A. Why do bees build honeycombs at all? What do they put in
the honeycombs? How do bees build them?
Do your own research to answer the above questions.
Go to the following website
http://www.islandnet.com/~yesmag/Questions/beewax.html to answer the
above questions.
B. Why do bees make a hexagon and not a square or triangle?
Justify your answer using mathematics.
Hint 1: Do bees want to maximize or minimize the amount of wax they use?
Hint 2: Bees want to maximize the amount of storage space and minimize the
amount of wax used. What part of the polygon (hexagon, square, triangle)
represents the wax? Colour it in red. What part of the polygon (hexagon, square,
triangle) represents the honey? Colour it in yellow.
Hint 3: The outside (the perimeter) of the hexagon is made up of wax, while
the inside (the area) is the available storage space. Since bees want to maximize
storage and minimize wax, the ratio of area: perimeter (storage: wax) needs to be
as large as possible.
Determine the area and perimeter of a square with sides equal to 1 cm.
Determine the area and perimeter of a triangle with sides equal to 1 cm.
Determine the area and perimeter of a hexagon with sides equal to 1 cm.
Complete the following table
# of sides of
Area
Perimeter
Area:Perimeter Ratio
polygon
3 (triangle)
4 (square)
6 (hexagon)
Compare the area: perimeter ratio of a hexagon to an octagon and decagon.
Which shape uses the least amount of wax for a given amount of storage space?
Why don’t bees make honeycombs out of that shape?
Hint: Try to draw as many hexagons as possible in a given space with no
gaps or overlaps. Do the same with decagons. Can you do it?
Hint: Using the hexagon cut-outs, arrange them next to each other so that
there are no gaps or overlaps. Try to do the same thing with the decagon cut outs.
Can you do it?
C. One of the reasons that bees use hexagons when making
honeycombs is because they tessellate. Complete Tasks 1 and 2
and at least one other task of your choosing.
Task 1: What is a tessellation? In your answer, be sure to include definitions of the
following terms: polygon, vertex, regular tessellation and irregular tessellation.
Task 2: Using a protractor, measure the interior angles at the vertex of the
following regular tessellations and complete the chart below.
1.
3.
Tessellation
1
2
3
2.
Sum of Interior Angles
Task 3: Measure the interior angles of regular polygons and complete the
following chart.
# of sides of regular polygon
Interior Angle Sum
3
4
5
6
7
8
Can you predict the sum of the interior angles of a regular polygon with 20 sides?
Hint: Try to determine a pattern between the number of sides and the sum
of the angles to create a formula.
Hint: Look for a pattern between the number of sides minus 2 and the sum
of the angles and create a formula.
Hint: The sum of the interior angles can be represented by the following
formula: sum = 180°(n -2 ), where n = the number of sides of the polygon.
Which shapes in the chart will tessellate? Why?
Hint: Think of the sum of the interior angles of the polygon and the sum of
the angles at the vertex of a tessellation.
Hint: The sum of the interior angles of polygons that tessellate is a factor of
360°, the sum of the angles at the vertex of a tessellation.
Task 4: Find the parent polygon in the following irregular tessellations.
Remember, to find the parent polygon, you must make a dot at a vertex,
wherever more than two shapes meet, and connect the dots with straight lines.
Task 5: Use the digital camera and take pictures of tessellations that you see in
the classroom. You may repeat this activity at home and outside. Create a
slideshow of your best images.
Task 6: Create your own
tessellation using one of the
following methods.
Method 1:
Source:
http://www.tessellations.org/diypapercut.htm
Method 2: Source: http://www.ehow.com/how_4535264_create-tessellation.html
Step 2: To create the
simplest type of
tessellation, the
translation, slide
each piece to its
opposite side. Tape
the edges together
to create this new
shape.
Step 1: To create any tessellation, start with
a square tile (thick heavy paper works fine).
Draw a line from one corner to an adjacent
corner on 2 adjacent sides. Cut these pieces
out.
Step 3: Trace the tile on a paper that is at least 4 times larger than
the tile, depending on how many repetitions you would like. Some
people like to draw a grid for a guideline as shown here to erase for
the final project, but it’s not necessary.
Step 4: Slide the tile to
the left to trace, as
well as above. Notice
they will interlock if
you did the prior steps
correctly. Repeat for
the left and below.
Continue tracing until
the page is filled.
Colour as you wish!
Task 7: Create a poem or rap to explain why honeycombs are hexagons.