Geometry in Nature

By Anne Marie Burdick

Walking around the Rory Meyers Children’s Adventure Garden at the Dallas Arboretum, I could not help but to turn into a child again. As I navigated through the different exhibits on nature I soon found myself in the Kaleidoscope gallery. Here I had the opportunity to discover how geometry can be found in nature. This particular display demonstrated how there are patterns, tessellations, and fractals throughout nature. During my visit to the arboretum, I decided to focus my attention on fractals in nature. A fractal is a pattern that constantly repeats itself. Fractals can be found in tree structures, snowflakes, and some leaf patterns. For example, the photo to the right shows a plant that has fractals in its leaves. It is amazing the amount of geometry that can be found just outside in your backyards. By providing students with examples of places geometry can be found in nature, they will start to notice all the mathematics that is around them.

Retrieved from: http://www.pbs.org/wgbh/nova

/physics/mandelbrot-fractal.html

Before I went to the arboretum, I completed a little research on my own and found some interesting ways geometry is found in nature through fractals.  Benoit Mandelbrot, the founder of fractals has many extensive interviews online detailing the importance of fractals and why it is essential to understand it in geometry. In a particular interview that I read from 2008, Mandelbrot emphasized how natural structures are not perfect shapes. For example, clouds do not have smooth sides, rather they are rough, and that is why fractals are important to understand.

An activity that teachers could have their students complete in the classroom is Fractals in High School: Exploring a New Geometry by, Randi Lornell and Judy Westerberg. This activity had students complete hands-on activities that led them to discover what fractal geometry was. This article gave four mini activities that could be completed. By integrating activities like these in the classroom, teachers will be having their student explore topics from a new approach. It allows students to make connections to geometry and the nature around them.

 

A second lesson that I found interesting was a lesson from PBS on fractals and the coastline. This lesson had students use fractals to find the length of a coastline. Just like other natural structures, a coastline is not a smooth, straight surface. It has ridges and edges. Therefore, fractals are a great way to investigate the actual length of different coastlines.

Through the integration of geometry and nature, students will be able to see the mathematics in the world around them. By giving students the opportunity to research where geometry is found in nature we are providing them with the chance to relate to the topic. The Dallas Arboretum is a great place to start if you want to get your students minds wondering outside of the classroom. 

Geometry in Nature

By Megan Hancock

What would our world look like if our students thought about Geometry every time they looked out the window? If we teach our students how Geometry connects to the world around them, this could become a reality.

The Rory Meyers Children's Adventure Garden at the DallasArboretum and Botanical Garden is a great way to show children of all ages how Geometry and nature are connected. While at the children’s garden, I was able to view mathematics in nature through children’s’ eyes. The Kaleidoscope Garden in the children’s garden has different stations that talk about mathematical concepts such as fractals, tessellations, rotations, reflections, and the Golden Ratio.  Children are able to read about these math concepts and see different plants that showcase these concepts.

In a high school geometry classroom, students learn about tessellations, rotations, rotations, and fractals.  Students often struggle with these concepts because they are not able to picture them or relate them to concepts they are already familiar with.  By taking students to the children’s garden, they are able to see these concepts come to life.

While visiting the Dallas Arboretum, I chose to focus on fractals and tessellations.  Lornell and Westberg (2014) described fractals as having the characteristic of self-similarity, which means the part of the whole closely resembles the whole.  Objects can be broken into small pieces that resemble the whole, and this process can be repeated multiple times. Trees, flower centers, and different plants are great representations of fractals in Geometry.  Fractals are often a difficult concept for students to understand, so by showing students examples of fractals in nature, they are able to better understand this concept. 

Their article, Fractals in High School: Exploring a New Geometry, goes into detail about the Koch curve and the Koch snowflake.  These concepts allow students to learn about the growth pattern of the perimeter of the snowflake under iterations.  In a higher level mathematics class, students can begin to think about what happens to the perimeter of the snowflake as it approaches infinity.  They support this understanding with examples of how to teach fractals in a high school classroom using these concepts. 

Retrieved from http://mathworld.wolfram.com/

To take students deeper into the idea of fractals and the Koch snowflake, Linda Bolte described an activity she did with her students that found the area and perimeter of a Koch snowflake.  In the article, Sharing Teaching Ideas: A Snowflake Project:Calculating, Analyzing, and Optimizing with the Koch Snowflake, Bolte provided readers with the activity, the assignment, assessment methods, and sample solutions.  This is a lesson that could be used in its entirety, or specific parts could be used depending on the focus of the assignment. 

 I also focused on tessellations while at the children’s garden.  The tessellation station in the garden described a tessellation as a “repeating pattern made of flat shapes that fit together with no spaces between them.”  At the garden, students are able to test their knowledge of tessellations and create their own.  Texas Instruments (TI) has created an interactive lesson for the TI-Nspire that allows students to use regular polygons to determine which will tessellate.  They use this information to determine why some objects will tessellate and why others will not.  Part 2 of the activity guides the students through dihedral tessellations, a tessellation using congruent copies of two different shapes.  They use this information to determine if their rules for tessellations still hold when two shapes are used. 

By taking students to the Rory Meyers Children’s Adventure Garden, they are able to gain insight into how concepts from Geometry are present in nature.  Making these connections ensures students will never look at nature the same way again.  They will see fractals when they look at trees and leaves and tessellations when they look at honeycomb, plants, and animal fur.  Students would begin to think about mathematics not only in their math classroom, but in the rest of their world. 

Bolte, L. (2002).  Sharing teaching ideas: A snowflake project: Calculating, analyzing and optimizing with the Koch snowflake.  The Mathematics Teacher, 95(6), 414 – 419. 

Lornell, R. & Westerberg, J.  (1999). Fractals in high school: Exploring a new geometry.  The Mathematics Teacher, 92(3), 260 – 265. 

Geometry in Nature

 

Geometry in Nature

As the weather gets warmer, I notice many of my students walk a little slower to my classroom. Not because they’re being lazy, but because the weather outside is simply more enjoyable. “Mr. T! It feels so nice out here! Can we please have class outside? Please!?” says one of my 7^th grade students as she walks to my portable classroom. Lately, my usual response of “Sorry, kiddo, not today. Maybe after the STAAR.” has made me feel a little guilty for not figuring out a way to actually teach outside my classroom. Every time a student asks me to teach outside, the first thing that comes to mind is how much I loved being outside as a kid. I loved going to the fields and exploring the different types plants, color patterns, and shapes.

Last year, I decided to take my kids outside the classroom, for one day, to collect the measurements of my portable for a project they needed to create on SketchUp. That was the only time I taught anything outside the classroom. Since then, even though the idea has always been there, I haven’t really made an effort to create another project that would take my students outside the four walls of my classroom.

One of the ideas I had in mind for outside the classroom activities was to connect mathematics to nature. However, I knew very little about those connections. After reading some articles on geometry in nature, and visiting the Dallas Arboretum (for the first time!), I realized that I can come up with some ideas to teach mathematics in a different environment. The articles were very informative on both animals and plants, but I was more interested in the plants, particularly in the Fibonacci sequence. I did not know much about it until I read the articles, and during my visit to the arboretum, I was able to see how it is found in nature.  

 

There is a particular part of the arboretum, in the children’s garden area where the kaleidoscope is located, where you can find mathematical concepts such as patterns, shapes, and structures. Inside this place, you can also find the Fibonacci sequence in many different plants and designs. As you walk around, you will also find fractal geometry in plants and other types of nature. There is also an area of geometric plastic plants that can be used to design a geometric garden. And in another area, I found tangrams and tessellations stations. Although the part of the garden dedicated specifically for math was small, compared to the entire arboretum, the math was very well presented.

As much as I would love to take my students to this place, it would be very difficult, logistically speaking. However, I know there are parks and green areas very close to our school that I can use as a setting to replicate some of the things found in the garden at the Dallas Arboretum. One activity I could do would be to find the grass areas in our school that do not have any flowers or designs, and create geometric designs that could be proposed to the principal as a way to “beautify” our school with mathematical themes. This activity would involve concepts like regular shapes, composite figures, symmetry, similarity, congruency, transformations, dilations, and of course area. 

 

Another activity I could create would be to find all the trees that can be cut into geometric shapes (if none are found, then this would work as a proposal project similar to the first activity). The students would have to come up with geometric designs for each tree found considering their individual measurements. In this activity, the students would create models of the tree designs using wooden stick and Styrofoam, using the scaled versions of the real measurements. Concepts such as measurements conversion, scaling, 3-dimensional figures, similarity, congruency, and volume would be deeply involved in this activity.

Although, not everything can be connected to nature in every way, we can always get a little creative when we connect geometry concepts to the real world our students see every day. Activities like the ones mentioned, not only would allow the students to learn outside the classroom, but would also give them a better understanding of geometric concepts and how they can be used in the world around them. These types of activities require good understanding of geometric properties, which would challenge the students to use everything they know about the geometry they are using or learn about what they don’t know. Experiences like these, are the ones that kids remember the most, mainly because they take place in a setting so different than the traditional classroom learning environment.

Here are some additional photos of geometry found in the garden's floor.

Links and Resources

Numbers in the Garden and Geometry in the Jungle

http://www.jstor.org/stable/41181193

Dallas Arboretum Adventure Garden

http://www.dallasarboretum.org/named-gardens-features/the-rory-meyers-childrens-adventure-garden

Learning Geometry through Astrology

By Sarah Alejandro

When it comes to Astrology, I instantly think of horoscopes but it is so much more than that. Astrology is defined by the OxfordDictionary as “the study of the movements and relative posi
tions of celestial bodies interpreted as having an influence on human affairs and the natural world.” Meaning it’s the study of movements by the sun, moon, stars, planets, and constellations to predict events and casting horoscopes. When I looked deeper into the subject, I found that “The first developments of mathematical astronomy came during the Mesopotamian and Babylonian civilizations, especially during the Seleucid Kingdom (ca. 320BC to ca. 620AD)” (Mathematical Techniques in Astronomy,1999) Since then, we have advanced so far with technology that we can now see these planets, stars, and galaxies. 

Using advanced telescopes, Astrologists can identify the
positions and movements of these planets and stars as well as their distance. They do this by using mathematics but specifically geometry.  ”In these cases, one has to know the size of an object, then one measures the angles at which one sees the two sides of the object, and from the difference between the angles, one gets the distance” (Feuerbacher,2003).

To find out more about this topic, I visited the Expanding Universe Hall in the Perot Museum located in Dallas. As I walked around the exhibit, I noticed that this would be a great source for teachers to use for teaching geometric concepts. Since my focus is elementary grades, I walked around to see if I could find something that an elementary teacher or even a middle school teacher would be able to use. There was a particular section of the exhibit that caught my interest. It dealt with identifying the visible distance  between stars in angles or degrees by simply using your hand. I thought this could be a great lesson on teaching distance and angles. The students could try doing this at night and come back the next day to discuss about the different distances between stars they discovered. Another idea would be to buy a star projector lamp, which aren’t too expensive, and use it in class to have the students demonstrate how to find the angles between the stars. This would be a fun and engaging
way for students to learn angles through Astrology. For elementary, the TEKS put geometry and measurement in the same category so there could be a lesson that focuses on measuring time. The students could discover how to tell time without a clock by using a shadow cast from the sun’s light. The students could create their own sundial and tell time during the day.

Astrology is more than just the horoscopes we read at the back of a magazine. It is the study of movements and positions of the stars, planets, and gallaxies. There are so many ways students can learn geometry through Astrology. We wouldn't know as much as we do now if it wasn't for Astrologists using mathematics to find out distances, size, and mass of stars, planets, and galaxies. Think about the fun activities your students could do just by going outside into the sunlight or looking up at the stars. 

Feuerbacher, B. (2003, October 26). Determining Distances to Astronomical Objects. Retrieved March 3, 2015, from http://www.talkorigins.org/faqs/astronomy/distance.html

Mathematical Techniques in Astronomy. (1999). Retrieved March 3, 2015, from http://www.hps.cam.ac.uk/starry/mathematics.html

Picture of measure- http://www.hyperhistory.com/online_n2/civil_n2/prehistory_n2/distances.html

Sundial picture- http://hilaroad.com/camp/projects/sundial/numbers.jpg

Finding Geometry in Art

Finding Geometry in Art

Teaching middle school mathematics in the state of Texas has become more challenging today than in the past. With the movement of TEKS up and down the grade levels (link to changes at the bottom), many gaps have been created in the students’ mathematical knowledge, and it has forced the teachers to become extra resourceful and strategic with their lesson plans and teaching approaches. As a consequence, many teachers do not see outside-the-classroom learning as the best option for teaching mathematics, instead, they opt for more traditional in-class learning. For this reason, I decided to go outside the classroom, and out of my comfort zone, to find meaningful experiences for my students that can enhance their learning and create a better understanding of the mathematics concepts they learn in my class.  

After looking at several options for my adventure, I decided to visit the Museum of Geometric and MADI Art, in Dallas, Texas. This museum is actually the only museum dedicated specially to geometric and MADI art in all of North America, and it’s conveniently located right in the middle of our great city. A short description of MADI art is art that is not representative of a hidden meaning, and it is focused on the object and the colors themselves. You simply enjoy the pieces for what they are instead of trying to figure out the emotions the artist was trying to express when he or she created it. 

I was pleasantly surprised when I found out that the MADI movement started in Buenos Aires, Argentina, by an artist named Carmelo Arden Quin, in the 1940s. I say pleasantly surprised because many of my students are from Latin America, and I’m sure I can find some cultural connections here and use them to get my students a little more interested into exploring this museum.

 

To say the least, I felt like a kid in a candy store. This place has so much art that can very easily be connected to many of the concepts I teach in my classroom. They are not only visually stunning, but the mathematics behind each peace is simply amazing. Most of the art are paintings, but there are also sculptures and moving art that are just as fascinating. Some of these pieces really make you wonder how they were created and make you think about all the mathematics you can easily see in them. I currently teach 7^th and 8^th grade mathematics, and I can tell you that I can definitely use some of the pieces I saw as models for my students to recreate, as they explore each of their geometric properties. 

 

 

Before visiting the museum I read some articles on cultural connections to geometry, and I was very lucky to find some of the examples the shown in the articles. The museum had a few Islamic art pieces, which beautifully show tessellations, and also African American quilts, which are not only rich in history, but also show interesting geometric patterns. I was hoping to find some Native American weavings or stained-glass paintings I also read about, but the museum did not have any.

 

Even though my research gave me a lot of ideas on activities I can do with my students to help them better understand geometric concepts, I found some on my own that I think my students would be very interested in.

One of the activities I would love to do is to have my students create their own versions of Salvador Presta’s MADI pieces (pictures below), which involve 3-dimensional shapes that create 2-dimensional figures when hung from the ceiling. This project would allow the students to create their own works of art, and at the same time, learn about the properties of each 3-D shape and the 2-D figures they create when they are precisely measured and aligned. Several relevant concepts such as similarity, congruency, symmetry, angles, dilation, reflection, translation and measurement could also be emphasized with is project.  

 

Another activity would be to connect geometry to art. Our school happens to have a very strong arts program, so it would be feasible to have students create their own versions of some of the paintings with the help of an art student. My students would be responsible for making sure a given number of geometric concepts are included in the painting, and the art students would help them figure out the artistic aspects of it. Perhaps my students will find a new passion for art, and the art students an interest in geometric painting. 

 

In a perfect world, I would love to take all my students to this museum so they can see how mathematics can be found in arts and culture. However, that is not very likely to happen because of logistics and other complications. Therefore, I believe it is up to me, the teacher, to go outside the classroom and find effective non-traditional ways to teach my students the given curriculum in a way that engages them and makes them excited about and interested in what their learning in my class. Creating these types of experiences build memories, which are easier to remember than formulas on a wall, rules on a chart, or problems in a workbook.

 

Informational Links

The Museum of Geometry and MADI Art

http://geometricmadimuseum.org/museum.php

Native American Weavings 

http://www.statemuseum.arizona.edu/exhibits/navajoweave/edu/navajo_weaving_secondary_education.pdf

The Beauty of Geometry

 http://www.jstor.org/stable/41181942

Changes in TEKS

http://www.projectsharetexas.org/resource/revised-mathematics-teks-side-side-teks-comparison

 Here are more photos of the great art this museum has!