Reflections

Classroom resources

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Students will be able to

• I can reflect a shape in the x-axis or the y-axis.
• I can say which coordinate changes sign for each axis.
• I can check a reflection by its distance from the mirror.

Timing

Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min

Before class

Do the Run board yourself first. Students need translations (previous lesson).

The one idea: a point and its image are the same distance from the mirror on opposite sides. In the x-axis the point crosses up or down, so y changes sign; in the y-axis it crosses left or right, so x changes sign. The common mistake is the swap (Mia's mistake).

Reflecting in other lines (such as y = x) is wider and is left for the stretch.

Seeing both reflections close together is deliberate: the Run board lets students flip between them.

Running the lesson

1. Predict: ask which way the point travels across the mirror.
2. Run: ask what stays the same for each mirror.
3. Investigate: ask how Mia could check her answer with the picture.
4. Modify: for the two mirrors, ask what single move does the same.
5. Make: ask for the rule written for both axes.

The hard part (on purpose)

Which coordinate changes. Ask students to say which way the point moves across the mirror, up and down or left and right.

When students get stuck

Answers to the checks

1. flip over the x-axis: Point A is at (2, 1). It is reflected in the x-axis. Where does A′ land?
   (2, −1). The point flips over the x-axis, to the same distance below it, so only y changes sign.
2. two mirrors: Reflect the triangle in the x-axis. Was your prediction right?
   
   In the x-axis, y changes sign: A (2, 1) becomes A′ (2, −1).
3. Which axis: B is at (4, 1). It is reflected in the y-axis. Where does B′ land?
   (−4, 1)
   In the y-axis, x changes sign: (−4, 1).
4. Find the mistake: Mia reflects P (4, 3) in the x-axis. Click the line where her reasoning goes wrong.
   line 2
   Line 2: reflecting in the x-axis moves the point up or down across it, so y changes sign. The image is (4, −3).
5. On the mirror: A point is on the mirror line, for example (0, 3) reflected in the y-axis. Where is its image?
   In the same place: (0, 3)
   It is zero distance from the mirror, so it does not move: −0 is still 0.
6. Reach (4, 1): Reflect the triangle so that A′ lands at (4, 1).
   
   In the y-axis, x changes sign: (−4, 1) becomes (4, 1).
7. Reach (−2, −4): Reflect the triangle so that C′ lands at (−2, −4).
   
   C is at (−2, 4). In the x-axis, y changes sign: (−2, −4).
8. Distance: A point is 3 units above the x-axis. After a reflection in the x-axis, where is the image?
   3 units below the x-axis
   The image is the same distance from the mirror on the other side.
9. Both mirrors: A point (3, 2) is reflected in the x-axis, and then the image is reflected in the y-axis. Where does it end?
   (−3, −2)
   (3, 2) goes to (3, −2), and then to (−3, −2): both signs changed.
10. y-axis: The point (−4, 6) is reflected in the y-axis. What is the x-coordinate of the image?
    4
    x changes sign: −4 becomes 4, so the image is (4, 6).
11. x-axis: The point (−4, 6) is reflected in the x-axis. What is the y-coordinate of the image?
    -6
    y changes sign: 6 becomes −6, so the image is (−4, −6).
12. The rule: Which rule describes a reflection in the y-axis?
    (x, y) → (−x, y)
    The y-axis is the vertical mirror: x changes sign and y stays.
13. Review: translations: Review: the point (2, 3) is translated 4 left and 1 up. What is the x-coordinate of the image?
    -2
    2 − 4 = −2, so the image is (−2, 4).
14. an arrow's mirror image: Reflect the arrow so that its point (corner C) lands at (−4, 2).
    
    In the y-axis, x changes sign: (4, 2) becomes (−4, 2).

Solutions

One possible solution for each task. Other correct programs are fine.

choose the mirror › Reach (4, 1)

Open task: any program that does what the step asks.

choose the mirror › Reach (−2, −4)

Open task: any program that does what the step asks.

choose the mirror › Distance

Open task: any program that does what the step asks.

choose the mirror › Both mirrors

Open task: any program that does what the step asks.

an arrow's mirror image

Open task: any program that does what the step asks.

Stretch and support

Stretch: Reflect in the line y = x. What happens to the coordinates? What about two reflections in a row, in the x-axis and then the y-axis: which transformation is that?

Support: Use tracing paper folded along the axis and press through the corners.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Reflections".

The idea: Reflections on axes: I can Students reflect a set of points on the x-axis and y-axis and describe the transformation using coordinate rules.

What understanding looks like:
- Correctly reflects a set of points across the x-axis and y-axis.
- States the coordinate rule for each reflection.
- Does not confuse which coordinate negates for each axis.

Mistakes students often make here:
- A student changes the wrong coordinate.
- A student changes both signs.
- A student puts the image on the same side.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Translations using coordinates), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Reflections".

Exit ticket

The point (5, −2) is reflected in the x-axis. Where is the image?

(5, 2): y changes sign.

What each step is for

• flip over the x-axis: Committing to the image of a point in the x-axis shows whether they change the coordinate that matches the mirror's direction or the one the point crosses.
• two mirrors: Trying both axes on the same triangle shows which coordinate changes sign for each mirror.
• which coordinate changes: The axis you reflect in decides which coordinate changes sign; the point crosses the mirror (Mia changes the coordinate along it); a point on the mirror stays put.
• choose the mirror: Choosing a mirror to reach a target and comparing the two together shows the rule for each axis; two reflections together make a half turn.
• reflections: Rules for both axes and a translation review, mixed; changing both signs is the planted slip; the review keeps left and down fresh.
• an arrow's mirror image: Choosing the mirror that puts the point of an arrow in a named place combines the rule and a check on each corner.

Where it leads

This continues AC9M7SP03.

Where it leads: rotations (next lesson); reflection and rotational symmetry in Year 8; combining transformations.