Logo Licenses

Getting Started

1. Fold a circle in half and cut pieces out of it in any way you like. Ask the students to anticipate the pattern that the "opened up shape" might have. Write down any mathematical vocabulary that might arise, in particular terms like symmetry, reflection, turn, etc. Open up the cut circle to confirm the students’ predictions. (The shape will have reflective symmetry along the fold line.) Ask why fold symmetry is referred to as mirror symmetry and flip symmetry sometimes. Each alternative term can be demonstrated. Holding a mirror along the fold line enables students to see an image of the whole shape even when one half is masked. Similarly, the amended (cut) circle can be refolded in half and traced around. Then turn the paper over the fold line and trace around again (an OHP is excellent for this). The traced figure will be that of the whole amended circle.
2. Students should then make their own shape by folding a circle in half and cutting out pieces (they must not cut all the way along the fold line). This will enable them to come to the generalisation that all such shapes have at least one line of reflective symmetry (some may have two depending on the cuts). These shapes may be displayed on a chart.
3. Move onto folding a circle into quarters (in half then in half again), and cutting pieces out. Ask what symmetry they anticipate the new cut circle will have. Have them create their own pattern in this way then compare it with a partner to find out what symmetry both shapes have in common. Most students will realise that there are two lines of reflective symmetry (the fold lines) but many will miss the half turn rotational symmetry. This is referred to as order 2 as the shape maps onto itself twice in a full turn.
4. Get the students to anticipate then investigate what symmetry the cut circle will have if folded in eighths or sixths before cutting. Sixths can be created by folding the circle in half and then looping the half into thirds (see diagram below). (The circle folded into eighths will have at least four lines of reflection symmetry and rotational symmetry of order 4; with sixths the symmetry is six lines and order 3). Ask the students if they can see a pattern in the "least number" of lines and orders.

5. Challenge the students to use paper circles to create a shape that has rotational symmetry of order 3 but no lines of reflective symmetry. Next ask them to produce a shape that has rotational symmetry of order 4 but with no lines of reflective symmetry. Then ask for one that has rotational symmetry of order 5 with no lines of reflective symmetry, etc. (For example, this symbol ∮ has rotational symmetry of order 2 but no lines of reflective symmetry).

Exploring

1. Car Logos
   Investigate the logos found on motor vehicles by looking at cars in the school car park. It may be possible to videotape television advertisements for motor vehicles. These can be used instead and paused at the end of the commercial to capture the logo. Car adverisements in magazines can be cut out and used. Most manufacturers use symmetry of some kind in designing their logos. For example, Audi uses four intersecting circles in a line. This pattern has one line of reflection symmetry. This logo is created by translating one circle three times.
   
   When in the car park tell the students that they will be required to recreate these logos back in the classroom so they will need to take careful note of their symmetry. If possible pencil rubbings can be taken of each logo by placing paper over it and gently rubbing a pencil over the paper to obtain an imprint.
   
   Discuss the symmetry of each logo found and compile a list of companies for future reference. (You may need to omit the manufacturer’s name from some of the logos in order to get any symmetry. For instance, removing ‘Ford’ from its logo gives an elliptical shape that has two lines of symmetry.)
2. Construct Car Logos
   Provide the students with drawing instruments such as rulers, protractors, drawing compasses and tell them to recreate the car logos they saw in the car park. Have the pencil rubbings or sketches available for them to refer to. At times it may be necessary to bring the students together in order to discuss construction skills. For example, for a logo involving rotational symmetry of order 3, a protractor will be useful. Since there are 360° in a full turn, one third of that is 120°, which gives the angle measured at the centre for dividing a circle in thirds. Construction skills like drawing a right angle by using a protractor or compass construction may be modelled if necessary.
3. Logos in the media
   As homework (see Homelink) ask the students to find other examples of logos from magazines or television commercials. Obviously not all logos have symmetry. Sporting goods manufacturers are good examples of this. Nike use a tick, Adidas use three stripes etc. This will illustrate to the students that logos have to be both aesthetically pleasing (i.e. often symmetric) and suggestive of the nature of the company. Share the logos they bring along and group them by symmetry discussing what message is suggested by the logo image. This has strong links to visual language in the English curriculum. For example, the Canterbury Clothing Company has a logo of three translating, overlapping C’s that resemble a ball. The three images give an impression of a single ball moving from left to right.
4. Creating Logos
   Set up the following scenario for the students: "You work for an advertising company as a logo designer. There are three new companies for whom you have to create logos. They have stipulated that the logo must have some kind of symmetry but must also suggest what goods and services they provide. (If you wish to develop their writing skills, for each logo you might get them to come up with a slogan that captures the message, e.g. "just do it",.)

Here are the companies:

Sweeties - the Company that make sugar-free lollies that taste great and don’t ruin your teeth.
Gadgets - who make neat construction gadgets (gears, blocks, wheels, etc.), so you can create your own toys.
Dudes - makers of cool clothes especially for primary school children.
Brainbuilders - the people who you see for extra help with schoolwork. You get one-on-one help so you are in a class of your own!

Give the students sufficient time to design logos for each company. They will need to present the logo in a report to each company that shows what symmetries are involved and how the design suggests what kind of goods or services they provide. You may decide to set up a post box for each company so that you can have a vote for the best logo. A vote will not be counted unless it explains why the voter has selected that logo. (You may wish to remove students’ names from the display to avoid personality issues.) Of course, as managing director and chief shareholder in each company, you reserve the right to select whichever logo you wish. This gives you an opportunity to reward effort as well as artistic flare.

Reflecting

1. Show students the OHT of the logos provided. Ask the students to each write down what symmetry each design has (this is useful for assessment purposes). Get them to share what they have written in pairs then bring the class together for a collective discussion.
2. For each logo get students to demonstrate what symmetry the design has using the OHP and flipping, rotating, or shifting the two transparency copies. Make a list of symmetries for each design.
3. Tell the students to look at their first list and add any information they may have missed. The additions may be done in a different colour if you wish to preserve their initial writings.