Week 05: Playing with words

Here we take an informal look at rotation and reflection, by considering words that have been written upside down or rendered as mirror writing. Intuitively one can think of mirror writing as the result of an action in three dimensions - flipping. However, mathematically, both kinds of action can be described as transformations of the plane!

Task 05A: The three given instructions are MIND THE GAP!, SLOW DOWN! and WATCH YOUR STEP! Mathematically, the phrases have been transformed, respectively, by means of
• a reflection in a horizontal mirror line
• a 180˚ rotation
• a reflection in a vertical mirror line.

However, if one thinks of this in a more concrete way, for example with the phrases written on a piece of glass, then one can think of the transformed phrases being the result, respectively, of
• flipping the glass over (through 180˚ about a 'horizontal' axis in the glass)
• rotating the glass (through 180˚ about an axis perpendicular to the glass)
• flipping the glass over (through 180˚ about a 'vertical' axis in the glass)

Some children will have difficulty reconciling the mathematical and the everyday interpretations of these transformations. However, the primary aim of this week's tasks is that children visualise the various transformations in whatever way they can. If they are more comfortable doing this in a concrete way, then you might want to leave the mathematical interpretation for now! (We refer explicitly to the mathematical transformations in Task 05C.) Also, if children find the task very challenging, you might want to go straight to Task 05B, where we apply the transformations to single words.

Children might notice that some letters are easier to transform than other, perhaps depending on the particular transformation. We look at this explicitly in Task 05D, but you might want to start discussing it here too.

This shows the transformations of HOLD TIGHT!, corresponding to the ways the original instructions were transformed.

Task 05B: Here we transform single words. The words are quite short, but they involve an interesting variety of letters - do children notice their various characteristics? You might also want to use some other words - including, for example, the children's own names.

These are the transformations of SIXTY, corresponding to those of TEN.

Task 05C: Here we describe the transformations mathematically and include images of the centre of rotation and mirror lines. However, if children prefer to stick to the everyday interpretations, then let them do so!

Here are the solutions:
 

Task 05D: Here we begin to look at the symmetry properties of the letters in a more thorough way, which we continue in Task 05E.

For the second part of the task, children might want to see what happens with the words AVA, VIV and BIX. Can they find a word that fits the third condition? 

Task 05E: Here we go through the letters of the alphabet one-by-one, noting whether they have each of the symmetries in turn.

The symmetries of the letters are shown here. Some letters have several symmetries, some have none. Note that the letters with rotational symmetry have either no lines of symmetry or two lines; none have just one line of symmetry. Conversely, the letters with two lines of symmetry all have rotational symmetry as well.
 

The letters fit nicely into five groups:
• those with rotational symmetry only
• those with rotational symmetry and 2 lines of symmetry
• those with only a horizontal line of symmetry
• those with only a vertical line of symmetry
• those with no symmetry.

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