The Lumenary · Grades 3–5

Symmetry: slides, turns and flips

Name: ______________________

Date: ______________________

You will learn to

Find the lines of symmetry and the order of rotational symmetry of shapes
Perform slides, turns and flips on a grid and describe them precisely
Find the move that undoes a given move, and explain what happens when two flips are done one after the other
Explain why a regular polygon with n sides has n lines of symmetry and rotational symmetry of order n

1.
How many lines of symmetry does a square have?
2.
A rectangle is 6 cm long and 3 cm wide. How many lines of symmetry does it have?
3.
Which shape has rotational symmetry (order 2 or more) but no line of symmetry?
- A.A parallelogram that is not a rectangle or a rhombus
- B.A kite
- C.An isosceles triangle
- D.A rectangle
4.
Which of these shapes have exactly 2 lines of symmetry? Choose from: square, rectangle, rhombus, kite, parallelogram, equilateral triangle. (Here the rectangle and the rhombus are not squares.) Write your answer like {kite, square}.
5.
A shape is given a quarter turn clockwise about the point P. Which single move puts it back where it started?
- A.A quarter turn anticlockwise about P
- B.Another quarter turn clockwise about P
- C.A flip in a line through P
- D.A half turn about P
6.
A flag is flipped in a vertical mirror line, and then flipped in a second vertical mirror line 3 squares to the right of the first. The result is a slide. How many squares does the flag slide?
7.
Amir says: "A flip in a vertical mirror line and a half turn always do the same thing." Which example shows that he is wrong?
- A.The letter F: flipped, it faces left with its arms still at the top; half-turned, it is upside down.
- B.The letter H: flipped or half-turned, it looks the same.
- C.A circle: flipped or half-turned, it looks the same.
8.
Two mirror lines cross at an angle of 60°. A shape is flipped in the first mirror and then in the second. The result is a turn about the crossing point. Through how many degrees is it turned?
9.
Lean proved flips_in_parallel_lines: flipAt d (flipAt c p) = slide (2 * (d - c)) 0 p for every point p. Here flipAt c flips in the vertical line x = c, and slide a b slides a squares right and b squares up. What single move is "flip in the line x = 1, then flip in the line x = 4"?
- A.A slide of 6 squares to the right
- B.A slide of 3 squares to the right
- C.A slide of 6 squares to the left
- D.A half turn
10.
Points on squared paper are pairs (x, y). Complete flip(p) (flip in the vertical line through (0, 0), so left and right swap) and quarter_turn(p) (a quarter turn anticlockwise about (0, 0), which sends (1, 0) to (0, 1) and (0, 1) to (-1, 0)). The tests check that two flips and four quarter turns undo themselves.
Computational — Practice

Answer key — Symmetry: slides, turns and flips

1.
A square has 4 lines of symmetry: 2 through the middles of opposite sides (up-and-down and side-to-side) and 2 along the diagonals, from corner to corner.
2.
The rectangle has 2 lines of symmetry: one through the middles of the two long sides and one through the middles of the two short sides. The diagonals are not lines of symmetry: folding along a diagonal makes the corners stick out. Only a square, whose sides are all equal, has its diagonals as mirror lines.
3.
A parallelogram (that is not a rectangle or rhombus) looks the same after a half turn about its centre, so it has rotational symmetry of order 2. But no fold line makes its two halves match, so it has no line of symmetry.
4.
Counting lines of symmetry: square 4, rectangle 2, rhombus 2 (its two diagonals), kite 1, parallelogram 0, equilateral triangle 3. So the answer is $\{\text{rectangle}, \text{rhombus}\}$ .
5.
To undo a turn, turn back through the same angle about the same centre: a quarter turn anticlockwise about P. (Three more quarter turns clockwise would also work, but that is not a single quarter turn.)
6.
Two flips in parallel mirror lines make a slide by twice the distance between the lines: $2 \times 3 =$ 6 squares, to the right (from the first mirror towards the second). For example, a point 1 square left of the first mirror goes to 1 square right of it, which is 2 squares left of the second mirror, and then to 2 squares right of the second mirror: 6 squares from where it started.
7.
The letter F works: a flip in a vertical line makes it face the other way, while a half turn turns it upside down. One counterexample is enough. For very symmetric shapes such as H or a circle the two pictures agree, which is why the mistake is easy to make — but the flip and the half turn are still different moves.
8.
Two flips in crossing mirror lines make a turn through twice the angle between the mirrors: $2 \times 60^\circ =$ 120°. (That is a third of a full turn — which is why an equilateral triangle, with mirror lines at $60^\circ$ to each other, has turns of $120^\circ$ as symmetries.)
9.
With c = 1 and d = 4 the theorem says the two flips equal slide (2 * (4 - 1)) 0, which is slide 6 0: 6 squares to the right and 0 up. The theorem is proved for every point p and for every pair of vertical mirror lines, not just this one.
10.
```
def flip(p):
    x, y = p
    return (-x, y)


def quarter_turn(p):
    x, y = p
    return (-y, x)
```
Flipping twice changes the sign of x twice, which brings it back. Two quarter turns give (-x, -y), a half turn, and four give (x, y) again.