visual modeling - Mr. L's Math

KenKen and Number Sense

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Mar 072012

KenKen 6x6 Number Sense

KenKen is a self-motivating puzzle that builds number sense. As an added bonus, using this teaching and learning tool builds creative logical thinking and patterns of self-reliance. It also develops strong focus for an extended period of time, a necessary skill in mathematical problem solving. This article spotlights some of the prime factorization, number trees, and number combinations found in KenKen.

KenKen and Sudoku share two of the same rules, which makes an easy transition for teachers and students: numbers in each row and column must not repeat. The third rule sets KenKen apart as being more mathematical than Sudoku. In KenKen there are Cages, which are heavily outlined areas with specific numerical properties.

The most commonly used puzzles for classroom use are 4×4, 5×5, and 6×6. As with Sudoku, KenKen comes in several levels of difficulty. This allows the instructor to introduce the puzzle at the elementary level, and then gradually ramp up the challenge level as students are ready.

Here is a 6×6 KenKen from the 3-7-12 issue of the NY Times at the Medium difficulty level:

What I suggest solvers do is write the possible numbers at the top or bottom as on a number line. This gives a sense of number magnitude and provides a great help when solving.

A. Look at the 9+ cage: How many different sets of three numbers add up to nine? Hint: there is more than one way.

B. Check out the 4+ cage: How many different sets of three numbers multiply to 4? Again, there is more than one way.

C. Now let your eyes rest on a similar region, the 24+ cage, with even more possibilities. Hint: what is the prime factorization of 24, and how many ways can you make a factor tree for 24?

By looking at numerical possibilities, along with other clues from rows and columns, it’s possible to solve this KenKen by logic and reasoning alone, with no guesswork. This is the hallmark of great mathematical thinking!

Every teacher that has talked/emailed me after using KenKen with students is enthusiastic in support of this methodology. Students always look forward to doing KenKens, and it’s always a great addition to a Friday workday, or useful as extra credit. Give it a try, and you will be a hero with your students!

KenKen puzzles can be found for free on several websites. Here are my two favorites, which publish six new puzzles daily: www.nytimes.com/ref/crosswords/kenken and www.kenken.com

The second site also has allows participants to receive regularly via email a nine page PDF file with KenKen puzzles in sizes from 3×3 through 6×6, including answers – something for everybody!

GeoGebra Has New Features – Even More of a Good Thing

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Aug 122009

GeoGebra, the award-winning free software that math teachers are raving about, now has even more neat features:

A Spreadsheet – numbers in a different view!
Automatic animation – your graphs come to life!
Statistics functions and graphs!
Matrix and complex number support!
. . . and more!

Find out about all the details in the GeoGebra 3.2 Release Notes.

All new features are described in the fully revised manual GeoGebra Help.

The school year is starting – a great time to get your students off to a flying start with this wonderful graphic brain food!

Google SketchUp – Add Spice to Your 3D Geometry Lessons

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Aug 122009

Google Sketchup Google SketchUp Add Spice to Your 3D Geometry Lessons

Google Sketchup - image from Wikipedia.org

Here’s another computer tool to bring alive 3D geometry in your classes – you may even encourage students to explore a career in design, construction, architecture, or landscaping!

You can use Google SketchUp to create, modify and share 3D models.

It was designed to be easier to use than other 3D Computer Assisted Design (CAD) programs. SketchUp can also save “screenshots” of the model in common graphics formats.

Dozens of video tutorials, an extensive Help Center and a worldwide user community mean that anyone who wants to make 3D models with SketchUp, can.

Every SketchUp model is made up of just two things: edges and faces. To build models in SketchUp, you draw edges and faces using a few simple tools that you can learn in a small amount of time. It’s as simple as that.

Because you’re working on a computer, everything you create in SketchUp has a precise dimensions. When you’re ready, you can build models that are as accurate as you need them to be. If you want, you can print scaled views of your model. You can use SketchUp’s Paint Bucket tool to paint your model with materials like colors and textures. You can use the Dimension and Label tools to add dimensions, annotations and other details to your work.

Here’s the website to take a look at Google Sketchup.

What Are the Odds of Losing Your Lunch? – A Probability Adventure

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May 172009

Here’s a neat problem recently under discussion in our school district: “Four kids, each with their own lunch, go on a trip. When it’s time to eat, the four identical-looking lunches are mixed up and the kids randomly choose one. what is the probability that exactly one kid gets his own lunch?”

This problem is nice because it can be solved several ways, allowing students to see how others solve the problem. By relating the different methods of solution, students build and strengthen their own problem-solving abilities.

It’s also a nice problem because it reminds us that probability is a subject full of “slippery concepts”; just when you think you’ve solved a problem, it can elude you. Many probability problems are simple, but can be made more difficult by adding/changing just a few words in the posing of the problem. Consider how many high-level mathematicians were fooled by the Monty Hall problem, made famous by Marilyn vos Savant.

One of my KenKen posts asks the question, “How Many Possible Puzzles Are There?” This has resulted in many reader responses and a lively discussion. The question in today’s post may elicit many solutions as well – will one of them be yours?

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Isosceles Triangles Make a Square – Video Project for the Geometry Class

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May 162009

Isosceles Triangles can make a square, as we all know, but here’s a way to do it with seven different size triangles. Measurement, the Pythagorean Theorem, and color make this a nice project for any class, and particularly for a geometry class. The video gives complete directions for the classroom. Pause it at any time to give your students time to finish each stage and to give yourself time to ask clarifying questions.

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Average Rate of Speed – a Simple Solution to a Confusing Problem

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Mar 302009

Ask people the following problem, and almost all of them will get it wrong: “If a car goes up a hill at 20 m.p.h. and comes down the same hill at 30 m.p.h., what is the average rate of speed for the whole trip?”

The “obvious” answer is 25 m.p.h., but that would be wrong. Take a quick look at a slideshow of some of the projects my students have done to see some nice explanations of confounding problems like these.

View more presentations from blombard.

Simply stated, since the car spends more time traveling at the slower speed, this has the effect of reducing the average rate of speed below the average of 20 and 30. The surprising answer to this poser is 24 m.p.h.

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Friday Afternoon Lifesavers – the Teacher’s Best Friend; Part 3, Pythagorean Theorem Cut-Up

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Jan 292009

pythgorean theorem cut up Friday Afternoon Lifesavers the Teacher’s Best Friend; Part 3, Pythagorean Theorem Cut Up

Friday Afternoon Lifesavers are exactly what they sound like: at the end of the week when you and your students are running out of gas, these activities come to the rescue. They are mathematical, engaging, fun, and give the teacher a much-needed break.

Here’s a puzzle that works anytime, but works especially well if your class is studying the Pythagorean Theorem. It will make a nice project as well for kids to make as a paper/pencil drawing or a computer project.

I first came across this at a math conference years ago. It came in the form of five pieces of plastic which could be rearranged to form one square by itself, with the other four pieces forming a square as well. The graphic above shows these five pieces. Then the directions said that it was possible to rearrange the five pieces to form another, larger square. This, of course, is a demonstration of the Pythagorean Theorem.

I’ve included a link to my puzzle file which gives a full-size ready-to-cut-out model of the puzzle. It’s sure to be a winner with your kids. You might consider making a transparency of this for an overhead projector model.

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Schools Turn Algebra into Child’s Play in Lebanon Oregon

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Dec 302008

This post features a video and some excerpts from the article “Lebanon schools turn algebra into child’s play”, by Betsy Hammond, The Oregonian.

While most high schools in Oregon and across the nation struggle to get freshmen to pass algebra, one school district is trying something very different. Lebanon, which educates 4,000 students in eight schools, is pushing algebra on students as early as first grade. And the kids are getting it.

Riverview elementary really teaches math

Visit a Lebanon elementary math class, and you will see:

First-graders set up and solve formulas such as 9 – x = 5, as they did when Raylene Sell talked with her class about “some teddy bears” walking away from the classroom rug, leaving five behind. Students don’t do worksheets, use flash cards or memorize multiplication tables. Yet by third and fourth grade, most of them add, subtract and multiply quickly and accurately.

Lebanon officials are loathe to proclaim their program perfect, noting that math instruction is evolving, that some teachers still use traditional methods and that the biggest payoffs are yet to come.

But they say they are confident that their new approach to teaching math is the way to go.

Among the key elements: Begin simple algebra and multiplication by first grade; have every child talk extensively about his or her mathematical reasoning; let students set up their own problems and equations and allow them to use big numbers if they choose; cover few topics in great depth; use lots of visual and hands-on modeling to make math ideas concrete.

Other tools in the Lebanon math toolbox include lots of visual modeling of math ideas. Plastic blocks represent hundreds, tens and ones; kid-sized balances show ways make both sides equal to balance the scale; number lines make it easy to see that 3/4 and 0.75 mean the same thing.

Lebanon’s approach is in line with recent national reports about what’s wrong with U.S. math classes and how to fix them.

The Center for Data-Driven Reform in Education at Johns Hopkins University reported this month that getting teachers to change their daily teaching practices does more to raise math achievement than buying new textbooks or computerized math programs.

Link to the original article:

Lebanon schools turn algebra into child’s play

Edward Tufte and The Visual Display of Quantitative Information – Data Displayed Properly

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Jul 272008

edward tufte photo1 Edward Tufte and The Visual Display of Quantitative Information Data Displayed Properly

the visual display of quantitative information1 Edward Tufte and The Visual Display of Quantitative Information Data Displayed Properly

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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