Abstract Thinking: Turns, Views and Symmetry

Abstraction keeps only the details that matter. This chapter uses it for turns and directions, clock angles, mirror images, folding and cubes.

Key ideas

Abstraction means keeping the important details and leaving out the rest, like a map that shows roads but not the colour of houses.
The 8 compass directions are 45° apart; add clockwise turns and take away anticlockwise ones to get one net turn.
On a clock, the minute hand moves 6° a minute and the hour hand 30° an hour plus 0.5° a minute.
A mirror held beside an object flips left and right; a water image flips top and bottom.
Each fold of a paper doubles the layers, so one punch makes a hole in every layer.
A painted cube cut into small cubes has painted corners, edges, face centres and unpainted cubes inside.

Remember

Turns	quarter turn 90°, half turn 180°, full turn 360°
Clock angle at h:m	hour hand 30 × h + 0.5 × m, minute hand 6 × m; take the difference (if over 180°, use 360° − it)
Mirror time	11:60 − the time shown (e.g. 3:20 → 8:40)
Fold k times, punch h holes	h × 2ᵏ holes when unfolded
Cube cut n along each edge	3 faces: 8; 2 faces: 12 × (n − 2); 1 face: 6 × (n − 2)²; none: (n − 2)³
Dice	opposite faces add up to 7
Squares in a 3 × 3 grid	9 + 4 + 1 = 14 (count each size)

Watch out

Mixing up clockwise (the way clock hands turn) and anticlockwise.
Forgetting the hour hand moves between the numbers as the minutes pass.
Counting only the smallest squares in a grid, and missing the bigger ones.

Worked example

(a) A child faces East and turns 90° anticlockwise, then 45° clockwise, then 180° anticlockwise. Which way is the child facing now?
(b) What is the smaller angle between the hour hand and the minute hand of a clock at 8:10?

Working: (a) net turn = 135° clockwise from East; (b) hour hand at 8 × 30° + 10 × 0.5° = 245°; minute hand at 10 × 6° = 60°; difference 185°, so 360° − that

Answer: (a) South-West; (b) 175°