Math Art and Craft
Patrick Honner’s Moebius Noodles guest post on mathematical paper weaving was very inspiring to me. Mathematical weaving employs one of my favorite making materials – colored paper! It was actually sort of challenging to get started, but after playing around I landed on some solutions which became a nice little unit of paper weaving and grid games with and for young children.
I am imagining that the weaving and the games can be completed in an enjoyable collaboration between adult and child over the course of a day or two. Here are some ideas for setting up the experience and playing the games.
After experimenting a little, a 3/4″ width for vertical and horizontal strips makes a more pleasing final product to my eyes than 1″. To make the vertical strips fold a piece of paper in half and use a paper cutter to cut 3/4″ strips from folded edge to about 3/4″ away from the open edges. Essentially, you are creating a paper warp that is still essentially one piece of paper.
As you can see, below, the horizontal strips weave in very nicely and don’t need any glue or tape to keep them in place if you focus on pushing them gently, but snugly, downward. For the young ones, at least, a basic over/under/over/under weave is challenging enough. Using two horizontal colors creates visual interest and perhaps even a conversation about the patterns you see: alternating colors both vertically, horizontally and diagonally. You can also make a connection to odd and even numbers. Yellow squares in the design show up 2nd, 4th, 6th… places. Green squares are 1st, 3rd, 5th…
The minute I finished the piece above I thought – A GRID! It’s a grid! Over the last couple years I have received mountains of inspiration from the Moebius Noodles blog especially as source of grid games (my favorite so far is Mr. Potato Head is Good at Math). As a result, grids are always in the back of my head. Here are some of the ideas I came up with using a newly woven paper mat/grid and one of my favorite math manipulatives — pennies!
Adult: Oh look! There are three different colors of squares in our woven grid. I’ve got some pennies — I wonder if we could make a square by putting pennies down on only one of the colors?
Adult: That does look like a square. Let’s count and see if there are the same number of little squares (yellow, blue, yellow, blue…) that make up each side? There are! How many little squares are there on each side?
Adult: But, wait! Look what happens when I push a corner penny in toward the center! Yep, it lands on a green square! Let’s do it with the rest of the corners and see what we get. Oh, lovely. A rhombus.
Adult: The corners on the rhombus are on the yellow squares. I wonder what would happen if we pushed them one square toward the middle? Ooooh, look! We have another square. Is it bigger or smaller than our first square? Each side on our first square was six little squares long. This square has sides that are…three little squares long. Cool.
Another exploration, this time growing patterns and a tale of some square numbers who also wanted to get bigger? What little kid doesn’t want to grow up?
And, here’s my favorite. It’s a ‘let’s make a rule’ kind of game. The first penny goes in the bottom left hand corner, and you start counting from there. The first rule here (pennies) was two over, one up. Each time you repeat the rule, you start counting from the last token on the grid.
You’re probably wondering about the buttons? Well, that’s a different rule: one over, one up. Isn’t it cool how they overlap, but not always? Kids can make up their own rules after a little modeling or you can challenge them to guess a rule you made up and keep it going.
And then, of course, the final thing would be to leave the pennies and the paper grid mat out to explore at leisure. Have fun making math!
p.s. After this first foray into mathematical paper weaving, I explored it a little more. Here are more posts on my blog: Weaving Inverse Operations, Multiples and Frieze Patterns - Weaving Fibonacci - Weaving Geometric African Motifs Part 1 and Part 2.
Sand has always been a good place to do geometry. In fact, ancient Greeks used it instead of the “modern” blackboards to show their ideas and schemes. It is also said that Archimedes died while drawing in the sand from the beach, disobeying a Roman order to stop.
At the beach, we can do lots of designs, but today we will focus on two ideas.
Ages two to five
What can we make? Sand polygons
How can we make it? Using a stick, draw a “path”. The kids should follow it. The paths can be curved lines or straight lines (forming polygons), they can be left open (with exit) or closed (to follow indefinitely, around and around).
Why make it? It is about experimenting with polygons, lines and angles. It is about feeling geometry in our bodies.
Five and older
What can we make? Gardener’s curves
How can we make it? We will need two large sticks in the sand, like poles from the beach umbrellas. Use those when sun is down, and they are not necessary anymore. We will also need a rope and another stick to draw the curve. Tie the rope to the two sticks so it has some slack. Pull the rope taut with the third stick, to form a triangle. Now draw with that stick, keeping the rope taut at all times. Changing the distance between the fixed sticks (the focus), we will get different ellipses. What happens if we put both sticks together and have only one focus?
Why make it? Conics (like ellipses) are known from ancient times. Those from the sausages are my favorites!
This post is from a series of everyday life activities to help kids (and grown-ups!) to discover how to look at the world with math eyes. They were published as “Mathematics is for the Summer” in “Today’s Women” magazine.
Today’s guest contributor to Moebius Noodles is Patrick Honner, an award-winning high school teacher and a passionate math enthusiast. My favorite part of Patrick’s beautiful blog is the Math Appreciation section, because you can adapt most ideas there for young kids. When I was four or five, I spent a lot of hours weaving with paper. I believe it helped me fall in love with mathematics.
Before starting on Patrick’s game, check out his TED talk on creativity and mathematics! - MariaD
Weaving is a fun and creative way to explore real mathematical ideas. Simple “mat” weaving offers a way to experience basic concepts in geometry and number theory, while encouraging the development of representation and modeling techniques– fundamental mathematical skills.
With some colored construction paper cut into long, thin strips, and some glue or tape, you can get weaving right away! Here are a few introductory activities. More examples and ideas can be found at my website: http://www.MrHonner.com/weaving/
A good place to start is the checkerboard pattern. It is simple, intuitive, and helpful in developing facility with the basic techniques of weaving. Start with two sets of strips of different color; align all of one color horizontally and all of the other color vertically. Now, a simple alternating over-under weave will create the checkerboard.
A more challenging activity with just two colors, each aligned horizontal and vertically, is to weave a tiling of the plane.
This activity definitely requires some planning. Once a type of “tile” is chosen, the weaver must figure out what kind of weaving pattern will produce the desired tilling of the plane. The orange-and-black weave above uses a “short L”-shaped tile and an alternating, 1-over / 2-under pattern. The orange-and-purple weave uses a “long L”-shaped tile and a similar 1-over / 3-under pattern.
Here’s where modeling and representation come into the process. With a blank grid, one can plan out the weave ahead of time, hopefully figuring out what kind of weaving pattern will produce the desired mat. A standard modeling approach can be used, or the weaver can develop their own representation—in both cases, the important mathematical skill of modeling is being developed. Here are some examples of different approaches to modeling various weaves.
Through trial (and error!), the weaver can refine their modeling process and their plans to produce the desired weave.
Once the basic techniques of simple two-color weaving are mastered, more interesting and challenging projects can be undertaken. Using more than one color in an alignment (horizontal or vertical) opens up new patterns, as does using more than two colors. More challenging patterns, tilings, and inversions can be attempted. Here are some examples.
A fun mathematical follow-up to introductory weaving is to consider the question “Which kinds of patterns are weavable?” For example, the following two mats weren’t really woven—some pieces were cut out and taped over other pieces. An interesting and highly mathematical question is, “Would it be possible to produce these mats through weaving alone?”
With some basic supplies and a few simple techniques, significant mathematical ideas can be explored through weaving. And once you’ve mastered the basics, you can start investigating circular weavings, hat and basket weaving, and even try your hands at mathematical knitting!
For more ideas, visit www.MrHonner.com
There’s a lot of talk about how playing with building blocks helps children develop math skills. But what about children that are too young to even “tote and carry” blocks? Have you thought about introducing them to Platonic solids? Ok, here’s a little refresher about Platonic solids:
A Platonic solid is a 3D shape where each face is the same regular polygon and the same number of polygons meet at each corner.
If the idea of introducing this concept to a small child sounds a bit over the top, here’s a surprise – your infant might already be enjoying one. After all, a cube is a Platonic solid. But why leave out the other four – tetrahedron, octahedron, dodecahedron, and icosahedron? That’s exactly what British mathematician Richard Elwes and his wife Haruka have done. Here’s Richard’s story:
When some friends told us they were having a baby, Haruka set to work making a soft cubic toy to give the child, by sewing together square patches of colourful cotton cloth left over from other projects, and stuffing it with cushion-filler. Being a mathematician, Richard immediately suggested the set should be expanded to include all five Platonic solids. (One challenge was to make sure that no two adjoining faces were made of the same cloth.)
These toys are intended for very young children, so it cannot be expected that they will ‘learn geometry’ in the usual sense. Instead, what we hope is that they will begin to foster a geometrical aesthetic, enjoying the symmetries of the toys, and developing a familiarity with these five solids, which will remain throughout their lives.
As the children grow older, we hope they will keep revisiting the Platonic solids in other forms, perhaps as wooden or plastic toys, maybe as dice or puzzles, later making them themselves out of paper or card. But there is no need to stop with the Platonic solids! As soon as practical, why not introduce shapes like prisms, antisprisms, and Archimedean solids (along with their duals: bipyramids, trapezohedra, and Catalan solids)?
For a small child meeting the Platonic solids for the first time, there is is one potential problem: apart from the cube, the names of these shapes fail to reflect their elegant simplicity. For a toddler, the word “icosahedron” is surely a bridge too far. So why not reduce them to their initial syllables: tet, cube, ock, dode, & ike? This will allow the child to have fun identifying and comparing the shapes, without getting bogged down in unnecessary Greek verbiage.
As we are getting ready for the Moebius Noodles display, we continue to be on high alert for great ideas that introduce grids to children. So I was really excited to see an art through math activity for young children on one of my favorite blogs, The Educators’ Spin On It.
The idea is to use grids to help make a copy of a picture. Inspired by a local chalk art festival, Amanda of the Educators’ blog decided to create chalk art with her children. The results are beautiful and Amanda documents the entire process with wonderful photographs (which she so generously allowed me to use in this post).
Amanda notes that even toddlers can participate in this activity. And the idea lends itself easily to customization based on your child’s interests. Amanda chose a picture of the beautiful St. Basil’s Cathedral in Moscow to reproduce. Your child might be more interested in something else (I’m pretty sure that mine is going to ask for either WALL-E or a Star Wars clone trooper).
You can also choose a different art medium – paints, crayons, markers, even thumb prints (hey, that would be a fun idea to try). Or, if your child has a favorite picture that’s very large (say, poster-size), you can try making a smaller version of it.
Thank you, Amanda!
If you haven’t yet, do read Amanda’s entire post, get inspired and try it this weekend! When you do this activity with your children, take pictures. You can upload them to Facebook and share them on our page. Or you can post them to your blog and link to the post on our Facebook page or in the comments.
Math games can be played any time anywhere. Here are some ideas for each day of the week. These games require very little, if any, advance prep. Give them and feel free to change them to make math more interesting for your children.
December 5 – Rhythm is Math
Does your child love drumming? Have a drum circle and come up with simple drumming sequences for him to repeat. No drums? No problem. Pots and kitchen utensils will do nicely or, for a quieter version, cardboard boxes and paint stirrers.
December 6 – Mathematical Poetry
You can find some mathematical rhymes, but why not write your own math-y poems? Does it sound intimidating? Then start with a cinquain. It has a prescribed form, but does not require you to count syllables which can be confusing to younger children. But cinquain’s structure allows even very young children be involved in the writing process, not to mention illustrating the completed poem.
December 7 – Winter Weather Day
Sure, you can play a game of matching mittens and socks. Or you can explore geometry with some mini-marshmallows and toothpicks.
December 8 – Evergreens are Everywhere
By now there’s a Christmas tree bazaar on every corner. Why not use this opportunity to practice some measuring? What can a tree be measured with? Can it be measured with a paper clip? How about a mitten and arm length? Find the smallest tree on the lot and measure its height, say, with a mitten. Now find a tree a bit taller and see if your child can estimate how tall this tree is in mittens?
If there is a Christmas tree farm nearby that you can visit, you can play a game of gradients, finding taller and taller (or shorter and shorter) trees and taking pictures of your child next to them. Then print the pictures and ask your child to arrange the trees from shortest to tallest.
December 9 – Pinecone Fibonacci
Go on a walk and collect some pine cones of different sizes. Let your child explore the pine cones. How are the pine cones alike? Show the whirls on the bottom of the pine cones. Your child might be interested in painting the whirls different colors or making pine cone prints with them.
December 10 – Start a Collection
Does your child have a collection? What does she collect? What other people collect? Can you have a collection with one item? Two items? Play a scavenger hunt in the house looking for items that can be grouped together into collections. Photograph or otherwise record your finds.
December 11 – Dicewalk
The idea is simple – walk around the neighborhood and every time you get to an intersection, throw a dice to decide which way to go. For detailed instructions, including how to make the dice, check out The Artful Parent Dicewalking blog post. If your kids are too young to walk a lot or you don’t live in a walkable neighborhood, you can play this game in the yard or even indoors. How about making a very simple map of your neighborhood (or your living room) and mapping the route while you’re at it?
Math games can be played any time anywhere. Here are some ideas for each day of the week. These games do not require any advance prep either. Give them a try this week and feel free to change them to make more interesting for your kids.
November 14 – Claude Monet’s Birthday
Monet would often paint the same subject at different times of the day as the light changed. Let’s create a color gradient collage today. All you need is a bunch of paint chips from your home improvement store. Suggest arranging different shades of the same color from lightest to darkest. Now try it with other colors. In case you don’t have time to run to a home improvement store, you can modify this game. Replace paint chips with liquid food coloring and give your child a dropper and several clear containers filled with water (glasses, clear jars or white ice-cube trays all work great).
November 15 – Children’s Book Day
There are quite a few wonderful children’s story books that go beyond basic counting and shapes. We are going to be reading Spaghetti and Meatballs for All by Marilyn Burns and Anno’s Magic Seeds by Mitsumasa Anno. If you or your child prefer to make up your own stories that include math, nothing beats another great book by Mitsumasa Anno, called Anno’s Counting Book.
November 16 – Talking Turkey Day
For this game you’ll need a marker, a piece of paper and a bag of bird seed. If you don’t have bird seed, a mix of 2 or more different pasta shapes or dried beans will do. First, trace your or your child’s hand on a piece of paper – that’s your turkey. Now, decide on a pattern, but don’t tell your child what it is. Let him guess which seed (or pasta shape) the turkey would like to eat next. Start with something simple, such as ABAB pattern. Then move to more complicated ones. Then let your child decide on a pattern and you’ll try to guess it.
November 17 – Bread Baking Day
Ah, kitchen is a perfect place for math! Let your children do all the measuring. Then let them experiment with estimating (i.e. how many tea spoons make a table spoon). The result is going to be some delicious math. And if you don’t have time to bake bread from scratch, there’s absolutely nothing wrong with picking up a muffins or cupcakes mix at the store.
November 18 – Mickey Mouse’s Birthday
Let’s watch a Disney cartoon today. How about this one – Donald Duck in Mathmagic Land (all three parts are available on YouTube). You can even try some of the math activities Donald tries during his adventure, starting with playing tic-tac-toe.
If you would rather stick with the Mickey Mouse’s theme, then how about revisiting November 14th idea of gradients, only using Disney Paint Chips.
Let’s start getting ready for the Pie Day!
November 20 – Pie Day
Nope, not the “pi day” which happens on March 14th (you know, 3.14). Instead, today is all about baking and enjoying pies! So why not do some more kitchen math. You can also cut a few pie shapes out of construction paper, let your child decorate them, then ask to share it with her toys (hello, fractions!).