Transformations on the Cartesian plane
Translating, reflecting and rotating a point, and describing each move with coordinates
About three lessons of 45 to 60 minutes
How do you tell someone EXACTLY how a shape moved?
"It moved a bit to the right and flipped over" is not precise enough for a video game engine, an animator, or a robot arm. Coordinates fix that: every move on the Cartesian plane can be described by an exact rule applied to every point's (x, y) position.
This unit covers three kinds of moves, called transformations: sliding a shape (translation), flipping it across an axis (reflection), and turning it about a fixed point (rotation). Each has its own coordinate rule, and once you know the rule, you can find the exact new position of any point without drawing anything at all.
- A game character sliding across the screena translation: every point moves the same distance in the same direction
- A mirror image in a lakea reflection: every point flips across a line
- A ferris wheel car turninga rotation: every point turns the same angle about a fixed centre
- Coordinates pinning down an exact positionbefore and after a move, so the transformation can be checked precisely
What students will be able to do
Students will translate a point by given horizontal and vertical steps, reflect a point across the x-axis or y-axis, and rotate a point 90, 180 or 270 degrees anticlockwise about the origin, finding the exact new coordinates in each case.
- I can translate a point by given horizontal and vertical steps and find its new coordinates.
- I can reflect a point across the x-axis or y-axis and find its new coordinates.
- I can rotate a point 90, 180 or 270 degrees anticlockwise about the origin and find its new coordinates.
- I can describe, using a coordinate rule, what each transformation does to a point.
Standards this unit teaches
- AC9M7SP03Australian Curriculum v9 (ACARA)Transformations on the plane
Describe transformations of a set of points using coordinates in the Cartesian plane: translations, reflections on an axis, and rotations about a given point.
- 8.G.A.3Common Core (US)Transformations with coordinates
Describe the effect of dilations, translations, rotations, and reflections on figures using coordinates.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 coordinate plane teaching unitplotting points in all four quadrants, including with negative coordinates
- Coordinates in the glossarya refresher on reading and plotting an (x, y) point before this unit's moves
- Integers worksheetsreflections and rotations often flip a coordinate's sign, so comfort with negative numbers helps
Words to teach and display
- Transformation
- a move that changes a point or shape's position, such as a translation, reflection or rotation
- Translation
- sliding every point the same distance in the same direction, without turning or flipping
- Reflection
- flipping every point across a line (an axis), so it faces the opposite direction
- Rotation
- turning every point the same angle about a fixed centre point
- Origin
- the point (0, 0), where the x-axis and y-axis cross
- Anticlockwise
- turning in the opposite direction to a clock's hands; the direction used throughout this unit
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Translation: sliding every point the same way
ConcreteA translation slides every point the same distance in the same direction. Given a horizontal step (dx) and a vertical step (dy), the coordinate rule is (x, y) -> (x + dx, y + dy): add dx to x, and add dy to y.
Translating triangle A(-4,-3), B(-2,-3), C(-3,-1) by (+5, +4) moves every vertex the same way: A(-4,-3) -> A'(1,1), B(-2,-3) -> B'(3,1), C(-3,-1) -> C'(2,3). The triangle's size, shape and angles never change, only its position.
Translate the point (2, -3) by (+4, +5). Find its new coordinates.
- The translation rule is (x, y) -> (x + dx, y + dy).
- Substitute: x + dx = 2 + 4 = 6, and y + dy = -3 + 5 = 2.
Answer: (6, 2).
- Why do all points of a shape move by the exact same (dx, dy) in a translation?
- Does a translation ever change a shape's size or angles? Why or why not?
2. Reflection: flipping across the x-axis or y-axis
PictorialReflecting across an axis keeps the coordinate ALONG that axis unchanged, and flips the sign of the other coordinate. Reflecting across the x-axis: (x, y) -> (x, -y). Reflecting across the y-axis: (x, y) -> (-x, y).
Reflecting triangle A(1,2), B(4,2), C(2,5) across the y-axis flips the sign of every x-coordinate, giving A'(-1,2), B'(-4,2), C'(-2,5). Each point's y-coordinate, its height, stays exactly the same, only its left-right position flips.
Reflect the point (5, -2) across the x-axis. Then reflect the point (5, -2) across the y-axis. Find both new coordinates.
- Reflecting across the x-axis flips the sign of the y-coordinate: (5, -2) -> (5, 2).
- Reflecting across the y-axis flips the sign of the x-coordinate: (5, -2) -> (-5, -2).
Answer: Across the x-axis: (5, 2). Across the y-axis: (-5, -2).
- Which coordinate changes sign when reflecting across the x-axis, and which stays the same?
- If a point is reflected across the y-axis and then reflected across the y-axis again, where does it end up? Why?
3. Rotation: turning about the origin
AbstractRotating a point anticlockwise about the origin follows a coordinate rule for each angle: 90 degrees is (x, y) -> (-y, x); 180 degrees is (x, y) -> (-x, -y); 270 degrees is (x, y) -> (y, -x). This unit always rotates anticlockwise unless stated otherwise.
Rotating triangle A(2,1), B(5,1), C(3,4) by 90 degrees anticlockwise about the origin uses (x, y) -> (-y, x): A(2,1) -> A'(-1,2), B(5,1) -> B'(-1,5), C(3,4) -> C'(-4,3). Every point turns the same 90-degree angle about the same fixed centre, the origin.
Rotate the point (4, -2) by 180 degrees anticlockwise about the origin. Then rotate the point (4, -2) by 270 degrees anticlockwise about the origin. Find both new coordinates.
- 180 degrees: (x, y) -> (-x, -y). Substitute: (4, -2) -> (-4, 2).
- 270 degrees: (x, y) -> (y, -x). Substitute: (4, -2) -> (-2, -4).
Answer: 180 degrees: (-4, 2). 270 degrees: (-2, -4).
- Why does rotating 180 degrees give the same result whether you go clockwise or anticlockwise?
- How could a 270 degree anticlockwise rotation be checked using three separate 90 degree rotations in a row?
Common misconceptions and how to address them
MisconceptionReflecting across the x-axis flips the sign of the x-coordinate.
Why it happens: Students assume the axis named in the instruction is the coordinate that changes.
How to address it: It is the OPPOSITE: reflecting across the x-axis moves a point straight up or down (perpendicular to that axis), so y flips and x stays the same. Reflecting across the y-axis moves a point left or right, so x flips and y stays the same.
MisconceptionTranslating a shape can change its size or angles, since the shape 'moves'.
Why it happens: Students conflate translation (a slide) with dilation (a resize), since both change a shape's position on the page.
How to address it: A translation only slides; every side length and every angle stays exactly the same before and after, only the position changes. Compare a translated shape's measurements directly against the original to confirm this.
MisconceptionRotations are always described clockwise unless told otherwise.
Why it happens: Students default to the clock-reading direction they use every day.
How to address it: This unit, and the Australian Curriculum wording, always rotates ANTICLOCKWISE unless a problem explicitly says clockwise. Always check the stated direction before applying a rule.
Guided practice (with answers)
1. Translate (3, -5) by (+2, +6). Find the new coordinates.
Answer: (5, 1), because 3 + 2 = 5 and -5 + 6 = 1.
2. Reflect (-4, 7) across the x-axis. Find the new coordinates.
Answer: (-4, -7), because reflecting across the x-axis flips the sign of the y-coordinate only.
3. Reflect (-4, 7) across the y-axis. Find the new coordinates.
Answer: (4, 7), because reflecting across the y-axis flips the sign of the x-coordinate only.
4. Rotate (6, -1) 90 degrees anticlockwise about the origin. Find the new coordinates.
Answer: (1, 6), because (x, y) -> (-y, x) gives (-(-1), 6) = (1, 6).
5. Rotate (6, -1) 180 degrees anticlockwise about the origin. Find the new coordinates.
Answer: (-6, 1), because (x, y) -> (-x, -y) gives (-6, 1).
Independent practice worksheets
Practise translating, reflecting and rotating points with computed, never-wrong answer keys.
Differentiation
- Use actual graph paper and physically slide, flip or turn a paper cutout shape before writing any coordinate rule.
- Colour-code the coordinate that changes versus the coordinate that stays the same for each transformation type.
- Provide a reference card listing every coordinate rule (translation, both reflections, all three rotations) to check against while practising.
- Start with translations only (no sign-flip logic) before introducing reflections and rotations.
- Combine two transformations in sequence (e.g. reflect, then translate) and find the final coordinates.
- Ask students to find the single transformation that undoes a given transformation (e.g. what undoes a 90 degree rotation?).
- Explore what happens when a point ON an axis, such as (5, 0), is reflected across that same axis.
- Investigate whether reflecting across the x-axis then the y-axis gives the same result as a single 180 degree rotation, and explain why.
Assessment: exit ticket
A three-question exit ticket sampling translation, reflection and rotation.
1. Translate (-2, -6) by (+5, +3). Find the new coordinates.
Answer: (3, -3), because -2 + 5 = 3 and -6 + 3 = -3.
2. Reflect (3, -8) across the y-axis. Find the new coordinates.
Answer: (-3, -8), because reflecting across the y-axis flips only the x-coordinate's sign.
3. Rotate (2, 7) 90 degrees anticlockwise about the origin. Find the new coordinates.
Answer: (-7, 2), because (x, y) -> (-y, x) gives (-7, 2).
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 translation (section 1), Lesson 2 reflection (section 2), Lesson 3 rotation (section 3) plus mixed review and the exit ticket.
- This unit assumes comfort with plotting points in all four quadrants, including negative coordinates (Grade 6 coordinate plane unit).
- Language to repeat: 'translation slides, reflection flips, rotation turns'; 'reflecting across an axis keeps the value along that axis unchanged and flips the other'.
- Curriculum note: AC9M7SP03 (Australian Curriculum v9) covers all three transformation types at Year 7. The Common Core crossover, 8.G.A.3, covers the identical coordinate-rule skill (plus dilation, not covered here) one year later in the US sequence.
- Present and print both work: use Present to build each transformed triangle live on the grid with the class, then print the worksheet for independent practice.