Translation & rotation

1Summary 重點摘要

Translation: slide; rotation: turn.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Translation slidesPush a chess piece on grid paper from (1,1) three squares to the right, arriving at (4,1). The whole time the piece faces one direction without any turning — such a move is a translation. Why does translation change only the position? Because every point of the figure moves the same distance in the same direction, and afterwards the shape, the size and the facing are completely unchanged; only the location is different. For example, a lift rising from floor 2 to floor 5 keeps facing upward the whole time, which is a translation. A common mistake is to think a slanting move is not a translation; as long as the move follows a straight line with an unchanged direction, a slanting move is still a translation, such as a piece going from (1,1) to (3,3).
2. Part 2: Say direction and distanceDescribing a translation needs two things made clear: which way to move and how many squares, like '3 squares right' or '2 squares up'. Why are both needed? With only the direction but no distance, nobody knows how far to move; with only the distance but no direction, nobody knows which way to go, and the new position of the figure cannot be fixed. For example, slide a triangle 4 squares right: every point of it moves 4 squares right, and afterwards its shape and size are completely unchanged. A common mistake is to mix up '3 squares right' with '3 squares up', or to draw the figure bigger or smaller after the slide; the figure before and after must be exactly the same, only in a different place, so count the squares one by one while copying.
3. Part 3: Rotation turnsThe minute hand moving from pointing at 12 to pointing at 3 turns clockwise around the centre by one right angle, that is 90°, and this turning around a fixed point is called rotation. Describing a rotation needs three things: around which point, in which direction, and through how many degrees, and all three are needed. Why must all three be stated? Because without any one of them the new position of the figure cannot be fixed: without the centre nobody knows which point to turn around, and without the direction nobody knows which way. For example, fan blades turning around the hub, or a door swinging on its hinges, are rotations. A common mistake is to omit the direction and say only 'turn 90°'; clockwise 90° and anticlockwise 90° give different results, so leaving out the direction causes errors.
4. Part 4: Feel a right-angle turn with your bodyStand facing the board and your body becomes a figure that can rotate: turn clockwise 90° and you face the window; turn another 90°, 180° in all, and you face the back of the room; a third 90° makes 270° and you face the other wall; after four right angles, 360°, you are back at the start. Why experience it with the body? Because angles are abstract, but turning yourself makes the relations among 90°, 180° and 360° concrete at once: two right angles make a straight angle, and four right angles make a full turn. A common mistake is to take 180° as a whole turn, although it is only half a turn; another is to think the figure changes after 360°, but a full turn brings it back exactly, identical to the original.
5. Part 5: Neither changes the shapeA triangle slid 5 squares to the right is still the same triangle; rotated about a point by 90°, it is still the same triangle, only facing differently. Why do translation and rotation never change a figure's shape or size? Because both transformations change only the position or the facing: the distance between any two points of the figure stays the same, so the side lengths, the angles and the area are all preserved. For example, a triangle of base 4 cm and height 3 cm keeps its area 4 × 3 ÷ 2 = 6 cm² wherever it slides and however it turns. A common mistake is to treat the rotated figure as a different figure, or to copy it on grid paper with edges drawn too long or too short; the figure before and after a transformation must be exactly the same.
6. Part 6: Sorting everyday examplesEveryday movements fall into two groups. Translations: a lift moving up and down, an escalator, opening a drawer, a sliding window. Rotations: fan blades spinning, a merry-go-round, a door swinging on its hinges, unscrewing a bottle cap. Why learn to sort them? Because naming the type of movement is the first step in solving problems about moving figures, and the key is how the object moves: sliding along a straight line with an unchanged direction is translation, while turning around a fixed point is rotation. For example, a pendulum swinging about a fixed point belongs to the turning group. A common mistake is to treat a mixed movement as pure translation, such as a rolling wheel, which moves along and turns at the same time; look at whether the object turns around some point.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Translation slides in a straight line: direction + distance
• Rotation turns about a point: direction + angle
• Clockwise matches clock hands
• Neither changes shape or size
• Lifts slide; fans and hinges rotate