Sequences and Series

CA STANDARDS

ALGEBRA II standard 22: Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

Image from http://heritagemath.files.wordpress.com/2012/01/fibonac_8.gif

Honeybees and Family trees

(this information was taken from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits)
There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee").

First, some unusual facts about honeybees such as: not all of them have two parents!

In a colony of honeybees there is one special female called the queen.
There are many worker bees who are female too but unlike the queen bee, they produce no eggs.
There are some drone bees who are male and do no work.
Males are produced by the queen's unfertilized eggs, so male bees only have a mother but no father!
All the females are produced when the queen has mated with a male and so have two parents. Females usually end up as worker bees but some are fed with a special substance called royal jelly which makes them grow into queens ready to go off to start a new colony when the bees form a swarm and leave their home (a hive) in search of a place to build a new nest.

Image from http://acm.tju.edu.cn/acm/image/2775_bee2.gif

Let's look at the family tree of a male drone bee.

He had 1 parent, a female.
He has 2 grand-parents, since his mother had two parents, a male and a female.
He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.
How many great-great-grand parents did he have?

If you look at the number of members of each generation, what sequence would it yield? Do you see any similarities with the image of the Nautilus shell above? This sequence is so famous it has a proper name - do you know what it is?

ASSIGNMENT DETAILS

As you complete the steps for this lesson, keep track of the information that you think is most important. You will be assigned a group to work with through Google Docs.

A member of your group will create doc and share with each group member AS WELL AS ME, using my email address kdumancic@rusd.k12.ca.us.
Your final product is going to be a slide presentation as a summary of the information that you have gathered through your studies.
The presentation need to have at least two slides per group member, plus a title slide and a reference slide.
For more detail on the assignment, make sure you check the rubric below the assignment steps.

GROUP DUTIES

1) Set up a document through Google plus and share it with your group members and me. Each group member will send the link to the document to my email. This is due by Friday at midnight.
2) Visit the document frequently to touch bases with your group members. Remember, communication is part of the assignment.
3) You may divide the work up any way you wish within the group, but you are expected to contribute to the presentation in a meaningful way.

Image from http://www.calu.edu/academics/colleges/eberly/math-computer-science/files/computer_lab.jpg

INDIVIDUAL DUTIES

1) Many sets of numbers have a pattern. Click here to study about the differences between the different types of number patterns. Then take the quiz to see how well you understood number patterns. Take a screenshot of your results and email it to me before Sunday at midnight.

2) What are sequences? Click on the link to learn about sequences, then take the quiz on the lesson. Take a screenshot of the results and email it to me before Tuesday at midnight.

3) What is the difference between a sequence and a series? Watch this video to help answer the question.

4) What is the difference between arithmetic and geometric sequences? Search the web to find the answer to this question. Send me three working links by Thursday at midnight to my email with a short paragraph answering this question.

Image from http://milan.milanovic.org/math/english/triangular/fig000292.JPG

RUBRIC for Sequence and Series Assignment

CATEGORY	20-25	15-19	10-14	0-9
Mathematical Concepts	Explanation shows complete understanding of the mathematical concepts used to solve the problem(s).	Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s).	Explanation shows some understanding of the mathematical concepts needed to solve the problem(s).	Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written.
Mathematical Terminology and Notation	Correct terminology and notation are always used, making it easy to understand what was done.	Correct terminology and notation are usually used, making it fairly easy to understand what was done.	Correct terminology and notation are used, but it is sometimes not easy to understand what was done.	There is little use, or a lot of inappropriate use, of terminology and notation.
Neatness and Organization	The work is presented in a neat, clear, organized fashion that is easy to read.	The work is presented in a neat and organized fashion that is usually easy to read.	The work is presented in an organized fashion but may be hard to read at times.	The work appears sloppy and unorganized. It is hard to know what information goes together.
Working with Others	Student was an engaged partner, listening to suggestions of others and working cooperatively throughout lesson.	Student was an engaged partner but had trouble listening to others and/or working cooperatively.	Student cooperated with others, but needed prompting to stay on-task.	Student did not work effectively with others.

Sequences and Series

Saturday, March 29, 2014