Reasoning · Grade 4-1 Transformations of Figures

Problem

Which glyphs survive a turn or flip

Six syllable cards are judged as shapes, not sounds. After the move every part must still be a legal letter. The parts must also sit in a legal arrangement. Find which cards survive each of the three moves.
1 2 3 4 5 6
Your answer
How to solve
Strategy Make a Systematic List — There are only 6 cards and 3 questions, so 18 small checks settle everything — a table is the right tool. But the work gets far lighter if I first simplify each move. A half-turn followed by a flip about a horizontal line is the same single move as a plain left-right mirror, and a quarter-turn clockwise followed by a flip about a vertical line is the same single move as a mirror in the diagonal that runs from the top-left corner to the bottom-right corner. So I build one small table showing what each move does to each letter part and to the layout, and then read all three answers off it.
1STEP 1

Replace each pair of moves by one move

Each pair of moves reduces to one left-right mirror or diagonal mirror.

turn 180° + flip up = left-right mirror; turn 90° cw + flip right = mirror in the ↖ ↘ diagonal
2STEP 2

Make the table of what happens to each part

Tabulate whether each part stays a legal letter.

3STEP 3

Track the layout as well as the parts

Not just the parts — the layout must be legal too.

4STEP 4

Answer (1): the half-turn

A half-turn leaves the three-layer cards 1, 3, 6.

5STEP 5

Answer (2): the half-turn plus an upward flip, that is, a left-right mirror

The left-right mirror leaves only 4 and 6.

6STEP 6

Answer (3): the quarter-turn plus a rightward flip, that is, the diagonal mirror

The diagonal mirror leaves 2, 4, 5.

7STEP 7

Check the count and the odd ones out

The three answers count 3, 2 and 3 cards.

Answer
1, 3, 6 / 4, 6 / 2, 4, 5 cards
Every answer is a subset of the six cards, and none of the three lists is empty or all six, which is what you expect from a question that is meant to sort them. The lists behave sensibly against the shapes: the half-turn keeps exactly the three-layer cards, because reversing a stack still puts consonants at both ends; the left-right mirror keeps exactly the two cards built entirely from left-right symmetric parts; and the diagonal mirror keeps exactly the three two-part cards, because it is the only move that changes a stack into a row. Each list also has a different size (3, 2, 3), so no two questions have accidentally been given the same answer.
Takeaway

Two moves in a row are really just one move — name that single move first, and then check the pieces AND where they sit.

  • Replace each pair of moves by one move
  • Make the table of what happens to each part
  • Track the layout as well as the parts
  • Answer (1): the half-turn
  • Answer (2): the half-turn plus an upward flip, that is, a left-right mirror
  • Answer (3): the quarter-turn plus a rightward flip, that is, the diagonal mirror
  • Check the count and the odd ones out