Welcome to Math Inside and Out

This blog was started in the time of COVID-19 social distancing in British Columbia, Canada. The founders of the blog are mathematics educators and parents who want to offer ideas for shared family/ household mathematical meaning-making activities that can take place inside and outside your home, connecting with the natural world and the body, and encouraging thoughtfulness and understanding of the patterns that give rise to big mathematical ideas. We hope that kids and adults can try these explorations individually and together.

We hope you will enjoy and contribute to this space where we can share ideas, experiences and activities that bring together mathematics and the living world. We would love to see images and read reflections of the things you explored and discovered, and to take them to the next level as we connect them with the knowledge and traditions of mathematics.

Photo credit: Susan Gerofsky

Interview with Megan Zeni: How to incorporate outdoor time in your teaching and learning

A great podcast interview with the amazing Megan Zeni about how parents, teachers and kids can start to teach and learn outdoors, in math, language arts, the arts, social sciences -- any aspect of curriculum. Megan has taught in outdoor elementary school classrooms for 22 years and is now doing her PhD at UBC. She has thoughtful, practical and highly engaging advice and ideas. 

Math is for the Plants 2

Author: Qiaochu (Joy) Xu

Let's see what happened in the last two weeks since we water plant the scraps!

Which plant burgeons first in your small garden?











What shape does lettuce grow into?

What is the earliest evidence of plant growing?

Which plant grow faster than you expected?

Can you graph the growing patterns? 

What do you notice about the growing patterns?

Math is for the Plants 1

Author: Qiaochu (Joy) Xu

It’s always exciting to check out how much the plants grown over a night/week, and I believe it would be so interesting to track and graph their growth! 

Don’t worry if you don’t have a backyard or garden to do this activity, you can either growing little seedings or water plant from kitchen scraps (such as green unions, leeks, carrots and more…) on your windowsill.

What plants do you want to grow...?

Avocado, hot pepper, lemon grass, mushroom, onion, basil, cabbage, potato, pumpkin, carrot greens, celery, cilantro, garlic sprout, ginger... or...you name it?

Here are some vegetables that you could easily grow from kitchen scraps:

Check out here:
19 foods you can grow from scraps

I would recommend lettuces and green onions, as they are easy to get and grow very fast. 

You could follow the steps in the picture, and track their growth within a week, to see:

1) Where do you notice when the growth occurring? 

(2) Do they volume out from inner leaves or grow into a pyramid shape?

(3)  How would you track the growth? 

(4) By taking pictures, making a chart with length they grow and the date, or by drawing it out?

You could start your garden journal, and let's talk about that after a week!!

Test your senses

From Ask a biologist

Vision

The eye is a fascinating and complicated organ that we can learn more about through some simple experiments. Check out the links below to have some fun exploring how your eyes work and why we may not actually see things as clearly as we might like to think!

Peripheral vision

When we see something “out of the corner of our eye”, we mean we are seeing something in our peripheral vision. Peripheral Vision | Science Snacks offers four short videos where you can learn more about peripheral vision and do an experiment to test just how well you can see things that are not directly in front of you.

The Bind Spot

You might have heard about the blind spot before. This is the part of our eye where the optic nerve leaves the eye; this means that there are no photoreceptors there to respond to light. For a quick way to “see” this, check out the image below from Neuroscience for Kids 

“Close your right eye. With your left eye, look at the +. You should see the red dot in your peripheral vision. Keep looking at the + with your left eye. The red dot will move from the left to the right and disappear and reappear as the dot moves into and out of your blind spot.”

There are lots more fun activities related to vision (an our other senses) at the 

Neuroscience for Kids page.

Averted Vision
This is when we purposely use our peripheral vision so that we can see better. 

Well, when it’s dark, looking directly at something doesn’t really work because the photoreceptors in our central vision are of the type that detect color so, if it’s dark, we may not see anything. In our peripheral vision, we use the photoreceptors that pick up black and white which is far better for seeing things in low light conditions. Pilots, hunters, and astronomers all use their peripheral vision to see things at night.

Follow this link for a full explanation: How to See – Averted Vision and Dark Adaptation

Puzzles

Calling all puzzle lovers!

Can we learn while having fun at the same time? YES! In fact, scientists say that having fun while learning is a great idea. And puzzles can be a lot of fun... We'll share some that we love and look forward to seeing the ones you love in your comments!

Many puzzles can be done on paper, not just Sudoku. You can do these puzzles online (from brainbashers.com or others) or printed out (from krazydad.com or others). I prefer to print them out so I can take them with me everywhere and puzzle over them... for hours if need be!

One of my favorites is... solitaire Battleships. Here's one that I've started solving:

Battleship rules:

• Your goal: find all the ships!
• The numbers tell you how many ship pieces there are in that row or column  
• The ships are placed horizontally or vertically, with water between them so that they don't touch (not even diagonally)
• Circle ships are submarines; bullet or diamond shapes are pieces of bigger ships. 

On this example, the ships are shown on
the right side of the puzzle, so you can
cross them off as you locate them.











According to this site, the Battleship puzzle was invented by an Argentinian named Jaime Poniachik in the 1980's. Imagine inventing a puzzle!

Inventing your own puzzle?

You could start by creating your version of a puzzle that already exists... for example, creating your own battleship puzzle. Maybe start from the end: draw a grid with all the ships, water and numbers and then erase more and more until you only have a few clues left... Will your puzzle be easy or hard? Will it have only one possible solution? Are you sure it CAN be solved?

In your own puzzle, you may want to change the rules... What if there are more types of ship? What if they can touch diagonally? What if there's water, ships, and...icebergs?

Maybe your puzzle will have castles instead of ships, or lava instead of water.
The sky's the limit! (Or is it?)

MATH GUY (Bad Guy Parody)

This Winnipeg math teacher is getting a lot of views with his great cover / parody of Billie Eilish's "Bad Guy" - worth a look.

Click here to see the video

More details below below:
Manitoba teachers parody pop songs, create Survivor challenges to engage housebound students
https://www.cbc.ca/news/canada/manitoba/parody-video-survivor-winnipeg-covid-19-1.5546788

MATH IS FOR THE BIRDS! Part 2

Crow photo on pixabay.com

 

A Murmuration of Starlings

******************************

A photo that I took of crows in January

As I sat on my back porch, I noticed a group of about fifteen crows flying around the top of a tall tree and making a lot of noise. I know that they do this when a racoon is up in that tree. But today, I heard the sound of a bald eagle. I wondered if the crows were working together to chase away the eagle. Sometimes a large group, or flock, of crows is called a murder of crows.


This made me think of another grouping of birds, a murmur of starlings. Starlings fly together in large numbers for safety; they are less likely to be prey for other larger birds. 

This video is taken near Vancouver by Pacificnorthwestkate. 

The flight velocity and direction of one bird affects 7 around it which multiplies throughout the group.

Like the movement of schools of fish, murmuration is an example  of scale free behaviour correlations. 

No matter how large the flock of birds is, the action of each bird will affect and will be affected by the action of all the other birds.

This means that each bird's range of perception is much larger than if they relied on direct interactions.

Murmuration has been compared mathematically  to magnets.

A three dimensional computer model shows an attempt to understand the mathematics in this flight patterning.

Dunlin birds can also form murmurations.  This video was created near Vancouver, BC for DVWildlife.

publicdomainpictures.net

Another reason that birds form patterns in flight is for migration. For example, Canada geese fly together in a V shape when travelling north in the spring and south in the fall.

JOURNAL ACTIVITY:

Observe groupings of birds. Record what you see with drawings, photos, or videos.
Why do these birds form groups?
Do all birds form flocks?
Do all birds form mating pairs?

Continue to identify and add more kinds of birds to your journal.

Do people form groups that are similar to flocks or murmurations? 
How has social distancing changed how people interact in groups?

MATH IS FOR THE BIRDS! Part 1

Black Capped Chickadee

An Invitation to 
Urban Birdwatching

********************************

Sitting on the back steps of my home in East Vancouver earlier this week, I was visited by four different kinds of birds. I identified them using a guide to birds in Vancouver.

European Starling

Bushtit

House Finches

The house finches appear in a pair. The male has salmon coloured feathers. The female is mainly brown and grey to help her go unseen when she sits on a nest. The two of them like to sit and feed on the top of my kale plants that are going to seed. 

I listen to the birds sing and call and when I see them. I want to try to identify them by their sounds even when I can’t see them.

ACTIVITY: START A BIRD JOURNAL

Start with three or four birds that you can see from your home or on walks close to your home. Using drawings and writing, record what you notice. 

Which birds do you see?

What do they look like?

Are the male and female different?

Identify them by name? 

What do you notice them eating?

What calls and songs do you hear?
Do they move about alone, in pairs, in groups?


Start a routine of recording when you see or hear these birds. How will you record over time?

Identify and describe more birds as you notice them.

Outside Listening: Mapping Soundscapes

In May the hummingbirds arrive on Haida Gwaii. One afternoon my good friend and I sat out on her deck and were entertained for hours by the hummingbirds. We counted 17 hummingbirds sipping the nectar from two feeders stationed on the deck corners. Listen … can you hear their calls? Can you hear the beating/buzzing sounds of their wings? 

 image: Amanda Kariella on Unsplash

Soundscapes, like landscapes, tell us a story about the place. Soundscape ecologists like Bernie Krause and Hildegard Westerkamp have recorded sounds in the natural environment for almost 50 years. Over time they have mapped how the sounds of the environment have changed. In some cases the hum or buzz of urban life on land, shipping traffic on sea, and air-traffic have transformed our soundscapes.

You can participate in being a soundscape ecologist by listening to and mapping the sounds outside in your backyard, or at a nearby park, or wherever you are.

Here’s one activity you can try by yourself or with others. You will need a piece of paper, pencil, and clipboard.

Find a spot outside to sit or stand. Draw a big circle that fills your page and place a dot in the centre. The dot represents you. Now along the edge of the circle mark nearby objects in front of you, to the right and left and to the back of you as a way to record your position. In this example the tree is front the building behind, mountains to the left and a deciduous tree to the right. 

Now close your eyes and listen. Listen. Listen. Mark on your map what you hear. How far from you is the sound and from what direction? Maybe you hear the wind rustling in the trees, or a bee nearby, or the sound of bicycle, or the call of an eagle. Mark both the distance from you and the direction of your sounds.

Try making soundscape maps at the same place over different days. What do you notice? Try making soundscape maps at the same place over different times of the day. What do you notice? What do you wonder?

Unexpected shapes

The Möbius^∞ strip

Source

   Consider a strip of plain old paper like, say, a long shopping receipt. Mathematically speaking, it’s kind of boring; it’s got the simple measurements of length and width from which we can calculate the area and a right angle at each corner but not much else. However, when we crumple it up into a ball, that same piece of paper becomes something different. All the new shapes and angles that are created make it far more interesting. The study of such twisted and scrunched up objects is called topology:

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects 1

wonderopolis.org/

   A simple and fun example of this is the Möbius strip. If you take a long strip of paper and make one half-twist (180°) and join the ends together (glue is best for this, but tape works too), then you’ve made a Möbius strip! So what? Well, first of all, let me tell you that the surface of the strip has only one side. To prove this, put a pen anywhere on the strip and draw a line down the centre; you’ll need to rotate the strip as you go. You’ll notice that you eventually make your way back to where you started without ever taking your pen off the paper; this is because the surface has only one side – try doing this with a regular piece of paper and you’ll find it can’t be done.

   Next, take a pair of scissors and cut down this central line all the way around the strip. It seems like you’re going to cut the strip in half and end up with two strips, right? But what happens?!

  

   You still end up with one strip! But, there’s a catch: it’s not a Möbius strip anymore. How can you tell? Well, take a pen of a different color and draw down the central line again. You’ll find your way back to where you started, but this time you’ll only have marked one side – this new strip has two sides!

But, wait, there’s more…

whatdowedoallday.com

    Make another Möbius strip but this time cut it along the point one-third of the width of the strip; see the example on the left. Keep cutting all the way around. What’s going to happen this time?

You now have a short Möbius strip intertwined with a longer twisted loop!

If that was a little hard to follow, check out this explainer video:

Cutting a Möbius strip in half  | AnimatedTopology |

Source

For more mindblowing unexpected shapes watch professor Tadashi Tokieda make a square out of two circles and some interesting conjoined love hearts.

Unexpected Shapes (Part 1) - Numberphile

For older learners, check out Dr. Tokieda's lectures on Topology & Geometry

This nifty Möbius Music Box is worth a look and this Möbius strip story telling demonstration is awesome.

 BONUS: Check out the Möbius Wall

∞  Named after German mathematician, and elderly Bilbo Baggins look-alike,  August Ferdinand Möbius

Fractals 1 of 5: Beauty, fun, and more

Jitze Couperus

A fractal is any shape that repeats itself over and over and over again at different sizes and scales.You can zoom in and find the same shapes forever.

Besides being beautiful and fun, fractals are very useful! For example, they can be used to predict or analyze parts of the human body, such as the inside of our lungs or the pattern of nerve dendrites. Another important use of fractals is in image compressing, so that some of our photos don't take up too much space in our computers! It’s mostly used if an image has patterns that are repeated (fractals!) .

Drawing, weaving, beading, building…

Indigenous people have used fractals since long before modern-day mathematicians:

Dreamcatchers can have fractal designs, like

https://shop.slcc.ca/learn/the-dreamcatcher/

Look at the fractal nature of parts of our universe in artist Margaret Nazon's beading work:
http://robertthirsk.ca/2018/10/19/the-spirit-of-innovation/
https://www.nwtarts.com/artist-profile/margaret-nazon

Many African cultures have long used fractals in their architecture, art, and other aspects of their culture, as explained in Ron Eglash’s book “African Fractals - Modern Computing and Indigenous Design” https://monoskop.org/images/f/fc/Eglash_Ron_African_Fractals_Modern_Computing_and_Indigenous_Design.pdf

Fractals 2 of 5: Drawing Fractals

We can draw fractals!

https://www.youtube.com/watch?v=PU1dwwB_Im0

Try drawing these... then, see if you can come up with your own fractal 'seeds' and draw them out to make your own original fractals... and share them with us in the comments!

www.study.com

   

https://en.wikipedia.org/wiki/Pythagoras_tree_(fractal)

Can you find some fractals outside? Or maybe even inside your fridge?

More references:

https://www.wikihow.com/Draw-the-Koch-Snowflake
https://www.es.com/Shows/FantasticFractals/FantasticFractals_EducatorsGuide.pdf

Fractals 3 of 5: Fractals in plants

I went outside fractal-hunting today. I was looking at plants, and here are a few of my findings:

A red cedar branch. It's Y's all the way down!

A camelia flower, its petals going round and round.

Each branch of this shrub comes from another (thicker) branch and so on until the trunk. Below ground, its root system is like a branch system that we could draw as a fractal.

Look at this really cool photograph (done by an artist) of an autumn leaf CLOSE UP:

PAUL OOMEN / FINEARTAMERICA.COM

Are there fractals in the plants around where you live, too? 

I wonder why plants grow with repeating patterns like this...does Nature just find fractals pretty or is there more to it? What do you think?

Fractals 4 of 5: What is the shape of a cloud?

Cloud photo credits: Susan Gerofsky

 Here's a lovely thing to do together, especially on a day when the sky is blue and those fluffy cumulus clouds are moving around in the wind. Try looking up (or better yet, lying on your back in the grass) and watching the clouds.

You might try photographing or drawing them, as I've done here. But how to draw a cloud? What shape is a cloud?

We have learned the names of shapes with straight-line edges like triangles, rectangles and octagons, and curved shapes with continuous line edges like circles, ellipses and parabolas. But none of these really describes the ragged-edged (and ever-changing) shapes of clouds.

You might notice that some clouds look a lot like the maps of islands, continents and shorelines. (The bottom photo here reminds me of a map of Europe and the Mediterranean...) Shorelines are also ragged-edged and changing, more slowly, because of forces of erosion and tectonic shift. So if we can figure out what shape clouds are, we might also figure out what shape islands, continents and shorelines are!

Try looking at a small bit of a cloud you have photographed. Is there a small section of the cloud that looks almost like a miniature version of the whole cloud? If so, then you may have discovered a fractal 'seed' that can be copied bigger and bigger (and/or smaller and smaller...) to create the shape of the whole cloud! That is what is meant by a fractal: a shape that repeats itself in more or less the same way on different scales ( really tiny, small, medium, large, really large...) to create a ragged-edged and evolving shape without straight-edged boundaries.

Here is a good explanation by a meteorologist at the University of Bonn, Germany about the ways that clouds are at least partly fractals (and not always completely so, as the big central parts of clouds are not as affected by wind turbulence as the edges).

 And here is a very nice film about natural fractals, featuring the mathematician who 'discovered' fractal geometry, the late Benoît Mandelbrot!

Try drawing the shape of bumpy or ragged edge of a cloud. It might be easiest to draw it fairly large on your page. Then, on each of the 'bumps' try drawing smaller bumps that have the same shape only smaller. Do that again, three or four times. Are you getting a shape that looks a bit like the edge of a cloud -- or a shoreline?