Folding and Symmetry: 1.5
Supporting materials
Indicator of Progress
Success depends on students recognising that certain shapes are made up of parts that repeat in some way.
For example, they see that a shape may have two halves that are 'the same' in the sense that one half is a back-to-front version of the other. They understand that this property can be checked by folding the shape in half. This is the beginning of understanding about line (or mirror) symmetry.
Students may also recognise that certain shapes look the same after being turned less than a full turn. This is the beginning of understanding about rotational symmetry.
At this stage, it is not expected that students be able to describe what makes a shape symmetrical, but they may 'know it when they see it'. They may describe the two halves of a symmetrical shape as being the same (when technically speaking, they are not). The recognition of rotational symmetry is more difficult than line symmetry.
More formal/advanced aspects of line symmetry are covered in material at progression point 3.5.
Illustration 1
Students recognise that the human body is made up of two halves. They can see that the left-hand side and the right-hand side are the same, in a sense. They start to appreciate the fact that the left hand and right hand (for example) are not exactly the same as each other, but that one is a back-to-front or reversed version of the other.
Illustration 2
Students start to recognise that there is something different about the shapes in the group on the left and those in the group on the right. They may not be able to describe exactly what it is about the shapes that makes the groups different, but they may be able to talk about each of the shapes on the left having two halves.
Illustration 3
There are some examples of shapes that are tricky when considering symmetry. A typical parallelogram actually cannot be folded in half even though it appears to have two 'equal halves'. For example, in the parallelogram on the left the dashed line seems to divide the parallelogram into two identical triangles. However, if the parallelogram is folded in half along this line the two halves do not match exactly, as seen in the second picture. Many students will be surprised that the halves don't match when the folding is done, because they will feel that the shape is symmetrical. In fact, it does have symmetry, because the two halves can be made to correspond by rotating by half a turn. This means that the shape has rotational symmetry, but it does not have line symmetry.
Teaching Strategies
These teaching activities are intended to help students understand that the symmetry that they have instinctively noticed can actually be tested.
- Folding is the key to investigating line symmetry, and is the main approach at this level. At this stage it may not be appropriate to explicitly emphasise the 'line of symmetry' terminology: what is important is the idea of finding whether the shape can be folded exactly in half.
- Rotating or turning is the key to identifying rotational symmetry; this aspect of symmetry is given less emphasis than the line symmetry/folding ideas at this level.
Activity 1: Making shapes with symmetry involves students in the hands-on creation of their own examples of symmetrical shapes.
Activity 2: Folding shapes to test for symmetry is a hands-on activity that allows students to explore the role of folding in showing that a shape is symmetrical.
Activity 3: Symmetry in the natural world encourages students to observe the world around them.
Activity 4: Shapes that turn provides some beginning experiences with the idea of rotational symmetry.
Students may learn the words symmetry and symmetric, but at this level it is not expected that they call the fold line the line of symmetry.
Activity 1: Making shapes with symmetry
This activity involves the hands-on creation of examples of symmetrical shapes, in which the fold lines (i.e. the lines of symmetry) are already evident.
A simple example of this is to paint a picture on one half of a piece of paper. Then, while the paint is still wet, fold the paper in half to create a mirror image of the original shape. The total picture is actually symmetrical and the fold line is the line of symmetry.
More complicated examples, with more lines of symmetry, can be created by making cutouts from square pieces of paper.
The paper is simply folded in half, and a design drawn and then cut out (making sure that the fold line is not cut away). The resulting shape will have one line of symmetry, as shown in Figure 1. The fold line is the line of symmetry.
Now fold the paper in half twice, to make a square. Cut out a design, again making sure that the fold line is not cut off. Unfold the paper to get a shape that has two lines of symmetry, as in Figure 2. Check by folding that each of them is a line of symmetry.
If the paper is folded in half twice and then folded on the diagonal to make a triangle, as in Figure 3, the resulting cut out shape will end up with four lines of symmetry. The four fold lines can be seen when the paper is opened up.
Students should be encouraged to fold their shapes in half along any of the fold lines, and confirm that the two halves match up exactly.
It is not expected that students at this level know how many lines of symmetry (ways of folding in half) there are, but they can see that this can be done along any of the fold lines.
Activity 2: Folding shapes to test for symmetry
The Fold Testing Activity sheets (PDF - 2.2Mb) can be downloaded, and cut into pieces so that the shapes are separated. The actual shapes could be cut out, especially for the first few, but this may not be necessary after students have done a few examples. Students should be asked to try to fold the shape in half in such a way that the two parts of the shape exactly coincide after folding. If they can make such a fold, then the shape is said to be symmetrical. In this case the fold line that they have made shows the 'line of symmetry'. However, at this stage the object is to see IF the shape can be folded in half, rather than to identify lines of symmetry per se.
If the shape is not actually cut out, the students should fold the shape so that the printing is on the outside, and then they can hold their shapes up to the light to see through the paper. They need to check to see if each half of the shape completely matches. If there are parts of the shape that don't match then that line isn't a fold line that shows symmetry, and they should try again to see if some other fold works.
Of course, some shapes are not symmetrical, in which case no matter what folds are tried the students will not be able to make two coinciding halves. On the other hand, for some of the shapes there may be two or more different ways of folding the shape to show symmetry.
A solution sheet (PDF - 778Kb) for teachers is available.
If mirrors are available, or the special MIRA reflective equipment, students can place a mirror on each of their fold lines to check that the shape viewed in the mirror is the same as the original.
Activity 3: Symmetry in the natural world
Encourage students to observe symmetry in the world around them, especially in animals, plants, leaves, flowers, crystals, etc. Many beautiful things are almost symmetrical.
Activity 4: Shapes that turn
When a six-pointed star like the one below, is given one-sixth of a turn, it looks the same again. It also looks the same again if it is given half a turn. The 'don't do it' sign needs half a turn to look the same again. This is the essence of rotational symmetry, which both of these shapes have.
A variety of shapes can be tested to see if they have rotational symmetry - ie if they can look the same if they are turned less than a full turn. Some good examples to use include fans, wheels, squares, rectangles, stars, circles, etc. Don't forget to include some non-examples, such as a love-heart or the tracing of a hand, as well.
Most commonly occurring shapes with symmetry have only line symmetry (e.g. the human body) or have both line and rotational symmetry (e.g. a square). There are very few examples of shapes that only have rotational symmetry. A good example is a pinwheel, which can be made from a square of paper (ideally with different colours on each side), as shown below.
What is significant about the pinwheel is that when it is rotated by only a quarter of a turn, it will appear to be in the same position as before it was turned.