At a glance. One lesson of about 60 minutes. It assumes Year 9 transformations and enlargement, and the scale factors from lesson 4. Students find images using coordinates, describe each transformation in full and recognize which are isometries.
Calculator: no calculator. The lesson is done by hand on a coordinate grid, so it carries no calculator label. Every coordinate is a whole number.
What students will learn
Know
- Translation: column vector (across, up). Rotation: angle, direction, centre. Reflection: mirror line. Enlargement: scale factor, centre.
- Translations, rotations and reflections are isometries.
- A point on the mirror line is its own image.
Understand
- A description is complete when it fixes the image uniquely.
- Isometries keep distances, so the image is congruent to the object; enlargement changes size, so the image is similar.
Do
- Find the image of a point under each of the four transformations.
- Describe a translation, rotation, reflection or enlargement fully from the object and image.
- Say which transformations change area.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: performing and describing the four transformations (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find the rules (x, y) to (y, −x) and (x, y) to (y, x) from the states.
- Criterion C. Communicating: writing each description in the full frame, with the vector, the centre and the angle named (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the mosaic tile in the hook.
The one thing to do with the figure
Settle the hook with the centre. Set Rotate, keep the angle and direction, and move the centre. Ask students to say what changed and what did not.
In the second figure, ask: which points do not move? Which mirror lines send the triangle onto itself, if any? None, but the line through the triangle's own vertices leaves those vertices in place.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tile. Collect votes and reasons. |
| 12 min | First figure; key idea; settle the hook. |
| 8 min | Second figure; worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the signs. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | (3, −5); (8, 6); 1 : 9. |
|---|---|
| Hook | The centre of the rotation. |
| Worked examples | top 7, bottom −3; (−1, −3); (6, 2); rotation of 180° about (1, 2). |
| Faded example | (6, −1). |
| 1. A column vector | B. A reverses both signs; C and D swap the positions. |
| 2. Image under a translation | 2. 6 − 4. |
| 3. Reflection in a vertical line | 10. 2 × 6 − 2. |
| 4. Rotation about the origin | −2. (5, −2). |
| 5. Rotation about another centre | 0. (0, 1). |
| 6. Enlargement from a centre | 7. 1 + 3 × 2. |
| 7. Finding a vector from two points | −3. −1 − 2. |
| 8. Reflection in a sloping line | 3. (3, 3). |
| 9. A full description | C. A has no centre; B fits only one point; D gives (2, 5). |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Reversed both signs, or put the numbers in the wrong places. | Say “across first, then up” and draw the arrow. |
| 10 or 3 in question 2 | Moved the wrong way, or gave the y-coordinate. | Write the new point as a pair before choosing one number. |
| 4 or 8 in question 3 | Gave the distance to the mirror, or measured from the origin. | Mark the mirror line, then count squares to it on each side. |
| 2 or 5 in question 4 | Turned anticlockwise, or gave the wrong coordinate. | Draw the point and rotate a piece of tracing paper. |
| 1 or 4 in question 5 | Used the origin as the centre, or turned anticlockwise. | Subtract the centre first; add it back at the end. |
| 9 or 6 in question 6 | Enlarged from the origin, or forgot to add the centre back. | Always write the vector from the centre first. |
| 3 or 1 in question 7 | Went from image to object, or added the coordinates. | Image minus object: say it aloud. |
| 5 or 7 in question 8 | Used y = x, or adjusted the wrong coordinate. | Check by the midpoint: it must lie on the mirror line. |
| A, B or D in question 9 | Left out the centre, tested only one point, or swapped without the sign. | Check the description on both A and B, and say the three parts of a rotation aloud. |
Questions worth asking
- Why does a rotation need a centre but a translation does not? A translation moves every point by the same vector; a rotation moves points by different amounts around a fixed point.
- Which transformations keep the area the same? Translations, rotations and reflections; an enlargement changes it by k².
- Can a reflection map a shape onto itself? Yes, when the mirror line is a line of symmetry.
- How do you check a reflection? The line joining a point to its image is perpendicular to the mirror and the mirror cuts it in half.
Differentiation
A way in for anyone stuck
- Give the description frame: “It is a translation by the column vector …”, “It is a rotation of …° clockwise about (…, …)”.
- Use tracing paper to turn and flip the shape.
- Restrict to centres at the origin before other centres.
Stretch
- Find the image of (x, y) when it is rotated 90° clockwise about (a, b), as a formula.
- A point is its own image under a rotation. Where must it be?
Students learning in English as an additional language
“Image” is the new shape after a transformation, and “object” is the original. “Clockwise” is the direction the hands of a clock turn. A “column vector” is two numbers written one above the other. Say “across, then up” every time.
| Word | Plain meaning | Example |
|---|---|---|
| translation | a slide: every point moves the same distance in the same direction | top 3, bottom −2 |
| rotation | a turn about a fixed point through an angle | a wheel turning on its axle |
| reflection | a flip in a mirror line | in the line y = x |
| enlargement | a change of size from a centre by a scale factor | scale factor 2 from the origin |
| isometry | a transformation that keeps every distance | a rotation |
| column vector | two numbers written one above the other for across and up | top 0, bottom 5: five squares straight up |
Sentence starters for explaining aloud or in writing:
- It is a translation by the column vector …
- It is a rotation of …° … about (…, …)
- It is a reflection in the line …
Two things not to say
“Just turn it.” A turn needs its centre; ask where the pin is.
“Write the translation as a point.” A vector is a movement, not a place; use the column form.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- TranslationFind the image of (1, 4) under the translation with top 5 and bottom −6.
(6, −2). - Earlier work (lesson 4)Two similar solids have a length scale factor of 2. Find the ratio of their volumes.
1 : 8.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tile pattern in Isfahan reflects a motif in the line x = 4. A corner of the motif is at (1, 3). Where is its image?
(7, 3). - Working backA reflection in a vertical line sends (2, 5) to (8, 5). Find the equation of the mirror.
x = 5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA triangle is rotated 90° clockwise about the origin and then 90° anticlockwise about the origin. What single transformation is that?
The identity: nothing moves. - OpenFind a rotation and a reflection that both map (3, 0) to (0, 3).
Rotation: 90° anticlockwise about the origin. Reflection: in the line y = x.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Squared paper, tracing paper and a pin are enough.