102i - Consolidation Transformations

Lesson 102 — Problem Solving and Consolidation: Transformations

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

Learning Intentions

• To consolidate performing, identifying and describing all six transformations.
• To apply transformations in design and pattern contexts.

Success Criteria

I can:

1. Perform any of the six transformations accurately.
2. Identify and fully describe a transformation from coordinates.
3. Combine transformations and find single equivalent rules.
4. Use transformations to create and analyse patterns.

Warmup

(6 minutes — rapid fire, mini whiteboards)

1. reflected in the -axis.
2. rotated anticlockwise.
3. translated by .
4. : name it.
5. : name it fully.

Answers: ; ; ; reflection in the -axis; none of our six — testing every rule fails. The mapping swaps the coordinates, , which is a reflection in the line .

Teacher note on Q5: this is deliberately outside the taught set. Students who test the rules rather than guess will find every one of the six fails. Reward the checking — and name the line as a legitimate mirror we simply have not studied.

Activities

Activity 1 — Mixed Skills Circuit (16 min)

Stations spanning Lessons 98–101.

Station A — Perform. For triangle , , :

1. Reflect in the -axis.
2. Reflect in the -axis.
3. Rotate anticlockwise about the origin.
4. Rotate about the origin.
5. Translate by .

Station B — Identify fully.

6. ;
7. ;
8. ;
9. ;

Station C — Combine.

10. Reflect in the -axis, then translate . Single rule?
11. Rotate acw, then rotate . Single rule and single description?
12. Translate , then translate . Single rule?

Station D — Work backwards.

13. After a reflection in the -axis, a point is at . Where did it start?
14. After a rotation of acw, a point is at . Where did it start?
15. After a translation, became . Apply the same translation to .

(Answers: 1. ; 2. ; 3. ; 4. ; 5. ; 6. reflection in -axis; 7. rotation acw; 8. translation ; 9. rotation ; 10. ; 11. : rotation acw; 12. ; 13. ; 14. undo with acw: ; 15. rule : .)

Activity 2 — Transformations in Design (14 min)

Pairs, grid paper. Transformations as a design tool.

Task 1 — the border pattern. Start with the shape — a small triangle.

1. Apply the translation repeatedly, five times, plotting each image. Describe the result.
2. Now alternate: translate , then reflect the new copy in the -axis, and repeat. Describe how the pattern changes.

Task 2 — the rotational rosette. Start with the shape .

3. Rotate it , and anticlockwise about the origin, plotting all four positions.
4. What symmetry does the finished figure have?
5. If you only used , what symmetry would you get instead?

Task 3 — reverse-engineer a logo. A design shows a shape at and a matching shape at .

6. Identify the transformation used.
7. Verify on all three vertices.

Socratic scaffolding for Task 3:

Prompt	Purpose
Did the coordinates swap?	Yes — so a rotation.
Which one became negative?	: the new first is — that is .
Test the second vertex.	✓
And the third.	✓
Describe it completely.	Rotation of anticlockwise about the origin.

(Answers: 1. a repeating frieze, evenly spaced, all facing the same way; 2. copies alternate up and down — a zigzag border; 4. rotational symmetry of order ; 5. order only; 6–7. rotation acw, verified on all three vertices.)

Activity 3 — Inquiry: what Stays the Same? (7 min)

Whole class, closing the block.

Across all six transformations, we have seen some things change and others hold fast.

1. List everything that is always preserved.
2. List what can change.
3. Why is “congruent” the right word for an image and its original?

Answers to draw out:

• Always preserved: side lengths, angle sizes, area, perimeter, shape.
• Can change: position (all six), orientation (reflections only).
• Congruent means identical in size and shape — which is exactly what “lengths and angles preserved” delivers. Every transformation in this block produces a congruent image.

The forward link: transformations that do change size (enlargements) exist and come later — this block was deliberately about the congruence-preserving ones.

Checks for Understanding

(7 minutes — exit ticket, collected)

1. Reflect in the -axis.
2. Rotate by about the origin.
3. Identify fully: and .
4. Find the single rule: reflect in the -axis, then translate .
5. After a translation, became . Where does go?
6. Reasoning. Which transformations could map a triangle onto a congruent triangle that is “flipped over”? Explain.

Answers: 1. ; 2. ; 3. Rotation anticlockwise about the origin; 4. ; 5. Rule : ; 6. Only the reflections — they alone reverse orientation; translations and rotations cannot flip a shape.

Common Misconceptions

Misconception	How to pre-empt it
Guessing a transformation instead of testing the rule.	The warmup’s out-of-set Q5 rewards checking.
Incomplete descriptions.	Marked at every station.
Applying the second transformation to the original rather than the image.	Station C requires tracking the point through both steps.
Undoing a acw rotation with another acw.	Station D Q14; the inverse is acw.
Believing transformations change size.	The closing inquiry names congruence explicitly.
Reading a design pattern without identifying the generating rule.	Task 3’s vertex-by-vertex verification.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A shape is rotated acw four times. Where does it end up?

Answer

Exactly where it started — four quarter-turns make a full turn.

E2 (AMC Junior style). A triangle at is reflected in the -axis and then in the -axis. Find the image, and name the single equivalent transformation.

Answer

; a rotation of about the origin.

E3 (Challenge). How many of the six transformations map the square onto the same set of points?

Answer

Reflection in -axis ✓, reflection in -axis ✓, rotations of , , ✓ — five of the six. Only a non-zero translation fails, since it would move the square away from the origin.

E4 (Challenge). A point is transformed by an unknown single transformation: . List every taught transformation this is consistent with, then find a second point-mapping that would distinguish them.

Answer

Consistent with rotation acw ( gives ✓) and with the translation ✓. Distinguish with : the rotation sends it to ; the translation to .

E5 (Challenge). A design has rotational symmetry of order about the origin. Through what angles can it be rotated onto itself?

Answer

Multiples of : and .

Homework

1. Triangle , , . List the image after: (a) reflection in the -axis (b) reflection in the -axis (c) rotation acw (d) rotation (e) translation .
2. Identify fully, citing evidence: (a) ; (b) ; (c) ; (d) ; .
3. Find the single rule: (a) reflect in -axis, then translate (b) rotate , then reflect in -axis (c) translate , then translate .
4. Work backwards: (a) after reflection in the -axis a point is at — where did it start? (b) after a acw rotation a point is at — where did it start? (c) after a translation, became — where does go?
5. Start with the shape and apply four times. Sketch the frieze and describe it in one sentence.
6. Start with and rotate it , , acw about the origin. Sketch and state the order of rotational symmetry.
7. Two shapes: and . Identify the transformation and verify on all three vertices.
8. Reasoning. List everything preserved by every transformation in this block, and the one thing only reflections change.
9. Reasoning. Explain why undoing a anticlockwise rotation requires a anticlockwise rotation, not another .
10. Challenge. A shape is transformed twice and ends exactly where it started. List three genuinely different pairs of transformations that could do this.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) reflection in -axis (b) rotation acw (c) translation (d) rotation . Q3 — (a) (b) — a reflection in the -axis (c) . Q4 — (a) (b) (c) rule : . Q6 — order . Q7 — rotation about the origin, verified on all three. Q8 — lengths, angles, area, perimeter and shape; only reflections change orientation. Q9 — the inverse must return the point the way it came; a second acw carries it further round to . Q10 — e.g. reflect in -axis twice; rotate twice; translate then ; rotate acw then acw.