Interest-Led Study: Origami and Math

Katie has been enjoying teaching James how to make origami creations. They became inspired by watching Between the Folds (on Netflix instant play).

Between The Folds

The artistic value of origami was pretty instant and obvious to them, but once I became interested in their project, I was amazed and fascinated by all the mathematical concepts that could be taught along the way, in a very pleasurable and engaging way.

To get you started:

symmetry

angles

mirror images

 area

 congruence

 volume

 geometric terms: faces, edges and vertices.

Finding area and volume

Percentages (What % of the exterior surface is (a certain color)?

And once you have explored these topics, you might want to go a bit further...

 Origami and Fractions

The Origami Cube with some math explorations (algebraic formulas)

Origami and Math

Origami Proof of the Pythagorean Theorem

Paper Folding Geometry

And a Little Further (Lesson Plans and Books)

• Origami Solutions: figures that will be useful for teaching geometry concepts, including reflections, rotations, and geometric solids.
• Symmetry Questions in a Wreath and Pinwheel
• The Geometry Junkyard

Books: 

• Origami and Math: Simple to Complex, John Montroll

And, by the way, Katie is half-way through her project of folding 1000 paper cranes.

The Islamic Empires in the Renaissance

(Previous posts: Marco Polo's Travels Through Persia, The Mongols, India ) 

map from Story of the World III Activity Guide

During the Renaissance there were three areas that rose as powerful Islamic Empires. At the beginning of the 1500's, Persia (now called Iran) regained independence under the Safavid dynasty. Persia soon became one of the leading cultures of the world. Safavid Persia was continually under pressure from the west and the Turks in the east, until Abbas I (1571-1629) came into power and created a cultural renaissance in Persia.

The Ottomans were Muslim Turks who built a large empire, with Istanbul as its capital. During the 16th century, the Ottomans expanded their empire, seizing land in the Middle East, North Africa , Russia and Hungary. In the mid-1500's they threatened Europe by attacking Vienna, and also a sea battle near the Greek coast. They also battled the Persians.

The Moguls were Muslims from the area now called Afghanistan. Like their ancestors, the Mongols, the Moguls were great warriors. Beginning in 1526, they began taking control of the land in India. They also built many beautiful buildings, the most famous being the Taj Mahal.

One of the things these Islamic cultures had in common was their love of beautiful designs. They loved tiled walls and floors and often make lovely patterns with the tiles often repeating the patterns in tesselations.

After looking at some of these patterns, we got out our pattern blocks and played around with making designs.  We looked at how the shapes fit together to make it easy to tesselate them. The hexagon, if cut in half forms two trapezoids. If you cut out a equilateral triangle from the end of a trapezoid, it forms a rhombus.

Quentin and I cut out these shapes in colored paper...

and James used these shapes to form a tesselation from a wall from the Book of Kings (Shah-nameh). He found the shapes difficult to align and was a bit frustrated at times,

but I was very pleased with how it turned out.

Can you see the stars that formed in the background around the hexagons?

Playing Pentominoes and Blockus

We played with pentaminos today. A pentomino is a geometric figure formed by joining five (Ancient Greek πέντε / pénte) equal squares connected edge to edge. There are twelve different ways to place the squares to form pentominoes, often named after the letters of the Latin alphabet that they vaguely resemble. The object of the standard pentomino puzzle is to fill a rectangular box with the pentominoes so that it covers it without overlap and without gaps. (A little like tesselations.) Since each of the twelve pentominoes has an area of 5 unit squares, the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20. You can buy the game, play it online, or you can make one.

For my younger students, I printed out this set.

For my older students, I had them figure out the 12 different pentaominos by gluing 5 squares together in different ways and then cutting them out. Tiffany at Child's Play had the unique idea of not telling her students what pentaminos were, having them put the five squares together and she would tell them whether it followed the rules for the 12 pentaminos (connected edge to edge) until they understood what the rule was.

The shapes reminded us of one of our favorite games, Blockus.

The differences are that Blockus' pieces come in many size combinations and that they are laid down corner-to-corner and not edge to edge. (BTW: There is a two-player Blockus game that has a slightly different strategy to it. I actually enjoy it more, but then perhaps it is because I got it first.)
For our first experience with pentominos, we just played around with trying to get all the pentominos in a grid, and it is more difficult than you would think.
Games can be played with the pentominos.
One of the games is played on an 8×8 grid by two or three players. Players take turns in placing pentominoes on the board so that they do not overlap with existing tiles and no tile is used more than once. The objective is to be the last player to place a tile on the board.
Another tie-in for this is the book Chasing Vermeer, which features the puzzle. I have not read the book yet myself, but I have heard positive things about it and it is on my list of books to check out.

Halloween Symmetry

Symmetry: (mathematics) an attribute of a shape or relation; exact reflection of form on opposite sides of a dividing line or plane.

 

Today we did a math lesson in symmetry and made Halloween decorations at the same time.
First  you fold your choice of paper in half. On one half of the paper, draw half your shape and then cut around the lines.

It could be bats or...

pumpkins or...

 cats or spiders or Frankenstein or

ghosts or whatever you fancy.
Now, open the paper up again to see how it looks!

  This activity seems to use the same sort of neurological path that handwriting uses, so if your child is learning or struggling with handwriting, this might be just as difficult for them...but then it is an additional time they can practice. For my youngest, I followed his directions and did the cutting for him.

For my struggling 9-year old, I guided him verbally, but had him do it himself. He struggled, but did not get too frustrated. He kept saying that it was really hard, but fun anyway.

And then we made symmetrical faces.

When you are finished they can double as Halloween decorations.

Halloween Symmetry Lesson from Child's Play

Symmetric Faces from Mathwire

More lessons on Symmetry.

And here.

Or if you would rather make an owl.

We have done symmetry activities with pattern blocks  and used paint to make symmetrical ladybugs.

Fold a round shaped piece of paper in half. Have the child put dots of black paint on one half. Fold over again and the identical spots will appear on the other side.

"If you read a child a math book" (with apologies to Laura Numeroff)

If you read a child a math book...(like Mummy Math)

he will want to make Platonic Solids models...

with toothpicks and gumdrops...

like tetrahedrons...

and octahedrons...

and that might make him a little confused...

but he'll keep working at it and he will get it...

 and then he might want make polyhedra models...

and that will remind him of one time we counted shape faces and he'll want to do it again...

and that will make him want to count the vertices and edges...

and that will lead the oldest child to remember when we did this before and that will lead to

Euler's Polyhedron Formula,
V-E+F=2...

and that will lead to counting in positive and negative numbers...and then he'll want you to read another.

Living math books with these geometric concepts:

• Mummy Math by Cindy Neuschwander (grades 1-4)
• Sir Cumference and the Sword in the Cone by Cindy Neuschwander (Grades 3-5)
• Mathematicians Are People, Too: Stories From the Lives of Great Mathematicians, Luetta Reimer (ages 8 and up)
• Leonhard Euler and the Bernoullis: Mathematicians from Basel, M. B. W. Tent (6th grade and up)

What is a polygon? 

The first time we visited the topic of polygons, we used some paper polygons. The we made a chart, counting the edges and vertices with a marker. The older students could plug in Euler's formula. While talking about it we discovered that the same formula can be applied to 2-dimensional figures with a constant of 1 instead of 2. Makes one think, huh? 

This time we used toothpicks and gumdrops, which we have done before. Toothpicks and gumdrops are good to work with to make regular polygon because the equal sized toothpicks make them automatically have all their sides the same length. This time when we counted the edges and vertices, they were easy to count because the edges are the sides, denoted by how many toothpicks are used and vertices are where two sides meet up, denoted by how many gumdrops are used. This time, because we wanted to look closely at the Platonic solids, they measured the angles with a protractor and (with a little adjusting) correctly determined that three of the five Platonic Solids are made out of equalateral triangles. I thought it was interesting that Tiffini at Child's Play, when she did this lesson, also brought the children's attention to Polyhedral Dice and their shapes, which made counting the sides even easier as that is how the dice are named. We use those dice all  the time for our math games.

What's Your Angle?

Euclid in The School of Athens by Raphael Sanzio

We started off our math day by reviewing some of the things we had done before.

We practiced squaring numbers 1-10 on the whiteboard.

Then I asked them if they could find three numbers in a row where the first number squared plus the second number squared equals the third number squared.

They found that 3, 4 and 5 squared fit that description.

Knowing that in advance, I had cut out squares in 3, 4 and 5 inches. I gave them to them and asked them to form a triangle out of them.

Which, of course, led us to a discussion of Pythagoras, which we had learned about before.

Which led back to a discussion of angles in triangles.

And so we measured some more angles, this time on triangles. I had them keep track of the angles they were measuring.

And after a bit of this, they discovered that all the angles of the triangles always equaled 180 degrees.

Well, actually they didn't get it at first because they didn't always add up to exactly 180; it is easy to get the measurement off a degree or two, but when they were off a bit, we measured again, and then got 180.

And all this led to a discussion of the differences between the different triangles.

And so we made little books with pages with the different triangles.

And, following the example at Daily Life of a Mom, we made these interactive pages.

Right angle triangle,
showing the other 2 angles make a right angle too.

Equilateral triangle,
showing all sides and angles are equal.

Isosceles triangle,
showing two sides are equal.

We could have also done an activity I saw at Free Play Life  in which the kids made the different angles with their hands, arms, fingers and even, getting together, their whole bodies, but they didn't seem to need it. I always try to have more activities planned than I could possibly do, so if one doesn't work, we can glide on into the next one without any problem, or in this case, if we don't need one at the end, we can just not do it, or save it as a warm-up for next time.

If you or your child likes worksheets, here are some that go along with this lesson.

Jimmie's Collage also has a similar lesson on angles.