Task

Part 1

1. Investigate what shapes can be used to tile a plane. As you work on this portion, post your tentative conjectures and emerging insights to the group so that you benefit from the insights and guidance of your mathematician mentor and the rest of your classmates.
2. Test your conjectures by creating a tiling. You can accomplish this in many ways. Make sure that you continue to bring your work and discussions to the rest of the members in your virtual classroom.
   • Simple constructed colored drawings work well.
   • Or you may choose to use colored construction paper, in which case you would need to carefully construct the shapes, cut them out,and glue them onto another surface.
   • Or you may choose to use a computer software program such as Geometer’s Sketchpad by Key Curriculum Press, GeoNext (The GNU General Public License), Cinderella or PlaneTiling Mathematica Package, by Xah Lee.
3. What have you learned about what shapes tile a plane?

Part 2

For the second part of this investigation you have a few choices. Choose to do 1 or all of the following tasks:

• create Penrose tilings
• create tilings of a sphere (completely)
• create tilings of a representation of a torus. For the torus, I would recommend using a rectangular piece of very flexible cardboard and gluing only with a touch of glue at the center of the tiles. Then, verify that the tiling wraps by bending the cardboard into a cylinder both ways.

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Theoretical or mathematical models are very common. Geometric ideas such as points, lines, planes, faces, edges, vertices, polygons, and diagonals can be used to represent physical objects. In the following investigation you will discover the rules that describe geometric situations.

Task

Part 1

Regular polyhedra have intrigued mathematicians for thousands of years. They were important to ancient Greek scholars who placed great emphasis on the study of science. Admired and exalted from earliest times, the five Platonic solids—tetrahedron, octahedron, icosahedron, cube, dodecahedron—are the most perfectly symmetric of all solids. How perfect? In nature, the cube, tetrahedron, and octahedron appear in crystals. The dodecahedron and icosahedron appear in certain viruses and radiolaria.

For the first part of this investigation you will be asked to make three-dimensional geometrical models by building ‘wire frame’ platonic solids using sticks and connectors.

A simple version of this is as follows: use clay or plasticine for the connectors and use toothpicks for the sticks. Then, build the five platonic solids.

I would like to make a suggestion here:

1. I would recommend that you use materials other than marshmallows.
   • Marshmallows are not robust. Perhaps use clay that dries, plasticine or use cut pieces of hoses with holes bored in them.
   • Use kebab sticks and cut them to the desired length.
2. As you build the Platonic solids keep track of your observations. While you are building these beautifully regular shapes you might want to consider:
   • what do all these five shapes share in common
   • why would they be called regular
   • what observations can you make about the angles of the adjoining faces on each solid
   • what observations have you made about the vertices, edges and faces of each of these solids

Part 2

If just one of the requirements for a solid to be regular is removed, a large collection of other highly symmetrical forms can be discovered. Using the rule below, make other shapes.

Explicitly construct the dual of each of the Platonic solids by adding ‘pyramids’ onto the faces

Part 3

1. If we continue to require that all vertices be identical and that the solid be convex, but we remove the requirement that only one kind of regular polygon be used, the family of solids that results is called the Archimedean Solids (also called the semiregular solids), of which there are 13. Seven of the Archimedean solids are derived from the Platonic solids by the process of “truncation”, literally cutting off the corners. For example, we can start with a dodecahedron, and trim off its corners to change each face from a pentagon to a decagon, leaving an additional small triangular face at each corner.
2. Construct as many Archimedean solids as you can.
   • As you build the Archimedian solids keep track of your observations. While you are building these solids you might want to consider:
     • what do all these solids share in common
     • how do they differ from the Platonic solids
     • what do they share in common with the Platonic solids
     • what observations can you make about the angles of the adjoining faces on each solid
     • what observations have you made about the vertices, edges and faces of each of these solids

What have you learned about the Archimedian solids?

Note: when I was a child (age 11), I did something like this purely for fun. I build out of sticks and cut up hose with holes in it a complete third frequency geodesic sphere. It was fun. My recollection of it was that the materials that I used were not great. Thus, I would recommend testing the materials prior to giving this project out. -Charles (creator of this investigation)

The post Geometric Models Investigation appeared first on Galileo Educational Network.

Part 1

1. Draw a circle.
2. Mark a number of equally spaced points on the circle (2,3,4,5,6,7,8, up to 17. Maybe try them all or just try quite a few of them).
3. With a ruler, connect the dots so that one could draw out the whole pattern without lifting your pencil (i.e. unicursally).

You are studying the relationship between the size of the jump, the number of dots, and the shape that they produce.

Investigate the following questions:

1. For a given number of points, how many ‘distinct’ ways are there to make a unicursal dot connecting pattern?
2. What counts as a ‘distinct’ pattern in the above? How else could you define ‘distinct’? What does this change in the definition do to your answer to 1 above?
3. For a given number of points, how many ‘symmetrical’ ways are there to make a unicursal dot connecting pattern?
4. Same as 2 but redefining ‘symmetrical’
5. Repeat 1-4 with the ‘unicursal’ condition lifted.

Gas, Water and Electricity

Geometry and the Imagination provides a nice extension to begin to answer the question, “So what?” “What are unicursal curves good for?”

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This investigation is best for Grades 5 and up

MATERIALS:

styrofoam cups, beans, a cardboard box. Set the styrofoam cups inside the box.

EXAMPLE:

OBJECTIVE:

• To involve students in a small group exploration of the concept of area for circles
• To use ratios of areas and probabilistic (Monte Carlo) simulation methods to determine some experimental estimates for the value of Pi.
• To extend students understanding of Pi.
• To explore the mysteries of Pi.
• To involve students in data collection.

LINKS TO THE CURRICULUM:

Shape and Space (Measurement, Geometry), Number Concepts (estimation, ratio and proportion), Number Operations (computation), Probability and Statistics (simulation)

Task

1. Find out what each person in your group thinks will happen if 100 beans are tossed randomly into the box you are provided.

• How many will go into the cups? Record everyone’s guesses.
• Why do you think that many should go in the cups?

1. Perform an experiment: Have everyone toss the beans from several steps away (to keep the distribution in the box more or less random).

• Count the number of beans that land in the cups.
• How many land in the box but not in the cups?
• How many land in the box, including those in the cups?
• Record your results.

1. What is the ratio of the total area of the cup openings (at the top) to the area of the box opening?

• What should this have to do with the proportion of beans that land in the cups?
• In what sort of units did you choose to measure areas?
• If the diameter of a cup at the top is two ‘andars’, what is the area of the cup opening? The box?
• Is there any advantage to using ‘andars’ here as units?

1. Express the relationship you found between the two ratios in (3) above as an equation.

• Your equation should involve , since the area of each circle is Pi times its radius squared. This equation expresses the fact that the ratio of beans in cups to the total beans in the box is an estimate for the ratios of the corresponding areas.

1. Solve the equation you set up in (4) above for.

• How does it compare with values you have seen for ?
• Is it too big an estimate, or too small?
• What might account for the error in the estimate?
• This type of estimation through experimental simulation using randomly generated data (our bean tosses) is called the Monte Carlo method. Where is “Monte Carlo” and why is this a good name.

Background

Mathematicians and scientists have always been intrigued by pi, but it acquired a whole new following when it foiled a diabolic computer in a Star Trek episode. wears different hats — it is the ratio of a circle’s circumference to the diameter, it is a transcendental number (a number that cannot be the solution of an algebraic equation with integral coefficients).

One of the most curious methods for computing is attributed to the 18th century French naturalist, Count Buffon and his Needle Problem. A plane surface is ruled by paralled lines, all d units apart. A needle of length at least d is dropped on the ruled surface. If the needle lands on a line, the toss is considered favorable. Buffon’s amazing discovery was that the ratio of favorable tosses to unfavorable was an expression involving pi. If the needle’s length is equal to d units, the probability of a favorable toss is 2/pi. The more tosses, the more closely did the result approximate. In yet another probability method to compute pi, R. Charles, in 1904 found the probability of 2 numbers (written at random) being relatively prime to be 6/pi.

It’s startling to discover the versatility of pi, crossing as it does the wide spectrum of geometry, calculus and probability.

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Some routine problems dealing with area and perimeter evoke unexpected responses. What to measure, how to measure it, and how to anticipate variations in area and perimeter are often difficult. “If we measure the area of rectangles by dividing them into squares, then what shape should we use to measure area for triangles?” Not such a bad question: Why don’t we use “triangular units” to measure triangular areas?

Part 1

A 3 x 5 index card has an area of 15 square units. You can see that the shape of this card is rectangular. If you want to create a square card with exactly the same area, 15 square units, the dimensions of the side will change. What this means is that the card has the same area as a square that also has 15 square units. It should be possible to cut up the 3 x 5 card into (several) pieces in such a way that these pieces can be put together to form a square card.

There are two problems now:

1. How do you construct a square whose sides are the square root of 15, given only a 3 x 5 card?

• The card allows you to measure 3 units and 5 units easily. Put two cards together and 2 units can be measured off quite easily. But how do you measure the square root of 15 units?
• More generally, how do you measure the square root of (x) units, where x can be anything?

1. Still more generally, what numbers can be ‘constructed’? For instance, can you draw a line that is pi units long? Or a cube root? Now that you    have made the square, how do you cut up the 3 x 5 card so that the pieces can be re-assembled to make the square? What are the minimum number of cuts that you need to make?

Part 2

1. Now what if the card isn’t 3 x 5, but some other dimensions. Like 7 x 9. Then how do you cut up the card so that the pieces form a square?
2. What’s the general recipe for cutting up a rectangle to create a square?

Part 3

1. What is the recipe for cutting up a triangle to make a square?
2. Create two squares S1 and S2. How do you cut up S1 and S2 to make a square whose area is the sum of the original areas? What does this have to do with the Theorem of Pythagoras?

Part 4

1. Now the big problem…take a convex polygon, how do you cut up the polygon so that when you put the pieces back together you get a square?

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