At a glance. One lesson of about 60 minutes. Enlarging and reducing from the origin, finding a scale factor, and the effects on length, angle and area.
Calculator: no calculator. The scale factors are 0.5, 2, 2.5, 3 and 4 so that every product can be done by hand.
What students will learn
Know
- An enlargement from the origin with scale factor k maps (x, y) to (kx, ky).
- Lengths are multiplied by k, angles are unchanged and areas are multiplied by k².
- An enlargement is a similarity transformation, not a congruence transformation, unless k = 1.
Understand
- Each image point lies on the line from the centre through its pre-image point.
- A scale factor below 1 gives a reduction and is still called an enlargement.
Do
- Enlarge or reduce a point or shape from the origin by a given scale factor.
- Find the scale factor from a pre-image and its image.
- Say which transformations keep size and shape and which do not.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying (kx, ky) and finding k (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to find how base, height and area change with k and to state a rule for each.
- Criterion C. Communicating: writing the rule with k, and explaining why area uses k squared.
- Criterion D. Applying mathematics in real-life contexts: photographs, maps, models and signs, where a scale factor links the copy to the original.
The one thing to do with the figure
Before moving the slider, ask for the image. Set scale factor 2 and ask where B (2, 1) lands. Then ask what changes if the factor is 3.
On the second figure, ask: “The base doubled. Why did the area not just double?” The height doubled as well, so the area is multiplied twice by 2.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 2 (simplifying a ratio). |
|---|---|
| 10 min | Hook, prediction for C, then the first figure. |
| 10 min | Key idea; the second figure, lengths, angles and area. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | (−3, −5); 3 : 2; −12. |
|---|---|
| Worked examples | (8, −12); (2, 2), (6, 2), (2, 8); (3, −2); scale factor 3; 15 cm by 10 cm, 150 cm². |
| Faded example | T′ (−12, 6). |
| 1. Enlarge a point | B, (6, 15). A adds; C multiplies only x; D divides. |
| 2. Enlarge, negative coordinate | −18. −6 × 3. |
| 3. Reduce | 5. 10 × 0.5. |
| 4. Lengths scale | 10. 4 × 2.5. |
| 5. Find the scale factor | 4. 8 ÷ 2. |
| 6. Congruent or not | C. Translation, reflection and rotation are congruence transformations. |
| 7. Area scales | 108. 12 × 9. |
| 8. Two enlargements | 6. 2 × 3. |
| 9. What stays the same | B. Lengths × 3, perimeter × 3, area × 9, angles unchanged. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Added the scale factor, multiplied one coordinate, or divided. | Test with (1, 1): where does (1, 1) go for k = 3? |
| −3 or 18 in question 2 | Added the factor, or lost the minus. | Write −6 × 3 on its own line with the sign in brackets. |
| 20 or 9.5 in question 3 | Inverted the factor, or subtracted. | Say it aloud: scale factor one half means halve. |
| 6.5 or 8 in question 4 | Added, or rounded the factor down. | Write 4 × 2.5 as 4 × 2 + 4 × 0.5. |
| 6 or 0.25 in question 5 | Subtracted, or divided the wrong way round. | Write “image ÷ pre-image” above the working. |
| A, B or D in question 6 | Chose a congruence transformation. | Ask: does any length change in this one? |
| 36 or 15 in question 7 | Scaled the area by k once, or added. | Draw a rectangle 2 by 3 and enlarge it by 2: count squares. |
| 5 or 1.5 in question 8 | Added or divided the factors. | Follow the point (1, 1): 2 then 3. |
Questions worth asking
- Why is the image of an enlargement still the same shape? Listen for “every length is multiplied by the same number, so the ratios stay the same, and angles are unchanged”.
- What scale factor would map the image back to the pre-image? The reciprocal: 1 ÷ k. For k = 2, one half.
- Which of translation, reflection, rotation and enlargement keep angles? Which keep lengths? All four keep angles; the first three also keep lengths.
Differentiation
A way in for anyone stuck
- Draw rays from the origin through each corner on squared paper and measure along them.
- Do one coordinate at a time on a table: x, then y.
- Keep k as a whole number until the rule is secure, then bring in 0.5.
Stretch
- A triangle is enlarged by scale factor k and its area goes from 5 to 45. What is k? (3, because 5 × 9 = 45.)
- An enlargement sends (2, 6) to (3, 9). What is k if the centre is the origin? (k = 1.5: both coordinates are multiplied by 1.5.)
Students learning in English as an additional language
Enlargement, scale factor, similar and congruent are new in this sense. Pair each with a picture: a small and large copy of one photograph for similar, and two identical stamps for congruent.
| Word | Plain meaning | Example |
|---|---|---|
| enlargement | a change of size from a centre, by a scale factor | a photograph printed larger |
| scale factor | the number every length is multiplied by | 2 doubles; 0.5 halves |
| reduction | an enlargement with a scale factor between 0 and 1: the image is smaller | scale factor 0.5 |
| similar | the same shape but not necessarily the same size | a photograph and its enlargement |
| congruent | exactly the same size and shape | two identical stamps |
Sentence starters for explaining aloud or in writing:
- The scale factor is …, so every coordinate is multiplied by …
- The image is similar, not congruent, because …
- The area is multiplied by … because both lengths are multiplied by …
Two things not to say
“Make it twice as big.” Twice as long, or twice the area? Say “scale factor 2”.
“The shape stays the same.” The shape is the same; the size is not. Say “similar”.
“Enlarge means bigger.” A scale factor below 1 makes it smaller.
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.
- EnlargeEnlarge (3, 5) by scale factor 4 from the origin.
(12, 20). - Earlier work, rotationsTurn (2, 7) through 180° about the origin.
(−2, −7).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ReduceReduce (12, −18) by scale factor one third from the origin.
(4, −6). - Scale factor(5, 2) goes to (15, 6) by an enlargement from the origin. What is k?
3.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA square of side 3 is enlarged by scale factor 4. By what factor does its area change, and what is the new area?
16; 9 × 16 = 144. - Earlier work, ratioShare 20 in the ratio 3 : 2.
12 and 8.
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 the slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper: draw the triangle and a ruler line from the origin through each corner, then mark the images.