The purpose of this unit is to engage the student in applying their knowledge and skills of geometric thinking to design and describe formations mathematically and the translations and/or rotations needed to create those formations.

Students develop their skills and knowledge on the mathematics learning progression, geometric thinking, to design and to choreograph synchronised swimming patterns and sections of a routine.

Students will apply their understanding of geometry to design and describe shapes and formations as well as the transformations that map one formation onto another.

### Structure

This cross-curricular, context based unit has been built within a framework that has been developed with input from teachers across the curriculum to deliver the mathematics learning area, while meeting the demands of differentiated student-centred learning. The unit has been designed around a six session focus on an aspect of mathematics that is relevant to the integrating curriculum area concerned. For successful delivery of mathematics across the curriculum, the context should be meaningful for the students. With student interest engaged, the mathematical challenges often seem more approachable than when presented in isolation.

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

The following five sessions are each based around a model of student-centred differentiated learning.

- There is a starting problem to allow students to settle into the session and to focus on the mathematics within the chosen context. These starting problems might take students around ten minutes to attempt and/or to solve, in groups, pairs or individually.
- It is then expected that the teacher will gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.
- The remaining group of activities are designed for differentiating on the basis of individual learning needs. Some students may have managed the focus activity easily and be ready to attempt the
*reinforcing ideas*or even the*extending ideas*activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the*building ideas*activity together, peeling off to complete this activity and/or to attempt the*reinforcing ideas*activity when they feel they have ‘got it’. - It is expected that once all the students have peeled off into independent or group work of the appropriate selection of
*building*,*reinforcing*and*extending*activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

### Introductory session

(This activity is intended to motivate students towards the context/integrated learning area and to inform teachers of each student’s location on the learning progression, geometric thinking):

Five synchronised swimmers spread their arms out straight (at an angle of 180° apart). They join hands making an obtuse angle between each adjacent pair of swimmers arms. There are no gaps in the formation and the swimmers arms have formed a regular polygon. Find the size of the obtuse angles made by the swimmers joining hands.

In this activity, the teacher(s) will be able to locate their students on the geometric thinking learning progression by observing their approach to solving this angle problem. Students may use reasoning, or practical techniques to work towards a solution. This activity integrates mathematical skills and knowledge with the arts learning area, in the context of dance formations and synchronised swimming. The problems could be acted out by students on land as dance formations. This unit of work could also sit alongside a focus on swimming and water safety.

Mathematical discussion that should follow this activity involves:

- What is the name of the regular polygon formed by their arms?
- Will the swimmers be able to make this polygon facing inwards or outwards?
- If their arms are
**not**held straight out (i.e., their arms**don’t**make an angle of 180°), what other regular polygon can these five swimmers make with their arms? What would the size of the interior angles need to be for this to be a regular polygon?

### Session two

**Focus activity**

- How many sides does this shape have?
- Name the shape.
- Describe the symmetry of this shape.
- Find the size of the interior angles of this shape.

- Is this shape a regular polygon?
- What approximations did you make to decide on the number of sides and the regularity of this shape?
- What are the physical requirements of the 8 swimmers to be able to form this figure?

**Building ideas**

- Name the shape made by the legs of the 8 swimmers.
- Is this a regular polygon?
- Find the lines of symmetry of this polygon.
- What is the rotational symmetry of this polygon?
- What is the order of symmetry of this polygon.

**Reinforcing ideas**

**Extending ideas**

- How would the shape made by the arms vary if there were only 6 swimmers?
- What would the internal angle of the shape made by the arms be with only 6 swimmers?
- Could only six swimmers make a star formation with their legs extended, like that made by the 8 swimmers? Give you answer in terms of angles and lengths of sides.

### Session three

**Focus activity**

- How many sides does this shape have?
- Name the shape.
- Describe the symmetry of this shape.
- Find the size of the interior angles of this shape.

- Is this shape a regular polygon?
- What approximations did you make to decide on the number of sides and the regularity of this shape?
- What are the physical requirements of the 8 swimmers to be able to form this figure?

**Building ideas**

- Name the shape made by the legs of the 8 swimmers.
- Is this a regular polygon?
- Find the lines of symmetry of this polygon.
- What is the rotational symmetry of this polygon?
- What is the order of symmetry of this polygon?

**Reinforcing ideas**

- Look at the shape made by the right legs of the 8 synchronised swimmers. The shape is (approximately) an octagon divided into 8 kites. The angle made by the bent knees of the right legs of the 8 swimmers are each 75°. What is size of the interior angles of the kite shapes within this formation?

**Extending ideas**

- How would the shape of the formation and the divisions vary if there were only 6 swimmers?
- What could the internal angle of the divisions made by the left legs be with only 6 swimmers? State any assumptions you have made.
- Describe the symmetry of the divisions and the overall shape of the formation that would be made by only 6 swimmers in this position.

### Session four

**Focus activity**

- How many swimmers are needed for this formation?
- How many petals are there in this flower formation?
- What is the angle between each of the petals in this flower? Explain how you found this angle.

- Could a pattern be formed using swimmers in the same position for a 5 or a 7 petal flower? What changes would need to be made?
- How could the rotational symmetry of the formation be used for effect in a synchronised swimming performance?

**Building ideas**

- Snowflakes have rotational symmetry of order 6. Which figure(s) could be used in a sequence about snowflakes?
- What is the order of rotational symmetry of the other figure?

**Reinforcing ideas**

- What is the order of rotational symmetry of this figure?
- Find the angle between adjacent swimmers.
- How many mirror lines are there in this formation?
- What is the total order of symmetry of this formation?
- Describe the formation that would result if only 6 swimmers were used in the same position and evenly spaced around a focal point. Give angles and a description of the symmetry of this new formation.

**Extending ideas**

- How many swimmers are needed for each formation?
- What is the angle between (the central axes of symmetry of) adjacent swimmers in each formation?
- Can you generalize the relationship between the number of swimmers (n) and the angle between the adjacent swimmers (A) when they're evenly spaced in formations?

### Session five

**Focus activity**

- What polygons can you see in this formation?
- Could a pattern be formed using an odd number of swimmers in the same position?
- How could the rotational symmetry of the formation be used for effect in a synchronised swimming performance?

**Building ideas**

- Locate these chevrons.
- If the angle between the swimmers’ legs are all 60° find the size of each of the interior angles of each chevron.

**Reinforcing ideas**

- Locate these chevrons.
- Use the symmetry of the shape to find possible sizes of the interior angles of the chevrons.
- Explain your working using geometric reasoning.

**Extending ideas**

- give instructions (using angles and leg lengths) that would allow these synchronised swimmers to transition from this formation to one that:
- Has all swimmers evenly spaced around a focal point,
- Has their arms extended and touching,
- Has their legs in the same position (30° apart).

- Describe what would happen if, once in the new formation, the swimmers holding hands pull their arms in by their sides. Include any limiting factors in your description.

### Session six

**Focus activity**

- What do you know about the interior angles of regular polygons that would help you to solve the problem?
- Could you find a practical way of solving this problem?
- How many swimmers are needed for this formation?

**Building ideas**

- has rotational symmetry of order 6,
- has 6 mirror lines,
- tesselates.

**Reinforcing ideas**

- has rotational symmetry of order 4,
- has 4 mirror lines,
- tesselates.

**Extending ideas**

- has rotational symmetry of order 3,
- has 3 mirror lines,
- tesselates.

- has rotational symmetry of order 6,
- has 6 mirror lines,
- tesselates.