Generating the Generations: Modeling Exponential Growth


Have you ever looked through old family pictures and constructed yourself, or looked at, a family tree? In this activity you will explore some mathematics related to family trees, and also the (sometimes tricky) question of how to properly apply mathematics to real-world situations.

I. For the purpose of the following questions, we will ignore divorces, second marriages, adoptions, etc. By ``how many great-grandparents did you have?'' we mean, ``while they were alive''

  1. With the above understanding, how many parents do you (or did you) have? __________

  2. How many grandparents -- i.e. two generations before yours -- do you (does anyone) have? __________

  3. Since each person has four grandparents, and since, also, each grandparent has two parents, this suggests that every person has ____ great-grandparents (direct ancestors "three generations above them"). Explain which arithmetic operation you used on the previous numbers and why:


  4. Identifying patterns and relationships

    i. As a power of two, how many direct ancestors did you have 7 generations ago? __________

    ii. How many direct ancestors did each of them have, 3 generations before them?_______(that is, as a power of 2, each of the ancestors in i. had ____ ancestors which came 3 generations before them).

    iii. The product of the two numbers above must equal the number of direct ancestors you have coming from 10 generations ago. But the number of "10 generations ago" ancestors you have is (as a power of 2) equal to _______

    iv. Write down the two mathematical expressions which must equal; write an equation with terms on each side: _____________

    v. What is an expression for the number of ancestors i generations ago? __________ . Using the fact that each of these ancestors had, j generations before them, _______ ancestors, and that the product of the previous two numbers equal the number of ancestors you have from i+j generations ago, what equation to you get? _____________

II. Assume that the average number of years per generation is 20. Estimate the number of direct ancestors you had:

  1. In the year 1750? __________

  2. In the year 1500? __________

  3. In the year 1450? __________

Before you go on, what do you notice so far? Write in your comments below:

Only after you've thought about and discussed your answer within your group, and written down your thoughts, should you continue