The purpose of this unit is to engage students in applying their understanding of geometric thinking to design and describe formations and the translations and/or rotations needed to create those formations. This is explored in the context of choreographing a synchronised swimming routine and related patterns.
In this unit, students apply their understanding of geometry to design and describe shapes, formations, and the transformations that map one formation onto another.
To ensure engagement and participation in this unit, you should consider your students' prior knowledge of angles, polygons, quadrilaterals and problem solving tasks involving these ideas.
This cross-curricular, context-based unit aims to deliver mathematics learning, whilst encouraging differentiated, student-centred learning.
The learning opportunities in this unit can be further differentiated by providing or removing support to students, and by varying the task requirements. Ways to differentiate include:
With student interest engaged, mathematical challenges often seem more approachable than when presented in isolation. Therefore, you might find it appropriate to adapt the context presented in this unit. For example, problems could be acted out by students on land as dance formations or used to make connections with a wider focus on on swimming and water safety. Students might also be engaged by the task of creating and interrogating formations of people in other sports teams and groups.
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.
Following the introductory session, each subsequent session in the unit is composed of four sections: Introducing Ideas, Building Ideas, Reinforcing Ideas, and Extending Ideas.
Introducing Ideas: It is recommended that you allow approximately 10 minutes for students to work on these problems, either as a whole class, in groups, pairs, or as individuals. Following this, 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.
Building Ideas, Reinforcing Ideas, and Extending Ideas: Exploration of these stages can be differentiated on the basis of individual learning needs, as demonstrated in the previous stage of each session. 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.
Importantly, students should have multiple opportunities to, throughout and at the conclusion of each session, compare, check, and discuss their ideas with peers and the teacher, and to reflect upon their ideas and developed understandings. These reflections can be demonstrated using a variety of means (e.g. written, digital note, survey, sticky notes, diagrams, marked work, videoed demonstration) and can be used to inform your planning for subsequent sessions.
The relevance of this learning can also be enhanced with the inclusion of key vocabulary from your students' home languages. For example, te reo Māori kupu such as koki (angle), taparau (polygon), koki roto (interior angle), and tapawhā (quadrilateral) might be introduced in this unit and then used throughout other mathematical learning.
This session introduces the context of synchronised swimming in an angle problem. You could adapt the context of this unit to reflect a different sports or dance team, or other group. It might also be valuable to investigate the given synchronised swimming context, before asking students to create their own problems (perhaps as an extension or paired task) relevant to a different context (e.g. kapahaka formations).
Introduce the following context to students: 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.
Support students to find the size of the obtuse angles made by the swimmers joining hands.
As your students work, observe their approach to solving an angle problem. Students may use reasoning, or practical techniques to work towards a solution. Use these observation to locate your students on the geometric thinking learning progression.
Discuss, drawing attention to the following points:
Introducing Ideas
Discuss, drawing attention to the following points:
Building Ideas
Reinforcing Ideas
Extending Ideas
Discuss, drawing attention to the following points:
Building Ideas
Reinforcing Ideas
Extending Ideas
Introducing Ideas
Building Ideas
Provide time for students to discuss the following points, in relation to the formations of red and purple swimmers:
Reinforcing Ideas
Extending Ideas
Introducing Ideas
Present students with the following image. Ask them to describe the symmetry of the formation made by the 8 swimmers:
Discuss, drawing attention to the following points:
Building ideas
Reinforcing ideas
Extending ideas
Introduce the following context to students: Synchronised swimmers spread their arms out straight making an angle of 180°. They join hands making an angle of 144° between each pair of swimmers arms. There are no gaps in the formation and the swimmers arms have formed a regular polygon.
Name that polygon.
Discuss, drawing attention to the following points:
Building Ideas
Provide time for students to work through the following tasks:
Design a formation that uses 6 synchronised swimmers and creates a closed regular shape that has rotational symmetry of order 6, has 6 mirror lines, and tessellates.
Find the sizes of the internal angles of the divisions within this formation.
Reinforcing Ideas
Provide time for students to work through the following tasks:
Design a formation that uses 8 synchronised swimmers and creates a closed regular shape that has rotational symmetry of order 4, has 4 mirror lines, and tessellates.
Find the sizes of the internal angles of the divisions within this formation. Give geometric reasons where possible.
Extending Ideas
Provide time for students to work through the following tasks:
Design a formation that uses 6 synchronised swimmers and creates a closed regular shape that has rotational symmetry of order 3, has 3 mirror lines, and tessellates.
Give instructions, using angles and leg lengths, that would allow your 6 synchronised swimmers could transition from this formation to one that has rotational symmetry of order 6, has 6 mirror lines, and tessellates.
Dear parents and whānau,
In class, we have been exploring the features of regular polygons, including the size of their interior angles, in the context of synchronised swimming formations. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/synchronised-swim-shapes at 8:54pm on the 26th February 2024