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← Level 3 – Fraction kid

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Coordinates: a map and a point

A coordinate system: how to write and find a point with two numbers.

Two numbers pin every place

“Battleships! B4!” — “Hit!” You know the game Battleships. And do you know what you actually do in it? You use coordinates: a pair of facts that pins a square exactly. A letter says the column, a number the row.

Maths does it almost the same, just with two numbers. Take two number lines (those from the integers lesson!) and cross them at zero: the horizontal one is called the x-axis, the vertical one the y-axis. The place where they cross is called the origin.

Every point in the plane you then describe with a pair of numbers in square brackets: A[3; 2] means: from the origin 3 to the right and 2 up.

The order is law: first x (horizontally), then y (vertically). Point [3; 2] and point [2; 3] are two different places — as if in Battleships you mixed a column with a row.

You meet coordinates everywhere: maps, GPS, graphs, computer games (every character on the screen has its x and y). Today you will learn to read them and write them.

Tereza and Filip play Battleships, but Tereza thinks of an upgrade: “Let’s play on squared paper with axes. Instead of B4 say [2; 4].” Filip tries the first shot: “[3; 1]!” Tereza goes with her finger: three to the right along the horizontal axis… and one up. “Miss!” After a few turns Filip fires “[1; 3]” and Tereza stops: “Wait, you already said that.” — “I didn’t! I said [3; 1] — three to the right, one up. Now I shoot one to the right, THREE up!” Two same numbers, different order, different square. From then on both read coordinates carefully: first x, then y.

💡

Remember the order by the alphabet and by movement: x is before y, like walking is before a lift — first walk the corridor (to the right), then go up in the lift (up).

How to find a point and how to write it

Plotting point A[3; 2] — method:

  1. Bắt đầu at the origin [0; 0] — the crossing of the axes.
  2. First number (x): go along the horizontal axis. Positive to the right, negative to the left. Here: 3 to the right.
  3. Second number (y): from that place vertically. Positive up, negative down. Here: 2 up.
  4. Make a dot and label it: A[3; 2].

Reading a point from a picture — the method backwards:

  1. From the point drop a vertical line to the x-axis → read the first number.
  2. From the point take a horizontal line to the y-axis → read the second number.
  3. Write [x; y].

Special places, so they do not surprise you:

Negative coordinates work exactly by the lines from integers: [−3; 2] is 3 to the left and 2 up; [−2; −4] is 2 to the left and 4 down. The axes split the plane into four quarters (quadrants) — top right both numbers are positive, bottom left both negative.

Coordinates are written in square brackets with a semicolon: A[3; 2]. A semicolon so it does not mix with a decimal point.

Plot points A[2; 5], B[5; 2] and compare

A[2; 5]: from the origin 2 to the right, 5 up. The point is high and close to the vertical axis.

B[5; 2]: 5 to the right, 2 up. The point is low and far to the right.

The same numbers, swapped order — completely different places. The line joining A and B would cut the plane on a slant.

This is the most important coordinates exercise: a pair is ordered, x must not be swapped with y.

Read a point from a treasure map

On a map with a coordinate grid a treasure is marked with a dot. How to write its place?

Vertically down to the x-axis: the line hits the axis at the number −3. First coordinate: −3 (the treasure is left of the vertical axis).

Horizontally to the y-axis: the line hits the axis at 4. Second coordinate: 4.

The treasure: P[−3; 4] — three steps left, four up. Who goes “three to the right” digs on the wrong side of the map.

A shape from coordinates

Plot points K[1; 1], L[5; 1], M[5; 4] and N[1; 4] and join them in order (and M back to K… actually N to K).

K and L sit at the same height (y = 1) → the bottom horizontal side of length 4.

L and M have the same x = 5 → the right vertical side of length 3.

A rectangle 4 × 3 comes out! And at once you can count: perimeter 2 × (4 + 3) = 14, area 4 × 3 = 12 little squares.

Coordinates and geometry shake hands — shapes can be written with numbers.

⚠️

Do not swap the order. [3; 1] is not [1; 3] — the first number is ALWAYS x (horizontally), the second y (vertically). Who reads coordinates backwards shoots in Battleships into their own water. Helper: x is earlier in the alphabet than y, you go right before you go up.

Play it

Before you go to the sheet, feel it with your eyes. This is not a test — it is a game. Tap until you see it.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

A point on the grid. First x to the right, then y up.

Practise the questions

On paper

Squared paper, a ruler, and happily an opponent for Battleships.

  1. Draw the x-axis and the y-axis (both from −5 to 5) and plot points: A[4; 1], B[−2; 3], C[0; −4], D[−3; −3]. Label each.

  2. Write coordinates: draw five dots anywhere on the grid and by each write [x; y]. Then give them to a friend or a parent to check the other way round.

  3. Plot points [1; 1], [6; 1], [6; 4], [1; 4], join them and say what appeared. Work out the perimeter and the area in little squares.

  4. Play Battleships with coordinates: an 8 × 8 grid, write ships as a list of points. After the game write three coordinates you hit with.

Now you 💪

  1. I can plot a point from the write-up [x; y]: first number right/left, second up/down.
  2. I can read the coordinates of a point from a picture using perpendiculars to the axes.
  3. I know that [3; 1] and [1; 3] are different points, and I can show it.
✅

Done when: You plot and read a point in a coordinate system including negative values and you keep the iron order: first x, then y.

What to take from this lesson

© 2026 Ing. Martin Polak / AlgoRhino · Điều khoản sử dụng nội dung