Bees and Polygons - Weebly and Polygons H oneybees build nests in the wild called hives. About...

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Bees and PolygonsHoneybees build nests in the wild called hives.About 60,000 bees live in a hive. Bees are fairly small insects, but packing a hive with 60,000 bees and their honey is tricky.

Bees store their honey in a honeycomb, which is filled with tubes. An interesting pattern appears on the face of a honeycomb. It is covered with a design of identical six-sided shapes that fit together like tiles on a floor.

You can also find many different shapes in art, architecture, and nature.

What shapes can you identify in Auguste Herbin’s painting below?

Investigation 1 Bees and Polygons 7

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In this unit, you will investigate properties that make two-dimensionalshapes useful. The unit will focus on polygons. First, let’s review some basicconcepts. A line is a familiar object. In mathematics, line means a straightline that has no end in either direction. You can use arrows to show that aline has no ends. A or segment, consists of two points of aline and all the points between these two points.

A polygon is a group of line segments put together in a special way. Forexample, some of the shapes below are polygons and some are not.

In order to be a polygon, what properties does a shape need to have?Draw a polygon that is different from the ones above.Then draw a shape that is not a polygon.

Polygons Not polygons

Line Line Segment

line segment,

8 Shapes and Designs

Getting Ready for Problem 1.1


The line segments in polygons are called The points where two sides of a polygon meet are called vertices. Polygons have special names basedon the number of sides and angles they have. For example, a polygon withsix sides and six angles is called a hexagon. The table below shows thenames of some common polygons.

You can label a polygon by using a single letter or numeral for the entireshape or by marking each corner, or vertex, with a different letter. To referto a polygon with lettered vertices, start with any letter and list the letters inorder as you move around the polygon in one direction. For the rectanglebelow, you could say rectangle CDAB or rectangle DCBA (but notrectangle ACDB).

A

A B

D CTriangle A Rectangle ABCD

Number of Sidesand Angles Polygon Name

3

4

5

6

7

8

9

10

12

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

dodecagon

Common Polygons

sides.



Problem 1.1

1.1 Sorting Shapes

Below are a variety of polygons. Many of these polygons have commonproperties.

Sorting Shapes

A. Sort the polygons in the Shapes Set into groups so that the polygons in each group have one or more properties in common. Describe theproperties the polygons have in common and give the letters of thepolygons in each group.

B. Take all the triangles and sort them into two or more groups.Describe the properties you used to form the groups and give theletters of the triangles in each group.

Shapes Set



C. Take all the quadrilaterals and sort them into two or more groups.Describe the properties you used to form the groups and give theletters of the quadrilaterals in each group.

D. Rose put Shapes R, O, and S into the same group. What properties dothese polygons have in common? Would Shape U belong to thisgroup? Explain.

Homework starts on page 17.

In Problem 1.1, you may have sorted the triangles according to the numberof equal-length sides they have. An has three sides thesame length. An has two sides the same length. A

has no sides the same length. (The small marks on thesides of each triangle indicate sides that are the same length.)

You may have sorted the quadrilaterals according to the number of sideswith the same length or the number of angles of the same measure. A

is a quadrilateral with four sides the same length and four angles of the same measure. A is a quadrilateral with opposite sides the same length and four angles of the same measure. A is a quadrilateral with opposite sides the same length and opposite angles of thesame measure. (Note: angles 1 and 3 and angles 2 and 4 are opposite anglesin the quadrilaterals below.) You will be seeing these shapes throughout this unit.

Square Rectangle Parallelogram

parallelogramrectangle

square

Equilateral Triangle Isosceles Triangle Scalene Triangle

scalene triangleisosceles triangle

equilateral triangle



1.2 Symmetries of Shapes

As you study the polygons in this unit, look for ways that differentcombinations of side lengths and angle sizes give different shapes. Inparticular, look for shapes that have attractive symmetries.

Reflection Symmetry Rotation Symmetry


line of symmetry

A shape with has two halves that

are mirror images of each other.If the shape is folded over its

the two halvesof the shape match exactly.line of symmetry,

symmetryreflection If you rotate any shape a full

turn, it will look like it didbefore you rotated it. When yourotate a shape less than a fullturn about its center point and it looks exactly as it did before it was rotated, it has

In the polygon shown above,there are three places in therotation where the polygon willlook exactly the same as when you started.

symmetry.rotation

Reflection symmetry is sometimes called line or mirror symmetry. (Can yousee why?) Rotation symmetry is sometimes called turn symmetry.


Problem 1.2

• Which of the following shapes have reflection symmetry?

• Which of the following shapes have rotation symmetry?

Symmetry

Use the Shapes Set from Problem 1.1.

A. Look at the triangles.

1. Which triangles have reflection symmetry? Trace these triangles anddraw all the lines of symmetry.

2. Which triangles have rotation symmetry?

3. Which triangles have no symmetries?

B. Look at the quadrilaterals.

1. Which quadrilaterals have reflection symmetry? Trace thesequadrilaterals and draw all the lines of symmetry.

2. Which quadrilaterals have rotation symmetry?

3. Which quadrilaterals have no symmetries?

C. Look at the remaining polygons (the polygons that are not triangles orquadrilaterals). What is special about these shapes?

D. Find shapes with symmetry in your classroom. Sketch each shape anddescribe its symmetries.



Getting Ready for Problem 1.2


1.3 Tiling a Beehive

A is a polygon in which all the sides are the same length and all the angles have the same measure. In an all sides are not the same length or all the angles are not the same measure. Theshapes below are regular polygons.

You can find an interesting pattern of regular hexagons on the face of ahoneycomb. The hexagons fit together like tiles on a floor.

irregular polygon,regular polygon



Problem 1.3

means covering a flat surface with shapes that fit together withoutany gaps or overlaps.

Which regular polygons can be used to tile a surface?

Tiling With Regular Polygons

Use Shapes A–F from your Shapes Set or cutouts of those shapes. As youwork, try to figure out why some shapes cover a flat surface, while others do not.

A. 1. First, form tile patterns with several copies of the same polygon.Try each of the regular polygons. Sketch your tilings.

2. Which regular polygons fit together, without gaps or overlaps, tocover a flat surface?

B. Next, form tile patterns using combinations of two or more differentshapes. Sketch your tilings.

C. The following tiling may be one that you found. Look at a point wherethe vertices of the polygons meet.

1. Look back at each tiling you made. Find a point on the tiling wherethe vertices of the polygons meet.

2. Describe exactly which polygons fit around this point and thepattern of how they fit together.

3. Is this pattern the same for all other points where the vertices of thepolygons meet in this tiling?


At this vertex, two octagonsand one square fit together.

Tiling


For: Tessellation ActivityVisit: PHSchool.comWeb Code: amd-3103



Tilings are also called tessellations. Artists, designers, and mathematicianshave been interested in tessellations for centuries. The Greekmathematician and inventor Archimedes (c. 287–212 B.C.) studied theproperties of regular polygons that tiled the plane. Beginning in the middleof the eighth century, Moorish artists used tessellating patterns extensivelyin their work.

The Dutch artist M.C. Escher (1898–1972), inspired by Moorish designs,spent his life creating tessellations. He altered geometric tessellating shapesto make birds, reptiles, fish, and people.

For: Information abouttessellations

Web Code: ame-9031


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