Ancestry of humans and bees

School Science and Mathematics, Feb 1998 by Flores, Alfinio, Birge, Lori

Grade Level: 7-12

Mathematics Skills/Concepts:

Sequence, geometric sequence, ratio, powers of 2, first differences of a sequence, Fibonacci sequence, and golden ratio.

Science Concepts/Processes:

Sex determination, X and Y chromosomes, haploid, diploid.

Prerequisite Skills:

Students should know the definition of ratio and be able to compute the ratio of consecutive terms of a sequence.

Basic knowledge about bees (differences and relations between workers, queens, and drones). Students should be familiar with the process of determination of sexuality in human beings in terms of an X chromosome received from the mother and either an X or Y chromosome received from the father.

Objectives:

Compare the differing number of ancestors of humans and bees which results from a dissimilar process of sexual determination. Study and compare the corresponding sequences of numbers (Fibonacci and geometric) that describe the numbers of ancestors of humans and bees and learn some of their properties.

Content background

Biology. A person's genotype determines whether that individual is a male or female. Every organism contains genetic material in all of its cells. The genetic material found in cells is deoxyribonucleic acid (DNA) and is organized in the form of chromosomes. Sex determination in most species is determined by two chromosomes called the X and Y chromosomes. In most animals, an individual with both an X and a Y chromosome is considered to be male, while someone with two X chromosomes is considered to be female. An individual's sex is determined by the father, because he can pass down either his X or his Y to his offspring, while a mother always passes down one of her two X chromosomes.

Bees, wasps, and ants have a different method of sex determination. Fertilized eggs develop into females while unfertilized eggs develop into males. The male bee begins life with only one set of chromosomes. A cell with only a single set of chromosomes is called haploid, while a cell with two sets is called diploid. Most normal body cells are diploid. Haploid or triploid (three sets of chromosomes) would not be viable in most species. The cells of male bees do not remain haploid, however. Although dividing cells appear to be haploid, chromosome duplication occurs in the absence of cell division in some but not all tissue. This leads to the presence of two copies of each chromosome in those cells. (Normally, chromosomal duplication only occurs during cell division.)

Bee eggs receive one set of chromosomes from the queen bee. Those eggs that are fertilized receive a second set of chromosomes from a male (the father) and the larvae become female bees. Those eggs that are not fertilized have the original set of chromosomes from the queen and develop into male bees. Note that unfertilized eggs in most species would be nonviable. Later, some of the cells in the male bees that have one set of chromosomes will undergo the chromosomal duplication described above and then have a second set of chromosomes. The second set that arises through duplication will be identical to the original set.

Mathematics. A sequence of the form a, ak, ak^sup 2^ , ak^sup 3^, ... is called a geometric sequence with constant ratio k. In the sequence 1, 2, 4, 8, ..., a = 1 and k = 2. The constant ratio is the quotient determined by dividing the new term by the preceding term.

The Fibonacci sequence 1, 1, 2, 3, 5, 8, ... is defined by letting the first two terms be equal to 1; the terms after the first two are obtained as the sum of the preceding two terms. The ratios of successive terms of the Fibonacci sequence, such as 8/5, 13/8 approximate closer and closer the number phi= 1.6180339.... This number, called the golden ratio, can be illustrated by means of a rectangle that has the following property. If the larger square is cut out from the rectangle, the remaining rectangle is similar to the original (see Figure 1).

Interrelationship of content. A situation that arises in biology, counting the number of ancestors in each preceding generation for humans and bees, gives rise to two different number sequences. Those sequences are then studied and compared from a mathematical point of view.

Research Background

Connections is one of the standards that run across all grade levels and across other content standards in the Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989). According to this standard, students should use and value the connections between mathematics and other disciplines. The National Science Education Standards (National Research Council, 1996) emphasize unifying concepts and processes such as models and measurement. Mathematical models, such as the number sequences used in this activity, can help students to model and count the number of ancestors for humans and bees.

Lesson Outline

Time needed: Two sessions of 45 - 50 minutes.

Materials: Graph paper, pencil, calculator.

Part One. Ancestry of humans.

Activity 1. Finding the number of ancestors of a human. Every human being has two parents. Each contributes one set of chromosomes to the process of forming a new human being. Human sexuality is determined by a chromosome received from the mother, which is always an X, and by one received from the father, which can be either X or Y. If the combination is XX, it's a girl; if it is XY, it's a boy. So in the case of humans, the number of ancestors is given by the sequence: 1 person has 2 parents, 4 grandparents, 8 great grandparents, etc. We can represent this situation in a diagram (see Figure 2).