Paper Weaving and Grid Games
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.
Posted in Make
“Little people, big play, and big mathematical ideas” by Robert Hunting
Robert Hunting’s paper I read this week is a part of MERGA (Mathematics Education Research Group of Australia) symposium on the role of play in mathematics (full text PDF). Robert makes a subversive statement about two types of big ideas:
To simplify, allow me to identify two poles or extreme positions representing this matter – what might be called the soft big idea and the hard big idea. The soft big idea is essentially to accept the status quo of school mathematics curriculum as we have experienced it for the past 100 years or so, and identify major curriculum topics that warrant attention. Examples might be: fractions, place value, long division, ratio and proportion, and so on. We call this meaning soft because of acceptance of the general belief that the selection and sequencing of school mathematics topics is the way we have always done it, based primarily on a logical analysis of elementary mathematics from an adult point of view, in the face of demonstrable overall failure to achieve success in teaching these. The hard big idea is to first ask what conceptual tools professional mathematicians have found to be fundamental and potent in the history of mathematics, and in their own mathematical education. Once established, attempt to develop ways and means to establish preparatory foundations at school level, mindful that children’s mathematics and mathematical thinking is not the same as that of the adult (NAEYC & NCTM, 2002). Examples of hard big ideas include variability and randomness in chance processes, the notion of unit system, scale and similarity, boundary and limit, function, equivalence, infinity, recursion, and so on. The intersection between soft and hard big ideas is by no means the null set.
The terms “hard” and “soft” are analogies to hard and soft sciences.
Each of the eighteen chapters of the Moebius Noodles book has a mini-map of big ideas. For example, this chapter deals with composition of functions:
Each chapter is a stand-alone piece. You can use them in any order, one at a time, because it’s important for each person to choose what math to do. But if you use more than one chapter, you will see more connections among ideas. Chapters “talk” to one another via shared deep ideas! The more chapters you use, the more connections you see.
Here is the map connecting all the big ideas in the book. The intersection with Robert’s list is not the null set.
What if we kept extending this map to more and more materials that make advanced math accessible to babies, toddlers and young kids? Imagine the possibilities!
Posted in Grow
Geometry in the sand
This is a guest post by Joseángel Murcia of TocaMates, translated from Spanish by Ever Salazar.
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.
Posted in Grow
Isobel’s thoughts on the Universe
Ask your children what they think about the life, the universe, and everything. Amazing stories can unfold. Here is what one seven-year-old has to say.
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Here’s how I came up with this theory. I saw the globe and was wanting to go on vacation and then I thought they should have a map of the universe but then I thought it would be impossible. But then this theory came to me. Since the universe doesn’t have an end it must be a sphere and that is why no one has ever reached the end of it.
You know a tube of toothpaste with toothpaste in it? Well, it might be like the universe with all the planets in it and the universe might be shaped like a sphere and part of something bigger. Just like the tube of toothpaste is part of our bathroom which is part of our house which is part of our town which is part of our city which is part of our country which is part of our world which is part of outer space. So, those images get bigger and bigger just like the universe is in something bigger.
Maybe our universe is just a galaxy drifting in another universe that we do not know the shape of, but I think it is shaped like a sphere.
See the solar system almost in the middle? Well, that’s ours. And that sphere is holding all the planets and asteroids and whatever is in space. That sphere is our universe inside another universe and that’s what I think is there.
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We need kids in adult communities – for inspiration! The story above sparked a big conversation (70+ comments; Facebook login required). People discussed hyperspheres, the curved space theory, the shape of the universe, measuring left-over background radiation from the Big Bang… And looked at pictures and stories from sources like NASA.
Posted in Grow
