Math Detours · Grade 5 · Sharpen · 12

Treasure maps: coordinates

Two numbers pin down a point. Along first, then up. Plot shapes, slide them, and find the missing corner.

Name ____________________Date ____________

Tonight's idea

A treasure map must say exactly where to dig. Draw a grid, then number the lines along the bottom and up the side. Start at the corner where both numbers are 0.

0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 4 along 3 up X (4, 3) Tree Rock Cave along up
From the corner: 4 along, then 3 up.

The treasure X is at \((4, 3)\). That means: from the start, go 4 along, then 3 up. The two numbers are the point's coordinates. Every point where grid lines cross has its own pair.

The order matters. The tree is at \((2, 5)\) and the rock is at \((5, 2)\). They use the same numbers but are in different places.

Coordinates are (along, up). Walk along the corridor first, then climb the stairs.

To slide a point to the right, add to its first number. To slide it up, add to its second.

Watch one: slide a triangle

A triangle has corners \((1, 1)\), \((4, 1)\) and \((1, 3)\). Slide it 3 to the right and 2 up.

Add 3 to every first number and 2 to every second number. The new corners are \((4, 3)\), \((7, 3)\) and \((4, 5)\).

The bottom side was 3 long and it still is: from 4 to 7. A slide moves every corner the same way, so the shape and its size stay the same.

Your turn

  1. Use the treasure map from the start of the sheet. (a) Write the coordinates of the cave. (b) What is at \((5, 2)\)? (c) You stand at the tree and walk 3 along and 2 down. Where are you, and what have you found?
  2. On the empty grid, plot \((1, 1)\), \((5, 1)\), \((5, 4)\) and \((1, 4)\). Join them in order and back to the start. (a) What shape is it? (b) How long and how tall is it?
    0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8
    Use this grid for questions 2, 3 and 4.
  3. Slide the shape from question 2 by 4 to the right and 3 up. (a) Write the four new corners without plotting. (b) Plot them to check. (c) One point is a corner of both shapes. Which?
  4. Three corners of a rectangle are \((3, 2)\), \((3, 7)\) and \((9, 7)\). Its sides run along grid lines. (a) Find the fourth corner without plotting. (b) Explain how the numbers told you. (c) How long is each side?

Stretch: a tilted square

A square has three corners at \((1, 2)\), \((4, 1)\) and \((5, 4)\), in order. Its sides do not follow the grid lines. Going from the first corner to the second is "3 along, 1 down". Find the fourth corner.

Answers

  1. (a) \((7, 5)\). (b) The rock. (c) \(2 + 3 = 5\) along and \(5 - 2 = 3\) up: \((5, 3)\). Nothing is there. The treasure is one step to the left, at \((4, 3)\).
  2. (a) A rectangle. (b) 4 long (from 1 to 5) and 3 tall (from 1 to 4).
  3. (a) \((5, 4)\), \((9, 4)\), \((9, 7)\) and \((5, 7)\). (c) \((5, 4)\): the top right corner of the old shape and the bottom left corner of the new one.
  4. (a) \((9, 2)\). (b) Corners above each other share a first number, and corners side by side share a second number. The numbers 9 and 2 have each been used only once. (c) 6 along (from 3 to 9) and 5 up (from 2 to 7).

Stretch. The fourth corner is \((2, 5)\). The side from \((4, 1)\) to \((5, 4)\) is "1 along, 3 up". The opposite side matches it, so start at \((1, 2)\) and go 1 along, 3 up. Check: from \((2, 5)\), "3 along, 1 down" lands on \((5, 4)\).

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