Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Mirror, Mirror...

Age 11 to 14
Challenge Level Yellow star
Secondary curriculum
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

Why do this problem?

This problem could be used as an extension task once students have learnt to draw reflections accurately. It can provide a valuable mathematical challenge for students who are ready to move on while others in the class need more practice with their basic drawing skills.


Alternatively, this problem and the two related problems ...on the Wall , and Who Is the Fairest of Them All? could form a unit of work on combined transformations. All three problems ask students to consider the effect of combining two transformations, and then challenge them to describe the single transformations that produce the same results.

Possible approach

This printable resource may be useful: Mirror, Mirror..., 

As an extension task, all that is needed is to provide the problem as a worksheet to a pair of students who make sense of it together. When they have established the combined transformation for one specific example, a teacher intervention may be appropriate, to move the focus to the general case - asking the key questions below.

With a full class, encourage different students to start with slightly different spacings of parallel lines and flag positions. The teacher intervention above could become a full class discussion. Students could regroup according to similarities/differences in the final combined transformations in order to acquire more information without needing to do a lot more drawing.
 

Key questions


What if the flag was in a different place?
What if you reflected in the other line first?
What if the lines were both at 45 degrees to the horizontal?
What if the lines were both at 60 degrees to the horizontal? (use isometric paper )
What, precisely, does the final position of the flag depend on?
Can you prove it?
 

Possible support


Do some introductory reflection work - filling in missing halves of shapes, reflecting patterns in vertical, horizontal and diagonal lines, etc.
Try Hidden Meaning and Reflecting Squarely

Ensure students use squared paper and encourage them to draw their reflections accurately.

Possible extension


Ask students to summarise their findings in exactly 20 words (!) - then ask what would happen if the lines were not parallel (ie. Move them on to ...on the Wall)
Ask (suitably experienced) pupils to create a dynamic geometry file that demonstrates their findings.
 

 

You may also like

Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Friezes

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo