100i - Rotations About a Point

Lesson 100 — Rotations About a Given Point

Strand: Space | Descriptor: AC9M7SP03 | Duration: 45 minutes

Equipment: grid paper, tracing paper (essential — one sheet per student), split pins or drawing pins optional.

Learning Intentions

• To rotate points and shapes about the origin through , and .
• To describe a rotation by its centre, angle and direction.

Success Criteria

I can:

1. Describe a rotation using centre, angle and direction.
2. Rotate a point about the origin using tracing paper and using coordinate rules.
3. State and apply the rules for , and rotations about the origin.
4. Identify the rotation that maps one shape onto another.

Warmup

(6 minutes — the cliffhanger resolved, mini whiteboards)

From yesterday’s inquiry: reflecting in both axes gave , and we said it looked like a half-turn.

1. Plot and . Draw the line joining them — what does it pass through?
2. Measure: is each point the same distance from the origin?
3. So what has happened to the point?
4. Predict the rule for a half-turn about the origin.

Answers: 1. The origin; 2. Yes; 3. It has been turned half a turn about the origin; 4. .

The class predicted a rotation rule before being taught one. Today formalises all three.

Activities

Activity 1 — Explicit Instruction: Describing and Performing Rotations (14 min)

A rotation needs three pieces of information:

1. Centre — the fixed point everything turns around (this lesson: the origin).
2. Angle — , or .
3. Direction — clockwise or anticlockwise. (Convention: anticlockwise is the positive direction.)

Missing any one, the instruction is incomplete. “Rotate ” is not enough.

I do — tracing paper method (do this before any rule).

1. Trace the shape and mark the origin with a cross.
2. Hold a pin (or pencil tip) firmly at the origin.
3. Turn the tracing paper through the required angle — using the grid lines as the guide: a quarter turn takes the -axis onto the -axis.
4. Mark the new positions through the paper.

Demonstrate with rotated anticlockwise: it lands at . Have every student verify by turning their own paper.

Now the rules, derived from several traced examples:

Check each rule against a traced example before trusting it. : acw gives ✓; gives ✓; acw gives ✓.

Two observations worth recording:

• needs no direction — clockwise and anticlockwise agree.
• anticlockwise and clockwise are the same transformation, so either description is acceptable.

We do:

1. Rotate by anticlockwise.
2. Rotate by .
3. Rotate by clockwise.
4. Which point is unchanged by every rotation about the origin?

(Answers: ; ; cw acw: ; the origin itself — the centre never moves.)

Activity 2 — Practice and Identification (14 min)

Grid paper and tracing paper. Verify at least the first of each set by tracing.

Set A — rotate about the origin.

1. by anticlockwise
2. by
3. by clockwise
4. Triangle by anticlockwise

Set B — identify the rotation. Each pair is an original and its image after a rotation about the origin:

Set C — which transformation? Translation, reflection or rotation?

9. and
10. and
11. and

Socratic scaffolding for Set B:

Prompt	Purpose
Compare the coordinates — were they swapped, negated, or both?	Diagnoses which rule applies.
For Q5: — swapped and the new first is negative.	Matches : anticlockwise.
For Q6: no swap, both negated.	.
For Q7: swapped, new second is negative… check: .	gives ✓ — anticlockwise ( clockwise).
Verify by tracing.	Rules are checked against physical turning, never trusted blind.

(Answers: 1. ; 2. ; 3. ; 4. ; 5. acw; 6. ; 7. acw (= cw); 8. ; 9. rotation acw; 10. reflection in the -axis; 11. translation .)

Activity 3 — Inquiry: the Transformation Detective (8 min)

Pairs — the block’s synthesis.

A flag shape sits at . Four images are given:

• Image 1:
• Image 2:
• Image 3:
• Image 4:

For each: name the transformation completely (with centre/axis/rule as appropriate), and say whether orientation was preserved.

Socratic scaffolding:

Prompt	Purpose
Image 1: what changed?	negated only — reflection in the -axis; orientation reversed.
Image 2?	negated only — reflection in the -axis; orientation reversed.
Image 3: both negated. Two candidates?	Could be rotation or a double reflection — and yesterday showed these are the same transformation. Orientation preserved.
Image 4?	Every point across, up — translation; orientation preserved.
The pattern across all four?	Translations and rotations preserve orientation; reflections reverse it.

The summary table to record (board):

Transformation	Rule (about origin / on axis)	Orientation
Translation		preserved
Reflection in -axis		reversed
Reflection in -axis		reversed
Rotation acw		preserved
Rotation		preserved
Rotation acw		preserved

All six preserve lengths, angles and area — every image is congruent to its original.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Rotate by anticlockwise about the origin.
2. Rotate by about the origin.
3. . Name the rotation.
4. Why must a rotation instruction include a direction, except in one case?
5. Reasoning. Which of the six transformations reverse orientation, and what does that mean physically?

Answers: 1. ; 2. ; 3. anticlockwise (or clockwise); 4. Clockwise and anticlockwise give different images — except for , where both give the same result; 5. Only the reflections; the shape is flipped over, so tracing the vertices in order reverses direction.

Common Misconceptions

Misconception	How to pre-empt it
Rotating in the wrong direction.	Tracing paper first, rules second; direction stated in every instruction.
Confusing with .	Verify each against a traced example; the physical turn settles it.
Giving an incomplete instruction (“rotate ”).	Centre, angle and direction required — marked.
Believing needs a direction.	Explicitly noted; both directions agree.
Thinking rotation changes size or shape.	Measure the image; congruence holds for all six transformations.
Assuming both-negated must be a double reflection (or must be a rotation).	Image 3 shows they are the same transformation, describable either way.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A point is rotated about the origin and lands at . Where did it start?

Answer

— a rotation is its own inverse.

E2 (AMC Junior style). What single transformation is equivalent to rotating anticlockwise twice?

Answer

: ✓

E3 (Challenge). A square has vertices . Which rotations about the origin map it onto itself?

Answer

, and (and ) — the square has rotational symmetry of order about the origin. Test: , already a vertex ✓

E4 (Challenge). A point is rotated anticlockwise, then reflected in the -axis. Find the single rule, and identify the transformation.

Answer

. Testing and : this is a reflection in the line . (Beyond the axis reflections taught, but a genuine transformation — worth naming.)

E5 (Challenge). After rotating anticlockwise about the origin, a triangle has vertices . Find the original.

Answer

Undo with a anticlockwise rotation, : .

Homework

1. Rotate about the origin by anticlockwise: (a) (b) (c) (d) .
2. Rotate about the origin by : (a) (b) (c) .
3. Rotate about the origin by clockwise: (a) (b) .
4. Name the rotation (about the origin): (a) (b) (c) .
5. Triangle is rotated anticlockwise about the origin. (a) List the image. (b) Plot both. (c) Verify one vertex with tracing paper.
6. Classify each as translation, reflection or rotation, and give its full description: (a) , (b) , (c) , .
7. A shape is rotated anticlockwise, then anticlockwise again. Write the single rule.
8. A point rotated about the origin landed at . Where did it start?
9. Reasoning. Explain why a rotation needs no direction stated.
10. Reasoning. Which transformations preserve orientation, and which reverse it? Explain what orientation means using a labelled triangle.
11. Challenge. A square has vertices . It is rotated anticlockwise about the origin, then translated by . Find the final vertices, and state whether the result could have been achieved by a single translation.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) . Q3 — using : (a) (b) . Q4 — (a) acw (b) (c) acw / cw. Q5 — (a) . Q6 — (a) rotation acw about the origin (b) reflection in the -axis (c) translation . Q7 — , i.e. . Q8 — . Q9 — both directions carry a point to the diametrically opposite position, so the images coincide. Q10 — translations and rotations preserve it; reflections reverse it; labelling a triangle and tracing the order shows the direction of travel flipping only under reflection. Q11 — rotated: ; translated: . No single translation works — the square has been turned, and translations cannot turn a shape (though for a square the final position happens to look like a square again, the labelled vertices have moved around it).