099i - Reflections on an Axis

Lesson 99 — Reflections on an Axis

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

Learning Intentions

• To reflect points and shapes in the -axis and the -axis.
• To express reflections as coordinate rules and recognise what they preserve.

Success Criteria

I can:

1. Reflect a point in the -axis or -axis using coordinates.
2. State and use the rules and .
3. Reflect a whole shape and describe how its orientation changes.
4. Identify the axis of reflection from an original and its image.

Warmup

(6 minutes — mirror intuition, mini whiteboards)

Fold a grid along the -axis in your mind.

1. lands where? (Encourage guessing before any rule.)
2. lands where?
3. Now fold along the -axis instead: lands where?
4. Which coordinate changed each time, and which stayed?

Answers: 1. ; 2. ; 3. ; 4. Folding on the -axis flips ; folding on the -axis flips .

Q4 is the rule, discovered before it is stated. Record both, then formalise in Activity 1.

Activities

Activity 1 — Explicit Instruction: the Two Reflection Rules (14 min)

Definition. A reflection flips every point to the opposite side of a mirror line, at the same perpendicular distance from it.

The two rules:

The memory hook, said aloud: you flip the coordinate you cross. Reflecting in the -axis moves points up-and-over the -axis, so changes sign. Reflecting in the -axis moves points side-to-side across it, so changes sign.

I do — a point in each axis. :

Plot all three and check by folding a transparency or ruler-measuring distances to the axis.

Points on the mirror line don’t move. reflected in the -axis stays at — it is on the mirror. Test with in the -axis. Such points are invariant.

I do — a triangle. , , reflected in the -axis:

Point	Image

What changes and what does not. Preserved: side lengths, angles, area. Not preserved: orientation — the image is “flipped over”. Trace the vertices on the original (say, anticlockwise); on the image the same labels run clockwise. This is the crucial difference from Lesson 98’s translations.

We do:

1. Reflect in the -axis.
2. Reflect in the -axis.
3. Reflect in the -axis. (Invariant.)
4. Reflect in the -axis. (Invariant.)

(Answers: ; ; ; .)

Activity 2 — Practice and Identification (14 min)

Grid paper; plot everything.

Set A — reflect in the -axis.

4. Rectangle

Set B — reflect in the -axis.

8. Triangle

Set C — identify the mirror. For each original/image pair, name the axis of reflection:

Set D — decide the transformation. Translation, reflection, or neither?

13. and and
14. and and
15. and

Socratic scaffolding for Set D:

Prompt	Purpose
For 13, what happened to each ?	All negated; unchanged — reflection in the -axis.
For 14?	Every point shifted — a translation.
For 15, check both mappings against one rule.	swaps; swaps too — coordinates swapped, which is a reflection in the line , not in an axis.
Is 15 within our two rules?	No — flag it as a reflection in a different mirror. Beyond this lesson, but worth naming honestly.

(Answers: 1. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. invariant; 8. ; 9. -axis; 10. -axis; 11. -axis; 12. -axis; 13. reflection in -axis; 14. translation ; 15. reflection, but in the line .)

Activity 3 — Inquiry: Double Reflections (8 min)

Pairs.

1. Reflect in the -axis, then reflect the result in the -axis again. Where do you end up?
2. Reflect in the -axis, then reflect that in the -axis. Where do you land? What single transformation does this match?
3. Try the same two reflections in the opposite order. Same result?
4. What single rule describes “reflect in both axes”?

Socratic scaffolding:

Prompt	Purpose
Q1: track it.	— back to the start.
So a reflection undoes itself?	Yes — reflecting twice in the same mirror is the identity.
Q2: track it.	.
Both coordinates negated. What does that look like on the grid?	The point has been turned half a turn about the origin.
Q3: the other order?	— the same destination.
Q4: the combined rule?	.
Looking back	Two reflections in perpendicular mirrors make a rotation — exactly tomorrow’s topic.

Deliberate cliffhanger: the class has just built a rotation out of two reflections. Lesson 100 names it.

Checks for Understanding

(5 minutes — exit ticket)

1. Reflect in the -axis.
2. Reflect in the -axis.
3. Write the rule for reflection in the -axis.
4. . Which axis was the mirror?
5. Reasoning. Name one thing a reflection preserves and one thing it does not, and explain the difference from a translation.

Answers: 1. ; 2. ; 3. ; 4. The -axis; 5. Preserves lengths, angles and area; does not preserve orientation — the shape is flipped, whereas a translation slides without flipping.

Common Misconceptions

Misconception	How to pre-empt it
Negating the wrong coordinate.	”Flip the coordinate you cross”; check by folding or by measuring distances to the axis.
Negating both coordinates for a single reflection.	That is the double reflection (Activity 3) — distinguish it explicitly.
Believing points on the axis move.	Invariant points demonstrated in Activity 1.
Thinking reflection changes size.	Measure the image’s sides; congruence holds.
Missing the orientation change.	Trace on both shapes and compare the direction of travel.
Assuming every coordinate-swap is an axis reflection.	Set D Q15 names the mirror honestly as out of scope.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A point is reflected in the -axis and lands at . Where did it start?

Answer

— reflection is its own inverse.

E2 (AMC Junior style). Triangle is reflected in the -axis. Find the image’s area and compare with the original.

Answer

Both square units — reflections preserve area.

E3 (Challenge). A shape is reflected in the -axis and its image is identical to the original. What must be true of the shape?

Answer

It is symmetric about the -axis — every point has a partner directly opposite, and points on the axis are their own partners.

E4 (Challenge). Point is reflected in the -axis, then translated by , landing at . Find .

Answer

Undo the translation: . Undo the reflection: . So .

E5 (Challenge). Which points are unchanged by reflection in the -axis? By reflection in both axes?

Answer

-axis: all points with — the whole -axis. Both axes: only the origin , since forces .

Homework

1. Reflect in the -axis: (a) (b) (c) (d) .
2. Reflect in the -axis: (a) (b) (c) (d) .
3. Name the mirror axis: (a) (b) (c) (d) .
4. Triangle , , is reflected in the -axis. (a) List the image vertices. (b) Plot both. (c) Trace on each — what do you notice about the direction?
5. Rectangle is reflected in the -axis. List the image and state its area.
6. Classify each as translation, reflection, or neither: (a) , (b) , (c) , .
7. A point is reflected in the -axis and lands at . Where did it start?
8. Reflect in the -axis, then reflect the result in the -axis. Write the single rule for the combination.
9. Reasoning. Explain why reflecting twice in the same axis returns every point to its start.
10. Reasoning. Explain the difference between a translation and a reflection in terms of orientation.
11. Challenge. A shape has vertices . It is reflected in the -axis, then translated by . Find the final vertices, and find the single translation that would give the same result if the reflection is ignored — explain why no such translation exists.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — (a) -axis (b) -axis (c) -axis (d) -axis. Q4 — (a) , , (c) the direction of travel reverses. Q5 — ; area . Q6 — (a) reflection in -axis (b) translation (c) neither — the two points follow different rules. Q7 — . Q8 — ; rule . Q9 — the first reflection sends each point to the opposite side at equal distance; the second sends it back along the same path. Q10 — translations preserve orientation (no flip); reflections reverse it. Q11 — reflected: ; translated: . No single translation works because the shape’s orientation has been reversed, and translations cannot do that.