Math // Geometry

Similarity and enlargement

Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids

Ages
12 to 14
Type
Concept
Builds on
3 topics
Leads to
0 topics
Standards
2 codes

Quick check

If the learner wants to enlarge a photo so it's three times bigger, can they work out the new dimensions — and explain that the shape stays the same but all lengths increase by the same factor?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Enlarge a shape by a given scale factor from a specified centre of enlargement
  • Determine the scale factor between two similar shapes by comparing corresponding sides
  • Explain why corresponding angles in similar shapes are equal while sides are in proportion

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

3 topics
  • Scale and similar shapesRequired

    Ratio & Proportion // Ages 10 to 11

    Enlargement extends KS2 scale factor work with similar shapes

  • Coordinate TransformationsRequired

    Geometry // Ages 11 to 14

    Similarity and enlargement build on understanding rigid transformations first

  • Ratio NotationHelpful

    Ratio & Proportion // Ages 11 to 12

    Similar shapes have sides in a constant ratio — connects to ratio notation

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Similarity and enlargement

Ages 12 to 14 // Concept

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Geometry

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Similarity means one shape is an enlargement of another: corresponding angles stay equal and corresponding sides share a single constant ratio (the scale factor). The central obstacle is additive reasoning — treating 'add the same amount to each side' as an enlargement. A 3×5 rectangle grown to 8×10 by adding 5 looks bigger but is not similar, because 8÷3≈2.67 while 10÷5=2. Learners need to see the structure as multiplicative: enlargement from a centre multiplies every vertex's distance from the centre by the scale factor while leaving angles fixed, and a genuinely similar rectangle (e.g. 6×10, each side ×2) must be contrasted with the additive fake.

Common misconceptions

  • Additive reasoning: treats 'add the same amount to each side' as an enlargement (3×5 → 8×10 by +5), when similarity requires multiplying each side by a constant
  • Believes angles get bigger when a shape is enlarged (scales the angles along with the sides)
  • Uses a different multiplier for different sides, or checks the scale factor on only one pair of corresponding sides and assumes the rest match
  • Enlarges the shape but does not scale each vertex's distance from the centre of enlargement, so the image is the wrong size or position

Useful representations

  • Centre of enlargement with ray lines through vertices (distances × scale factor)
  • Corresponding-sides ratio table (each pair reduced to the same scale factor)
  • Two overlaid shapes with corresponding angles marked equal and sides labelled with their ratio
  • Diagonal test: extending the diagonal of the original passes through the corner of a true enlargement but misses a fake (additive) one

Teaching practices

  • Confront the additive misconception head-on: present a shape 'enlarged' by adding the same amount to each side and have children test whether corresponding side ratios match — the diagonal of the original, extended, misses the corner of the fake enlargement
  • Enlarge from a centre of enlargement using ray lines: draw rays from the centre through each vertex, and multiply each vertex's distance from the centre by the scale factor — makes visible that enlargement scales distances, not adds to them
  • Build a corresponding-sides ratio table so children check the scale factor on EVERY pair of corresponding sides, not just one
  • Keep angles explicit throughout: measure or mark corresponding angles before and after enlargement to establish they are invariant while sides scale
  • Connect to ratio and to the multiplier: scale factor is the constant ratio between corresponding sides, linking similarity to y=mx proportional reasoning

Activities to try

  1. 1

    Grow it by adding, then test whether it stays the same shape

    Why it works: Adding a constant to each side does not in general preserve shape (the only exception is when the sides being compared are equal, as in a square); similarity requires every pair of corresponding sides to reduce to the SAME multiplier. The 8×10 fails the ratio test; the 6×10 (×2) passes.

    1. Start with a 3 cm by 5 cm rectangle and predict, in writing, whether making it '8 by 10 by adding 5 to each side' keeps it the same shape just bigger — say yes or no and WHY before checking
    2. Divide each pair of corresponding sides and build a ratio table: 8÷3 ≈ 2.67 beside 10÷5 = 2
    3. Place the 3×5 rectangle on the 8×10 rectangle at a shared corner, with the 3 cm side along the 8 cm side and the 5 cm side along the 10 cm side. Extend the diagonal of the 3×5 that starts at the shared corner and observe whether it passes through the opposite corner of the 8×10 rectangle (it misses it)
    4. Now compare the 3×5 against a 6×10 rectangle: check both ratios (6÷3=2 and 10÷5=2). Then, with the 3 cm side along the 6 cm side at a shared corner, extend the diagonal from that shared corner to see it pass through the opposite corner of the 6×10
    5. State which rectangle is similar to the original and justify using a single shared scale factor
  2. 2

    Enlarge from a centre with ray lines

    Why it works: Enlargement scales distances from a centre by a constant multiplier and preserves angles; the one shared ratio IS the scale factor, which is why the image is similar to the original.

    1. Mark a centre of enlargement and draw rays from it through each vertex of the 3×5 rectangle
    2. Choose a scale factor and multiply EACH vertex's distance from the centre by that factor to place the image vertices
    3. Measure or mark the corresponding angles before and after enlarging, confirming they are unchanged
    4. Fill a ratio table for every pair of corresponding sides, confirming they all reduce to the same scale factor
  3. 3

    Decide similar or not, and find the scale factor

    Why it works: The multiplicative test — all corresponding-side ratios equal, angles unchanged — can be applied consistently to classify and quantify similarity.

    1. Judge whether 3×5 is similar to 8×10 or to 6×10, giving a reason — and reject 'yes, you added 5 to each side' as a justification
    2. Find a scale factor by dividing a new side length by its corresponding original side
    3. Match corresponding sides between two similar shapes and sort several shape-pairs into 'similar' and 'not similar'
    4. For a stated enlargement by scale factor 3 applied to a triangle, say what happens to its angles (unchanged) and its sides (×3)

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
8.G.4
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Geo.9

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