Symmetry Squares

Introduction

In this lesson, you will be learning about reflection and rotational symmetries. (Click on the hyperlinks if you are unsure what they are.)

3 x 3 squares

How many reflection and rotational symmetries does the diagram on the right have?
By shading in different boxes of the 3x3 square (PRINT ME), how many other ways could exactly 2 squares be shaded so that the shape has exactly one line of symmetry? How many different “types” of solution are there?
How many other ways could the 2 squares have been shaded so that the shape has exactly two lines of symmetry?
Repeat points 2 and 3 above if you can now shade in 3 squares. What about 4 squares?
Can you write down the answer for 5, 6, 7 squares without doing any work? Why/ why not?

5 x 5 squares

Now look at the 5x5 square on the right, Shade one more square so that the array has reflection symmetry.
How many solutions are there? What are they? Where are the mirror lines?
Take a look at this worksheet (PRINT ME).

Find the two different solutions for each puzzle.
Include the mirror line (line of symmetry) on your solution.
For each question there are two grids so draw one solution on each grid.
When you have finished, for each solution write down the number of dots that there are on the mirror line.
What can you say about the number of dots on the mirror line? Why?

4x4 squares

Look at the 4x4 squares on the right, describe the symmetries in each of these patterns.

Now, use these 4x4 squares to create shapes that match these rules:

Use 8 dots and make four different patterns that only have rotational symmetry (not line symmetry).
Produce four patterns that have reflection (lines of) symmetry but no rotational symmetry.
Use an odd number of dots to produce four patterns, each one with four lines of symmetry.
Come up with your own interesting rules to investigate.

Further Questions and Challenges

In the 5x5 square task above, all the problems have exactly two solutions. That is, there are two different places to put the last dot if you want to leave a symmetrical pattern. Can you develop a problem that has:

0 solution
1 solution
3 solutions
4 solutions

What is the greatest number of possible solutions for a single problem?

Use these 5x5 squares (PRINT ME).

Further Practice

In this lesson, we learnt about rotational and line symmetries. Practise the relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.

Transum

Look at this video if you need further support.

Starters:

Dice Reflections: A dice is reflected in two mirrors. What numbers can be seen?
Freemason's Cipher: Find symmetric words in this ancient cipher.
Match Fish: A classic matchstick puzzle designed to challenge your spacial awareness.
Mirror Maths: The bottom half of some symmetrical calculations are shown above. Can you work out the answers?
Reflective Cat: On squared paper copy the drawing of the face then reflect it in three different lines.
Rotational Symmetry: Draw a pattern with rotational symmetry of order 6 but no line symmetry.
Wrapping Paper: Find the order of rotational symmetry of the repeating pattern.

Activities:

Pattern Clues: An interactive activity challenging you to reproduce a pattern of coloured squares according to given clues.
Symmetry Table Challenge: In how many cells can you draw symmetrical shapes with the given row and column headings?
Xmas Symmetry Pairs: Match the pictures with the description of their symmetry.
Snowflake Generator: See how the hexagon can be transformed into a snowflake with some basic translations.
Polygons: Name the polygons and show the number of lines and order of rotational symmetry.
Rotational Symmetry Pairs: The traditional pairs or pelmanism game adapted to test knowledge of rotational symmetry.

For more goodies on Symmetry on Transum, click on the hyperlink.

Extension

Take a look at these two interesting videos:

Symmetry, reality's riddle: Marcus du Sautoy' TED talk illustrating some aspects of symmetry.
The science of symmetry: An illustrated talk about the notion of symmetry and how it applies to nature.