, `(6, 2)`, `(6, 5)`, `(2, 5)`. Name the shape.
Answer
A rectangle (sides `6-2 = 4` across and `5-2 = 3` up). Joining the four points in order and back to the start closes it.
Art hook Snap-to-grid drawing tool: clicking the listed points in order rubber-bands a rectangle you can then recolour or duplicate.
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Translate the point `(2, 2)` **right 4 and up 1**. Where does it land?
Answer
Add `4` to x and `1` to y: `(2+4, 2+1) = (6, 3)`.
Art hook One dot with two arrow-key sliders (left/right, up/down); the coordinate label updates every step as it moves.
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A square has corners `(1, 1)`, `(3, 1)`, `(3, 3)`, `(1, 3)`. Translate the **whole square right 2 and up 2**. Write all four new corners.
Answer
Add `2` to every x and `2` to every y: `(3,3)`, `(5,3)`, `(5,5)`, `(3,5)`. Still a square of the same size, just shifted diagonally.
Art hook Draw the square, then stamp translated copies at +2/+2 repeatedly to make a diagonal staircase of squares marching across the canvas.
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Emma plotted `(5, 2)` but her friend plotted `(2, 5)` for the same instructions. They are in different places! Explain the mistake and say which rule keeps you right.
Answer
Someone swapped the numbers. `(5, 2)` means 5 right and 2 up; `(2, 5)` means 2 right and 5 up — different points. The rule: always the x-coordinate (across) first, then the y-coordinate (up).
Art hook Toggle button that swaps a dot between `(a,b)` and `(b,a)` so you watch it jump to a mirror position across the diagonal line — a first taste of reflection symmetry.
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On a map with scale `1 cm = 3 km`, the harbour is `5 cm` due East of the lighthouse. How far is that in real kilometres, and in which direction?
Answer
`5 x 3 = 15 km`, heading East.
Art hook Compass-rose map: type a map distance in cm and pick a direction; the app draws the arrow and prints the real distance using the scale.
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Point `B` at `(1, 2)` is translated to `(1, 6)`. Describe the translation. Did its x-coordinate change?
Answer
Up `6-2 = 4`, and `0` left/right. The x-coordinate stayed `1`, so the point moved straight up.
Art hook Show that when only y changes, the dot travels along a vertical line; sweep several such dots to reveal the vertical grid line as a column of points.
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A flag is drawn with corners `(2, 1)`, `(2, 4)`, `(4, 3)`. Translate it so that the corner at `(2, 1)` moves to `(7, 1)`. What is the translation, and where do the other two corners go?
Answer
`(2,1)->(7,1)` is right `5`, up `0`. Apply the same slide to the others: `(2,4)->(7,4)` and `(4,3)->(9,3)`.
Art hook Grab-and-drag one corner; the whole flag follows by the same translation, corners re-labelling live as it slides.
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Plot these seven points in order and join them: `(1, 2)`, `(2, 4)`, `(3, 2)`, `(4, 4)`, `(5, 2)`, `(6, 4)`, `(7, 2)`. Describe the pattern you see.
Answer
A zig-zag (up-down-up-down) line: the y-coordinate alternates `2, 4, 2, 4...` while x climbs by `1` each time. It makes a row of connected peaks.
Art hook Generate a zig-zag by rule: y = 2 when x is odd, 4 when x is even; add a slider for peak height and one for how many peaks, drawing a whole mountain range.
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Give a list of at least 5 coordinate points (all numbers from `0` to `10`) and the order to join them so a friend can copy your picture. Then explain why the join order matters.
Answer
Any valid figure works, e.g. a house: `(2,2),(6,2),(6,5),(4,7),(2,5)` joined in order and back to the start. Order matters because joining the same points in a different order draws a different shape.
Art hook A 'coordinate portrait' maker: paste a point list, watch it draw, then shuffle the join order to see how the picture changes — the seed of connect-the-dots generative line art.