As part of our Waldorf geometry unit, we are learning how to create shapes by shading them from the periphery, from the center and from intersecting lines. Join me as I make three chalk drawing to accompany this unit.

Cascading triangles are a beautiful representation of triangles in motion. You can simplify this project and just draw it with pencil for a stunning image that appears to be moving or you can add color in with watercolor pencils or color pencils.

Today we are making triangles from intersecting lines. We'll only color in the triangles formed and leave the other shapes blank.This is a beautiful representation of triangles created by intersecting lines. You have to make sure not to color in other shapes, so it encourages you to pay attention to the angles and sides of each shape.

How can you shape a curved image from triangles? Watch this video to find out. We cut small triangles out of watercolored paper and turned it into a beautiful organic image that resembles a fern or droopy leaf.

Today we are making symmetrical images using triangles. We are using a Waldorf curriculum from Live Education. Other materials used for this project: watercolor paper, spray adhesive, watercolors (we used distress inks), and scissors or the Silhouette Cameo.

As part of our geometry unit we decided to piece together a mandala using watercolor paper we painted in beautiful autumn colors and cut into diamonds and triangles using our Silhouette Cameo. You can find the watercolor paper, watercolors (we used Distress Inks by Tim Holtz) and spray adhesive from Blick Art Supplies.

After reading the book The Warlord’s Puzzle, we decided to make our own set of tangrams. I took a picture of the tangrams from the book and imported it into my Silhouette Cameo‘s program. I traced the image and reconfigured them and then used the Cameo to cut the shapes.

While reading the book “String, Straightedge, and Shadow” by Julia E. Diggins, we came across the chapter that explains the six-part division of a circle and we decided to recreate the project. For this project you’ll need a compass, watercolor paper, watercolors, embossing pen, embossing powder and an embossing tool.

While working with our Live Education Waldorf curriculum from we decided to do the project on “Angles and Degrees” three different ways. We’ll do this project with watercolors, watercolor pencils and color pencils. We will divide the circle into acute angles, a right angle and an obtuse angle.

This quick and easy projects demonstrates how the interior angles of a triangle equal 180 degrees. All you need is a piece of paper and a pair of scissors.

How did Pythagoras show that the square of two sides of a right triangle equal the square of the hypotenuse? Watch as we recreate the proof from the 6th century.

While reading "String, Straightedge and Shadow" by Julia Diggins, we came across a chapter on constructing a square using 'string' and 'straightedge'. Watch as I show you two methods for creating a perfect square, just as Pythagoras himself would have done...only we use a compass instead of string :)

Watch as we make three dimensional ‘solids’ from paper. We used 90lb. watercolor paper and our Silhouette Cameo to make the five regular solids as discovered by Pythagoras in the 6th century BC. We were inspired by the book “String, Straightedge and Shadow” by Julia Diggins which we are reading for our geometry unit for homeschool.

This short demonstration show how to construct a hexagon and an equilateral triangle using just string and straightedge, or in today's terms a compass and a ruler. We are reading the book "String, Straightedge and Shadow" by Julia Diggins for our geometry unit for homeschool. We learned how to make six equal divisions in a circle to construct a hexagon.

We used the kit “Plato Glo Mobile” from Rainbow Resource to make sturdy three dimensional shapes. The five regular solids were discovered by Pythagoras and his Secret Brotherhood in the 6th Century BC. We recreate them using bamboo sticks and the rubber connectors. If you don’t have this kit, don’t worry!

You can make this string art as part of a geometry unit or Ancient Rome/Greece unit. We are using the book called "Classical Kids An Activity Guide to Life in Ancient Greece and Rome" by Laurie Carlson as inspiration for this project.

Math is beautiful and arguably the most beautiful math is geometry. Geometric designs are all around us, and when we pause and recognize them, their mysteries are revealed. This lesson, the 12 division of a circle, is one such project. When a circle is divided into 12 sections with circles of the same diameter, the results are nothing short of stunning.

It's Pi Day again. On March 14th (3.14) mathematicians around the world celebrate the beauty and mystery of the circle. Today, I'm sharing three projects that we did for a geometry lesson as part of our Ancient Greece main lesson block.

Today's lesson is a LIVE geometry lesson I did with my 13-year-old son. He and I work through 3-fold symmetry of a circle using the book Drawing Circle Images. We (or I should say I) encountered a problem. I did the design different than the book, and it took me a few minutes to recognize my error.

Circle symmetry is one of the most beautiful geometric projects you can easily do with your students. While some symmetries are more complicated, 3-, 4-, 6-, 8- and 12-fold are fairly straightforward and can be achieved with moderate effort and skill, in my opinion and experience.

Today's project is a tutorial for fourfold symmetry. You can explore this design and come up with creative ways to paint or color in the shapes. In this video I show you two ways to make 4-fold symmetry using the book Drawing Circle Images. You can find links to the materials I used in the blog post that accompanies this video.

Fivefold symmetry is the first challenging shape to explore in my opinion. You begin with fourfold symmetry, then re-adjust your radius to to be the distance between the two arcs on the circle and place your compass on one point and draw a new arch. Connect the arcs with a ruler and now you have the two points which create the distance between each of the five arcs to get five-fold symmetry.

Sixfold symmetry is one of the easiest of the complicated designs, in my opinion :) I find this one to be one of the most beautiful designs because from sixfold symmetry comes the hexagon and that shape is seen in tile work across the world. The design opportunities are endless. I love the way this one turned out and can't wait to explore this design again.

I LOVE 7-Fold symmetry. It's my favorite of all the fold symmetries we did during this series. I love the way the center overlaps. Depending on how you paint it, you can get some spectacular results. For this tutorial we needed to follow the example in the book which included a table with measurements on what to set your compass to achieve the symmetry.

To construct eightfold symmetry, you need to start with fourfold symmetry which we did in an earlier tutorial. Once you draw arcs on the outside of the circle and and connect them to make a square, you can draw arcs between each of the four corners to achieve 8-fold symmetry. You can see two different designs emerge depending on how you color in each shape. I love the way this one looks in the end in part because of my color choices.

Initially I thought ninefold symmetry would be simple because achieving 3- and 6-fold symmetry is fairly simply, but I was wrong. I attempted this several times and I show the challenges in the video, so you know this one isn't as easy as it seems. For this shape, we need to follow the chart provided in the book

Tenfold symmetry is stunning! I love the designs in which the arcs don't cross the center of the circle by spiral just around it instead. For 10-fold symmetry, you construct your circle as you would fivefold symmetry. Then at each of those arcs, you draw a line from the arc through the center of the circle to get a point on the opposite side of the circle. In this way, your original five arcs become 10 and now you can begin drawing small and large circles around each point.

Elevenfold symmetry is the last in the series where we needed to use the book's chart to construct the divisions of a circle. In truth, I prefer the symmetries that can be achieved through geometry rather than through a chart that gives the measurements for the each circle. In the end, this design is beautiful and reminiscent to 12-fold symmetry which we have done before.

Twelvefold symmetry is very satisfying to make. You can construct 12 divisions of a circle using basic geometry. Once the 12 divisions are complete, you can choose which areas to paint or color in to give your final design a unique feel. I definitely like this symmetry, and what's fun about it, is that you can continue to extend it beyond the page for additional spectacular designs. The book does suggest a design we opted not to use and if you buy the book, I highly suggest you try some of the designs we didn't do.

Exploring Quadrilaterals is a lesson from the Live Education Waldorf curriculum for year 5. For this lesson I'm using Lyra Color pencils from A Child's Dream

Nesting Squares is a lesson inspired by the Live Education Waldorf curriculum for year 5. For this lesson I'm using Lyra Color pencils from A Child's Dream

Curved lines from Straight lines is a lesson inspired by the Live Education Waldorf curriculum for year 5. For this lesson I'm using Lyra Color pencils from A Child's Dream

We continue with our Waldorf Geometry main lesson block with this lesson on bar graphs and pie charts. We used data on the tree population of California as it was relevant to our Botany lessons rather than using the data that was provided in the main lesson book by Live Education Waldorf curriculum. We constructed a bar graph with this data as well.

We continue our Waldorf Geometry lessons with constructing the ‘perfect’ hexagon using six division of a circle, then connecting the points to construct a hexagon. From that we connect every other point to construct two equilateral triangles. You will see a new hexagon emerge in the center again! Connect every other point to construct two equilateral triangles and once again another hexagon will emerge.

For this lesson on six division of a circle, we look at flowers that exhibit six fold symmetry. This lesson fit perfectly with our botany Main lesson block as we are studying bulbs and lilies. For this lesson we constructed a circle with a radius of 4 inches (2.5 inches would have been better) and divided our circle into 6 equal parts by placing our compass on the circumference and swinging the compass around to intersect the circumference with two arcs.

The first several lessons in this geometry main lesson block focuses on 6 division of a circle. By this point the student should be proficient at constructing a circle, placing the point of the compass on the periphery of the circle and swing the arcs to intersect with the circle.

This lesson was phenomenal! How exciting to learn how to make a three dimensional looking hexagon and then experiment with variations of making quartz crystal drawings from different angles.

I underestimated the difficulty of this lesson. I thought it would be easy, fun and quick. And at first it wasn't and then later, it was only slightly more fun, but still challenging and slow. Firstly, make your radius about 2.5 inches or less, as a big radius makes for too big of a snowflake and much harder to get lines and shapes straight.

Making snowflakes is a great family project that works well for almost all ages. While working with 6-fold ‎ symmetry with your middle schooler, you elementary ages students can enjoy cutting and designing snowflakes. You may start with any size circle, but we worked with a radius of about 2.5 inches.

I love when botany and geometry come together in a lesson. For this geometry lesson, we continue with 6-division of a circle, but as opposed to using the 6-fold symmetry, we will concentrate on the triangle.

For this lesson, you want to start with a 9 cm circle. Draw a line through the center. Extend the compass and place your point on the diameter at the circumference and swing the arc to the right and left.

This is the first in a series of lessons devoted to five-division of.a circle. This is a low step by step tutorial on how to construct 5-fold symmetry. Once you've mastered this lesson, the remaining lessons for five-fold symmetry will be easier to complete. For this lesson, you'll need a compass, ruler and pencil.

This is the second a series of geometry videos on 5-division of a circle. This one is a beautiful geometry expression of five fold symmetry. We used the book Drawing Circle Images for this lesson. We used our General's Chalk Pastel pencils and Matte Finish to seal our work.

We continue with our five-fold symmetry and five-division of a circle with this easy, quick and beautiful lesson exploring five-fold symmetry in the world of botany. I love how the Waldorf curriculum by Live Education naturally overlaps with previous lessons or previews upcoming lessons so the whole year works holistically together.

We conclude or botanical geometry lesson with this project on pentagons and leaves. We use the book Botanicum and illustration inspiration and copy the leaves of the trees: White Mulberry tree, Oregon Maple, Japanese Maple, and Scarlet Oak.

The Golden Triangle Lesson is our experiential introduction to the Golden Section. Known by many names like the Golden Mean, The Golden Ratio and more recently as Phi, the Golden Ratio is a relationship in which a lesser portion is to the larger portion as the larger portion is to the whole.

For this lesson, we used some additional books and resources to help us complete this design. We used the books: Islamic Geometric Patterns by Eric Broug, Islamic Geometric Patterns by Eric Broug, Islamic Design Workbook by Eric Broug and Islamic Design: A Genius For Geometry by Daud Sutton.

While we continue our work for our Waldorf Main Lesson Block for Geometry, we departed from the lessons to include some projects that are culturally significant for us and historically relevant for our upcoming history blocks.

For this math lesson we are exploring the quality of numbers 1-10 using the book A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science by Michael S. Schneider.

For the tetrad (the number 4), we explore the 3-Dimension with volume and the construction of the 5 regular solids: The Tetrahedron, the hexahedron, the Octahedron, the icosahedron and dodecahedron. Though we've made the solids several times before in our homeschool lessons, this is the first time we are making them from 'scratch' with the construction of the circle. the triangle, square or pentagon.

For these projects I used Fabriano hot press 90lb. watercolor paper measuring 9"x12", but you may use a smaller piece of heavy weight construction paper as I only set my compass to 1.5"-2" for the radius of the circle. While we've made these shapes larger in the past, I actually prefer the smaller shapes that fit snuggly in the palm of your hand.

On this lesson for the Fibonacci sequence, we depart from learning about Fye and Fee (the Golden Ratio) to see how the numbers in the Fibonacci sequence (the ratio of two numbers in the sequence) approach Fye or Fee. This lesson is very simple and rather quick, but it is part of a more extensive series of lessons and isn't intended as a stand alone lesson.

We have been getting a tremendous amount of inspiration from the book A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science by Michael S. Schneider as we work through the quality of numbers 1-10. We have rested on the pentad for several lessons spanning weeks as this lengthy chapter is rich with information and examples.

It may be simple to construct a Golden Rectangle, but the types of tools you use to construct it will play a large role in the accuracy of your final result. After many trials and errors, I found that using a pencil and ruler was not as precise as using a mechanical pencil, calipers and a straightedge. While you may be able to get one Golden Triangle with fair accuracy with a ruler and pencil, your accuracy will increase with a mechanical pencil (or very sharp pencil) and calipers.

To make expanding Golden Rectangles, draw straight lines from the long side of your rectangle using a straightedge. Then mark them with the measurement of your long side to create a square attached to your original Golden Rectangle. Connect the points and erase the extra line and now you can keep going to make even more Golden Rectangles! You make need to add more paper as the expanding Rectangles get big fast.

If making a 3-Dimensional model isn't your preference, you may use the same increments and draw out the spiral on grid paper or with a ruler making a 90 degree turn for each increment. This is the easiest yet still profound demonstration to illustrate how quickly a Golden Spiral grows compared to an Archimedean Spiral.

While we have become proficient in constructing a pentagon, pentagram, Golden Triangle, and Golden Rectangle, we hadn't yet made a Golden Spiral or compared it to the Archimedean Spiral. Making a Golden spiral by way of constructing a Golden Rectangle is the most involved of the three versions we did.

The Golden Spiral is unlike any other spiral. While it is often shown in relation to the nautilus or ammonite shell, those shells are not in fact the Golden Ratio. Those shells exhibit a logarithmic spiral, though you can see some similarities. While reading and putting together lessons from the book A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science by Michael S. Schneider.

For our Waldorf Main Lesson Block in Geometry, we have explored 3-division all the way to 12-division of a circle. For this lesson, we are doing a variation on 12-division of a circle to create double hexagons rather than a floral motif. I used 140lb, Strathmore watercolor paper measuring approximately 22"x 16". I used Stockmar pan watercolors for this project, but have also done it on drawing paper with color pencils and I think that might have turned out even better.

We continue our geometry lessons with 12-division of a circle with inscribed squares. This time we are skipping every 2 all the way around the circle giving us three perfectly formed squares. You can stop there or continue making squares into the center. If you continue, I suggest numbering your points 1-12 on the outside of your circle as well as every level you making moving towards the center.

In this lesson, we explore skipping 3 points on a 12-pointed dodecagon. We've previously made 2 inscribed hexagons, then 3 inscribed squares and now we are skipping three points to make 4 inscribed triangles. I love the way this one turned out and can't wait to make a larger one with more concentric triangles.