Showing posts from 2008

Lesson 3.6 Interpolation & Extrapolation - Guessing Between and beyond

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 l…

Globs Contest meets Parabolas (Lesson 3.4)

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…

Halloween Globs Challenge

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 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,

Linear equations attack the green globs on Halloween!

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 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 I will enter your equations into the Globs program to …

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

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 fo…

Lesson 3.3 Functions with Line Graphs

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 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 ineq…

3.2 Descartes and the Coordinate Graph

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 a…

What are functions?

Image 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 …

Re: Running late - phew!! - an update

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 Hallo…

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…

A Fibonacci Number Trick

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 lis…

Welcome to Jacobs Math Study Group - Part 2.

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 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.