### Chapter 3 - Functions and their Graphs - Kickoff

After an all too short, but hopefully productive hiatus known as summer, I thought I would bring us back to Harold Jacob’s world and his fascinating look at mathematics, which indeed is a human endeavor.

Last year we completed the first two chapters of his book where we explored mathematical ways of thinking and number sequences (chapters 1 and 2.) In Chapter 3 we’ll take a look at the powerful idea of functions, their graphs and the stories they tell.

Since I’ll act as your guide for at least this next chapter I’ll try to capture what made Mr. Jacob’s book (including his teacher’s guides) so special in my growth as a teacher. More about that as we go along.

Here’s the lesson outline:

Functions and their Graphs

1. The idea of a function

2. Descartes and the coordinate graph

3. Graphing linear functions

4. Functions with parabolic graphs

5. More functions with curved graphs

6. Interpolation and Extrapolation: Guessing Between and Beyond

Chapter 3/Review/Problems for further exploration

Culminating activity: The Great Green Globs Contest. (This is not in Jacobs book. More about it later.)

Please let the group know that you are interested in participating in our Chapter 3 journey. Since we don’t have a specific time limit on this I thought we would try to complete this chapter in a month’s time just so we a group have a sense of progress. Let's make Friday, October 17th the last day for Chapter 3. OK?

So, let’s get started.

--------------------------------

From the lesson outline above, this chapter can appear to be quite intimidating for students. That’s because they don’t have a lot of familiarity with concepts like functions, linear, interpolation etc. But defining these terms for the students at the outset is not necessarily the best way to start off either. Rather a more informal (let’s get to know the big ideas) approach might work better. That’s what Jacobs does by introducing the Tower Puzzle as an opening activity in lesson 1 (edition 2) not only to motivate students but also to get the big idea here. Note: You will find a scanned copy of this activity as it appears in Jacobs' teacher’s guide (p. 30-32) in the files folder on our site. Here’s my take on it.

He starts off by telling story about a legend of how the world will come to end. It goes something like this:

In a temple in the city of Benares, India are 64 discs of precious diamonds piled up to a tower.

Some holy men, monks, have the god given task to move this tower to a new place inside the temple. By doing this they have to obey some holy rules. The discs are just allowed to be placed at three marked places inside the temple. The first place is the one where the tower was before they started to move it, the second place is the place of the destination, the third place is right between start and destination. The discs are so heavy and precious that the holy rules allow just a movement of one disc at a time. The last rule says that it is at no time allowed to place a disc on top of a smaller disc, while it is always allowed to place a disc on any disc with a greater diameter. At the time when the monks have finished their work and the whole tower is moved from its starting place to its destination, at that very time, the tower will collapse and turn to dust and with this tower the whole earth will cease to exist.

If you believe the story is there any reason for concern? In other words how long will it take before our earth self destructs?

Although this is an amusing story, the puzzle is actually a very famous one that usually goes by the name of the Tower of Hanoi. Here is a photo

of one you can find in the Tower wing of the Puzzle Museum.

Jacobs writes on page 31 (of the teacher’s guide – edition 2):

To find out how to solve this puzzle solving a simpler version of the puzzle is a good place to start. If you don't have the actual puzzle, you can experiment with a virtual version of the puzzle.

Try to move all the disks from Tower 1 to Tower 3. You many only move one disk at a time. You must never allow a bigger disk to go on top of a smaller disk.

Start with the easiest tower first. 3 is the minimum for the puzzle. Increase tower size by 1. Keep track of moves. Look for a pattern. Make a table like this:

Number of disks in tower | Number of moves (it takes to move the tower to another peg following the prescribed rules.)

1

2

3

4

5

Can you make a prediction about how many moves it will take for a given number of disks? For example, 8 is the most disks you can have in the applet. But can you make a prediction for the minimum number of moves it would take for a tower with 9 disks? 10 disks? 20 disks? Any number of disks?

Or do it dynamically using Google Docs – Spreadsheets. http://spreadsheets.google.com/ccc?key=pUglwXqBtKmlo1Ifn1hyFQQ&hl=en

Here’s the completed spreadsheet. http://spreadsheets.google.com/ccc?key=pUglwXqBtKmn1EF2Jpg1TPg&hl=en

More about the puzzle after we hear back from you that you are on board for the upcoming journey. Let me know if you have any questions or concerns.

Last year we completed the first two chapters of his book where we explored mathematical ways of thinking and number sequences (chapters 1 and 2.) In Chapter 3 we’ll take a look at the powerful idea of functions, their graphs and the stories they tell.

Since I’ll act as your guide for at least this next chapter I’ll try to capture what made Mr. Jacob’s book (including his teacher’s guides) so special in my growth as a teacher. More about that as we go along.

Here’s the lesson outline:

Functions and their Graphs

1. The idea of a function

2. Descartes and the coordinate graph

3. Graphing linear functions

4. Functions with parabolic graphs

5. More functions with curved graphs

6. Interpolation and Extrapolation: Guessing Between and Beyond

Chapter 3/Review/Problems for further exploration

Culminating activity: The Great Green Globs Contest. (This is not in Jacobs book. More about it later.)

Please let the group know that you are interested in participating in our Chapter 3 journey. Since we don’t have a specific time limit on this I thought we would try to complete this chapter in a month’s time just so we a group have a sense of progress. Let's make Friday, October 17th the last day for Chapter 3. OK?

So, let’s get started.

--------------------------------

From the lesson outline above, this chapter can appear to be quite intimidating for students. That’s because they don’t have a lot of familiarity with concepts like functions, linear, interpolation etc. But defining these terms for the students at the outset is not necessarily the best way to start off either. Rather a more informal (let’s get to know the big ideas) approach might work better. That’s what Jacobs does by introducing the Tower Puzzle as an opening activity in lesson 1 (edition 2) not only to motivate students but also to get the big idea here. Note: You will find a scanned copy of this activity as it appears in Jacobs' teacher’s guide (p. 30-32) in the files folder on our site. Here’s my take on it.

He starts off by telling story about a legend of how the world will come to end. It goes something like this:

In a temple in the city of Benares, India are 64 discs of precious diamonds piled up to a tower.

Some holy men, monks, have the god given task to move this tower to a new place inside the temple. By doing this they have to obey some holy rules. The discs are just allowed to be placed at three marked places inside the temple. The first place is the one where the tower was before they started to move it, the second place is the place of the destination, the third place is right between start and destination. The discs are so heavy and precious that the holy rules allow just a movement of one disc at a time. The last rule says that it is at no time allowed to place a disc on top of a smaller disc, while it is always allowed to place a disc on any disc with a greater diameter. At the time when the monks have finished their work and the whole tower is moved from its starting place to its destination, at that very time, the tower will collapse and turn to dust and with this tower the whole earth will cease to exist.

If you believe the story is there any reason for concern? In other words how long will it take before our earth self destructs?

Although this is an amusing story, the puzzle is actually a very famous one that usually goes by the name of the Tower of Hanoi. Here is a photo

of one you can find in the Tower wing of the Puzzle Museum.

Jacobs writes on page 31 (of the teacher’s guide – edition 2):

To find out how to solve this puzzle solving a simpler version of the puzzle is a good place to start. If you don't have the actual puzzle, you can experiment with a virtual version of the puzzle.

Try to move all the disks from Tower 1 to Tower 3. You many only move one disk at a time. You must never allow a bigger disk to go on top of a smaller disk.

Start with the easiest tower first. 3 is the minimum for the puzzle. Increase tower size by 1. Keep track of moves. Look for a pattern. Make a table like this:

Number of disks in tower | Number of moves (it takes to move the tower to another peg following the prescribed rules.)

1

2

3

4

5

Can you make a prediction about how many moves it will take for a given number of disks? For example, 8 is the most disks you can have in the applet. But can you make a prediction for the minimum number of moves it would take for a tower with 9 disks? 10 disks? 20 disks? Any number of disks?

Or do it dynamically using Google Docs – Spreadsheets. http://spreadsheets.google.com/ccc?key=pUglwXqBtKmlo1Ifn1hyFQQ&hl=en

Here’s the completed spreadsheet. http://spreadsheets.google.com/ccc?key=pUglwXqBtKmn1EF2Jpg1TPg&hl=en

More about the puzzle after we hear back from you that you are on board for the upcoming journey. Let me know if you have any questions or concerns.

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