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''

- With the above understanding, how many
parents do you (or did you) have? __________
- How many grandparents -- i.e. two generations before yours -- do you (does anyone) have? __________
- 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:

______________________________________________________________________

**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? _____________

- In the year 1750? __________
- In the year 1500? __________
- In the year 1450? __________

Only after you've