The California Math Council - Southern Section Conference, Palm Springs math conference will be held this year on November 7-8, 2008 at the Palm Springs Convention Center.

As one of the Featured Speakers, I’ll be presenting two sessions, one on Friday and one on Saturday. These sessions cover two of the topics I’m currently focused on in my classroom:

STAR TESTING: PREPARE STRESS-FREE STUDENTS & TEACHERS EASILY

Focus your curriculum while preparing students to perform well on the STAR by incorporating simple, easy-to-use techniques. Free, ready-to-use resources for students at, below, and above grade level.

VISUALIZE ALGEBRA & GEOMETRY CONCEPTS WITH GREATEST OF EASE

Experience this easy-to-use way to help students visualize algebra & geometry concepts with GeoGebra. Free ready-to-teach extensive resources will amaze students with the beauty and structure of math.

My partner, Brad Fulton, and I will be sharing a booth with one of Brad’s friends this year. Our books and DVD’s will be available for sale, and we look forward to seeing our many friends and meeting new ones at this year’s conference.

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Front cover and author, Bruno Ernst

Mathematician Bruno Ernst visited Escher every week for a year, creating a friendship that allowed Ernst an inside look at the creative process of one of the world’s most amazing creative geniuses. Escher edited the book’s materials to ensure an accurate depiction of his work.

This book contains biographical data of Escher’s life, 250 illustrations, and thorough explanations of the mathematics involved in creating many of his stunning images.

Chapters include: Explorations into Perspective, Creating Impossible Worlds, Marvelous Designs of Nature and Mathematics, and An Artist’s Approach to Infinity. There are discussions and images of limits, crystals, cubes, icosahedrons, moebius strips, spheres, spirals, and square limits. Something for every mathematics/art fan.

For those looking for an online experience: on the Official Website of M.C. Escher you can find information about the use of M.C. Escher’s work, a short biography, bibliography, links and some fun stuff like a Virtual Ride through some of his works. You can also order posters, sculptures, ties, t-shirts, watches.

Enjoy the fabulous works of a modern master!

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Header image from Math in Daily Life website

The byline for this nice website is “How do numbers affect everyday decisions?” The site explains how math can help us make important decisions and perform everyday tasks. The language of numbers will also show up in common situations that we may all face when you’re trying to get things done. The treatments of all topics are done very well and can be used for Teacher Resources or Student Projects. Some of the topics covered:

• Statistics and Probability
• Savings, Credit, and Managing Your Credit Cards
• Buying vs. Leasing a New Car - Rates of Interest
• Population Growth (exponential study)
• Put Your Money to Work - Simple vs. Compound Interest
• Communicating Data Visually - Charts and Graphs
• Home Decorating - Areas of Rectangles and Circles
• Cooking by Numbers - Ratio and Proportion
• The Metric System and Conversions from Metric to U.S. Measurement
• Mathematics, the Universal Language Shared by All Human Beings

There is also a rich source of Related Materials. The website materials from Math in Daily Life are inspired by programs from the text For All Practical Purposes: Mathematical Literacy in Today’s World.

Related Post: Edward Tufte and The Visual Display of Quantitative Information - Data Displayed Properly

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Edward Tufte wants to “change the way people see,” and he certainly did with this landmark book, which is said to be “pictures of numbers.” He wants data to speak for itself with as little ornamentation as possible. He has a lot to say (and show!) about how teachers can help their students display information.

Edward Tufte’s Home Page showcases the seven books he has written and self-published. These books have won more than 40 awards for content and design. He has taught courses in statistical evidence, information design, and interface design at Yale University, where he is Professor Emeritus.

The story behind the creation of this first book is fascinating: Tufte published his Visual Display in 1982 himself and financed the work by taking out a second mortgage on his home. He had 5,000 copies printed up, rented a post office box, and placed an ad in Scientific American. He and his wife went away for a few days. “The mailbox filled up just like that,” he says. “We came back Saturday morning, and it was stuffed.” The money, and accolades, kept rolling in.

Some things said about Mr. Tufte and The Visual Display of Quantitative Information:

“A visual Strunk and White.” Boston Globe

“A classic, as beautiful physically as it is intellectually.” Optical Engineering

“The Leonardo da Vinci of data.” The New York Times

“The world’s leading analyst of graphic information.” Martin Kemp, Nature

For a real visual treat, take a look at Edward Tufte’s work. Visit his website or use the Google Image search engine to look at some of his graphics.

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My co-author, Brad Fulton, and I have created hundreds of articles and activities, and we share them with the mathematics community on our websites and at conferences.  We encourage others to use them freely with their students and share them with fellow teachers. Many teachers ask if they may present our materials at conferences, or adapt them for their own use.  This post covers some of my thoughts about the conditions upon which others may use materials from mrlsmath.com.

The following text is from the Creative Commons website.

“Creative Commons is an organization which provides a collection of free content licenses that you may apply to your work.

A content license is a document that states the freedoms and limitations that you apply to your work- an explanation of what someone can and cannot do with what you make.

Creative Commons offers 6 different licenses so that you can share your work exactly how you want to.

All Creative Commons licenses require attribution. This means that others may share your work so long as they credit you. There are some other things to consider when you choose to share your work with the world.

Do you want to allow others to make changes (derivatives) to your work without having to ask? Prohibiting derivatives might prevent others from making something really cool out of your original. But, this also means they may not use your work in a way you disagree with.

If you allow others to make changes to your work, you also need to think about whether or not you will require them to use the same license as you (”Share Alike”). A ShareAlike condition ensures that the terms you chose for your original creation are preserved, but also may limit how much the derivative work can be shared (and in turn, how likely it is for someone to use your original work in a derivative). On the other hand, Share-Alike ensures that your work is always used under the terms you want, even after many generations of copies and derivatives.”

I have decided to use the Creative Commons license called Attribution Non-commercial (by-nc).

This license lets others remix, tweak, and build upon my work non-commercially, and although their new works must also acknowledge me and be non-commercial, they don’t have to license their derivative works on the same terms. I read, enjoy, and benefit from others’ works online; crediting them with the original sourcework is something I always try to do so my readers can do follow up work as they wish. Please visit the Creative Commons website for more information.

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The June 2008 issue of the California Math Council ComMuniCator journal has posed the following problem in the student problem section:
“Use each of the numbers 3, 4, 5, 6, and 7, and any operations to express as many counting numbers as possible, beginning with 1.  All five of the numbers must be used in each expression and each of the five numbers can only be used once in each expression.”

Many teachers are aware of the similar puzzle called “The Four 4’s”, in which only four 4’s may be used to create numbers from one through one hundred. It’s a challenging puzzle that develops lots of creativity and number sense in students. The puzzle is also great for introducing a puzzle that at first looks overwhelming, but becomes manageable after focusing on number strategies.

My students had previously worked with The Four 4’s and realized that the new puzzle from the CMC had many of the same features. At first they thought the new puzzle was harder since it had different numbers, but then they realized it was actually easier since they had five digits to work with.

This is a puzzle contest for students and the deadline for submission of entries is still open. Therefore I don’t want to supply any answers, but rather indicate the building blocks my students have found. Some of the discoveries they have made to date include:
4! = 1•2•3•4 = 24
5! = 1•2•3•4•5 = 120
sqrt (4) = 2, and therefore sqrt (7 – 3) = 2
⁶/₃ = 2, and therefore ⁶/_.3 = 20
The discoveries involving parentheses and exponents are too numerous to list, but students have become quite proficient and creative using these tools.

Some students originally look for a numerical expression equaling 1, then look for an expression equaling 2, then 3, etc. This normally is not the best and most efficient way to find solutions to this puzzle. I encourage students to “just play around” with the numbers and combine them in every way they can. When they do this, they are rewarded with “numerical treats” and get solutions for unexpected numbers.

We have worked on this puzzle for about a week. The first day I introduced it and discussed the rules, one of which is that using place value was not acceptable for a solution. In other words, the number 35 could not be used, but 5•7 could be used to achieve the same number. Squaring a number, such as 5², was not allowed since it involve using the digit 2, but 5^(6-4) was allowed since there was not a 2 in the expression. We spent about 20 minutes on this puzzle in class, and the homework that night was to find expressions for the numbers from 1-50. The following days we spent about 10 minutes per day summarizing the discoveries made overnight. This allowed us to work with this puzzle while still continuing with our regular curriculum. After four days on this puzzle the students have expressions for all the numbers from 1-114. Our goal is to compute all the numbers from 1-150 by the end of the school year (we have 4 days remaining).

Opportunities for transitioning from number sense to algebra thinking abound with this puzzle. Order of operations is reinforced for almost every number created. Many times students substitute different, but equivalent expressions when trying to compute a number. Number properties show their strength in supporting algebra throughout this puzzle activity. Give it a try and watch the enthusiasm and skills grow in your students!

Related Posts:

Backwards Math, Part 3 - Use Four 5’s to Create Expressions from 1 to 100

Backwards Math Extension- Four Fours Creating Many Equivalent Expressions

Backwards Math - An Activity for All Operations and All Levels of Students

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The June 2008 issue of the California Math Council ComMuniCator has posed the following problem in the student problem section:
“What four positive integers have a sum that is equal to their product?  Meaning: a+b+c+d = a•b•c•d.” And if you can find a, b, c, d, then see if you can find five such positive integers.

This sounded like an intriguing challenge, so I wanted to pose these questions to my classes. Pedagogically, it would be better to start with simpler problems, so I first asked my students to solve this question:
“Find two positive integers whose sum equals their product,” which means finding a and b such that a+b = a•b. First I asked them to not tell their classmates once they found a solution, so all students would have enough time to reason out the solution to this problem. Keeping quiet is a hard thing for students to do once they have the exhilaration of making the discovery of a solution. They were told to pat themselves on the back quietly and repeat the poem, “Zip the Lip,” which is used to remind them not to give answers away. After enough time passed, students shared their solution, which is 2+2 = 2•2. The solution shows that a and b could actually equal the same number; nothing in the problem statement requires them to be different.

After discussing the solution to the first problem, students were asked to find a solution to a+b+c = a•b•c, a simple extension of the first problem. This problem is only slightly more difficult than the first one, and many students found a solution: 1+2+3 = 1•2•3.

Now they were ready for the original problem posed by the CMC ComMuniCator. This is a puzzle contest for students and the deadline for submission of entries is still open. Therefore I don’t want to supply any answers, but rather indicate some results my students have found.

I assigned the ComMuniCator problem for homework, and the next day a number of students shared their solution. We talked about how they found answers, and most said they “just played around with numbers” until it worked out. I placed a grid on the board that showed solutions to the first three problems, and suggested they look for a pattern. Then I assigned the second ComMuniCator problem, a+b+c+d+e = a•b•c•d•e, which asks for five positive integers.

Next day in class was interesting, to say the least. When students shared their solutions for five integers, I was stunned to find two different ways to solve this problem. I had worked the problem beforehand and assumed there was only a single way to answer the question posed.

Not one to leave a problem without pushing it to its limits, I asked for an extension to six integers. After seeing the pattern for the prior solutions, almost the entire class solved this six-number puzzle. Then I stretched them and asked for the solution for 15 numbers; they solved this also. Then we went for the home run: can a solution be found for n integers?
Students successfully were able to represent the solution for an arbitrary number of integers. Once they had done this, they had crossed the line from arithmetic, which likes answers, to algebra, which likes structure.

After listening to the students’ solutions, I spent some more time with the problem and its extensions. There are a number of variations and patterns that can be found for those who wish to pursue this further with students.

To the CMC ComMuniCator staff: Thanks for a great problem. You gave my students a real math lesson-how to create things like a mathematician, get the thrill of discovery, and then generalize the result!

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Kakuro is a number puzzle that is gaining popularity recently, and in some parts of the world it is more popular than Sudoku. Let’s take a look at this phenomenon, see why people like it so much, and how you can use it in the classroom.

First of all, let’s compare it with its fellow puzzle, Sudoku. People playing Sudoku use logic only to place digits according to the game’s three rules. The game can actually be played without numbers; any set of different symbols would work as well.

Kakuro, on the other hand, requires using the digits from 1 through 9 in order to play the game. The rules of Kakuro are as simple as Sudoku’s, but there is a mathematical side to the game because the numbers involved must have a certain sum. It is this focus on the sum of the numbers that gives Kakuro its appeal for mathematics teachers.

It’s possible to design a Kakuro puzzle that is easy enough for young children to do, but the structure of the game allows puzzles to be created of any difficulty level. Number sense is one of the things we want our students to possess, and the design of Kakuro promotes the development of this.

Let’s gain some simple skills that will help us solve Kakuro puzzles. Find ways to write these sums using the following rules:
1. Each sum must have a different combination of numbers.
2. You may not repeat the same number in a combination.
3. Use only the digits from 1 through 9.

Combinations using two numbers:
3 = __ + __
4 = __ + __
5 = __ + __ = __ + __
6 = __ + __ = __ + __
7 = __ + __ = __ + __ = __ + __
8 = __ + __ = __ + __ = __ + __
9 = __ + __ = __ + __ = __ + __ = __ + __
10 = __ + __ = __ + __ = __ + __ = __ + __

Combinations using two numbers:
6 = __ + __ + __
7 = __ + __ + __
8 = __ + __ + __ = __ + __ + __
9 = __ + __ + __ = __ + __ + __ = __ + __ + __
10 = __ + __ + __ = __ + __ + __ = __ + __ + __ = __ + __ + __

Here are some websites teachers can use; they give a nice introduction to the game for beginners.
This site has a video tutorial:
http://www.yourkakuro.com/howtoplaykakuro.php
This site has a flash tutorial:
http://www.nikoli.co.jp/en/puzzles/kakuro/

Related Posts:

Sudoku - A Size for Every Body and Every Mind

For Puzzle Enthusiasts: Do It Yourself Sudoku Solver - a Great Visual Showing Progress Toward the Solution
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I move the upper transparency around until a student yells, “Stop.” Then the students race me to find the total of the numbers inside Algebra Man. The total for the placement shown is 580. I always win this contest and boast that it is because I am so talented at math. The students insist there is a trick and demand a rematch. After a few times, some of the students see a pattern and discover the trick. (Turns out I’m not a rocket scientist after all!) Once a few discover my secret, we explore the problem using algebra.
The students notice that for any number on the chart, the number to the right is one more and the number to the left is one less. Also, the number beneath the given number is ten more and the number above it is ten less. This is always true no matter where Algebra Man runs. If we think of the number in his waist as n, then the following diagram shows the values of all ten numbers inside Algebra Man:

Many of my students say that they got the answer a different way. Often they simply look two spaces to the right of his waist and put a zero after the number. I express this algebraically on the board. The number two spaces to the right of his waist would be called n+2. Putting a zero after it is the same as multiplying by ten. Thus their method is 10(n+2). I show them that these are equivalent being the distributed and factored forms of the same expression:

10n+20 = 10(n+2)

Algebra Man works on any hundreds chart, even one that begins with -49 and ends with 50. He also works on a calendar, but the formula for his sum is slightly different. Challenge your students to find the formula when Algebra Man runs around on a calendar. This is just one of many algebraic explorations that can be made on the hundreds chart. If your students are like mine, they will want to explore all the “what if’s” they suggest. One time my students wanted to find out what happens to Algebra Man’s formula when he is standing on his head. Now I was curious.
Related Posts:

Hundreds Magic - a Simple Diagram Produces Patterns, Number Sense, and Algebra Proofs

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This is a follow up post to Kali: Symmetry Groups which shows a wonderful java app that creates any of the 17 different wallpaper groups, as well as frieze and rosette patterns.

My students were given the assignment of creating a pattern that intrigued them and writing a short story about the pattern. They were able to explore angle relationships and learn about kaleidoscopes as well. I always like empowering students with new tools, especially ones involving technology.

Here is the link to several of the projects on my Picasa website - Kali: Symmetric Kaleidoscope Art - Student Sample Work with Stories.

Related Post -

Kali: Symmetric Wallpaper Sketching - Math and Art Create Fun

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