The Bee Lab

A tessellation is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.
Bees have not studied tessellations theory, however, the patterns they use to construct the hive can be explained mathematically. Bees know by instinct that the hexagon is the best polygon to build a hive because it has the smallest perimeter for a given area. So, when bees are constructing hexagonal prism cells in the hive, they use less wax and do less work to enclose the same space than if using one of the other two self-tessellating polygons with square or triangular bases. The honeycomb walls are made up of cells which are 1/80 of an inch thick, yet can support 30 times their own weight. A honeycomb roughly the size of a sheet of paper can hold more than five pounds of honey. The bees are creating hexagonal prisms in three rhombic sections, and the walls of the cell meet at exactly 120 degree angles. What is even more amazing is the fact that the bees work simultaneously on different sections forming a comb with no visible seams. It is built vertically downward and the bees use parts of their bodies as measuring instruments. In fact, their heads act as plum lines to keep them on plane.

Take a walk in nature and ask your child if they can identify any other patterns and what characteristics they might have. Are they repeating? Are they structurally strong? What are they made of? Now invite them to draw the patterns on paper using any drawing tools they would like.

Exercise 1
Use the following information about European honey bees and Africanized honey bees to answer these math questions:

European queen bees can lay a maximum of 2,500 eggs per day and the queens of Africanized bees can lay a maximum of 4,000 eggs per day. It only takes 19 days for an Africanized worker bee to complete all developmental stages from egg to adult, while it takes 20 days for a European worker bee to develop to adult. European worker bees live an average of 42 days during the spring and summer. Africanized worker bees only live 24 days during the spring and summer. During the winter, European worker bees live an average of 135 days, while Africanized worker bees live an average of 90 days.

1a. How long would it take for a colony of Africanized honey bees to lay 10,000 eggs? 1b. How about a colony of European honey bees?

2a. How much longer does an adult worker European honey bee live than an Africanized honey bee worker during the spring and summer? 2b. How about during the winter?

3a. How many days does an Africanized honey bee live from egg to the end of adulthood in the summer? 3b. In the winter?

4. The maximum number of worker bees in a well-managed hive is about 80,000. How many days would it take for an European honey bee queen to lay that many eggs?

Exercise 2
The number of workers found in a honey bee hive can be estimated using the following information:
About one-third of the worker bees in a hive forage every day. Based on the average number of flights per day by a single bee and the amount of time spent foraging, the following formula can be used to calculate the number of bees in a hive:
N = 3 x (f/0.0138)
N = number of bees in the hive
f = number of bees leaving hive per minute

If Joe observes 35 bees leaving a hive in one minute, how many bees are inside? Round off the answer to the nearest whole number.

Note: The value 0.0138 is based on average amount of time spent foraging for an average honey bee colony on an average day. This value will actually change considerably with amount of food available, weather conditions, etc.
Each beehive cell is a six-sided shape called a hexagon. Invigorate your study of bees with the following bee hive math activity related to the number six.

In order to build a “Hive of Sixes” you will need:
• Shoebox
• Lots of toilet paper rolls
• Yellow paint
• Strips of paper
• Markers

On strips of paper, write equations representing the factors of six in addition, subtraction, multiplication, and division, including the answer for each on the back. Paint a shoebox with yellow paint and fill it with toilet paper tubes, standing the tubes on end. Insert equations into each cell. Children choose a slip from each cell and solve the problem. They can check their work by comparing their answers to the back of each strip. Challenge students to solve the whole hive!

Answer Key
Exercise 1
Answer 1a: 4,000 eggs laid the first day + 4,000 eggs laid the second day + 2,000 eggs laid in the first half of the third day = 2 1/2 days.
Answer 1b: 2,500 laid the first day + 2,500 laid the second + 2,500 laid the third + 2,500 laid the fourth day = 4 days
Answer 2a: 18 days. So, Africanized worker bees die almost three weeks earlier than the European worker bees.
Answer 2b: 45 days. Bees live longer in the winter because they stay in the nest where they are protected from enemies and harsh conditions.
Answer 3a: 19 + 24 = 43 days in the summer
Answer 3b: 19 + 90 = 109 days in the winter
Answer 4: 80,000 eggs/ 2,500 eggs per day = 32 days

Exercise 2
Answer: 35/0.0138 = 2,536 bees foraging per day. This is about one-third of the hive, so 2,536 x 3 = 7,608 bees in the hive.