Jacobs Math Study Group - part 2

Lesson 3.6 Interpolation & Extrapolation - Guessing Between and beyond

2008-12-22T21:06:00.046-04:00

Something that your math teachers may have left out when teaching about graphs is that graphs tell stories. And it's the stories that make the math more interesting. In lesson 3.6 Mr. Jacobs tells the story of Mark Twain’s prediction about how the length of the shrinking Mississippi River will end up. Read the page on the left. (Click on it for a larger image.)

Back in 1992 I worked with an 8th grade teacher who told this story of Mark Twain's prediction to her students and asked them if what he said was plausible. I captured that part of her lesson on video. Here's a 5 1/2 minute clip. Notice how the teacher responds when the first girl supports Twain's prediction. As you will see her goal was to keep the discussion open without judging the student's response. Video is not great, but the audio captures it well.

Here's the cover of the June, 1966 Scientific American journal showing an aireal view of the "shrinking" Mississippi River.

Also, see a lesson on predicting
• Postal rates
• The 3 1/2 minute mile

Globs Contest meets Parabolas (Lesson 3.4)

2008-11-13T11:28:00.028-04:00

This is the glob array I posted for the contest on 10/31/08 and here is my video that describes one solution using linear equations. In order to get the highest score possible you need to to maximize the number of globs you hit with each equation.
I was hoping to get 5 globs with one and I came close with my first shot, but it missed (1,-1). The same was true for shot #2. Here’s all 5 of my shots. (I made the figure below with Geometer’s Sketchpad.) So I managed to get 37 points. I’m not sure that I can do better using linear equations. But what if could use parabolas? Mr. Jacobs introduces us to parabolas in lesson 4. He defines it as the graph of y equals x squared. In set 1 he explores the patterns that help us to move the parabola along the y axis and also how to draw it “upside down” by making the coefficient of the x-squared term negative. In the next video clip I describe more about graphing parabolas.
Here’s the three parabolas I used to get my score of 61 points. I used the vertex form of the equation to draw each parabola. To draw the parabola so its orientation is "perpendicular" to the original parabola switch the x and y in the equation.

To help you use the vertex form of the equation to draw parabolas more effectively try out the Interactive parabola applet at http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php

PS. If you go to the website above and want to see a perfectly awful example of a math video game click on their banner promoting the game or click here.

Halloween Globs Challenge

2008-11-01T12:46:00.011-03:00

Here is the Glob array you will use to try to get the best possible score for this year's Halloween Globs challenge. Here's what you do:
1. Download the score sheet for this array.
2. If are using the Green Globs program download the game #2 file (PC or Mac).
3. Submit the equations used and a user name in an email to globscontestcentral@mac.com by NOV. 7th.
4. Once your equations are received, your score will be tabulated and posted on November 8th at the Contest home site. An email with the URL for where the results are posted will be sent to you.

I hope you give it a try. Let me if you need help. I can walk you through it.

Best wishes,
Ihor

Linear equations attack the green globs on Halloween!

2008-10-29T15:57:00.020-03:00

In lesson 3.3 you learned about linear functions and graphs. As an extension of that lesson I thought it would be useful to try the Great Green Globs challenge because you will need to use the skills you learned in this lesson. And if you didn't learn them all that well, this will be a great opportunity to learn them a fun way. Now using the Globs program ($20 from greenglobs.net) is the best way to go, but you can still compete anyway. Try this practice game to find out what you will be doing in the contest. Here are the instructions:
1. Download the score sheet and print it out.
2. Watch my Youtube video clip (3:40) to get an idea of how to play the game.
3. Graph the function* and write down the equation as shown in the figure below.

4. Continue drawing and recording equations until all 13 globs have at least one line going through them.

When finished send your list of equations in an email to globscentral@mac.com. I will enter your equations into the Globs program to determine your score and post the results on my blog.

On Friday, October 31st I will be posting the actual contest score sheet and you will have a week to submit your results. After that I will post all results.

I look forward to joining us for this activity.

After you complete lesson 3.4 Functions with Parabolic Graphs. We'll use Globs again to see how you can get a very high score with a parabola!

For more details on how to write linear equations in preparation for the contest see this article.

*This is the same function that is shown in the Youtube video above.

Everything You Ever Wanted to Know about Equations, Lines, Slopes, and Graphs but Were Afraid to Ask

2008-10-12T23:02:00.025-03:00

The next lesson in chapter 3 of your math book is on graphs, lines, slopes, and equations. Why would you ever want to learn this you might ask? One very good reason is that in the very near future you will be participating in the Great Green Globs Contest and you will need to know all about coordinate geometry. You will learn by playing the Globs game either on the computer* (which is the best way) or with a printout of the score sheet along with a pencil and ruler.

Green Globs: The Game (a description)
Thirteen randomly scattered “green globs” are displayed on a coordinate grid. The goal is to "explode" all the globs by hitting them with the graphs of equations entered on the keyboard (or paper). The scoring algorithm encourages you to hit as many globs as possible with each equation in order to get the best possible score.

The following lesson will help you to learn all you need to know about graphing equations to succeed in Green Globs. Have a good time, but don’t forget to look for the big picture.

To get started you will need to print out:

(1) the lesson and
(2) the score sheet.

If you have the software you should download the saved game (as described in the lesson) and open it with the Green Globs program.

The Globs array for the contest will be posted on October 31.

If you questions or problems let me know. Also let me know if are getting the software. The lesson is written to be used with the software but you can also follow along by using a couple of printouts of the score sheet.

Good luck! - Ihor

You can also watch a short video intro to using Globs which I made and is on Youtube.
* The software program Green Globs and Graphing Equation is now available from http://www.greenglobs.net for only $20.00.

Lesson 3.3 Functions with Line Graphs

2008-10-03T14:38:00.008-03:00

So far we have seen that functions can take on different forms such as an equation, a table or a graph. In lessons 1 and 2 the “guess my rule” function machine metaphor helped to show the connection between the rule and the table. Lesson 3 graphing functions (more specifically, linear equations) is the focus.

As usual, Mr. Jacobs likes to present a motivational activity to start and he uses in Edition 3 the story about Robert Wadlow who at one time was the tallest man in the world to show the relationship between age and height. There is a marvelous website about Mr. Wadlow at http://www.altonweb.com/history/wadlow/ which is a fascinating read.

Note: The word rule is used interchangeably with equation and even sometimes with function. This can cause confusion because a rule doesn’t always have to be an equation and not all equations are necessarily functions depending on how it’s defined. For example a mathematical rule (which is what we are talking about here) could be an inequality like y > 3. Functions in higher math can become very complex so I tend to avoid using the term especially with younger students.

Let me know if you have any problems with Sets I and II. His Set III questions are usually a bit more challenging (and interesting), so you can skip them if you feel it would be appropriate.

3.2 Descartes and the Coordinate Graph

2008-09-29T14:54:00.025-03:00

This lesson leads off with an intro to Rene Descartes the 17th century philosopher who is credited with coming up with the idea of coordinate geometry (not surprisingly called the Cartesian coordinate system.) There's a cool, but most likely apocryphal story of how Descartes discovered the idea. My quick Google search came with this description:

One morning [], Descartes found himself watching a fly on the wall (or so the story goes) and suddenly discovered that he could define the fly’s position using only three numbers: the perpendicular distance of the fly from each wall and from the ceiling. Generalizing from this realization, he discovered that any point in space could be defined in a similar way by measuring their distances from perpendicular lines or planes. These numbers have commonly become known as “Cartesian coordinates” and the perpendicular lines as the x- and y-axes. That discovery led to the development of analytical geometry, the first mathematical blending of algebra and geometry. The discovery of the coordinate plane, alone, is a huge contribution to psychology, for without it, defining the relationship between independent and dependent variables, calculating correlations, performing tests of significance, and other quantitative analysis would not be possible. (Snipped from this site.)

Note: You can find an interesting introduction to this topic at “Descartes and the Fly.”

If you are a bit hazy about how coordinates work, I would suggest that before you tackle the Set I problems you try this fun way to introduce (or review) exploring the X-Y plane with Billy the bug!
Set II questions 1 to 6 start to make the connection between ordered pairs and functions. (We will return to this idea in lesson 3.)

Set III (editions 1 and 2 only) is a bit tedious (and dated?) but my students loved doing it! (Spoiler: see attached file on the Jacobs site.) Set III in edition 3 is more textbookish.

How did you make out?

What are functions?

2008-09-26T13:25:00.015-03:00

Wikipedia.com defines a function in mathematics as an abstract entity that associates an input to a corresponding output according to some rule.

A good mental model for this idea is a function machine. My colleague at CIESE (where I worked until last October) Jason Sayres created this nice example which he calls The Mystery Box Game* which is really version of "Guess my Rule". For example, let's say

2 goes through the box and becomes 5
5 goes through the box and becomes 11
9 goes through the box and becomes 19
Can you predict what rule is being used here? (Spoiler alert: click here)

A rule is a mathematical way of explaining a pattern if one exists. Since mathematics can be thought of as the study of patterns, rules or functions are foundational for understanding mathematics.
Hopefully, the rest of lesson 1 (Chap 3) will be easier to understand. (Note: Bob has posted the answers for the first lesson of the first edition on our Jacobs group site.)

So what's the tower puzzle got to do with functions?
Julie wrote this:
Thanks Bob. You said:
Instead of comparing the number of moves to the number of discs, try comparing the number of moves for "n" discs to the number of moves for "n-1" discs.

I now say:
We now have an exponentially increasing difference 2, 4, 8... so I would guess it continues 16,32, 64, 128 ...

Let's try ( 2^n) -1 as a function. This means that when I replace: n with 1 the function rule produces: (2^1) = 2 -1 = 1. Continuing in the same with n = 2, 3, 4, my outputs are:
(2^2) = 4 -1 = 3
(2^3) = 8 -1 = 7
(2^4) = 16 -1 = 15
etc
am also seeing that the 1 disc is moved every alternate move in the pattern CBA although I have no idea how to express the moves mathematically. There is also something like 2,3,2,4,2,3 which looks patternish happening but I get very muddled trying to rermember what to do next physically when listing it this way.

Ihor replies:
You certainly got most of it. I assume that you increased the number of disks by 1, the number of moves jumped to the next power of 2 (minus 1). You also wrote the formula to get you the number of moves if the number of disks is N (or any number.) As you might recall, the monks in the tower of Benares started working on a tower with 64 disks. And when they are done the earth will become a pile of rubble. Is that something we should be worried about? In terms of our function machine, the 64 tower suggests the following:

Input: 64
Rule: Take 2 to power of 64 and subtract 1
Output: Whatever you get after you multiply 2 by itself 64 times and subtract 1! Quite a formidable task. (If you would like to see how a spreadsheet does it click here.)

As you probably can guess we don't have to worry that the monks will finish anytime soon and the worry about the earth's (unlike the stock market) demise is premature.

More about the way the disks move

For a 1 disk tower, it takes one move.
For a 2 disk tower, it takes three moves.
For a 3 disk tower, it takes seven moves.
Check out this animation of 4 disks:
Since 2^4 - 1 is 15, that's how many moves it takes. Now if you look at the 4-tower, you notice there is a 3-tower on top of the 4th disk. We first need to move the 3-tower to another place before we can free up the 4th disk to move. So this will take 7 moves. Now we can move the bottom disk which is the 8th move. Next we have to rebuild the 3-tower on top of the relocated 4th disk. So that will take another 7 moves and we are done. So in sum we need to move the 3-tower twice and the bottom disk just once so that's 2 x 7 + 1 = 15 moves.

We know there are 31 moves for 5 disks. And we know that we can't move the bottom disk until we move the 4-tower out the way... Hmmm. (Can you finish what I'm thinking based on what we did with the 4-tower?)

Coming this weekend: Descartes and his marvelous discovery (Lesson 2). Stay tuned...

-Ihor

*Black box mystery game is explained in detail here: http://ciese.org/math/activities/blackbox/

Re: Running late - phew!! - an update

2008-09-24T13:55:00.006-03:00

Karen writes:
Don't know about anyone else but I, too, am not having much luck at getting to the chapter [3] so far. I am delighted that you are running late!!

Ihor replies:
May I suggest you put a bookmark wherever you are and join us in Chapter 3 if that's OK. One of the things I really like about HE (Human Endeavor) is that each chapter potentially could be done separately though I'm pretty sure Harold didn't intend that since he was writing it for high school teachers in conventional classrooms. I think a lot about Web 2.0 and math and I'm sure that the direction we're heading towards is short chapters on topics that can stand alone. Or I should say short STORIES rather than chapters since chapters usually implies a long, boring textbook.

I'm rethinking my timeline for Chapter 3 since this is my favorite chapter and I hope you will join us in participating in the culminating activity which is the Great Green Globs Contest. (Sounds perfect for Halloween, doesn't it?) It's actually an experiment I'm working on to make the on-line math learning experience more engaging for students of all ages. More about that as we go along. So I'm thinking of extending the timeline to the end of October. Let me know what you think about that.

Yesterday, I went to the library in my area that has the first edition of the textbook and borrowed it. It’s the only version I don’t have. The one I used in 1970s with my 7th and 8th grade classes I managed to lose. (I’m still trying to find a replacement. My occasional Google searches come up empty.) So now I have all the versions at least till the end of October.

More about Chapter 3 lesson 1 and 2 in my next entry.

-Ihor

Chapter 3 - Functions and their Graphs - Kickoff

2008-09-15T02:24:00.021-03:00

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.

A Fibonacci Number Trick

2008-07-20T22:00:00.041-03:00

Summer is definitely getting the best of me. I’ve been distracted by the usual summer pastimes like surfing and bungee jumping - only on TV, of course. Mowing grass is more my speed… I can’t believe how fast grass actually grows. This current east coast hot spell should slow it down a bit so I can have more time for watching the surfing channel.

One of the things I said I would do is to share with you some neat activities that Mr. Jacobs has "hidden" in his treasure trove Teacher’s Guide (Editions 1 and 2) and Instructor’s Guide (Edition 3). The following activity made the cut for all three editions and was one of my favorite activities to do with my students back when I was earning an honest living.

It's a number trick based on the Fibonacci sequence that Harold uses as his opener to Chapter 2, lesson 6. Below you will find his write up of it taken from Edition 1.

Here's my summary of the activity:
Tell your students that you have a special talent for adding lists of numbers in your head. To show your talent ask them to:
1. Number a piece of paper column-wise from 1 to 10.
2. Write a 2 digit number next to number 1.
3. Write a different 2 digit number next to number 2.
4. Add the two 2 digit numbers and write your answer next to 3.
5. Add the numbers next to 2 and 3 and write the sum next to 4.
6. Continue adding the previous 2 numbers and write down the answer on the next line until you have a number next to each of the numbers 1 to 10.
Your list should look something like this:
7. Add up the 10 numbers you wrote.
8. Hide the answer so that no one else can see it.

At this point you announce that you already know what the sum is. All you need to know is the 7th number on the list which you can easily see if you look at the student's paper before they finish their list. In the example above the 7th number is 387. Simply multiply this number by 11. (There is a shortcut way to multiply by 11.)

Have the students try 2 other numbers to see if the trick still works.

Click here to play this video of me showing how a Google Docs spreadsheet can be used to demonstrate the trick.

Can you explain why this trick works? (Hint: a little algebra can be of help here. For a further hint read Jacob's description of the activity below.
Google Docs is available at this site.

Chap. 2, Lesson 6 – The Fibonacci Sequence

(Click above for larger image.)

Welcome to Jacobs Math Study Group - Part 2.

2008-06-10T20:37:00.000-03:00

Due to some technical difficulties I had to change the address of my Math Endeavor blog. I now call it Jacobs Math Study Group - Part 2. The new address is jacobsmath.bogspot.com. You can still find these previous entries: Number Guessing Trick Dr. Spock Clock Gauss's Challenge The Famous Jinx Puzzle at the old blog site.