The enlargement transformation: a week of ready-to-teach maths

Five lessons for Year 9 Space. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus 12 scale factor cards, 12 property cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: sheet 1 has the scale factor cards (Day 3), sheet 2 the property cards (Day 5). Cut them before Day 3.
4. Each student needs a ruler, a sharp pencil and a calculator; on Day 5 a computer or tablet with a free dynamic geometry app helps.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

An enlargement has a fixed point, the centre, and a scale factor k. Every point moves along the ray from the centre until it is k times as far away: image = centre + k × (point − centre). A factor above 1 enlarges, between 0 and 1 reduces, and exactly 1 keeps the shape. The image is similar to the original: angles, ratios of sides and parallel lines stay the same, while lengths and the perimeter multiply by k, areas by k² and volumes by k³.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

Points are coordinates (x, y); an image point is named with a dash, so A′ is the image of A. Scale factors are fractions or decimals, as in k = 1/2 or k = 0.5. Lengths keep their units and areas their square units; three decimal places at most, with ≈ when rounded.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 enlarges by drawing rays on a grid; Day 2 turns the same idea into a rule for coordinates; Day 3 adds reductions and scale drawings; Day 4 finds what happens to lengths, areas and volumes; and Day 5 sorts every property into what stays and what changes. It builds on similar triangles and scale factors from earlier in Year 9 and on Year 8 transformations, and it supports scale drawings, maps and the proportional reasoning used across the curriculum.

Materials for the week (one trip)

• From the classroom: rulers, sharp pencils, calculators, scissors, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 scale factor cards (Day 3).
• Cut-out sheet 2, one per pair: 12 property cards (Day 5).
• If you have them: computers or tablets with a free dynamic geometry app (Day 5). No other equipment to buy.

Words for the word wall

transformation, enlargement, reduction, centre of enlargement, scale factor, image, ray, similar, invariant, corresponding, perimeter, area, volume, scale drawing, dynamic geometry

Dear families

This week in maths, Year 9 is enlarging and reducing shapes from a centre and working out what changes and what does not. Lengths grow by the scale factor, areas by its square, and angles not at all.

Try this at home

• Shine a torch at a hand in front of a wall and move the hand closer to the torch. The shadow is an enlargement with the torch as the centre.
• Ask: a photo is printed twice as wide and twice as tall. How much more paper does it cover? (Four times as much.)
• Ask: on a plan drawn 1 : 100, a table is 1.5 cm long. How long is the real table? (150 cm, or 1.5 m.)

My scale week

Fill one row a day. Write something enlarged or reduced that you saw, its scale factor, and one thing that stayed the same.

Enlarging from a centre

To enlarge a shape from a centre O by a scale factor k, draw a ray from O through each corner and mark the image corner k times as far from O. On a grid, count from O: a corner 1 across and 1 up becomes k across and k up. Joining the image corners gives the image, a shape similar to the original.

We are learning to

• enlarge a shape from a centre by drawing rays,
• find the centre and the scale factor from a shape and its image,
• enlarge with the centre on the shape itself.

Success criteria

• I can draw the image of a triangle with k = 2 on a grid.
• I can find a centre by joining matching corners and extending the lines.

You need

The worksheet; a ruler and a sharp pencil each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Counting k squares on from the corner instead of k times the distance from O.
• Using the bottom-left of the grid as the centre when the centre is somewhere else.
• Joining the image corners in a different order from the original’s.

Answers

• Part A: A′ is 2 across and 2 up from O; B′ is 6 across and 2 up from O; C′ is 2 across and 6 up from O; every side is twice as long.
• Part B: the lines PP′, QQ′, RR′ meet at the centre, 1 across and 9 up from the bottom-left corner of the grid; k = 4 ÷ 2 = 2.
• Part C: the centre corner stays where it is; the other corners go to (8, 1) and (2, 7), counting from the bottom-left of the grid.
• Part D: the image is the same size and shape as before but in a different place: with the centre at C its corners are (2, 0), (6, 0), (2, 4) instead of (3, 3), (7, 3), (3, 7).

Enlarging from a centre

C. Centre on a corner

Enlarge the triangle by 3 with the centre at its corner (2, 1). Coordinates count squares from the bottom-left of the grid.

D. Stretch: a new centre

Enlarge the Part A triangle by 2 again, this time with the centre at C. What is the same about the two images, and what is different?

The mapping rule

With the centre at the origin, an enlargement multiplies both coordinates by k: (x, y) maps to (kx, ky). With the centre at (a, b), find how far the point is from the centre, multiply that by k, and add it back on: (x, y) maps to (a + k(x − a), b + k(y − b)).

We are learning to

• find image coordinates from the origin by multiplying by k,
• plot a shape and its image on a coordinate grid,
• enlarge from a centre that is not the origin, and work back to the centre and k.

Success criteria

• I can write the images of four corners for k = 2, 3 and 1/2.
• I can enlarge a point from the centre (1, 1).

You need

The worksheet; a ruler and a sharp pencil each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C.

Watch for these ideas

• Multiplying only the x-coordinate.
• Adding k to each coordinate instead of multiplying.
• Multiplying the coordinates by k when the centre is not the origin.

Answers

• Part A: k = 2: T′(4, 2) U′(8, 2) V′(8, 6) W′(2, 6); k = 3: T′(6, 3) U′(12, 3) V′(12, 9) W′(3, 9); k = 0.5: T′(1, 0.5) U′(2, 0.5) V′(2, 1.5) W′(0.5, 1.5).
• Part B: the image has corners (4, 2), (8, 2), (8, 6), (2, 6), every side twice as long.
• Part C: (3, 2) goes to (5, 3); (4, 4) goes to (7, 7); (1, 5) goes to (1, 9); in general (x, y) goes to (1 + 2(x − 1), 1 + 2(y − 1)).
• Part D: k = 4 ÷ 2 = 2; the lines through matching points meet at the centre (0, 1).

The mapping rule

A. From the origin

Write the image of each corner for each scale factor, with the centre at the origin.

B. Plot it

Plot the image for k = 2 and join its corners. Measure one side of each shape.

C. A centre at (1, 1)

Enlarge each point by 2 with the centre at (1, 1): subtract the centre, multiply by k, add the centre back.

D. Stretch: work back

An enlargement takes (2, 2) to (4, 3) and (4, 2) to (8, 3). Find the scale factor and the centre.

Enlarge, reduce or keep

The scale factor is the image length divided by the matching original length, in the same units. Above 1 it enlarges, between 0 and 1 it reduces, and exactly 1 keeps every length. A scale drawing or a plan is a reduction: a 1 : 100 plan has k = 1/100, so a real length is 100 times the plan length, and a real area 100² times the plan area.

We are learning to

• find a scale factor as image length over original length and say what it does,
• reduce a shape from a centre with a factor between 0 and 1,
• read real lengths and areas from a scale drawing.

Success criteria

• I can sort the twelve cards and give each scale factor.
• I can find the real size of a room from a 1 : 100 plan.

You need

The worksheet; the 12 scale factor cards (cut-out sheet 1) per pair; a ruler and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B.

Watch for these ideas

• Dividing the original by the image, which turns an enlargement into a reduction.
• Comparing lengths in different units (7 cm and 70 mm).
• Multiplying a plan area by 100 instead of 100².

Answers

• Part A: the card key is with the mini-check answers.
• Part B: the corners go to (7, 2), (10, 2), (10, 4), (7, 4); every side is half as long, and the image is 0.5 of the way from the centre to each corner.
• Part C: 4.2 cm × 100 = 420 cm = 4.2 m and 3.5 cm × 100 = 3.5 m; the plan area is 14.7 cm² and the floor 14.7 m² (14.7 × 100² = 147 000 cm²).
• Part D: back where it started: the two factors multiply to 3 × 1/3 = 1, which keeps every point.

Enlarge, reduce or keep

A. Sort the scale factor cards

For each card, find k = image length ÷ original length (in the same units). Write each letter and its k in a pile.

B. Reduce by 0.5

Draw a ray from O to each corner. Mark each image corner halfway along, and join them.

C. A 1 : 100 floor plan

On a 1 : 100 plan, a room is 4.2 cm by 3.5 cm. Find the real length and width in metres, the area on the plan, and the real floor area.

D. Stretch: there and back

A shape is enlarged by 3 from a centre, and the image is then enlarged by 1/3 from the same centre. Where does it end up? Explain.

Lengths, area and volume

Under an enlargement by k, every length multiplies by k: sides, diagonals and the perimeter. An area is a length times a length, so it multiplies by k × k = k², and a volume, three lengths multiplied, by k³. Doubling the sides of a rectangle makes room for four copies of the original inside the image.

We are learning to

• multiply lengths and perimeters by k,
• show by counting squares that area multiplies by k²,
• use k² and k³ in real problems.

Success criteria

• I can show with squares why doubling the sides makes the area four times as large.
• I can decide whether a bigger pizza is good value by comparing areas.

You need

The worksheet; a ruler and a calculator each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with Part C.

Watch for these ideas

• Multiplying an area by k instead of k².
• Multiplying a perimeter by k², as if it were an area.
• Using the map scale once for an area instead of squaring it.

Answers

• Part A: k = 2: sides 4 and 6 cm, perimeter 20 cm, area 24 cm², 4 times the area; k = 3: sides 6 and 9 cm, perimeter 30 cm, area 54 cm², 9 times the area; k = 0.5: sides 1 and 1.5 cm, perimeter 5 cm, area 1.5 cm², 0.25 times the area.
• Part B: the 4 by 6 image holds 24 squares, 4 copies of the 6-square original.
• Part C: k = 30 ÷ 20 = 1.5, so the large pizza has 2.25 times the area; at the small one’s price per bite it would cost $27, so $24 is good value. The map rectangle is 1 km by 1.5 km, so 1.5 km².
• Part D: edge 6 cm; surface area 24 × 9 = 216 cm²; volume 8 × 27 = 216 cm³.

Lengths, area and volume

A. One rectangle, three factors

A 2 cm by 3 cm rectangle is enlarged. Fill the table, then compare each area with the original 6 cm².

B. Count the squares

Enlarge the rectangle by 2 from the marked corner. How many copies of the original fit inside the image?

C. Pizzas and maps

D. Stretch: a cube

A 2 cm cube is enlarged by 3. Find the new edge, surface area and volume, and say what each was multiplied by.

Same and changed

An enlargement keeps every angle, so the image is similar to the original: ratios of sides, parallel lines and the number of sides stay the same too, and so does the centre. Lengths and the perimeter multiply by k, areas by k² and volumes by k³. A dynamic geometry app shows this live: drag the scale factor and watch which measurements move.

We are learning to

• sort properties into those an enlargement keeps and those it changes,
• measure a shape at several scale factors, on paper or in a dynamic geometry app,
• explain why angles stay the same while lengths and areas change.

Success criteria

• I can sort the twelve property cards and say what each changing one is multiplied by.
• I can explain, with my measurements, why the image is similar to the original.

You need

The worksheet; the 12 property cards (cut-out sheet 2) per pair; computers or tablets with a free dynamic geometry app if you have them; the unit’s self-check quiz.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the sort; start the second with Part B and the quiz.

Watch for these ideas

• Putting area in the same pile as length, multiplied by k.
• Thinking the image of an angle grows with the shape.
• Forgetting that the centre of enlargement itself does not move.

Answers

• Part A: the card key is with the mini-check answers.
• Part B: k = 1: angle at A 90°, AB 2, perimeter ≈ 6.83, area 2; k = 2: angle at A 90°, AB 4, perimeter ≈ 13.66, area 8; k = 3: angle at A 90°, AB 6, perimeter ≈ 20.49, area 18; k = 0.5: angle at A 90°, AB 1, perimeter ≈ 3.41, area 0.5 (lengths in grid squares).
• Part C: the angle stays 90° because the image is similar; AB and the perimeter multiply by k; the area multiplies by k².
• Part D: the areas are in the ratio 45 : 5 = 9 = k², so k = 3.

Same and changed

A. Sort the property cards

For an enlargement by k, sort the 12 cards. Write each letter in its pile.

B. Measure at four factors

Enlarge the Day 1 triangle A(2, 2), B(4, 2), C(2, 4) from the centre (1, 1) in a dynamic geometry app, or on grid paper. Measure each image.

C. Explain

Use your table: which measurement stayed the same, which multiplied by k, and which by k²? Why?

D. Stretch: work back from area

Two similar shapes have areas 5 cm² and 45 cm². What is the scale factor for their lengths?

Scale factor cards

Day 3, one set per pair: 12 cards. Find k = image length ÷ original length, in the same units, then sort: enlarge, reduce or keep.

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

Property cards

Day 5, one set per pair: 12 cards. A shape or solid is enlarged by a scale factor k. What happens to each property: does it stay the same, multiply by k, or multiply by k² or k³?

Teacher note: the key is with the mini-check answers, so this sheet can go to pairs whole.

What we know: The enlargement transformation

Work on your own. Show your working.

1. The point (3, 2) is enlarged by 4 from the origin. Where does it map to? ______________________
2. A side of 7 cm is enlarged by 0.5. How long is the image side? ______________________
3. A shape of area 10 cm² is enlarged by 3. What is the area of the image? ______________________
4. Does a scale factor of 0.8 enlarge or reduce a shape? ______________________
5. From the origin, the point (2, 5) maps to (6, 15). What is the scale factor? ______________________
6. A shape with a 40° angle is enlarged by 5. What size is that angle in the image? ______________________
7. A cube is enlarged by 2. What is its volume multiplied by? ______________________
8. Two similar shapes have areas 4 cm² and 100 cm². What is the scale factor for their lengths? ______________________

Answers and marking guide

Answers with working

1. (12, 8).
2. 7 × 0.5 = 3.5 cm.
3. 10 × 3² = 90 cm².
4. Reduce: 0.8 is between 0 and 1.
5. 6 ÷ 2 = 3 (and 15 ÷ 5 = 3).
6. 40°: angles do not change.
7. 2³ = 8.
8. k² = 100 ÷ 4 = 25, so k = 5.

A quick three-level guide

Eight questions, four ideas. A student at standard finds images and scale factors and knows that areas use k².

Card keys

Scale factor cards (Day 3): enlarge: A (k = 1.5), D (k = 2.5), F (k = 3), K (k = 40), L (k = 3.5); reduce: C (k = 1/50), E (k = 0.75), H (k = 0.25), J (k = 0.5); keep: B (k = 1), G (k = 1), I (k = 1).

Property cards (Day 5): stays the same: A, D, F, I, K; multiplies by k: B, E, H, J; multiplies by k² or k³: C, G, L.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: enlarging from a centre, the mapping rule, scale factors, lengths and areas, and same and changed.