Math Archives - Page 2 of 3 - Eva Varga

December 14, 20152
I received this product in exchange for an honest review. I also received monetary compensation for my time spent in reviewing the product. All opinions expressed are true and completely our own. Please see my disclosure policy for more information.

computerprogrammingWhy limit your child’s exposure to computer programming to just a single hour annually?  Fischertechnik provides the materials and curriculum support to engage students in basic computer programming skills year round. 

Last month I shared how excited we were to discover the Introduction to STEM I Kit in my post Play and Learn with Fischertechnik. We have since spent weeks building the various models and discovering how everyday technology and simple machines actually work.

How Does It Work In Our Homeschool?

When the Introduction to STEM I & II Kits arrived at our home, we immediately opened all of the materials and of course Jeffrey was SO excited to just BUILD something. I let him experiment with all of the pieces for a few weeks and he built model after model. It wasn’t long before the models he was building caught the attention of his older sister. She wanted in on the fun as well.

Over the Thanksgiving holiday, we sat down with our cousins (the more the merrier!) and installed the software. This was as easy as putting the disc in the computer.

I promise.

programfischertechnikComputer Programming with Fischertechnik

The girls jumped right in and immediately began to CONSTRUCT a merry-go-round model without any problems. The instructions took them through the process step by step. The process is very achievable for this age level.

They then spent a little time with the software and soon figured out the basic PROGRAM to get the wheel to operate a series of simple commands. The ROBO Pro light software programming is drag and drop, so it’s very easy to learn! A sample is given on screen as well as in the accompanying guide book.

After the model had been built, it was time to CONTEMPLATE why things worked they way they did and for further experimentation. Once the girls were familiar with what each of the different buttons did, they were ready to begin experimenting in all kinds of ways.The curriculum guide provides suggestions for assessment questions to aide students in reflecting on the process.

Finally, students are invited to CONTINUE their learning with more experimentation and a challenge to build and program more complex models and interactions.  My daughter is now motivated to learn MORE.

The Introduction to STEM II kit allows elementary classrooms of different ages to enter the exciting world of STEM and learn the basic skills of computer programming – an exciting activity for young people. 

fischertechnikMaking the Investment in Fischertechnik

If your child is interested in engineering and computer programming, I strongly suggest you invest in Fischertechnik products.

While the cost may at first seem high (the kits begin at $99), you receive a versatile set of bricks, sensors, motors, and a USB connection, along with the software and teacher’s guide. This enables your student to build the 12 models, but from there the creativity is LIMITLESS.

The Introduction to STEM I kit contains 500 pieces with instructions for building 40 different models in all. Students can build a centrifugal governor, manual transmission, block and tackle, wind turbine, a beam balance, and so much more.

The Introduction to STEM II contains 200 additional fischertechnik parts, including:

  • ROBO LT Controller (USB interface/USB power supply)
  • ROBO PRO Light software
  • An XS motor
  • 2 lights
  • A lens tip lamp
  • Photo-transistor
  • 2 switches

Of course, my kids love the models they have made with the instructions, but they also loves to be creative. They have now begun experimenting with the variety of components.

The beauty of this is that they are thinking through problems and solutions; asking good questions. Skills that will serve them later in life.

Interested? Order a free sample activity kit today! 

October 5, 20151

STEM has always been a major part of our homeschool curriculum. As my children have gotten older, their interests have only strengthened in relation to math and sciences. They have both expressed an interest in computer programming and we have dabbled a little with different curricula (Homeschool Programming) and online resources (Khan Academy).

Exploring Digital Animation with Pixar & Khan Academy @EvaVarga.netIn addition to computer programming, my daughter has also become interested in digital animation and graphic art. For Christmas she received a Wacom tablet and has spent many hours learning how to create elaborate graphics. Her work has even won two different t-shirt design contests!

She also loves Japanese anime – some of her favorites are Sword Art Online, Hunter X Hunter, and Fairy Tale. It was no surprise when she expressed interest in learning how to animate her own illustrations.

We were thereby very excited to learn of Pixar in a Box, a collaboration between Pixar Animation Studios and Khan Academy, This free course sponsored by Disney, is a series of video lessons, interactive exercises, and hands-on activities.

The course materials will enable students to discover how the academic concepts they learn in school enable Pixar filmmakers to create new worlds, animate unique characters and tell stories through animation. Although designed especially for students in middle and high school, these resources are available to learners of all ages, completely free of charge.

There are presently six topics in the course:

  • Environment Modeling
  • Character Modeling
  • Animation
  • Crowds
  • Sets & Staging
  • Rendering

Each topic begins with a design focused lesson that doesn’t require math concepts beyond the elementary level. The goal is for people of all ages try these lessons – to first get you creating things with interactive tools while exposing the connections to mathematics.

The second lesson dives deeper into the concepts involved and are designed to increase in difficulty so that students can attempt more advanced material. Each lesson is designed to take approximately one hour to complete and includes hands-on activities to extend the lesson.

Exploring Digital Animation with Pixar & Khan Academy @EvaVarga.netWhat I love best about Khan Academy is that the course material is self-paced. We can pick up a lesson at any time and work through the material at a pace with which we are comfortable. There is no deadline and thereby no pressure.

We look forward to integrating this course into our curriculum more fully in the near future. She’s already begun a story board for an animated show of her own.


May 3, 2014

Like most little boys, my son has been interested in trains since he was just a toddler.  However, unlike his peers, he was never fond of Thomas the Train. His interest first began upon watching The Polar Express and though his taste has evolved over the years, his fascination with real locomotives has never waned.

I believe this is due in part to the lifetime hobby of our dear friend, Albert, whom I mentioned in my post earlier this week when I wrote about the importance of adult mentors for children (see my post Mentors and Role Models).  Like my son, Albert developed an interest in trains as a young boy himself and has been collecting HO model railroads ever since.  He now has an incredible collection and with a little artistic help from his wife, he has put together an amazing display.

model railroads

His collection of 12 locomotives, 250 rolling stock (cars or carriages), numerous buildings, and scenery take up the entire second story of their home in an awe-inspiring display.  Each model has been meticulously crafted, every detail attended to, in an effort to recreate working train and lumber yards.  Albert has worked tirelessly to recreate the route of the Union Pacific from Veneta to Coquille, Oregon. While his display is aesthetically pleasing, one could spend hours immersing oneself in all the work that has gone into the display, what really struck me were the train time tables pinned to the ceiling and the routes sketched on maps on clip boards.

Model Railroads – Math

HO is the most popular scale of model railway in the world; in North America 3.5 mm (0.1378 in) represents 1 real foot (304.8 mm); this ratio works out to about 1:87.1.  The name HO is derived from the fact that its 1:87 scale is approximately half that of O scale which was the smallest of the series of older and larger 0, 1, 2 and 3 scales introduced by Märklin around 1900.

In the past few years, Albert’s focus has turned his attention to programming his trains according to a timetable.  While this is an advanced skill, and my little guy is not even interested in this aspect of model trains (yet anyway), I love that this hobby is one that can grow with him.

There are a variety of programs available to aid railway modelers in creating and operating their layouts. Many timetable operators use stringline graphs to schedule their trains. These can be constructed using a spreadsheet such as Excel. For more information on building a model railroad timetable, I suggest a series of videos by Model Railroad Hobbyist Magazine.

Model Railroads – Science

Modern HO trains run on two-rail track, which is powered by Direct Current (varying the voltage applied to the rails to change the speed, and polarity to change direction), or by Digital Command Control (DCC – sending digital commands to a decoder in each locomotive).

On simple, usually temporary layouts, power is supplied by a power pack consisting of a transformer and rectifier, a rheostat or potentiometer for regulating voltage supplied to the track (and thus train speed), and a switch to control train direction.

On permanent layouts, multiple power supplies are traditionally used, with the track divided into electrically isolated sections called blocks; toggle or rotary switches (sometimes relays) are used to select which power supply controls the train in a particular block. With the advent of digital command control, block divisions are largely eliminated, as the computerized controllers can control any train anywhere on the track at any time, with minor limitations.

Model Railroads – History

HO scale trains were developed in response to the economic pressures of the Great Depression. The trains became very popular in the United States, where it took off in the late 1950s after interest in model railroads as toys began to decline and more emphasis began to be placed on realistic details in response to hobbyist demand. While HO scale is by nature more delicate than 0 scale, its smaller size allows modelers to fit more details and more scale miles into a comparable area.  Currently, HO is the most popular model railroad scale in both continental Europe and North America.

Model Railroads – A Lifelong Hobby

Over the years, Buddy has shown a wavering interest in his own HO collection.  He’ll be really engaged in trains for a few months (usually sparked by a visit to Albert’s house) and then something else will capture his attention.  In his room, he has two card tables that support a 4′ x 6′ plywood frame.  Presently, his Lego City occupies this space (in which his Mærsk train and container ships are proudly featured), but in the past it has displayed his own growing collection of HO trains.


April 11, 20143

While visiting friends in Bend, Oregon recently, we had the opportunity to visit a local printshop and book making studio that had a small collection of MC Escher prints on display.

Escher was a prolific artist and master printmaker, producing more than 400 original prints in his lifetime. The exhibit we visited showcased Escher’s skill in wood cut, wood engraving, lithography, and linocut.

MC EscherThe prints spoke to Escher’s fascination with illusion, whether created through tessellations, impossible architectural constructions, or reflections. “Escher’s images appear logical at first, but the deeper you look, the more improbable they become. Each print is a puzzle that asks to be solved, but the solution is elusive,” explains Ron Schultz, a volunteer who contributed to the exhibit catalog. “Escher’s work endures because of its mystery.”

So let us then try to climb the mountain, not by stepping on what is below us, but to pull us up at what is above us, for my part at the stars; amen.


Escher-Hand-with-Reflecting-Sphere-1935Maurits Cornelis Escher (1898-1972) is one of the world’s most famous graphic artists. His art is enjoyed by millions of people all over the world. He is most famous for his so-called impossible constructions but also created many wonderful, more realistic works.

Born on 17 June 1898 in Leeuwarden, Friesland, he was the youngest son of civil engineer George Arnold Escher and his second wife, Sara Gleichman. Although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old. He briefly studied architecture, but later switched to decorative arts. In 1922, Escher left the school after having gained experience in drawing and making woodcuts.

He lived and worked in Italy for a time, where he met and married his wife, Jetta Umiker. They lived for a time in Switzerland but eventually settled in the Netherlands.

During his lifetime, Escher created 448 lithographs, woodcuts and wood engravings, and over 2000 drawings and sketches. Like some of his famous predecessors (Michelangelo, Leonardo da Vinci, Dürer and Holbein), Escher was left-handed.

In his early years, Escher sketched landscapes and nature. He also sketched insects, which appeared frequently in his later work.

Bring it Home


A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. There are three main types of tessellations: Regular (a pattern made by repeating a regular polygon), Semi-regular (made of two or more regular polygons; whereby the pattern at each vertex is the same), and Demi-regular.

Working with tessellations is a great way to integrate math and art.  Tessellation activities incorporate many important skills, most noteably: positive/negative shapes, spacial rotation skills, and spacial relations. 

Visit Juliana Kunstler for a great art lesson on Tessellations.

Möbius Strips

Mathematicians know all about mobius strips. A mobius strip is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

I don’t grow up. In me is the small child of my early days.

M.C. Escher was fascinated by the Möbius strip. Escher’s Bond of Union is well-known for its portrait of a couple using a Möbius strip. The man and woman are infinitely intertwined.

Create a Möbius strip of your own by cutting a long strip of paper, twisting once, and taping the ends together.  Cut the strip lengthwise down the center.

  • What do you think will happen if you cut the strip lengthwise down the center? Try it! What really happens?
  • What happens when give the paper strip a double twist, then tape and cut down the middle?
  • What happens when you cut the strip 1/3 of the way in from the edge? The first time around, do your scissors line up with where you started cutting? What kind of shape do you get in the end?
  • Play around with different kinds of paper. Our favorite: paper that is a different color on each side.

March 3, 20147

What is your favorite color of Skittles® candy? Do you want to know what dyes were used to make that color? Check out this science project to find out how you can do some scientific detective work to find out for yourself.

Using a simple scientific technique called chromatography, you can separate and identify the various compounds in a complex mixture or solution.

RainbowCandiesIn this activity, water is used as the mobile phase (or solvent), a fluid the solution is dissolved in.  The stationary phase, the material the fluid moves through, is filter paper (a coffee filter cut into strips will work well). The water is absorbed into the fibers of the paper by capillary action.  As the water travels through the paper, it picks up ink particles (the solute) and carries them along. This same process that spoils a perfect print-out can also be put to good use.

The components in the dye mixture move at different speeds as they travel through the stationary phase due to the different properties of the solution’s components, such as their molecular sizeelectrical charge, or other chemical properties. In paper chromatography, different pigments can be separated out from a solution based on the solubility of the pigments. A pigment that is more soluble (or more hydrophilic) than another pigment will generally travel farther because it will be easier for it to dissolve in the mobile phase (water) and be carried along the stationary phase (filter paper). A pigment that is less soluble (or more hydrophobic), or interacts more with the filter paper than the water, will generally travel a shorter distance.

Candy Chromatography

How are the dyes used in hard-shelled candies similar? How are they different?


  • Candies with a colored coating ~ I recommend testing four different colors and five identical candies of each color
  • Filter paper ~ I used a white coffee filter cut into rectangles of approx. 1″ x 3″
  • Petri dish (or small plate)
  • Pipet or eyedropper
  • Cup of water
  • Skewer or chopstick
  • Tape


  1. Prepare the test strips by making a faint line on each strip with a pencil about 1″ from the bottom of the strip.
  2. Tape the other end to the skewer so that the strip hangs freely.
  3. Place the candies in a shallow dish and add a few drops of water atop each candy.  In a few moments a small colored puddle should form.
  4. With the pipet, place a drop of the colored candy solution onto the line you drew on the test strip.
  5. Carefully place the strip into the cup containing the water.  The pencil mark should NOT be in the solution but rather be about 1/2 inch above the water.  Make sure, though, that the end of the paper is in the water.  Watch the water as it moves up the strip of paper (due to capillary action), and see what happens as it comes in contact with the candy solution.  Leave the strip in the solution until the dye no longer travels up the strip with the water.
  6.   When this is complete, remove the strip of paper and place it somewhere on your desk so that it can dry thoroughly.  Continue to test the remaining strips.

Integrating Math

Rf Example 1

Why do different compounds travel different distances on the piece of paper?  In paper chromatography, you can see the components separate out on the filter paper and identify the components based on how far they travel. To do this, we calculate the retention factor (Rf value) of each component. The Rf value is the ratio between how far a component travels (the dye) and the distance the solvent (the water) travels from a common starting point (the origin).  To calculate the Rf value, divide the distance traveled by the sample component by the distance traveled by the solvent. For example, 2.5cm ÷ 5.0cm = 0.5

You can use Rf values to identify different components as long as the solvent, temperature, pH, and type of paper remain the same. This ratio will be different for each component due to its unique properties, primarily based on its adhesive and cohesive factors.

Take it Further

  • Try this project with a variety of candies— for example, does the red in Skittles® look the same as the red in M&Ms® when processed with chromatography? Is the average Rf value nearly the same? Look in the ingredients on each package – were the same dyes used?
  • Try this experiment again but this time use different kinds of solvents (e.g., salt water, vegetable oil, isopropyl rubbing alcohol, etc.). Does a dye travel different distances depending on the solvent you use? What do you think this tells you about the solubility of that dye in the different solvents?
  • Do the dyes you tested travel differently on different kinds of filter paper (lightweight paper towels, heavyweight paper towels, white coffee filter papers, etc.)?

You can probably now imagine how chromatography can be used to separate specific components from a complex mixture and identify chemicals, for example crime scene samples like blood, drugs, or explosive residue. Highly accurate chromatographic methods are used for process monitoring, for example to ensure that a pharmaceutical manufacturing process is producing the desired drug compound in pure form.

December 22, 20124

We love snow.  When we lived in Central Oregon – we were delighted with each snowfall.  Living now in Northern California, we don’t get to enjoy as often.  Mount Shasta is just far enough away that it would be a full day outing if we desired to go skiing whereas Mount Bachelor was near enough, we could go for just a few hours and still have time enough in the day for other errands or activities.  The first year we lived here, sadly, there was very little snowfall – even at Shasta.

Mt. Shasta shrouded in clouds as viewed from our deck.

When the snow fell this week – we were both delighted and heartbroken.  My father, brother, and sister-in-law were planning to drive down to spend the weekend with us for Christmas.  I-5 was closed off and on all through the end of the week and through the weekend. Even with chains, the delays and closures forced them to stay home.  As a small consolation, the kids and I chose to curl up with a few holiday books and do a little nature study to take advantage of the snow fall which coordinated perfectly with the Outdoor Hour Winter Snowflake Challenge.

Science Literature

One of the books we read was Snowflake Bentley by Jacqueline Briggs Martin. We’ve read it before but I learned that Buddy didn’t remember it so it was a new discovery for him (not surprising when you read my previous post and discover what what he was focused on that day).  Winner of the 1999 Caldecott Medal, it tells the story of Wilson Bentley.

From the time he as a small boy, he saw snowflakes as tiny miracles and was determined to capture these small wonders on film. His work and dedication revealed two things – no two snowflakes are alike and each one is based on a six-pointed crystal.

After the story, we watched a Brain-Pop video on snowflakes and googled snowflake images.  We had wanted to capture a few flakes to observe under the microscope, but it had stopped snowing by now and the snow on the ground was wet, melting off by early afternoon.  We noted, therefore, that we had an easier time illustrating snowflakes than Bentley would have.

Nature Journaling

We recorded in our journal that since the angle that the individual atoms in water form is 104°, ice freezes into a roughly hexagonal molecular lattice. This six-sided crystalline shape is reflected into the snowflake’s overall shape, causing snowflakes to have a 6-fold symmetry.  Here is a fun video from Khan Academy to help illustrate this fact, Snowflakes, Starflakes, and Swirlflakes.

Our nature journal entries … sadly, the image is not very clear. 

Coincidentally, we had created numerous paper snowflakes earlier in the week to send to Sandy Hook Elementary School.  Members of the school PTA are coordinating an effort to convert the new school students and staff of Sandy Hook will be housed in after the holiday break into a winter wonderland.

I love that as unschoolers, we were able to “seize the moment” and thoroughly integrate our curricular studies:  nature, science, writing, literature, biography, math, art, and service learning.  It was indeed a full day of learning but took only an hour or so of actual clock time.