At a glance. One lesson of about 60 minutes. It assumes coordinates and enlargement. Students find images under fractional and negative scale factors, centres other than the origin, a centre from two points, and the effect on area.
Calculator: no calculator. The arithmetic is small and exact, and the lesson is about the structure P' = C + k(P − C). A calculator is not needed.
What students will learn
Know
- P' = C + k(P − C).
- Negative k puts the image on the opposite side of the centre.
- Area is multiplied by k², lengths by |k|.
Understand
- Subtract the centre first, because the scaling is about the centre.
- k = −1 is a half turn, so the family of dilations contains rotations of 180°.
- Two dilations about one centre multiply their factors.
Do
- Find the image of a point or a shape under a dilation.
- Find the scale factor and the centre of a dilation from a point and its image.
- State the area factor of a dilation.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: dilations by rational and negative scale factors (questions 2 to 8).
- Criterion B. Investigating patterns: the what-if, two dilations about different centres.
- Criterion C. Communicating: the three-line layout of subtract the centre, multiply, add the centre in the worked examples.
- Criterion D. Applying mathematics in real-life contexts: the pinhole camera, where the image is inverted.
The one thing to do with the figure
Predict the image of P before moving the slider. Set the centre to (0, 0) and the scale factor to −1, then ask for the image of P and Q before reading them. Then change the centre and ask what changes.
Ask: where is the image when the scale factor is 0? Move the scale-factor slider to 0 and see.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the pinhole camera. Collect predictions for side and size. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask which side of the centre the image is on. |
| 18 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 | (12, 8); 45 cm²; (2, 5). |
|---|---|
| Hook | The image is on the far side of the pinhole and inverted, so the scale factor is negative. It is smaller when the screen is nearer the pinhole than the candle is. |
| Worked examples | (5, 5); (0, 3); scale factor 1.5, centre (−2, 1). |
| Faded example | (−4, −4). |
| 1. Image under a negative scale factor | C. A uses +2, B divides, D slips on −1 × (−2). |
| 2. Image with a centre that is not the origin | 8. 2 + 3(4 − 2). |
| 3. Negative fractional scale factor | −5. −2 − 3. |
| 4. Finding a scale factor | −1.5. −9 ÷ 6. |
| 5. Finding the centre | −1. 8 = 6 − 2c. |
| 6. Area under a negative factor | 2. 18 ÷ 9. |
| 7. Working back to the original point | 4. 1 + (−6) ÷ (−2). |
| 8. Two dilations from one centre | −6. 2 × (−3). |
| 9. Scale factor from an area and a side | B. A uses the area factor as k, C loses the sign, D inverts the ratio and takes no root. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used a positive factor, divided, or slipped on a negative times a negative. | Multiply both coordinates by the factor and keep the sign. |
| 12 or 6 in question 2 | Ignored the centre, or did not add it back. | Write the three lines: subtract, multiply, add. |
| −2, −4 or 1 in question 3 | Ignored the centre, added it wrongly, or lost the sign. | Subtract the centre first. |
| 1.5 or −0.67 in question 4 | Lost the sign, or inverted the ratio. | Scale factor = image ÷ original. |
| 1 or 3 in question 5 | Used the formula with the subtraction reversed, or used the shift. | Solve the equation for c. |
| 6 or −2 in question 6 | Used k instead of k², or kept the sign. | Square first. |
| −11 or −2 in question 7 | Multiplied instead of dividing, or lost the sign. | Undo the dilation: divide the vector from the centre. |
| −1, 6 or −1.5 in question 8 | Added, lost the sign, or divided the factors. | Multiply the factors. |
| A, C or D in question 9 | Used the area factor as k, lost the sign, or inverted the ratio. | Write k² = image area ÷ original area, then decide the sign from the side. |
Questions worth asking
- Why must the centre be subtracted first? The scaling is about the centre, so the distances are measured from it.
- What is the image of the centre itself? The centre, for every scale factor.
- When is the image congruent to the original? When k = 1 (the same) or k = −1 (a half turn).
- Can two different centres give the same image of a shape? Not for the same non-zero k other than 1, but a dilation plus a translation can match another.
Differentiation
A way in for anyone stuck
- Give a ray diagram from the centre with the distance marked in squares.
- Provide a three-column table: P − C, multiplied, plus C.
- Use centre (0, 0) for the first questions, then move the centre.
Stretch
- A dilation with centre (2, 3) and scale factor k sends (5, 7) to (−1, −1). Find k.
- Show that a dilation with scale factor −1 about the point (a, b) sends (x, y) to (2a − x, 2b − y).
Students learning in English as an additional language
“Dilation” means a change of size from a centre. The “centre” is the fixed point. The “image” is where a point goes. Say “on the same side” and “on the opposite side” while pointing.
| Word | Plain meaning | Example |
|---|---|---|
| dilation | a transformation that multiplies every distance from a centre by the same scale factor | a photo enlarged from its centre |
| centre of dilation | the one point that does not move | the pinhole of a camera |
| scale factor | the number the distances from the centre are multiplied by | 3 makes every distance from the centre three times as long |
| image | the point or shape after the transformation | P′ is the image of P |
| opposite side | on the other side of the centre, on the same line | (−1, 0) is on the opposite side of the origin from (3, 0) |
Sentence starters for explaining aloud or in writing:
- The vector from the centre to P is …
- The scale factor is negative because the image is …
- The centre is where the lines through each point and its image …
Two things not to say
“Multiply the coordinates by the scale factor.” That works only about the origin.
“Negative means smaller.” The size is |k|; the sign decides the side.
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.
- ImageFind the image of (3, 5) under a dilation about the origin with scale factor −2.
(−6, −10). - FactorA dilation about O sends (4, 8) to (1, 2). Find the scale factor.
1 ÷ 4.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA lamp at (0, 0) casts the shadow of a card corner at (2, 3) onto a screen so that the shadow corner is at (5, 7.5). Find the scale factor.
2.5. - Earlier work (Year 9)Translate (4, −2) by the vector (−3, 5).
(1, 3).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFind the image of (5, −2) under centre (3, 1) with scale factor −4.
(−5, 13). - ProofShow that two dilations about different centres with scale factors 2 and ½ combine to a translation.
With centres A and B, P goes to A + 2(P − A), then to B + ½(A + 2(P − A) − B) = P + ½(B − A). Every point moves by the same vector ½(B − A), so it is a translation.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works 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 calculator is needed. Every answer is reached by hand.