# LESSON PLAN 11

- Symmetry ( concept and different symmetries)

- Arts (includes music, architecture ...), ICT (includes technical education, computer science ...), Mathematics (includes geometry, proportionality, finances ...), Science (includes physics, biology ...)

- ISCED 3 = Upper Secondary Education

## Duration

4 learning units, 8 hours, 4 activities

## Topic

Identify symmetry in nature and explore its mathematical background and theory

## Synopsis

Students are encouraged to find examples of symmetry in nature. Examples can be found in living organisms, architecture, music, furniture design … Inductive reasoning is used to derive theoretical foundations of symmetry in mathematics

## Contributors

Parents who are engineers, scientists…, teachers from other related subjects ( NS, ICT, Arts…)

## Framework

**Teacher’s cooperation and division of work:**

Teacher 1 (T1) – mathematics

A3: Collects natural and abstract models found and leads discussions towards identifying their mathematical models and deriving the true nature of symmetry.

A4: Explains mathematical models and explores their symmetries in a formal way.

Teacher 2 (T2) – sciences

A1: Leads discussion in class or during a field visit of surrounding symmetrical objects. Once line symmetry is identified, encourages discussion of space symmetry. Explains appearances of crystals in nature and human made.

Teacher 3 (T3) – arts or language, ITC

A2: Discusses symmetries in literature, music, architecture, painting. Provides examples and non-examples. The teachers can work together in arbitrary configuration or can teach their parts individually.

Activities A1 and A2 can be carried out in arbitrary order. Organization Working with examples, field visit: architecture, painting gallery, jewelry shop. Making explicit measurements of natural objects.

**Action plan formulation:**

There are 4 activities involved. A1 and A2 are interchangeable. A3 and A4 are interchangeable as well. They serve as the sum-up of the plan.

## Material / Equipment

## Previous knowledge and skills

This plan does not require any prior knowledge. The central concept of symmetry will be explained from the scratch (ICT) or on the other way (debate, provocative questions, e – platform, outside of classroom…) and illustrated with everyday examples.

## Learning Objectives

**Learning goals and objectives:**

Upon completion of the class students should **know**:

- Understand the meaning of symmetry;
- Distinguish and name various basic types of symmetry;
- Name symmetric patterns present in the nature;
- Comprehend symmetry role in arts and literature;
- Be able to present results of the project in written and oral form.

** **

**Learning outcomes and expected outcomes:**

** **Students should **be able to:**

- Remember definition of symmetry and basics of their classification.
- Understand role of symmetry in everyday live, arts and physical systems.
- Apply gained information towards versification of the presence of symmetry in new situations and objects.
- Tell difference between less and more symmetry.
- Generalize gained knowledge to other concepts, e.g. similarity.

## Methodology

Students shall comprehend real live roots of an important mathematical concept. They will be confronted with inductive method in science which starts with collecting data and seeks for an abstract generalization, theory. Students may discuss issues such as: is the symmetry understanding learned or are we born with this concept. Why is the linear symmetry the most common, etc. The goals of the plan can be achieved with an inquiry-based, an active learning that starts posing questions, problems and scenarios, or project-based approach, e.g., an in-depth project on finding symmetry in architectonical objects in the given area or even in the school itself. Supervision and light guidance of discussions is required throughout the project.

Connections with e-platform ( general ideas):

On the platform, the teacher publishes the content that will be covered in the next week’s teaching, along with several models / methods offered for the implementation of the lesson (short videos, presentations,…). Students choose how they want to realize the contents of the lesson voting on the platform.

**(For teachers/parents**)For each of the offered methods, a checklist is also offered that every student should fill out and serves to find out why the student chooses the method or does not choose the method. There are students who cannot learn in any of the offered learning methods (visual, game, programmed, problem-based, etc.). There are students who will not initiate cooperation, nor will they accept cooperation with other students.

A part of the checklist will also include prior knowledge about the contents that will be implemented, so the student will be able to mark which of the prior knowledge they master, for which they need support, and for which they can, if they feel capable, provide support to another student. (What do I need to learn from the other / what can I learn from the other).

This information is vital for the teacher to be able to prepare activities and apply appropriate forms of work for the lessons and provide appropriate manipulatives and other teaching aids.

**The teacher/s** can organize work on a project (formative assessment, the conception favors it) in the class: “Students – tutors”, in which not all students must be involved, but those who need support. In this way, both successful students are motivated (they keep diaries about the specific instructions and directions they give to solve a specific problem) and those who need support (they are freer to collaborate with someone who has now gone through and overcome the same or similar difficulties). The work of the project is evaluated according to previously determined criteria. On the positive side, the project activities do not have to be carried out during class, and the platform can also be used for communication, journaling and tutoring. The teacher/s should provide links and tools that can help the tutors (videos, applets, applications, assignments, literature).

Another idea is to assign homework to pairs, but it is realized on the platform, so that a log of events is kept (it can also be in the form of chat communication) and “evidence” is attached to support the homework.

The platform can also be used as a Help Center, a place to post a question, a task that needs support (something like a forum), so that in order to stimulate students to answer and write answers, the answers will be counted and voted for best explanation through likes/stars. Perhaps the best incentive for activity on the platform will be the ranking list of students who have given the most support (TOP HELPERS), so that all offered answers to questions per student will be counted. You can also rank students who asked for help, but were also active helpers.

## Preparation & Resources

Preparation, Space Setting, Troubleshooting Tips:

The work should start with identifications of symmetry in the real world. This can be achieved in the classroom and outside of it. The discussion can be triggered by using a mirror. A mirror can be replaced by a smartphone with a camera facing its user switched on. One can multiply the effect facing two cameras and discuss why facing three cameras is not possible. Axial and planar symmetries are the easiest to identify. Then one can search for more complex symmetries and objects having high number of symmetries (even infinitely many, e.g. ideal mathematical objects such as a circle, a line, a torus etc.). Theoretical, mathematical framework should be taught in the classroom.

Resources, Tools, Material, Attachments, Equipment

It is possible to use objects naturally creating symmetry, e.g., mirrors, cameras. Pictures of symmetric object might be helpful. One can consider natural crystals and go to platonic solids and their symmetries. Possible diversion here is to let students grow their own crystals at home. There are instructions available online, which explain how this can be done effectively, e.g., https://www.youtube.com/watch?v=kKLga-8IMiY

Inspirations for symmetry in the literature can be found here: https://www.sciencedirect.com/science/article/pii/0898122186901513

Symmetries in music are considered here: http://www.mi.sanu.ac.rs/vismath/visbook/apagyi/index.html

Nice introduction to symmetry in the architecture, already organized in form of a lesson is available here: http://www.mi.sanu.ac.rs/vismath/visbook/apagyi/index.html

Additional material is presented here:

## Implementation

This plan requires 4 units, 90 minutes each. It can begin either with T2 or T3 explaining symmetry in the real world or in the world of arts. T2 seeks with students’ examples of symmetry in the real world, collects them and classifies types of encountered symmetry. (2 hours) T3 provides initial examples of symmetry in arts. Most likely the students will come up with examples in painting, sculpture and architecture first. It is less obvious to come up with examples in the literature and the music and this might require some instruction. (2 hours) 17 T1 provides theoretical, mathematical background for the concept of symmetry. More importantly, T1 explains how examples from the real life are abstracted to the ideal world of mathematics. This provides opportunity to recall similar process for numbers (e.g. passing from concrete 2 cars to the abstract number 2). Some classification of planar and/or special symmetries should finish the course. (4 hours)

## Outline of the Lesson

## Extension Activities

This work is designed to be done in groups of 2 or 3 students, groups of 3 students will have a student with educational needs

## Assessmement

Evaluation: Each activity can be accompanied by quick tests checking the comprehension of discussed concepts and examples. A possible way to test is to use a hand-out with a picture exhibiting a number of symmetries and asking to identify them. A funnier way to test, which increases considerably, the focus of students is by peer questions with immediate presentation of results using, e.g., https://pingo.coactum.de/ query tool. If prepared well, one can the opportunity to discuss the Gauss curve and its symmetry https://www.sciencedirect.com/topics/engineering/gaussian-curve At the end students should complete a short (5-10 minutes) multiple choice test, which can be carried out using electronic devices.

Presentation – Reporting – Sharing

Students might be required to collect examples of symmetric objects. If they come from real world they can create an exhibition, if they have only pictures of them, they can create a gallery. It is important that some examples come in the form hands on rather than digital.

Extensions – Other Information

A more general concept than symmetry is similarity. This is, in a sense, symmetry with a scale. Or just a scale. Again multiple examples are present in the real world and can motivate passage to abstract approaches.

## References

Rochelle Lueders: Symmetry from nature to the Classroom, https://www.bemidjistate.edu/academics/honors/symmetry-from-nature-to-theclassroom-lueders-rochelle/