Section 1.1: Makedo Windball

Makedo Windball is a creative geometry project that encourages students to assemble cardboard into an art piece. This project illustrates an abstract concept: the idea that perfect spheres do not exist in our universe.

To begin, it is crucial to understand that no perfectly spherical object exists—planets, raindrops, marbles, and coins all have imperfections. This can be visually demonstrated by zooming in on these objects to observe their molecular structures. Veritasium on YouTube offers a compelling explanation of this phenomenon.

The Makedo Windball project utilizes 32 twelve-inch squares. As students connect these squares, they will notice gaps resembling pentagons and triangles. To minimize these gaps, smaller squares can be added. While students can strive for a nearly perfect sphere, it’s important to remember that tiny imperfections will persist.

The Makedo Windball project also promotes essential skills such as design, prototyping, recycling, and, most importantly, fun! You can find the instructions [here](link).

• Project Type: Abstract, Geometry, Engineering
• Supplies: Makedo Cardboard Construction Toolset for Whole Classroom or Makedo Cardboard Construction Toolkit and 12 x 12 Cardboards.

Section 1.2: Harmonograph

A harmonograph is an intriguing drawing machine that creates stunning geometric designs through the motion of swinging pendulums.

This project is particularly enjoyable for students interested in design and engineering. While constructing a harmonograph, students can observe the energy transfer between the pendulum and the drawing instrument, showcasing a real-world application of physics. Paul Borke provides a detailed exploration of the mathematics involved on his website.

Karl Sims has authored an excellent guide on crafting a harmonograph. I’ve updated some of the materials to enhance the project, such as using magnetic balls for greater movement flexibility. You can also find plywood and wooden dowels to assist in construction.

• Project Type: Art, Geometry, Engineering
• Supplies: Micron Pen, Magnetic Balls, Foldable Support Brackets, Plywood, Wooden Dowels.

Section 2.1: Euler's Disk

Euler's disk is an impressive scientific toy that every educator should have in their classroom. It serves as an excellent tool for teaching concepts such as magnetism, momentum, and gravity, while effectively demonstrating the conservation of energy.

When students see the disk spin, their curiosity is piqued, even those who are typically disinterested in physics. This is an excellent opportunity to delve into scientific principles and the natural physics that allow the Euler's disk to spin indefinitely. A fascinating video on this topic can provide further insights.

• Project Type: Science
• Supplies: Euler's Disk.

Section 2.2: Visual Proof of Pythagoras' Theorem

While students engage with Legos, I am continually impressed by their creativity. To encourage innovative thinking in geometry, I introduced Legos as a medium for mathematical projects, allowing students of all ages to express their creativity through construction.

One of our initial projects involved visually demonstrating the Pythagorean theorem. Utilizing a visual aid is crucial, as purely verbal explanations may only resonate with mathematically inclined students.

For this project, you can order Legos online to save costs. Decide on a triangle, such as the 6-8-10 triangle, and gather the necessary 200 Legos to build your proof.

• Project Type: Geometry, Construction
• Supplies: A Lego set & Base Plate.

Section 2.3: Sierpinski Triangle & Pyramid

Many students are unaware of fractals, which are intricate repeating patterns with numerous applications in various fields.

The Sierpinski triangle is a simple fractal that can be constructed using Legos, making it an engaging project for those interested in geometry. This project can turn mathematics into a fun and artistic endeavor, especially when discussing triangles and fractions.

To create the Sierpinski triangle, you’ll need approximately 243 2x2 Legos, which can be ordered to help reduce costs. Instructions for additional fractal projects like the Menger Sponge can be found [here](link).

• Project Type: Geometry, Algebra, Construction, Art
• Supplies: A Lego set.

Section 2.4: Estimate Pi by Dropping Toothpicks

Many students only associate pi with circles, but it has a much broader significance in mathematics. The Buffon's needle experiment exemplifies this relationship through statistics.

This project involves dropping toothpicks on a marked board to estimate pi. Students will measure the length of the toothpicks and space the lines accordingly. By counting how many toothpicks cross the lines, they can derive an approximation for pi.

Researchers at the University of Illinois have created simulations to further illustrate this experiment.

• Project Type: Probability & Data Science
• Supplies: A board, toothpicks, a marker.

Section 2.5: Pencil Star

Mathematics allows for the creation of various parabolic figures using straight lines, making it an accessible project for students regardless of their design skills.

Hexastix, a popular design, involves intersecting hexagonal prisms. To construct a challenging version, gather 72 unsharpened pencils and elastic bands, ensuring a safe and colorful building experience.

You can explore more designs, including stars or intricate structures, using the provided instructions.

• Project Type: Geometry & Art
• Supplies: Unsharpened Pencils, Elastic Bands.

Section 2.6: Lego Macintosh

For those interested in technology, the Lego Macintosh project combines creativity with coding. Jannis Hermans created a functional design using Raspberry Pi and e-ink display.

This project is a fantastic way to ignite students' imaginations and foster a love for programming. You can find various designs online and start building with your students.

• Project Type: Geometry, Design & Coding
• Supplies: Lego Architecture, Raspberry Pi, e-ink Display.

Section 2.7: Pi Poster

Pi is an irrational number without an apparent pattern. By visualizing its digits through color coding, students can illustrate the randomness of pi.

Inspired by Martin Krzywinski, students can use M&Ms to represent each digit's color and create a vibrant poster. This project enhances data analysis skills while exploring the characteristics of numbers.

• Project Type: Algebra, Design & Data Analysis
• Supplies: Black Poster Board, M&Ms, Gorilla Glue.

Section 2.8: Volume by Slicing

This project introduces calculus by showing the relationship between area and volume. Instead of merely presenting formulas, students can engage with the concept through hands-on experiences.

By slicing a vase into circles, students can approximate its volume, bridging the gap between two and three-dimensional objects.

• Project Type: Calculus & Design
• Supplies: A ceramic vase, cardstock paper, beads.

Section 2.9: Archimedes' Trammel

The Archimedes' Trammel, known as the "nothing machine," is a fascinating tool for drawing ellipses.

While constructing this device may be time-consuming, it is rewarding and helps students enhance their engineering skills. Various instructions are available on YouTube to guide you through the process.

• Project Type: Calculus, Geometry, Engineering, Construction & Design
• Supplies: A ceramic vase, cardstock paper, beads.

Section 2.10: Honeycomb Wall Art

Honeybees are exceptional mathematicians, utilizing hexagons in their honeycombs for optimal efficiency.

To illustrate this concept, students can create honeycomb wall art with ice cream sticks, showcasing the area-to-perimeter ratio of regular polygons.

• Project Type: Geometry, Algebra, Construction & Design
• Supplies: Ice cream sticks, wood glue.

Section 2.11: Design Snowflakes

Snowflakes are captivating geometric shapes ideal for teaching symmetry.

Using Perler beads and pegboards, students can craft unique snowflake designs. This project emphasizes transformations and the relationship between mathematics and nature.

Instructions for this project can be found [here](link).

• Project Type: Geometry, Construction & Design
• Supplies: Perler beads & pegboards.

Section 2.12: Leaf Tessellation

Tessellations occur when shapes repeat to create a pattern without gaps.

Students can create leaf tessellations by following specific rules of translation, rotation, and reflection. Basic supplies include paper, pencils, scissors, and tape.

• Project Type: Geometry, Construction & Design
• Supplies: Perler beads & pegboards.

Section 2.13: The AxiDraw

The AxiDraw is a versatile tool for exploring mathematical design.

I was fortunate to convince my principal to purchase one for our classroom, enabling us to create intricate designs for exhibitions. This tool can inspire students to appreciate coding and mathematical design.

• Project Type: Geometry, Coding & Design
• Supplies: AxiDraw Drawing Robot, Bristol weight paper, Cutting Mat, Prismacolor Markers, Micron Pen.

Section 2.14: Sand Pendulum

Sand pendulum machines are popular in science classrooms due to their ability to produce fascinating patterns.

Enhancing this project with a magnetic sheet and iron filings, as demonstrated by Mr. Yeany on YouTube, can yield even more impressive results.

• Project Type: Geometry, Construction & Design
• Supplies: Iron fillings, magnetic sheet.

Section 2.15: Mirror Symmetry

Children are naturally drawn to reflections and symmetry, making mirrors an ideal tool for exploration.

This project allows students to create intricate patterns with wooden blocks and mirrors, fostering their understanding of symmetry and reflection.

• Project Type: Geometry, Construction & Design
• Supplies: Rectangle Mirror, Connector, Geomagic Mosaics.

Note 1: Beyond Euclid! is my weekly newsletter that curates high-quality math and science content. I encourage you to subscribe for a weekly dose of inspiration.

Note 2: I earn commissions for purchases made through links in this post.