Teaching Sequence

Work through this resource material in the following sequence:

25 minutes – Part A: Beauty in nature?
30 minutes – Part B: Beauty in art?
20 minutes - Part C: Beauty in patterns?
50 minutes – Part D: Beauty in maths?
15 minutes – Reflection

Part A: Beauty in nature?

Step 1.

Introduce the topic of the lesson by projecting the images below (images also available on the Student Worksheet). Direct students to the following questions (also available on the Student Worksheet) and invite them to discuss and record their ideas in response to these questions:

Is beauty a mathematical concept?
Do you think there is a mathematical formula or element behind what we consider to be beautiful?

Step 2.

Direct your class to the following video on the Fibonacci sequence. Explain that students will need to record the first 15 numbers in the Fibonacci sequence, so encourage them to pay close attention to these details (numbers available below).

The magic of Fibonacci numbers - Arthur Benjamin (https://www.youtube.com/watch?v=SjSHVDfXHQ4)

Step 3.

Now provide your class with some items that have natural swirls on them including:

Pinecones.
Pineapples.
Artichokes.
Romanesco broccoli.
Sunflowers.

Direct students to use glitter glue, metallic markers, paint and paintbrushes or washi tape to identify and mark the swirls on the items they have been given. This activity will work best on the larger items such as pinecones and pineapples and could be difficult to do on the smaller spirals found on sunflowers and romanesco broccoli. You could provide your class with images of these instead, which they could draw on.

• Don't try and identify swirls from the centre of the pinecone. This can become confusing pretty quickly. Encourage students to find the next swirl from this point. Students will be able to trace the swirls inward to the centre once they identify the pattern.
• Alternative colours for each swirl to make them stand out.
• Once students have identified the swirls in one direction, look for swirls in the opposite direction.

Example images can be downloaded here: Pine Cone Example Images.

Alternatively, you could take your class on a short tour of the school and identify Fibonacci numbers in flowers by counting the number of petals or spirals within the centre of the flower. Your students may be able to find other natural swirls in the plants at school, particularly succulent plants. Encourage students to look for swirls in the leaves as well (i.e. swirls that travel upwards on the stem from leaf to leaf). You may want to consider bringing a camera and allowing students to photograph swirls, particularly those found in the centre of flowers, and use these in class later on.

Step 4.

Direct students back to the Thought Starter: Is beauty a mathematical construct? and ask your class to discuss whether or not they have changed their previous opinions.

Part B: Beauty in art?

Now that students have had a good look at Fibonacci numbers, your class will examine the Golden Ratio.

Step 1.

Direct your class to Table 1 on the Student Worksheet. Ask students to complete the Fibonacci sequence and begin dividing each pair of numbers by the smaller number in the pair, to obtain the Golden Ratio.

The solution is available to download here.

At this point, you could extend the task by finding the ratio for the next 5-10 numbers in the Fibonacci sequence (answers available in the solution sheet above). You could simply write this on the whiteboard, or project this from your computer. This task may give additional proof to students that although the Golden Ratio is irrational, it approximates to 1.618.

Step 2.

Explain to your class that the ratio that was approximated in Step 1 is called the Golden Ratio or Phi ɸ and happens to show up in a number of places including plant and animal proportions, DNA, the Solar System, art, music and even business. Explain further that the Golden Ratio is often claimed to appear in human anatomy as a measure of beauty.

Advise your class that they will be investigating these claims in pairs, whereby each team will take the following measurements, calculate ratios and make comparisons to the Golden Ratio.

Students should complete the following table (also available on the Student Worksheet):

Claim	Measurements	Lengths	Ratio (smaller ÷ larger)
The Golden Ratio can be found between your arm, forearm and hand.	Arm (inner elbow crease to middle fingertip)		Arm to Forearm ratio: Forearm to hand ratio:
	Forearm (inner elbow crease to crease beneath hand)
	Hand (inner crease beneath hand to the tip of middle finger)
The Golden Ratio can be found between your face length, eye height and nose height.	Face length (top of head to the tip of chin)		Face length to eye height ratio: Eye height to nose height ratio:
	Eye height (tip of the chin to pupil height)
	Nose height (tip of the chin to the bottom of the nose)

Step 3.

Once students have finished calculating their ratios, allow the class to share and discuss their results to determine whether the claims were mostly justified or refuted. To do this, you could:

Encourage students to discuss as a class whereby you might lead the discussion.
Setup a short rotational activity whereby you will instruct one student from each pair to move to the next group and continue to rotate through all groups.
Collect class data on the board and calculate a class average for each ratio category.
For example:
- (1.63 +1.89 + 1.3 + 1.45) / 4 = 1.56
- The class average for arm to forearm ratio is 1.56, which is similar to the Golden Ratio

Step 4.

Now ask students to apply the Golden ratio to popular artworks (or even celebrity faces) to see if there are any correlations between the Golden ratio and what is considered to be beautiful. To complete this task you could:

Use printed images of popular artwork including (WikiArt is a useful place to source images):
- the Mona Lisa or The Last Supper by Leonardo da Vinci.
- Bathers At Asnieres by Georges Seurat.
- Girl With A Pearl Earring by Johannes Vermeer.
- The Creation of Adam by Michelangelo.
Project images outlined above onto the board and complete as a class.
Use images from magazines or books.
Use student devices to search images and measure the image carefully on the screen.

Direct students to measure their image and calculate its ‘Golden lines’ using the table below (also available on the Student Worksheet):

	Length (mm)		Golden lines
Height of painting	65 mm	÷ 1.618	65 ÷ 1.618 = 40.1 ~40 mm
Width of painting	141 mm	÷ 1.618	141÷ 1.618 = 87.14 ~87 mm

Then, direct your students to use their Golden line calculations to draw a ‘grid’ on their artwork as seen in the example below. Students will need to measure their Golden lines from the left and right/top and bottom of their image to draw these lines.

Example: Red lines are 87mm from the left and right sides, blue lines are 40mm from the top and bottom.

Image source: https://www.wikiart.org/en/michelangelo/sistine-chapel-ceiling-creation-of-adam-1510

Evaluation - the Golden lines do seem to indicate pivotal components of the artwork, especially since the iconic ‘hands almost touching’ section occurs on a golden line. The position of Adam’s eyes and chest and God’s chest on golden lines seems important as well.

Step 5.

To conclude this section, ask students to think about whether they think the Golden Ratio is, in fact, a good tool to measure beauty. Encourage them to describe when the Golden ratio may and may not be applicable.

Part C. Beauty in patterns?

Step 1.

Distribute a copy of these images, Fractals in Nature - Examples, to pairs or small groups of students, explaining that these represent simplified concepts of fractals. A fractal is a pattern that repeats itself over and over, so it can look the same up close and far away. Ask your class to look at these images and explain why they represent fractals in nature.

Explain to your class that abstract artwork can often be used as a different way to represent something real and require the viewer's imagination and experiences to create meaning. Ask students to discuss what real-life representations they can see in these abstract artworks.

Step 2.

Now distribute a copy of these images, Fractals in Art - Examples to the same pairs or small groups. Invite students to compare the images of fractals in nature with the fractals in art examples.

Step 3.

Now you can give your class some time to have a go sketching their own fractal using a geometric or circular pattern. Choose one of the videos below to inspire your class:

Fractal image tutorial (https://vimeo.com/265113621/5065227038)

Cool Sketch Doodle Technique - Drawing a Random Pattern (https://www.youtube.com/watch?v=J6CYwBeQMvM)

Part D: Beauty in maths?

Note: During this section of the lesson you will be guiding students through the creation of an artwork, using a combination of Fibonacci numbers, the Golden Ratio and fractals.

Step 1.

The first task is to lead students through the creation of a Fibonacci spiral. They should use this spiral and others to create their artwork. Provide your class with graph or grid paper. You may choose to show the video below to complement the instructions.

Fibonacci spiral photo tutorial (https://vimeo.com/265113843/0d46f4de6e)

In addition, written instructions are available here. You could print and distribute these instructions to pairs or small groups of students.

Step 2.

Now ask your class to repeat the process of drawing a Fibonacci spiral but using bigger or smaller units. It may help if you assign the students within each group to all produce a differently sized spiral. You could also encourage your class to use the same dimensions, but draw longer spirals (i.e. extending past the 13x13 square).

Once they have finished, explain to the class that they will now be using these spiral to create a ‘mathematical’ work of art! Explain that students will be creating a fractal of Fibonacci spirals to develop an abstract artwork, that itself complies with the golden ratio. The artwork will mostly be made using maths but will allow for some creativity too.

Just before moving on, allow your class some additional time to create stencils of each spiral by using tracing paper or carefully cutting out a stencil shape.

Step 3.

Hand out paper, poster board or canvases and explain to your class that they will now need to apply ‘golden lines’ to their canvas as they did in Part B, and begin to sketch out their design. They should consider the location of these lines when deciding where to place large or small spirals in their artwork. They should also decide whether they may like to include a theme or idea in their painting. Ideas for these are listed below:

Spirals could depict movement (e.g. wind or water).
Spirals could be the leaves and branches of a tree or plant.
Spirals could be arranged in different ways and use different colours to make recognisable shapes or animals (e.g. shark and heart images below).

All markings should be made in lead pencil which should remain visible through most medium to light coloured paint, so students should not need to worry about losing the sketch once they paint over the top.

If students are choosing a black or dark background for their painting, allow them to paint their background first and mark their golden lines and sketches on top using a chalk marker or a washable paint pen.

Step 4.

Once your class has finalised their sketches, allow them to start their artwork.

Reflection

Step 1.

On completion of this task, ask students to evaluate their own artwork and those produced by other students. You may choose to use the following rotational activity to provide students with peer feedback:

Split the class into groups and provide all students with at least 3 post-it notes.
Assign groups to start with different artworks and allow them 1-2 minutes to view each one.
Tell students to write feedback and thoughts on the artwork on a post-it note, and to place it beside the artwork before rotating to the next piece.
Ask students to read the post-it notes other students have posted and add a tick if they agree, a cross if they disagree or write a new post-it note if they have a different idea.
Rotate groups through all artworks.

Step 2.

Now that your class has thoroughly covered how mathematical patterns and concepts appear in nature and in artwork, ask students to think back to the original Thought Starter: “Is beauty a mathematical construct?’. Use the following points as topics of discussion and have students record their main ideas on the Student Worksheet:

Why do you think these concepts appear in nature?
Why do you think these concepts appear in art?
You created an artwork that was planned out using maths - is it beautiful?
Can we create beauty using maths?

Extension - Robotic Art

If your class already has basic robotics skills, then you may choose to complete the project from Part D using robotics instead of making a painting. Given limited space allowance for each group, the activity may work better if the whole class works towards producing one large fractal whereby each group produces a small part.

If your class has not used robots before or is at an introductory level, this task can be simplified to just incorporating the Fibonacci sequence as outlined below.

This task would be best completed in a large space where robots are able to move around freely. Attaching chalk or chalk pens to the sides of robots would work well in a concrete space. If using paint or coloured markers, you will need to cover an area with paper, fabric or other useful materials (e.g. flat sheets, lino). Robots and programs that are suitable for this task are outlined below, as well as tutorial videos that may be useful if you have little to no experience with robots.

Edison Robot

Attach chalk, chalk pens or coloured markers to the side/s of the robot with sticky tape and ensure the pen will drag along the paper as the robot moves.

Program: Edison EdBlocks

This program is free and accessible through a web browser: https://www.edblocksapp.com/
Tutorial video: https://meetedison.com/robot-programming-software/edblocks/

Sphero, Sphero SPARK or Sphero Mini

Allow the robot to be covered with washable paint before starting the program, or to roll in puddles of paint laid out on the paper/canvas. Ensure paint is safe to use on the robot or wrap the robot in plastic film securely.

Program: Sphero Edu App

Lego NXT or EV3 Robots

Attach chalk, chalk pens or coloured markers to the side/s of the robot with sticky tape and ensure the pen will drag along the paper as the robot moves.

Program: Lego Mindstorms for EV3 Home edition (is compatible with NXT robots).

The program is free to download onto a computer from https://www.lego.com/en-us/mindstorms/downloads/download-software.

NOTE: Students should use the top download button (ARROW) and not the second download button that is shown in the video as their robots will need to travel further than the download cable allows. Students can access their program through the robot menu under Software/Program/‘Program Name’ and by pressing RUN.

Step 1.

Explain the idea of the task to your class and encourage them to use the following ideas in their artwork:

Keep it geometric - curves may be tricky for beginners and straight lines with sharp turns can look just as good.
Begin with one main element (e.g. a geometric Fibonacci spiral) as the feature for your artwork, adding in secondary elements to complement the first.
Start simple - programming a robot can be tricky and take a lot of perseverance to get it right. Keep your program simple and increase the complexity once you feel you have mastered the basics..

Step 2.

Once your class has designed their artwork, you can start to show your class how to use their robots and programs. Explain the basic control functions of your chosen robot using the associated manual or online tutorials below:

Sphero: https://support.sphero.com/article/83gb4cqieu-sphero-mini-getting-started.

Edison:

NXT: https://www.generationrobots.com/media/Lego-Mindstorms-NXT-Education-Kit.pdf.

Now show your class the tutorial video for your chosen robot. Ensure that each group has their robot and student device setup before you play this tutorial so that students are able to follow along with the instructions. Pause the video in places if you need to. Alternatively, you could allow students to watch the tutorials on their own devices (with headphones) so they may start and stop at their own pace.

Step 3.

Once each group has completed the tutorial exercise, allow students time to program their robot to complete their artwork. To ensure success in this activity, ask students to do the following:

Program one movement at a time and test the program after each change.
Keep written notes of your program, keeping a list of numbers or coding that failed or needed adjustment.
Save different versions of the program regularly (e.g. Version 1, Version 2) at various stages through the program as a safety net in case the current program fails and cannot be fixed easily.
Ensure the tiny details are perfect before moving on to the next stage.
Persevere! Robotics can be tricky at times, but you need to keep trying!

Step 4.

Once groups have perfected their programs and completed several trial runs, allow students to complete their artwork using the required craft materials. It may be a good idea for students to do a practice run on newspaper or scrap paper before setting up their final run. Ensure that each group presents their program to the class, so they may share ideas and jointly celebrate their successes.

Differentiated Learning

Group work is ideal throughout most of this lesson, but the project can be completed individually if materials allow it. Higher ability students can be tasked with creating a precise mathematical artwork (similar to a scientific sketch) whereas lower ability students may engage more with the task if precision is less important. Higher ability students may also be challenged to create a more complicated pattern while lower ability students can focus on simpler ideas.

Provisional Learning Support - Depending on support requirements, students requiring support can work in pairs or groups and adjust scaffolding where necessary. Provide given examples (as found above) as support materials where necessary and reduce the steps within each task.

Teacher Reflection

Take this opportunity to reflect on your own teaching:

What did you learn about your teaching today?
What worked well?
What didn't work so well?
What would you share?
Where to next?
How are you going to get there?

STEAM - Maths = Beauty?

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