Educator How-To: Teaching tessellation, symmetry & point reflection

Tessellations — tiling a plane using geometric shapes without overlaps or gaps — are a pretty fun way to teach students about shapes, symmetry, reflection and rotation. Plus, they require the most minimal of supplies!

Materials:
•    Plain 3″ x 5″ index cards
•    Scissors
•    Scotch tape
•    Blank white paper
•    Optional: colored pencils/crayons, etc.

Procedure:
1.    Draw a simple design from one corner of the sheet to an adjacent corner. (Do not draw diagonally). Stress to the students that they must draw from corner to corner on this first attempt. Do not stop halfway across!

1

2.    Cut on the design line, being sure to have two pieces when done. NO TRIMMING ALLOWED.

2

3.    Slide the cut piece across the sheet to the opposite side and tape the straight edges together. The corners of the original card and the cut piece should match perfectly.

3

4.    Repeat the procedure for one of the short side sides. Key points to remind students:

4

  • Do not flip the cut pieces of your index cards
  • Do not overlap the edges of your cut pieces
  • Make sure you only cut one long side and one short side. The cut pieces will attach to the opposite straight edges
  • Cut exactly from corner to corner
  • When taping your cut piece to a straight edge, make sure everything lines up

5.    Once you have a completed pattern for your tessellation, place it anywhere on a piece of paper and trace around it. This example is squared up with the original index card edge and the paper’s edge, but this is not necessary.

5

6.    Move your pattern so that its edges match up with the outline you drew. If done correctly, they should fit snugly together like a puzzle piece, leaving no gap. Trace around your pattern when it is in position.

6

7.    Continue moving and tracing your pattern until your page is filled with your pattern’s outlines.

7

8.    For extra fun, and in homage to M.C. Escher, you can add some details to your artwork!

8

 
Background:
A tessellation is created when you cover a surface with a repeating pattern of shapes, making sure there are no gaps or overlapping pieces. The word “tessellation” comes from the Latin word “tessella.” Tessella were tiny squares of stone that were used to make large mosaics. The patterns for these mosaics were usually intricate and often geometric. This tradition continued in many different cultures, but is particularly well recognized in the Moorish artwork of Morocco and southern Spain. More recently, M.C. Escher applied the concept of tessellations to less geometric and more fantastical shapes. You may also recognize tessellations in nature in the form of bee hives, snake scales, and the outside of a pineapple.

Regular tessellations are made by repeating squares, hexagons and equilateral triangles – all regular polygons. If you figure out the interior angles of these three shapes, you might note something interesting…

9

All of these numbers — 60, 90 and 120 — are evenly divisible into 360 degrees. This means that they fit together extremely well and make filling a space easy, either while touching themselves, as in triangle to triangle, or each other, as in triangle to hexagon. When you combine these three shapes to make a pattern, you are creating a semi-regular tessellations.

10

When creating tessellations, it is important to understand reflection symmetry, rotational symmetry and point reflection.

Reflection symmetry occurs when you can easily divide an item into two and the two halves are identical. The dividing line is referred to as the line of symmetry.

With rotational symmetry, the polygon is rotated around a central point, without leaving the surface you are working on.

Point reflection occurs when you create a mirror image of a shape. Reflection is not terribly important if you are using the shapes in a regular tessellation as they are all perfectly symmetrical when divided in two. Reflection becomes much more important when you are using fluid shapes. This concept is important for students to understand as they make their tessellations. Their shapes won’t fit together correctly if they have been reflected!

13

Educator How-To: Prep for Pi Day with a circle-folding exercise

Pi Day is all about circles, circumference and diameter. Pi (approximately 3.14) is delicious is the ratio of a circle’s circumference divided by its diameter. This ratio is the same for all circles.

In the spirit of Pi Day, let’s see what we can do with the fabulously fun circle and what we learn along the way by making some simple folds.

You will need the following supplies:

•    Ruler
•    Pencil
•    Paper
•    Compass
•    Markers
•    Scissors
•    Scotch Tape™
•    Small piece of candy

Educator How-To: Prep for Pi Day with this exercise on circles

Procedure:

1. Use a compass to draw a 7-inch circle. Carefully cut the circle out.

Educator How-To: Prep for Pi Day with this exercise on circles

2. Describe the properties of the circle.

3. Can any other shapes be made using this circle? Let’s find out.

4. First, fold the circle in half and open the circle back up. Each half of this circle is called a semi-circle. Notice that both halves of the circle are identical. We say the halves are symmetrical, or have symmetry.

5. Can you think of anything in the classroom that is a semi-circle?  A protractor is a semi-circle.  How many degrees are on a protractor?  If you don’t know, investigate. There are 180˚ on a protractor, so, how many degrees in a full circle?

6. Now, fold the circle into fourths, then unfold the circle and locate the center. Mark the center using a marker or pencil.

Educator How-To: Prep for Pi Day with this exercise on circles

7. Using a ruler, draw a line from one side of the circle to the other, making sure to pass through the center. This line is diameter of the circle.

8. Using a ruler, draw a line from the center of the circle to one point on the edge of the circle to create a radius.

Educator How-To: Prep for Pi Day with this exercise on circles

9. Next, fold one side of the circle down so the edge meets the center point. Unfold and use a marker or pencil to darken the line of the fold. This line is called a chord.

Educator How-To: Prep for Pi Day with this exercise on circles

10. Re-fold the circle along the chord line and fold an additional edge to the center of the circle to form an ice cream cone-like shape.

Educator How-To: Prep for Pi Day with this exercise on circles

11. Fold the remaining edge of the circle to the center to form an equilateral triangle.

Educator How-To: Prep for Pi Day with this exercise on circles

12. Make a new shape by folding one vertex of the triangle down so that its tip touches the center of the side opposite to it. What is the resulting shape? The shape is a quadrilateral and a trapezoid.

Educator How-To: Prep for Pi Day with this exercise on circles

13. Let’s make another shape: Fold one acute vertex so that it meets one of the obtuse vertices. What is the shape created as a result of this fold? You should come up with the terms: parallelogram, quadrilateral, and rhombus.

Educator How-To: Prep for Pi Day with this exercise on circles

14. Unfold the shape until you get back to the larger triangle. Then, fold each of the three vertices to the center point. The new shape that is created is a hexagon.

Educator How-To: Prep for Pi Day with this exercise on circles

15. Again, unfold the shape to the original triangle. Fold the triangle so that all of the vertices touch at a single point to form a triangular pyramid. Is this shape a polyhedron?

Educator How-To: Prep for Pi Day with this exercise on circles

16. Lastly, fold all of the top halves of the triangles down so they cross each other. Use tape to secure the sides. This creation is a truncated triangular pyramid. It can be used as a space to put a special treat.

Educator How-To: Prep for Pi Day with this exercise on circles

Vocabulary

Property – an attribute common to all members of a class

Semi-circle – a half circle, formed by cutting a whole circle along a diameter line. Any diameter of a circle cuts it into two equal semicircles

Symmetry – when one shape is identical to another

Diameter – the length of the line through the center and touching two points on its edge; sometimes the word ‘diameter’ is used to refer to the line itself

Radius – the length of the line from the center to any point on its edge. The plural form is radii.  The radius is half the length of the diameter.

Chord – a line segment that only covers the part inside the circle. A chord that passes through the center of the circle is also a diameter of the circle.

Equilateral Triangle – a triangle in which all three sides are congruent (same length)

Vertex – typically means a corner or a point where lines meet; every triangle has three vertices.

Polygon – a number of coplanar line segments, each connected end to end to form a closed shape

Quadrilateral – is any 4-sided polygon

Trapezoid – a quadrilateral which has at least one pair of parallel sides

Acute – an angle less than 90°, or to a shape involving angles less than 90°

Obtuse – an angle greater than 90° or a shape involving angles of more than 90°

Parallelogram – a quadrilateral with both pairs of opposite sides parallel

Rhombus – a quadrilateral with all four sides equal in length

Hexagon – a polygon with 6 sides

Triangular Pyramid – a pyramid having a triangular base; the tetrahedron is a triangular pyramid having congruent equilateral triangles for each of its faces

Polyhedron – a solid figure with many plane faces, typically more than six
Truncated Triangular Pyramid – is the result of cutting a pyramid by a plane parallel to the base and separating the part containing the apex

For more Pi Day fun, join us at the Museum from 10 a.m. to 1 p.m. on Thursday, March 14 for an event celebrating all things Pi! Expect crafts, Einstein-themed goodies and pies of every variety from Pi Pizza Truck and Oh My Pocket Pies. Click here for more info!