A reflection flips a shape over a mirror line. In the x-axis, (x, y) becomes (x, −y); in the y-axis, (x, y) becomes (−x, y). Each point lands as far on the other side as it started.
A face in a mirror, a butterfly's wings, a lake showing a mountain. The mirror line sits exactly halfway between each point and its reflection.
A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror, on opposite sides, and the mirror cuts the line joining them at right angles.
Reflect in the x-axis: the point moves up or down, so y changes sign: (x,y)→(x,−y).
Reflect in the y-axis: the point moves left or right, so x changes sign: (x,y)→(−x,y).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Flip over the x-axis
Picture the point flipping straight down over the x-axis.
TryTry it on the board and compare the result with your prediction.
Two mirrors
Press the buttons to choose a mirror, and read the list of corners.
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Reflect the triangle in the x-axis. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Which coordinate changes
Which direction does the point move?
Read Mia's working. Which way does the point cross the x-axis?
Mia reflects P (4, 3) in the x-axis. Click the line where her reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
Choose the mirror
Need a hint?
The triangle is on the left of the y-axis.
Another hint
A has x = −4. Flip it across the y-axis.
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Reflect the triangle so that A′ lands at (4, 1).
Use the board. This ticks when you get there.
Reflect the triangle so that C′ lands at (−2, −4).
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Reflections
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The point (−4, 6) is reflected in the y-axis. What is the x-coordinate of the image?
The point (−4, 6) is reflected in the x-axis. What is the y-coordinate of the image?
Review: the point (2, 3) is translated 4 left and 1 up. What is the x-coordinate of the image?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
An arrow's mirror image
Then write the rule for each mirror in your own words, and say which coordinate stays the same.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The point of the arrow is at (4, 2).
Another hint
To get x = −4, change the sign of x.
Reflect the arrow so that its point (corner C) lands at (−4, 2).
Use the board. This ticks when you get there.
For teachers
Students reflect shapes in the x-axis and y-axis and state which coordinate changes sign (AC9M7SP03, node 2). About 35 minutes.
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.
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
Predict: ask which way the point travels across the mirror.
Run: ask what stays the same for each mirror.
Investigate: ask how Mia could check her answer with the picture.
Modify: for the two mirrors, ask what single move does the same.
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
A student changes the wrong coordinate. Ask: Does the point cross the mirror by going up and down, or left and right? Fix: The coordinate that measures the distance to the mirror changes sign.
A student changes both signs. Ask: How many mirrors did you use? Fix: One mirror changes one coordinate; two mirrors change both.
A student puts the image on the same side. Ask: Which side of the mirror is the image on? Fix: The image is on the opposite side of the mirror.
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.
Where it leads
This continues AC9M7SP03.
Where it leads: rotations (next lesson); reflection and rotational symmetry in Year 8; combining transformations.
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.
Now you can
Reflect a shape in the x-axis or the y-axis.
Say which coordinate changes sign for each axis.
Check a reflection by its distance from the mirror.