IO Project

Activity

The Bricklayer's Dilemma: Inviting Students In

There was a bricklayer famous for his exceptional designs. His latest customer wanted him to show all designs possible using bricks that were all 2 by 1.

He started by drawing designs for 8 bricks, but he wasn't sure how he would know that he had recorded all possible designs. So he started to draw possible designs. But it wasn't long before he had a long list of designs, in fact there were so many that he couldn't tell whether he had repeated any of them. It was then that he started to wonder, just how many designs are possible to create with 8 bricks? How would he know whether he had all the possible designs using 8 bricks? Was it possible to figure this out if he started with a smaller number of bricks, say 3 or 5 bricks?

Is there a pattern that our bricklayer can use to help him out? He really needs one, since a job realistically would have 5000 bricks - not just 10!

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INVITING YOUR STUDENTS IN:

While it is true that a bricklayer would not probably actually ever do such a problem that is not the point of the problem. You are creating an experience for the students that will put them into the rich territory of Fibonacci numbers. The Bricklayer's Dilemma problem is a good problem to kick off this inquiry because:

- It can be successfully explored at many levels.

- It encourages students to come up with innovative, insightful solutions.

- It opens quite simply and then evolves into some profound mathematics.

- It explores worthy mathematical ideas that intrigue the mathematical community.

- It deals with important, useful mathematics.

- While it leads to a unique answer, it provides opportunities for interpretation, multiple correct solutions.

This is a wonderful and exciting place to begin a project on Fibonacci Numbers and Nature.

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THE PLACE FOR TECHNOLOGY IN THIS ACTIVITY:

When solving this problem, have students use a spreadsheet, such as Excel to create a table of values. Spend time examining the emerging pattern of possible brick designs.

As you look down the column of possible designs, you will notice that something seems to be going on. Is there a pattern or structure to these numbers? What is the pattern? How many possible patterns could you make with 5000 bricks? How about with n number of bricks?

Our bricklayer and you, too, have stumbled into one of the most glorious number sequences in mathematics: the Fibonacci sequence.

Using the spreadsheet create a line graph using the data from the table. What type of line is created? How is it generated? What numbers go on which axes? Does it matter?

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Task

Fascinating Fibonacci

We don't only see the Fibonacci sequence in our bricklayer's designs but many designs in nature. Which ones? What does our bricklayer have in common with rabbits, honeybees, sunflowers, pinecones, flowers and pineapples?

You will the connections among three apparently different problems in order to understand the structure that underpins them. With a partner look for another example of the Fibonacci sequence in the world. Show that it is the Fibonacci sequence, find a interesting and intriguing way to present and share what you and your partner have found and speculate what further investigations this might lead to? Keep a journal to describe and analyze what you are figuring out about the Fibonacci sequence.

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WHY DO THIS TASK?

These problems allow students to explore and develop conceptual understanding of Fibonacci's original problem. The task requires that students bring forward their understanding by applying to a new and novel situation. It requires a wide range of approaches, different abilities, different problems and different contexts.

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THE PLACE FOR TECHNOLOGY IN THIS TASK:

Students may spend time online finding new problems, accessing the Fibonacci Quarterly, finding new contexts that have arisen which the Fibonacci sequence structure. They can use a spreadsheet, word processor or project management application to keep their task focused and on time. They can use spreadsheets, concept maps as modeling tools. They can use digital cameras or digital video in their presentations. They could use shared workspaces on local servers, dot mac. They can use a variety of synchronous and asynchronous technologies such as msn or text message to communicate throughout the task. Students will keep their reflective log a class blog.

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Activity

Fibonacci's Rabbits

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive? Have the students model the problem and organize the information. What was Fibonacci's answer to the original question? How many rabbits will there be in 1 year?

1. At the end of the first month, they mate, but there is still one only 1 pair.

2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

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WHY DO THIS PROBLEM?

This problem allows students to explore Fibonacci's original problem. This is important because it helps student build connections between the original ideas that created this number sequence. It helps to connect them to the ancestry of the idea. Students have a chance to establish a connection with this very human side of mathematics--that people create mathematics.

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THE PLACE FOR TECHNOLOGY IN THIS ACTIVITY:

When solving this problem, have students use a spreadsheet, such as Excel to create a table of values. Spend time examining the emerging pattern of reproducing rabbits.

Using the spreadsheet create a line graph using the data from the table. What type of line is created? How is it generated? What numbers go on which axes? Does it matter?

Have the students compare this spreadsheet with the one they created when solving the Bricklayer's Dilemma problem. They should notice that while the problems are quite different, the table of values and the resulting graph are identical. What's going on here? How are the Bricklayer's Dilemma and Fibonacci's Rabbits related?

Have students select another way to model this problem, other than a spreadsheet. This is a good time to introduce students to a tree diagram. A concept mapping application such as cmap or inspiration can be used to create a tree diagram of Fibonacci's Rabbits.

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Activity

Drones and Family Trees

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. Honeybees live in a colony called a hive. And while there are many interesting and unusual things about honeybees, we want to determine if we can create a family tree for male honeybees?called drones.

First, some unusual facts about honeybees such as: not all honeybees 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.

Now, for this problem we are interested in the male drone bee. We want to figure out what a family tree for a male drone might be:

1. He had 1 parent, a female.

2. He has 2 grand-parents, since his mother had two parents, a male and a female.

3. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.

4. How many great-great-grand parents did he have?

5. How many great-great-great grand parents did he have?

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THE PLACE FOR TECHNOLOGY IN THIS ACTIVITY:

Tree diagram and connecting to the other two problems.

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Activity

Pinecones, Sunflowers, Daisies and Pineapples

Bring in pinecones, sunflowers, daisies and/or pineapples. Create a chart to keep track of what you find as you start to examine each of these more closely.

Count the number of petals on the daisy. You will find that it is one of Fibonacci numbers. Examine the daisy more closely by looking at the yellow centre. You will notice that there are two sets of spirals: a clockwise set and a counterclockwise set. How many spirals are there? Now, it might be too difficult to count the spirals, so how about counting the spiral scale patterns of the pinecones. The pinecone has 5 spirals in one direction and 8 in the other. What about the spiraled seed patterns of sunflowers, spiraled bumps on the pineapples?

Record and describe what do you notice in each item you analyze? Speculate about what you have found.

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IMPORTANCE OF THIS TASK:

But just a minute, plants don't know about the Fibonacci sequence - they just grow in the most efficient ways. So nature isn't trying to use the Fibonacci numbers. Fibonacci numbers seem to be appearing as a by-product of a deeper physical process. The Fibonacci numbers are applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. Take some time to explore this idea with the students.

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THE PLACE FOR TECHNOLOGY IN THIS ACTIVITY:

Take digital pictures of the various items, show by drawing on the picture the connection to the Fibonacci sequence. Create a written explanation of what you have found.

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Fascinating Fibonacci – (This an analytic trait rubric.)

	1	2	3	4
Connections	The student provides a correct explanation of the Fibonacci sequence for each of the individual problems but is unable to recognize the connections between the various problems.	The student provides a correct explanation between the various contexts in which the Fibonacci series appeared.	The student provides a clear, concise, in depth explanation between the various contexts in which the Fibonacci series appeared and built on one another to produce a cohere	The student provides an elegant and insightful explanation of the connections between the various contexts in which the series appeared, applying and adapting these to other contexts and explaining the series meaning and importance.
Reasoning & Proof	The student makes no attempt to formulate mathematical conjectures.	The student formulates an accurate simple mathematical conjecture.	The student explores, justifies and uses mathematical conjectures.	The student develops and evaluates mathematical conjectures, arguments and proof.
Communication	The student’s communication is disjointed with no logical flow to the ideas being expressed.	The student communicates their mathematical thinking in a logical but sometimes lengthy manner.	The student communicates their mathematical thinking coherently and clearly.	The student communicates their mathematical thinking in a clear, compelling and insightful manner using the language of mathematics to express ideas precisely and elegantly.
Blog Entry	The student presents a version of one of the given problems.	The student presents a new example of the Fibonacci sequence	The student presents a new example of the Fibonacci sequence in an interesting and intriguing manner.	The student presents an unusual example in a manner that brings new clarity and insight to the Fibonacci sequence.
Reflection	The student’s reflections are either incomplete or lack details or both.	The student’s reflections are complete with a number of supporting details.	The student’s reflections are comprehensive with good supporting details and examples.	The student’s reflections are thorough and thoughtful with insightful details and examples.
Project Management	The student follows a project management plan to manage time, gather resources and follow the given tasks for each team member	The student follows a project management plan to manage time, select resources and decide on which tasks the team members will undertake.	The student creates and follows a project management plan to manage time, determine resources and allocate tasks to team members.	The student creates and follows a comprehensive, detailed project management plan to manage time, determine resources and allocate tasks to team members.

Problem Solving Rubric (Generic Rubric)

	1	2	3	4
Algebraic Conceptual Understanding	The student recognized patterns and relationships.	The student recognized some patterns and relationships.	The student recognized important patterns and relationships in the problem.	The student created a general rule or formula for solving related problems.
Representation	The student used a model or representation that was insufficient or inaccurate.	The used a model or representation to explore the relationships in the problems.	The student used a variety of models or representations to explore relationships in the problem.	The student used a model or representation that helped to clarify and extend the problem?s meaning.
Strategies	The student's strategies were not appropriate for the problem. The student didn't seem to know where to begin.	The student used an oversimplified approach to the problem. The student offered little or no explanation of the strategies used.	The student chose appropriate, efficient strategies for solving the problem. The student justified each step of their work.	The student chose innovative and insightful strategies for solving the problem. The student proved that their solution was correct and that their approach was valid.
Reasoning	The student's reasoning did not support the student?s work.	The student sometimes made leaps in their logic that were hard to follow.	The logic of the student's solution was apparent.	The student provided examples and/or counter examples to support their solution.
Computation and Execution	The student made errors in computation serious enough to flaw the solution. The student gave no evidence of how they arrived at their answer.	The student made minor computational errors. The student gave evidence for their solution that was inconsistent or unclear.	The student's computations were essentially accurate. The student gave evidence for their solution that clearly supported their solution.	All aspects of the student's solution were completely accurate. The student showed multiple ways to compute their answer.
Technology	The student worked with an identified technological application to analyze a pattern using a limited number of representations.	The student worked with an identified technological application to analyze, and generalize a pattern using a limited number of representations.	The student selected appropriate technological applications to analyze, and generalize a variety of patterns using tables, diagrams, graphs, words, and, when possible, symbolic rules.	The student selected a variety of appropriate technological applications to analyze, and generalize a variety of patterns using tables, diagrams, graphs, words, and, when possible, symbolic rules.
Communication	The student's explanation rambled without a clear focus.	The student's explanation was redundant or hard to follow in places.	The student's explanation was well organized, flowed logically and was easy to follow.	The student's explanation was clear, concise providing an in depth explanation of their reasoning.
Insight	The student found no connections to other disciplines or mathematical concepts.	The student's solution hinted at a connection to an application or another area of mathematics.	The student connected their solution process to other problems, areas of mathematics or applications.	The student related the underlying structure of the problem to other similar problems.