101i - Describing Transformations with Coordinates

Lesson 101 — Describing Transformations Using Coordinates

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

Learning Intentions

• To describe any transformation precisely using coordinate rules.
• To identify a transformation from an original and its image.

Success Criteria

I can:

1. Write any of the six transformations as a coordinate rule.
2. Identify a transformation from coordinate evidence alone.
3. Describe a transformation completely, with all required information.
4. Combine two transformations and find the single equivalent rule.

Warmup

(6 minutes — the rules from memory, individual then check)

Reproduce the summary table from Lesson 100, then check against your notes.

Transformation	Rule	Orientation
Translation
Reflection in -axis
Reflection in -axis
Rotation acw
Rotation
Rotation acw

Answers: preserved; reversed; reversed; preserved; preserved; preserved.

Today’s skill is the reverse direction: given the coordinates, name the transformation.

Activities

Activity 1 — Explicit Instruction: the Identification Protocol (14 min)

The four diagnostic questions, in order:

1. Did the coordinates swap? No → translation or reflection. Yes → rotation ( or ).
2. If no swap: were they both changed by adding? → translation. Were signs flipped? → reflection or rotation.
3. Which coordinate flipped? Only → reflection in -axis. Only → reflection in -axis. Both → rotation.
4. If swapped: which one is negative? → acw. → acw.

Always check on a second point. One point can fit several rules; a transformation must work for every point.

I do — three worked identifications.

Case 1: and .

No swap; only flipped, consistently → reflection in the -axis.

Case 2: and .

Swapped; first coordinate is the negated → → rotation anticlockwise about the origin.

Case 3: and .

No swap; both flipped → rotation about the origin. (Could also be described as a double reflection — same transformation.)

The completeness standard. A description must include everything needed to reproduce it:

Transformation	Must state
Translation	the rule, or ” right and up”
Reflection	which axis (or mirror line)
Rotation	centre, angle and direction (except )

“It’s a rotation” is incomplete. “Rotation of anticlockwise about the origin” is complete.

We do — identify and describe fully:

1. and
2. and
3. and
4. and

(Answers: 1. reflection in the -axis; 2. translation ; 3. rotation acw ( cw) about the origin; 4. rotation about the origin.)

Activity 2 — Identification Circuit (14 min)

Pairs. Every answer states the transformation completely and cites the evidence.

Set A — identify. Each gives two point-mappings:

1. ;
2. ;
3. ;
4. ;
5. ;
6. ;

Set B — apply and describe. For triangle , , :

7. Reflect in the -axis. List the image and describe the orientation change.
8. Rotate about the origin. List the image.
9. Translate by . List the image.
10. Rotate anticlockwise. List the image.

Set C — the impostor. One of these is not a single transformation. Find it and explain.

11. ; ;
12. ; ;
13. ; ;

Socratic scaffolding for Set C:

Prompt	Purpose
Test each mapping against one candidate rule.	Systematic checking, not eyeballing.
For 12: the first two suggest what?	Reflection in the -axis, .
Does the third obey it?	should map to , but it went to ✗
So what happened?	Two different rules were applied — not a single transformation.
Why does this matter?	A transformation acts on every point identically; that is its definition.

(Answers: 1. reflection in -axis; 2. rotation acw; 3. translation ; 4. reflection in -axis; 5. reflection in -axis; 6. rotation acw; 7. — orientation reversed; 8. ; 9. ; 10. ; 11. translation ✓; 12. the impostor; 13. reflection in -axis ✓.)

Activity 3 — Inquiry: Combining Transformations (8 min)

Pairs.

Apply each pair of transformations in order to the point , then find the single rule that does both.

1. Reflect in the -axis, then reflect in the -axis.
2. Rotate acw, then rotate acw again.
3. Translate , then reflect in the -axis.
4. Reflect in the -axis, then translate .
5. Compare 3 and 4. Does order matter?

Socratic scaffolding:

Prompt	Purpose
Track through Q3.	.
Track it through Q4.	.
Different destinations — so?	Order does matter when a reflection meets a translation.
But Lesson 98 said translations commute.	Two translations commute; mixing types generally does not.
Q1 and Q2 both give what?	— the rotation, by two different routes.
Looking back	Some combinations commute, some do not — checking is required, not assuming.

(Answers: 1. ; rule ; 2. ; same rule; 3. ; rule ; 4. ; rule ; 5. No — order matters here.)

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Identify fully: and .
2. Identify fully: and .
3. Identify fully: and .
4. Why must a rotation’s description include a direction?
5. Reasoning. and and . Is this a single transformation? Justify.

Answers: 1. Rotation acw about the origin; 2. Reflection in the -axis; 3. Translation ; 4. Clockwise and anticlockwise give different images (except at ); 5. No — the first two shift by but the third by . A transformation must act identically on every point.

Common Misconceptions

Misconception	How to pre-empt it
Identifying from one point only.	Every diagnosis checks a second point; Set C punishes single-point reasoning.
Confusing with .	The diagnostic question “which one is negative?”; verify by tracing paper.
Incomplete descriptions (“a rotation”).	The completeness table, marked on every answer.
Assuming all combinations commute.	The inquiry’s Q3/Q4 contrast.
Believing a rotation and a double reflection are different transformations.	They produce identical images — same transformation, two descriptions.
Treating a “shape landed on a shape” correspondence as automatically a transformation.	Set C Q12’s impostor.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which single transformation maps to for every point?

Answer

Reflection in the line — beyond the axis reflections taught, but the coordinates name it exactly.

E2 (AMC Junior style). A point is rotated acw, then reflected in the -axis. Find the single rule.

Answer

— a reflection in the line .

E3 (Challenge). A triangle is transformed to an image with the same orientation. List every transformation this could have been, and one it could not.

Answer

Could be: any translation, or any rotation (, , ). Could not be: either axis reflection — reflections always reverse orientation.

E4 (Challenge). Find a transformation such that applying twice returns every point to its start, other than “do nothing”. How many such transformations have we met?

Answer

Three: reflection in the -axis, reflection in the -axis, and rotation . Each is its own inverse. (A translation works only if .)

E5 (Challenge). Point is reflected in the -axis, then rotated acw, landing at . Find .

Answer

Undo the rotation with acw, : . Undo the reflection: . So .

Homework

1. Identify each transformation fully, citing your evidence: (a) ; (b) ; (c) ; (d) ; (e) ; (f) ;
2. Triangle , , . List the image after each: (a) reflection in the -axis (b) rotation (c) translation (d) rotation acw.
3. State which of these preserve orientation: translation, reflection in -axis, rotation acw, reflection in -axis, rotation .
4. Find the single rule for: (a) reflect in -axis, then reflect in -axis (b) rotate , then translate .
5. Which of these is not a single transformation? Explain. ; ; .
6. Point is translated by then reflected in the -axis, landing at . Find .
7. Reasoning. Explain why checking a single point is not enough to identify a transformation.
8. Reasoning. Explain why a reflection can never be replaced by a translation, however cleverly chosen.
9. Challenge. A square has vertices . Find two different transformations that each map the square onto exactly the same set of four points, and explain how the labelled vertices differ.

Answers: Q1 — (a) reflection in -axis (b) rotation acw about origin (c) reflection in -axis (d) translation (e) rotation (f) rotation acw. Q2 — (a) (b) (c) (d) . Q3 — translation, both rotations. Q4 — (a) (b) . Q5 — the third mapping shifts by while the first two shift by — not a single transformation. Q6 — undo reflection: ; undo translation: . Q7 — many different rules can agree on one point; only checking further points distinguishes them. Q8 — translations preserve orientation and reflections reverse it, so no translation can produce a flipped image. Q9 — e.g. rotation about the square’s centre , and reflection in the line : both leave the square occupying the same region, but the vertex labels end in different positions.