What does your children’s summer math time look like? Perhaps it is filled with worksheets and arithmetic drills similar to what they get to do the rest of the year. If that’s the case, it is time to get outside, and try mathematics that is playful, adventurous, creative, and lively! Try these ideas from our growing collection.

Follow Natural Math’s board Outdoors & body-scale math on Pinterest.

Sign up for twice-a-month newsletter. It has Natural Math games for kids, questions and answers, child-friendly math art and photography, open course information, and stories of math adventures. Send us your questions, comments, and stories. Join the Natural Math community!

This week was all about Flatland. Download the Math Spark that invites you to travel to Flatland with us.

Before new activities, we sat and reflected on what math we had noticed last week. Check out what we call “mathematical faces” – when children think about mathematics, their faces look special.

“Sometimes, these thoughts can make me gigglish!”

Maria is taking the tools (papers, sticky notes…) out for the hands-on creative activities on the table. Do your children have trouble transitioning into activities? Make it a ritual to set the table for mathematics with colorful materials, and whatever you need to feel good – snacks, drinks, and toys. Rituals like this give children time and a framework for transitions.

Maria: “Look at this 2D hairband. It is curved but also can be straightened out.” We can manipulate some 2D surfaces to be either flat or curved.

Priyesh cut out a 2D shape (bound by curves) for his football project.

Maya is in the process of making and playing a 2D orchestra – two drums so far, a circle and a square.

Maria is listening to the story of Owen’s project: his (2D) coffee filter represents a (3D) ice block with cuts representing cracks. We use black or bright construction paper for dramatic backgrounds to models. It helps with storytelling.

Maria is about to show how to make a shape similar to Owen’s with just one cut of a (folded) coffee filter.

“What do we have here?”

Another example of how a 2D flat paper can turn into a model of a 3D spiral snake.

Charles is making a random 2D decoration for his wall, with each piece uniquely, irregularly shaped. Another math or art term for random shapes is “abstract.”

For his next project, Charles triangulated a turtle-bobcat, then added dots for more regularity. This is similar to a technique from Australian Aborigines.

Lori constructed a curved shape (a circle) arranging LEGO blocks that are straight squares. Then she commented how the shape becomes smoother and the triangular wedges between the squares become smaller when you use more blocks. You get a more precise approximation of your circle with a polygon that has many, many sides.

Can you cut a hole in a piece of paper that is large enough for you to fit through? Amy shared a way… Besides, you can use the resulting zigzag to form curves, like a big goofy smile or a circle.

Now that the 2D ring is cut out and formed, let’s see if we can go inside the ring! First comes Serrin.

Then, Eashan…

…Finally, Maria sneaks in for a photo-op.

Eashan is busy making a mosaic of a waterfall (2D, curved) out of sticky notes (also 2D, but straight).

Maria is showing a 2D face Serrin made with 1D details out of pipe cleaners (very curved).

“Can *your* eyebrows curve like this?!”

Our eyebrows can form many different kinds of shapes – but pipe cleaners can do even more. Serrin shaped and attached a 1D shiny pipe cleaner on Maria’s right eyebrow for comparison. We can be playful even as grown-ups, but there is also a serious purpose here. People of all ages are very good at recognizing features of human faces and bodies. So any time you can use your own body to make a math point, do so. And physically laying one shape over another is the most immediate, hands-on, embodied way to compare curves, angles, sizes, and other math or art elements.

Meanwhile, Amy created an even bigger paper loop out of the same standard piece of printer paper. Amy’s science teacher said if you made cuts fine enough, you could wrap your loop around the Earth. Can you imagine doing really, really, really fine cuts to wrap loops around the house or the Earth? Half the kids said they could and half said they could not, pointing out all the problems (slippery roof, the paper will tear, etc.) Kids who point out practical problems have good engineering minds. You can invite them into the human-made world of abstract math only if you specify it’s unreal – it’s all about fantasy and imagination!

Our whole group was able to fit into the paper circle!

Discussing paper craft techniques.

Everyone gathered around the computer to watch string spin toy (on a flat, 2D screen we might add – but it fakes 3D views). http://www.zefrank.com/string_spinv2/

More math faces: Charles is totally engaged, observing and smiling.

Hanna is trying the String Spin Toy with Maria.

Owen enjoys predicting what will happen next… You ask children to predict things – but don’t judge predictions! Let them experiment and see for themselves.

Boys are happily anticipating how their conjecture would come out.

More portraits of math focus and math excitement.

It’s a spaceship! It’s a castle! It’s a cannon! It’s a viking boat cross-section! Wait, what?

Check out parents thinking and anticipating together with their children.

A sketch of the swing we discussed last time. What shape does this triangle make when you rotate it? What shape do you need to rotate to make a cylinder?

The strings are 1D, the shapes we see seem 3D, but we are really watching everything on a 2D screen. Pet the picture on the screen to feel it for yourself!

Each hair is a 1D line, but together all the hairs are integrated into a 3D waterfall (or a cone-like shape). Maya values engineering precision: “But really, each hair is a 3D cylinder.”

We did a bit more of scavenger hunting for shapes out of shapes in many dimensions. The ceiling is made out of planks; can you imagine them infintely thin, like 1D instead of 2D?

And this colorful creation approximates a curve with (straight) bottoms of triangles, for a colorful sun. The semi-random colors and angles of the triangles make the sun look artistic.

Photos by Erin Song, captions by Erin Song and Maria Droujkova, Math Spark by Kalid Azad, Shelley Nash, and Maria Droujkova.

This is a photo gallery for a math circle called Calculus for Kids, in Cary, NC. Math circles are informal groups where children, their families, and teachers explore math together. This session was our first introduction to making shapes out of shapes. Here are two pieces of supporting materials for the topic:

One-page Math Spark for building models.

One-page sheet with relevant terms

Emma is drawing on (2D) paper and also creating a 3D model out of pipe cleaners, inspired by a 3-dimensional dolphin.

A fictional narrative: “A jellyfish taking a ride on a dolphin.” Check out the evolution of the pipe cleaner model from the photo above.

Maddie is using (1D) pipe cleaners to create a wide, 3D ice cream cone.

Mark is inspired by Wall-E and fractals. He brought a 3D printed fractal to share.

Inside the fractals, there are squares, lines, empty spaces…

Sydney is adapting a story of Frozen recognizing the symmetry of sun.

The song “Let It Go” from “Frozen” popularized fractals:

One of the main characters Elsa is skating; the whole composition is also symmetrical. 3D Elsa is similar to the 2D version.

A 2D balloon with circular patterns made from playdough.

Eli is miming one of the things he likes. Maria’s not getting it.

Eli finally his wish (without breaking it) by writing. Children need to have multiple modes of expression so they can choose, and we must respect their choices.

Maggie and mom Rebecca are gathering 1D pipe cleaners to build a 3D spiral bird’s nest.

“Eggs are growing in the nest…”

And the (potentially) endless spiral reminds us of infinity.

Yash is creating a laser torch project – finding squares, circles, and cylinders in LEGO blocks.

“I built a birthday cake, a sky scrapper with two elevators, and some stairs!”

The stairs represented the counting sequence going up and also down (5, 4, 3, 2, 1). Later, another example of a sequence – 2, 4, 8… – was built as well.

Julie made a 2D spiral glasses using 1D pipe cleaners.

Allison made a ring consisting of a small sphere and a large circle.

How can we make numbers out of ourselves? Each of us is 1 (The One?) – then parent+kid makes 2, or we can use fingers, arms, legs, point out the eyes…

Again, things develop and grow (a bunny ear out of Mark’s head). And number 3 is formed by 3 boys standing in a row.

Could we try making a spiral with all of our hands together?

Let’s make a human square on the body pillows… hmm, 1, 2, 3… seems like we are missing something… we need one more person to make 4 sides!

We need to hold our arms and legs as straight as possible; a square has 4 sides, 4 angles attached to each other, like this.

A cylinder please! Imagine 2 circles and a folded rectangle between, holding us on all sides, as we extend our arms circularly and spread our feet.

Now, this 3D shape tapers nicely from a flat base to a point, the apex there our feet stand together; a cone upside down!

Julie and Allison are making a human pentagon; raising one side of their arms (2 sides) attached by the tips of their fingers, bodies (2 sides) and the ground as the last side.

Yash and Mark are making a 2D roof top by attaching their hands and bodies facing each other. The surface is supplied by the imagination.

Maria: “Grown ups, try this! Feels sooo good.”

Yeah! Jumping on two over-sized cushiony square pillowy mats: so relaxing and comfy!

What have we here on the ceiling? Lots of patterns, as children suddenly point out!

We discovered these hidden math words from our own imaginary creations (hands-on-craft) during the one hour of our math circle. For now, most words are shapes. Let’s see what words appear by the fifth meeting!

Photos by Erin Song, captions by Erin Song and Maria Droujkova, Math Spark by Kalid Azad, Shelley Nash, and Maria Droujkova.

This is a photo gallery for a math circle called Calculus for Kids, in Cary, NC. Math circles are informal groups where children, their families, and teachers explore math together. This session was our first introduction to making shapes out of shapes. Here are two pieces of supporting materials for the topic:

One-page Math Spark for building models.

One-page sheet with relevant terms

Calculus for Math Circle begins!

Ice breaking time: Who are you? If you’re happy and you know it raise your hand… We take turns naming one thing we love, and whoever else loves it too – waves.

Bonk on the head to show whose turn it is to introduce themselves.

If Maria bonks everybody on the head, who bonks Maria on the head? All together now…

Julianne loves to draw.

Maya wants to make a 2-dimensional fan!

I want to make a pine cone in 3D shape!

An hour later, many different models of the cone:

Serrin making tree-trunk as a 3D cylinder using orange 1-dimensional felt.

Cylinders are popular: here they serve as legs for a future doll:

Let’s make a cylinder out of ourselves. Or is it a circle? It’s both, kids say: depending on how you look at it.

What kind of shape is this? Triangle? Half-circle?

Maria pointing shapes she found in the backyard

How do we make a square? Team building time. A square has 4 sides and 4 angles … “Let’s see, how many people do we have here? 1, 2, … 8. Okay, so a square has 4 sides if we divide by four that’s two people for each side.”

The group square is finally made! Smile!

We found another shape! The table is made out of 2-dimensional glass circle! But it’s also a 3D cylinder, children say.

Here are the list of math words we collected from our creative craft makings on the board. Let’s see how the list growth from meeting to meeting.

A snail and a 3-dimensional diamond shape Lego block.

Making 3-dimensional pine cone with 1-dimensional pipe cleaners can’t possibly be this fun, right?

Snack time with mommy – and her 3-dimensional LEGO approximation of a… sphere!

A colorful circular sphere out of rice-based packaging beans.

Made a line out of pieces using beans…

Turn a model into a scene (here, grass under a tree) for added interest – and added analysis. The three are discussing how connectors are used in different projects:

A spaceship among the stars. Another whole scene! Check out the lateral thrusters. “This spaceship does not need a nose cone, because it’s for deep space.”

Photos by Erin Song, captions by Erin Song and Maria Droujkova, Math Spark by Kalid Azad, Shelley Nash, and Maria Droujkova.

**Subscribe and read archives**

**Pinterest | Twitter | YouTube | Facebook | Google+ **

In this newsletter:

- Join us online: Calculus for Kids course for adults+children starts Monday April 13
- Spots in our Cary, NC circle starting Sunday April 12th in Cary, NC
- Join the Natural Math crew: film a reaction video and be a star

Join us for an intensive 4-week course for parents or teachers with their children ages ~5 and up. We designed the course with Shelley Nash, the organizer of Monarch Academy, a homeschool group that meets online and in physical space. The materials are created by Kalid Azad from Better Explained, an online project helping to grow math intuitions, and Maria Droujkova and Yelena McManaman of Natural Math.

This is the first time we offer such a course, so we will work very closely with every participant to learn what helps you best.

In this 6-minute talk from a festival, Maria Droujkova shares some of the big WHYs behind this project.

**What do you get from Calculus for Kids?**

- 4 weeks of mentoring from Maria! (Invaluable if you really want to “do math”)

- Each week you’ll have one live online class for the grown-ups, and one live online sharing session for grown-ups with kids.
- Sparks to get you thinking about math before the class.
- Resources after class to help you try out the Calculus activity with your children.
- Access to the Natural Math online forum to ask questions, even long after the class is over.
**Most importantly, you’ll get the confidence in your own ability to do math differently in your family!**

Update: to get on the waiting list, email maria@naturalmath.com

We’ve never ran circles on Sundays before, but some people requested that day. Still, Sundays are less popular than weekdays, so we have a few spots left. Below is the information from the previous newsletter. Skip if you have already read it. But if you want to join us Sundays, starting April 12th, **click here to sign up**!

Here is a tiny sample of parent and teacher materials, called a Math Spark. We’ll have Sparks like that to go with every activity. The goal is to start thinking about ideas – to spark curiosity.

We will be running two math circles for children ages ~5-10 and their grown-ups in Cary, NC (conveniently close to Raleigh and Durham). The groups will meet on Sundays at 1 pm, starting April 12th and ending in May. These will be small mixed-age groups where children and adults come to have good times exploring mathematics together. Our Math Circles are for curious, inquiring, and playful families. Here is what participants get:

*Children*: make calculus their own by making models, doing art, sharing ideas, and seeing the world from the new (calculus!) point of view.*Parents*: experience math as beautiful, useful, and fun; discuss ideas with peers; get a playful and fresh view on a key area of mathematics.*Natural Math crew*: play-test our upcoming book*Calculus for Kids*and prepare for an online course too (coming very soon – stay tuned).

We usually take shapes at face value, as a single entity. Calculus gives us superpowers to dig deeper! You don’t just see the tree. You know it’s made of rings, and that the rings are growing. Just imagine having X-Ray vision to use at will. That’s what our calculus is about!

Calculus for Kids is not just for children – it’s for parents and teachers too.

We will only have 8 children in each circle, so if you want to participate, sign up soon. If the space runs out by the time you try to register, email maria@naturalmath.com to get on the waiting list.

Update: the circle is full.

We are growing. Join our adventures!

Video is one of the most requested things on Natural Math, but we haven’t done much of it. What videos do people want? *Children doing Natural Math activities, and grown-ups showing what children’s actions mean.*

One big change activities bring is that children “grow their math eyes” and start seeing mathematics everywhere. We want to capture that in a short reaction video. If you and your children want to be stars in our video, do this:

- Together with children, do a short activity about fractals and film it.
- Together with children, watch the video Cows&Cows&Cows (original or Minecraft version) and film reactions.
- Send us both videos.

We will give you more tips if you volunteer.

Here is a “kids react” video (not ours) so you have a sample of what we mean.

**Email maria@naturalmath.com to join our video reaction project.**

*See you online!*

*Dr. Maria Droujkova and the Natural Math crew*