My favorite subject of all of the math blocks is geometry taught the Waldorf way. It might even be my favorite main lesson block of all! Though Botany and Astronomy are right up there in the top three. What makes the Waldorf approach so engaging in this subject area is all the art that accompanies the lesson. If that last sentence sent a way of anxiety through your being, take care, I’ll ease your fears about Waldorf geometry with plenty of tutorials and lots of encouragement.

### Geometry Resources for Elementary, High School and Beyond

Geometry is one of the most fascinating fields of math a student will encounter, in my opinion. There’s so much you can do when children are young with simple math centered on identification of shapes and solids. While introducing geometry this way and with picture books may not reflect the Waldorf approach authentically, it has been our approach in our homeschool for years. I love adding living books into our lessons because I’m often at a loss for stories of my own. Plus, I am not a teacher by profession and rarely solely rely on a curriculum, so I appreciate the breadth and depth of the books and resources available.

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### How to Put Together a Geometry Main Lesson Block

Over the years, I’ve put together several math main lesson blocks and a number of geometry blocks. What sets these blocks apart from other math blocks is the level of engagement the teacher and student experience with each lesson. There are no dry lessons. The upside is obvious: full engagement means maximum learning and high possibility of retention. The downside is that it takes a lot time, a fair amount of planning and the occasional (sometimes frequent) disappointment when lessons don’t go as planned or you (both teacher and student) struggle with achieving the forms you are striving to make.

When putting together a unit, I look for a number of resources to enrich our experience. While math blocks are different than other blocks in that I’m not likely to have a set of recipes like a geography unit or a historical fiction for a history unit, I will plan for us to cook and alter recipes so that fractions can come to life real applications and I will include a number of math books, especially picture books, when possible. One thing I won’t go without with putting together a math unit are games! I find math games to be some of the best ways to truly understand math concepts and master basic math skills. New for this unit are Islamic Geometric Design books that were gifted to our family from Siraj Bookstore.

### 12 Division Of A Circle | Chalk Drawing

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Geometry chalk drawings are my favorite and most satisfying because they are simple (though not easy) to make and look so beautiful when complete. This 12-division of a circle is a geometric shape we’ve done before in our geometry blocks before using Distress Inks, and it looks spectacular on a chalkboard. You will need a Chalkboard Compass to do this drawing as I can’t think of an alternative. But the investment is well worth it. For this and all my chalk drawings, I use Sargent Art Chalk Pastels. I like these chalks because they are highly pigmented and I like the vibrant look of the final results. I also used the workbook Drawing Circle Images as inspiration for this project as well as the Grade 6 Live Education Waldorf curriculum. You can find more Geometry tutorials on this blog post as well as video tutorials for our Math Main Lesson Block.

### Sacred Geometry

Waldorf geometry is one of the most beautiful main lesson blocks. I love the Waldorf approach to teaching math and especially geometry. In the following series, I go from 3-fold to 12-fold symmetry of a circle in short tutorials. I use the book Drawing Circle Images as inspiration for this series. I use Fabriano hot press 90lb. watercolor paper and Distress Inks to paint in our final images. For these projects, you’ll also need a compass, ruler and pencil. I love the way our projects turned out and even tried a few chalk drawings of circle symmetry on our front porch and driveway.

#### LIVE Homeschool Lesson | Waldorf Geometry | Threefold Symmetry

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. In so doing, we discovered a new design that looks just as beautiful. That’s the thing about these designs, once you see the pattern, you can experiment with arcs of differents radii to discover new final designs. Then of course, you can still choose one of many ways to color in your design for a magnificent end result.

#### How To Draw 3-Fold Symmetry | Geometry Tutorial

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 threefold symmetry. While you can explore this design and come up with creative ways to paint or color in the shapes, it’s not my favorite. I like the symmetry starting at 6-fold symmetry. But, I’d love to see what you come up with. Maybe your creative way of painting this design will change my mind. Honestly, the possibilities are endless and that’s exciting.

#### How To Draw 4-Fold Symmetry | Waldorf Geometry Tutorial

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. I use Distress Inks as my watercolor medium. I like the way these inks dry and the range of colors offered.

#### How To Draw 5-Fold Symmetry | Waldorf Geometry Tutorial

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. This one took a while for us to achieve and there was a fair amount of starting over and frustration which I have edited from the video. I highly recommend that you buy Drawing Circle Images so you can have a step by step set of instructions because it does take a bit of practice for some of these. You can find links to the materials I used in the blog post that accompanies this video. I use Distress Inks as my watercolor medium. I like the way these inks dry and the range of colors offered.

#### How To Draw 6-Fold Symmetry | Waldorf Geometry Tutorial

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. This is the easiest of the symmetries because the arcs to divide the circle into 6 parts is the radius. So all you have to do it position your compass on the circumference and mark 6 arcs. How you do your secondary circles and which parts you decided to fill in is where the uniqueness of each design comes through.

#### How To Draw 7-Fold Symmetry | Waldorf Geometry Tutorial

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.

#### How To Draw 8-Fold Symmetry | Waldorf Geometry Tutorial

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. What I like about this one is the intricacies of of the final result.

#### How to Draw 9-Fold Symmetry | Waldorf Geometry | Tutorial

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. I adjusted and readjusted my compass multiple times but the divisions were not equal. Finally, after several attempts I got it. I’m glad I persevered because this turns out to be a stunning design in the end. Of all the designs so far, 7- and 9-fold symmetry are my favorite!

#### How To Draw 10-Fold Symmetry | Waldorf Geometry Tutorial

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. This design creates a unique circle interior which supports many choices in how to color in the design. Tenfold symmetry is a fun one to play around with.

#### How To Draw 11-Fold Symmetry | Waldorf Geometry Tutorial

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.

#### How to Draw 12-Fold Symmetry | Waldorf Geometry Tutorial

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.

## More Geometry Videos!

#### 12 DIVISION OF A CIRCLE | WALDORF GEOMETRY

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. Once the math is complete, it’s time to color or paint in the sections, allowing magnificent designs to emerge.

I’m using the Live Education Waldorf Curriculum for grade 6 for lesson inspiration and Josie Lewis Art for color inspiration for this project. Art supplies are from A Child’s Dream Stockmar watercolor paints, Paint Jars, Paint brushes are a mix of cheap synthetic fiber brushes and high quality all natural fiber brushes. I used 90lb. 14″x11″ watercolor paper by Fabriano and Resist medium.

#### The Beauty of Math | Waldorf, Geometry & Pi Day | March 14

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.

The first project today is similar to one we did recently in which I walk you through constructing the 12 divisions of a circle. For this project I used Distress Inks by Tim Holtz in a range of colors from yellow to red to green. I used a resist medium to keep the watercolors from bleeding into other sections. I finished off the project with Mr. Huey’s Shimmer Spray in Shine. For the second project, I used my Silhouette Cameo to make an original design. I then cut out the pieces on kite paper and on adhesive backed cardstock to make templates. I placed the whole design on contact paper with I also cut on my Cameo. I did an alternative and arguably easier design by just layering twelve circles in rainbow color. Both look nice, but were really time consuming. If you didn’t have a Silhouette Cameo, you could use a cup to trace a circle. Cut a total of 13 and arrange 12 of them symmetrically around the first circle. Then remove the original circle and you’re left with a similar design. It looks great on kite paper or tissue paper so the light can shine through.

### Pi Day Math Books, Activities and Crafts

What’s Pi Day? It’s the 14th day of March 3.14! It’s just a fun day to celebrate this amazing number. Do you even know how to find pi? It’s actually pretty easy to find, and we got pretty close to pi. How close? Check out the video to find out! I also share 3 projects, 1 game and 4 books…get it? 3.14 😉

Don’t forget to check out the other channels participating in the Pi Day festivities: Pi Day Playlist

If you like the game, here’s a tutorial I made on how to play it as well as others like it. 24 game is a fun, challenging and educational math game. I also share my Geometry projects on my channel. These are creative, artistic math projects that are both educational and enjoyable to do. We got a lot of our ideas from our Waldorf curriculum as well as the book String, Straightedge and Shadow geometry book.

How To Do A MATH Unit Study | GEOMETRY

Geometry is one of my favorite main lesson blocks and I have a few videos on how I put together a Math and Geometry unit. Enjoy this first look at how I put together a unit from the YouTube archives! Along with the math picture books and Waldorf curriculum you can also see a number of math games we love.

#### Division of a Circle: Geometry

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. You can skip watercoloring this image and simple do the six division of the circle using a compass. You can color in the image with color pencils or leave it blank.

#### Chalk Drawing: Geometric Shapes

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. These drawings are for the first couple lessons in geometry and serve as an introduction to shapes. We depart shortly from the lesson plan to spend more time and the terminology of quadrilaterals and explain the various types (rectangle, square, parallelogram, etc.) and some attributes of each before moving on to the next lessons.

#### Triangles | Geometry & Art

Today we are making triangles from intersecting lines. We’ll only color in the triangles formed and leave the other shapes blank. We are using a Waldorf curriculum from Live-Education. Watercolors and watercolor paper can be found at Blick Art Materials. Embossing pen, powder and heat tool can be found at local craft stores. 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. This project was for my 8th grader, but my 4th grader joined in and did a great job too. Alternatively, you could do this project with penciled lines filled in with color pencil.

#### Geometry: Symmetry with Triangles

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. After cutting and coloring multiple sets of triangles, we assembled mirror images on a paper sprayed with adhesive. The whole project can be laminated to keep the pieces in place. This an artistic representation of symmetry and geometry.

#### Geometry: Organic Shapes from Triangles

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. We are using a Waldorf curriculum from Live Education. Materials for this project: Colored paper, paper, scissors, spray adhesive or glue stick. Materials can be found at local crafts stores.

#### Geometry: Cascading Triangles

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. Materials for this project include paper, ruler, pencil and color pencils. Duration: 10-20 minutes Level of difficulty: easy

#### Interior Angles of a Triangle

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.

#### Geometry: Degrees of a Circle

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. We will color in each section with a different color, then mark the angle. he first project uses watercolors, embossing powder and an embossing pen. This method involves special tools and materials that can be found at local craft stores or from Blick Art Materials. Materials for this project: watercolor paper (or drawing paper), watercolors (we used Distress Inks), embossing pen, embossing powder, and heat tool. You can also do this project with watercolor pencils or color pencils. Tools needed: protractor, compass and a ruler.

#### Pythagorean Theorem

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. First we will construct a square with only a compass and a straight edge, just as Pythagoras would have done (only he might have used string instead of a compass). Then we will construct a smaller square within that square. Further divisions of the square will reveal that squares of the smaller sides of the triangle will equal the square of the hypotenuse. This is an amazing demonstration that you will be fascinated by. This visual demonstration clearly shows the Pythagorean theory.

#### How to Make a Perfect Square: A Geometry Demonstration

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 🙂

#### Geometry: How to Make a Hexagon and an Equilateral Triangle

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. I’ll also show you how to make a single equilateral triangle.

#### Geometry: How to Make the Five Regular Solids from Paper

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. If you don’t have a Silhouette Cameo to cut out these shapes, don’t worry! Just go to my website and download the template and make them on your own. All you need is a printer, scissors and tape!

#### Tangrams

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.

#### How To Make A Mandala

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 Materials. We found inspiration for this project from the image on the cover of our Live-Education Geometry curriculum. We sprayed our paper with spray adhesive and then assembled the mandala with triangles and diamonds. The pieces adhered well, but we sent the finished pieces through the laminator to ensure that none of the pieces would fall off. These make nice decorations for fall and can be easily hung in a window.

#### How To Make The Five Regular Solids: Plato Glo Mobile Kit

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 use toothpicks and wax (or Play-Doh or clay). I show you how easy it is to use the materials you probably already have around the house.

## Waldorf Geometry Lessons : Nature, Number and Geometry

#### Pie Chart

*September 2024*

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. This is the data we used for the pie chart and bar graph: Conifers make up 51.1% of the trees of California with a division within this category as follows: Other conifers 17.2%, Ponderosa 14.1%, Jeffrey Pine 6.3%, Douglas Fir 5.6%, White Fir 4.5%, Red Fir 3.4% Hardwoods make up 26.4% of trees with a division within this category as follows: Other hardwoods 9.6%, Coastal live oak 6.3%, Tanoak 4.5%, California black oak 3.9%, Interior live oak 2.6% Mixed conifers-hardwoods 14.5% and other 8.9% We constructed a bar graph with this data as well. For this lesson you will need a compass, protractor, ruler, calculator, Main lesson book and pencil. The math is proportion math with 360 degrees/X = 100%/ (insert % of tree)

#### The Perfect Hexagon

*September 2024*

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. We used our thick Lyra colored pencils and General’s Chalk pastel pencils to color in the triangles in blue and green. The colors were too close on the color wheel, and it would have been better to use contrasting colors like blue and yellow. For this lesson you will need a compass, ruler, Main lesson book, colored pencils and pencil.

#### Geometry Lesson 6-Fold Symmetry with Bulbs and Lilies

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. We then placed our compass on the arc and swung it again. In this way we get six equal divisions. Once complete, we used our ruler to connect the opposite points to make a 6 pointed star. On each of those lines we constructed our onion flower which is a bulb plant whose flowers exhibit six fold symmetry. For this lesson you will need a compass, ruler and colored pencils.

### The Hexagon | Geometry Lesson

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. Then place the point of the compass on one of the arcs and swing it again to intersect the circle. Repeat until you’ve divided the circle by six. For this lesson, we work in an angular way as opposed to our previous lesson in which the image was more free form and fluid while still retaining six-fold symmetry when doing the onion flower (or other bulb flower).

In this lesson, you will connect the points to make a hexagon which you may then gently shade with pencil or chalk to give a dimensional look. Then you may practice free-handing an amethyst crystal. This lesson is fairly short, though the free-hand crystal took several tries to get right. For this lesson you will need a compass, ruler and colored pencils.

### Geometry Lesson | Hexagon and Rose Quartz

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. This lesson was only challenging in constructing an oval. You can easily construct an ellipse by placing a tied string around two points (with pencils for instance), and using a third pencil to trace the ellipse by anchoring the pencils in place and swinging the string and pencil around the double center.

For this lesson we free-handed the oval and labeled the six points and traced parallel lines. We also free-handed a point vertically from the oval then connected the six points to the single vertical point. Seeing the image go from flat hexagon to 3-dimensional hexagon was thrilling.

Next we practiced again with a smaller hexagon and oval, and finally we free-handed a quartz crystals with the techniques we just learned with immediate success. For this lesson you will need a compass, ruler and colored pencils.

### Six Division of the Circle | Snowflake Geometry

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. Once you construct your circle and divide it by six (see previous lessons), you may add some concentric circles at equal increments to make constructing your snowflake easier.

Looking at inspiration pictures of snowflakes (or photographs of actual snowflakes) will be helpful. As snowflakes are each unique, you won’t find two exactly the same, but you will see similar elements like structures on the 6 radiating arms, or hexagon shaped snowflakes and you may even find triangle and 12-pointed snowflakes, though for this exercise, I recommend sticking to a traditional 6-sided snowflake.

Once your circles are constructed, it’s time to have creative fun in constructing your unique snowflake. Use pencil, make sure each arm is the same and be mindful of expected shapes, as everything is quite angular and not circular. You may even lightly erase your initial circles.

### Structure of a Snowflake

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. We used our onion skin sheets of paper that separate the pages of our Main lesson book. I used a double sheet I folded in half, so when I cut out my circle, I had four circles. We marked off six-division of a circle by placing our compass point on the circumference and swinging an arc on either side. We then placed our compass point on the arc and swung it again repeating until we completed the circle.

Now we can fold our circle by connecting opposite points to fold in half. Next we fold in thirds. And finally we fold in half. You can cut your design on the open flap (not folded side) and try cutting angular designs rather than curved shapes. Then comes the truly exciting part, unfolding your snowflake! We had a great time doing this part of the lesson and started keeping our onion skins to make more.

### 6-Division of a Circle | Calla Lily

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. Measure your radius to 9 cm or 3.5 inches. Divide your circle into 6 equal parts by setting your compass to the circumference and swinging the compass to make an arc on either side crossing the circle. Then place your compass point onto the arc and swing again. In this way you can divide the circle into 6 parts. Label them 1-6 and then connect every other point.

The sides of your triangle should measure 16 cm. Make 2cm divisions along each side of the triangle. Now connect each point to the center of the circle. Already this is a beautiful geometric design. Next draw your calla lily within the triangle. We chose to color our calla lily using Lyra colored pencils, but this is optional. We did a little research online to find colors and patterns we liked and each chose one to copy.

### 16-Division of a Circle and Water Lily

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. Place your compass on the top point and swing arc to right and left. Where the arch cross, you may draw a line through your center to make four divisions of the circle. Adjust your compass to the radius of the circle and pace compass on two adjacent points and swing arcs between and once again draw a line from the intersecting arcs and the center. Repeat for all angles in the hemisphere until you’ve completed 16 divisions.

Label the lines 0-8 on either side from the top. Now you may begin drawing your water lily. You could stop at the pencil drawing, but adding some color with Lyra Colored Pencils brought this lesson to life.

### Designing Historical Geometric Tile Patterns

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. We also used large heavy weight watercolor paper by Strathmore, but Fabriano watercolor paper would have been sufficient for this project.

We ended up not completing the whole page with six-division of a circle as it got too tedious, and I noticed that some of my daughter’s circles had become smaller, making the design misaligned. Once we completed our six-divisions and filled most of the page, we began our design. There are several ways to create designs and we included 2-3 within one project. Once we erased extra lines and arcs, we used our watercolor paints to finish off the project.

### Six Division of a Circle | Islamic Geometric Patterns

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 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. We also used large heavy weight watercolor paper by Strathmore, but Fabriano watercolor paper would have been sufficient for this project.

We ended up not completing the whole page with six-division of a circle as it got too tedious, and I noticed that some of my daughter’s circles had become smaller, making the design misaligned. Once we completed our six-divisions and filled most of the page, we began our design. There are several ways to create designs and we included 2-3 within one project. Once we erased extra lines and arcs, we used our watercolor paints to finish off the project.

### How to Construct a Pentagon

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. One note to be mindful of is that precision and accuracy are necessary for this lesson. Double check your compass once you’ve found 1/5 of the circle by lightly drawing arcs around the circle to be sure that they are evenly spaced. If not, re-adjust and try again. Even being a hair off will have significant outcomes.

### 5 Division of a Circle

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. This is a simple step by step tutorial on how to construct a pentagon and five division of a circle.

### 5-Fold Symmetry | Botanical | Buttercup

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. For this lesson we used our Lyra Colored Pencils, a compass, ruler and pencil. The duration of this lesson was less than 30 minutes, especially if you’ve already practiced 5-fold symmetry.

### 5-Division of a Circle | Botanical | Pentagons and Leaves

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. We drew one circle on the page before this lesson, measuring the radius to 2 inches. When then drew four circles even spaces on the next page. Then we worked through our five division of a circle. Once our compass was set to 1/5 of the circle, we returned to our four circles and drew arc around the circle with the compass. We then connected each point to make our pentagon. We used Lyra Colored Pencils to draw our leaves and label each drawing.

## Opening Activities for Geometry Block

### Math Picture Books

I love adding picture books to our main lesson blocks, and I’m so pleased with the math picture books that are available now. For a complete list of math picture books that we’ve used over the years, check out this blog post.

### Math Games

We love our math games! In fact, I made a whole blog post just on the math games we love and play regularly, as well as some we only moderately love and play occasionally. Check out math games blog post here.

### Mental Math

Mental math is the process of doing math problems in your head. Generally it’s math you already know, but are aiming for proficiency and mastery. Mental math calls on capabilities of memory, math ability and the ability to work out math problems in your head without a paper and pencil or calculator.

Mental math the Waldorf inspired way (and with my own additions), involves two part questions (more so as the children grow, less so when they are first starting out), in which two different math operations are completed in one problem. One other significant difference you’ll find with mental math the Waldorf way is that sometimes the answer is offered and the solutions are discovered. It looks something like this: “What makes 12?” You ask a student. “3 x 4,” a student offers. “4 x 3,” another student says. “6 x 2” another chimes in. “24 divided by 2,” yet another student says. You can quickly see how many answers there can be. While not many questions are posed this way, they are scattered throughout the worksheets.

Thank you for the detailed Waldorf Math ideas and lessons shown on Youtube. Very appreciated!! 🙂