Here’s a beautiful combination of math and art as well as demonstration of power and versatility of GeoGebra created by John Golden. He writes: “Thinking about ways to make a nice polygon spiral. This one is created by rotating and dilating the point that determines the first rotated polygon. (Using matrix exponentiation to rotate around the origin.) I love the spiral and 3D effect of the transformation.” This is found on the GeoGebra Tube site. Here’s the direct link to John’s creation. NOTE: the double arrow at the upper right of the app resets the figure.
Here’s a simple GeoGebra applet created using the “RandomBetween” function. You can refresh the screen with a new problem by using the double arrow in the upper right hand corner. Credit for this idea is due Steve Phelps from the Ohio GeoGebra Institute.
Here’s a quick intro to GeoGebraTube, a nice resource for Common Core teaching materials.
GeoGebra is dynamic mathematics & science software for learning and teaching from elementary school to university level, which contains interactive geometry, algebra, statistics, and calculus abilities. There are, as of July, 2014, almost 100,000 free materials available. You can search by type of material, age, and language.
After using the search function, you can select an entry, find some info about the sketch, see related sketches, and check out the author. If you find an author you like, you can visit his/her page for more materials.
You can display and play with any of the materials, and can also embed the GeoGebra applets in your own website as well. It’s also quite easy to customize any applet for use in a classroom setting; sometimes there are extra features that are not needed and can be turned off thus simplifying the experience for the user.
Here’s a proof of the Pythagorean Theorem showing a way to use translation as a visual demonstration by Steve Phelps, who is the chairperson of the GeoGebra Institute of Ohio:
Pythagorean Theorem Proof Without Words
Drag each colored point to the other point of the same color.
How does this demonstrate the Pythagorean Theorem?
Would this constitute a PROOF of the Pythagorean Theorem?
Note: the double arrow in the upper right lets the user reset the applet.
Voronoi Diagram Animation using GeoGebra
Here’s a GeoGebra animation shared by Michael Borcherds on GeoGebraTube. It’s the type of visual imagery that stimulates students to want to be able to create the same thing, and asks them to uncover the coding steps behind the colorful patterns. You can stop/start the animation using the button at the lower left of the diagram, and relocate the starting (seed) points for more creativity. Reset the animation using the double arrow at the upper right. Enjoy!
For those looking for a little more of the mathematics behind this, here’s what Wikipedia has to say:
In mathematics, a Voronoi diagram is a way of dividing space into a number of regions. A set of points (called seeds, sites, or generators) is specified beforehand and for each seed there will be a corresponding region consisting of all points closer to that seed than to any other. The regions are called Voronoi cells.
Here’s a great reason to use KenKen Puzzles
Here’s an edited version of an email I received today:
This month, KenKen is being featured by the National Council of Teachers of Mathematics (NCTM) for its powerful attributes as a math learning tool. The NCTM’s December 2013 Mathematics Teacher Journal explains how KenKen puzzles allow students to explore basic operations, factors, parity, symmetry, algebraic thinking and various problemsolving strategies….all while keeping students interested and making math fun! Here’s the link to the seven page article: http://www.nctm.org/workarea/downloadasset.aspx?id=40092
As a KenKen advocate, you know that KenKen puzzles can be solved in multiple ways and used for a vast variety of purposes, which is why they are so powerful in the classroom. Read this impressive article written by three leaders in the field of mathematics and education. And show your nonKenKen’ing colleagues. They should be playing KenKen too!
SHARE! Your colleagues can join KKCR for free at:
http://www.kenken.com/
(sent every Friday morning)
Today I updated mrlsmath.com to use the WordPress Suffusion theme.
This is a free theme and can be found here:
http://aquoid.com/news/themes/suffusion/
Here’s the blurb from the WordPress.org site: http://wordpress.org/extend/themes/suffusion
An elegant, responsive and versatile theme with a powerpacked set of options and semantic HTML5based markup. It supports MegaMenus, custom layout templates, advanced support for custom post types, customizable dropdown menus, featured sliders, tabbed sidebars, a magazine layout and lots of enhanced widgets for Twitter, Flickr, Google etc. It has 19 widget areas, onecolumn, twocolumn and threecolumn responsive, fixedwidth and flexiblewidth layouts, 9 predefined templates and 19 predefined color schemes. Responsive capabilities are switched on by a single click. RTL support is builtin and translations in many languages are available. Custom Post Types and Custom Taxonomies are integrated. Compatibility packs for BuddyPress, bbPress, Jigoshop and WooCommerce are available as plugins for smooth integration. Support forum at http://www.aquoid.com/forum.
I came across a terrific website today, http://www.openculture.com/ and spent quite a bit of time there. This is a treasure trove of materials for educators and lovers of knowledge. This post serves as an introduction to the site, giving its Mission Statement and the background of its lead editor. From the website:
What is Open Culture’s Mission?
Open Culture brings together highquality cultural & educational media for the worldwide lifelong learning community. Web 2.0 has given us great amounts of intelligent audio and video. It’s all free. It’s all enriching. But it’s also scattered across the web, and not easy to find. Our whole mission is to centralize this content, curate it, and give you access to this high quality content whenever and wherever you want it. Free audio books, free online courses, free movies, free language lessons, free ebooks and other enriching content — it’s all here. Open Culture was founded in 2006.
Who is Behind Open Culture?
Dan Colman, the lead editor, is the Director & Associate Dean of Stanford’s Continuing Studies Program. Before that, he served as the Managing Director of AllLearn, an elearning consortium owned by Stanford, Oxford and Yale, and as the Director of Business Development and Editorial Manager at About.com. He received his PhD and MA from Stanford, and his BA from the University of WisconsinMadison. The common thread running through his career is his interest in bringing relevant, perspectivechanging information to large audiences, often with the help of the internet. Get his full bio here. You can reach Dan at mail@openculture.com.
The site offers many free books, courses, and videos for educators and students. I hope you enjoy your visit there as much as I did!
The Structure is the Standards
This is taken from an essay by Phil Daro, William McCallum, and Jason Zimba, February 16, 2012
“You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you
get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards.”
The full essay can be found here. It’s a good read and does a nice job explaining where we are, and where we’re heading in math teaching.
Many (most?) teachers present material straight from the book, and may not realize how this method fragments the material that is already challenging for students. If there is an overarching theme and approach to the methodology of teaching mathematics, more students would probably understand and succeed at mathematics better. Here’s hoping . . .
NOTE: The following post, with modifications, was submitted to the California Math Council ComMuniCator for the June 2012 issue with the title: “GeoGebra Promotes Equity in the Mathematics Classroom.“
I had been looking several years for good classroom mathematical modeling software; when I discovered GeoGebra, it was an memorable day. I sat with a laptop in my recliner and didn’t get up for three hours, just playing and exploring. Whatever I asked GeoGebra to do, it did with ease and precision. Since then I’ve used GeoGebra in all my classes, given several conference presentations and workshops for districts, and become a Resource Manager and Trainer for the California GeoGebra Institute.
GeoGebra is well suited to promote equity in the math classroom:

It is free, opensource software

No internet connection is necessary

No programming experience is needed

It is intuitive, userfriendly, and easy to use

GeoGebra has been translated into 52 languages

It can run on all platforms since it is written in Java

There is a version for new/younger learners and users

There is a worldwide network of support from a user forum

It creates multiple representations as equations, graphs, tables

It can be used from primary grades through college level courses
In spite of affordability (FREE) of this software, implementation of GeoGebra is still not widely seen. When a teacher uses a software tool, it is much more likely that students will use it. Here’s my story about introducing GeoGebra in my classroom: I started the program and let students view it using my LCD projector; then I asked a student to come up and sit at my desk. I instructed her to press certain icons to create a diagram, and within two minutes she had stopped listening to me and was running the program on her own – it’s that easy! After giving a basic introduction to the program, all my classes were then able to visit the campus computer lab and create, explore, learn, and play with the mathematics in multiple representations.
Students easily use calculators because they are familiar with the mathematics underlying the keys they press, and so I’ve been able to teach graphing calculator skills and mathematics concepts at the same time. It’s possible to do the same with GeoGebra, since there are few barriers to entry. Teachers accept and promote the use of calculators in learning many concepts in mathematics; we need to promote GeoGebra as well in a similar, but stronger fashion, since the potential for learning is much greater.
GeoGebra was designed specifically for teaching mathematics, and its rich multiple representation environment invites exploring and creating virtual models and simulations. There is a large international support system of educators and an increasing number of articles, videos, books, and readymade materials for classroom use. Those interested in the growing STEM education and collaboration movement will find this technology to be a welcome breath of fresh educational air. To give you an idea of the popularity of this software, there were more than 7 million visits to the main GeoGebra website, www.geogebra.org, in 2011; the visitors came from 226 countries and territories.
We need to provide free and open access to powerful mathematical modeling tools and strong mathematics curricula to all students and educators without regard to age or background. My verdict: GeoGebra is a sure winner, and I strongly encourage all teachers to get comfortable with this software so their students will be better able to develop mathematical minds with this easily accessible and friendly tool. Enjoy!
KenKen and Number Sense
KenKen is a selfmotivating puzzle that builds number sense. As an added bonus, using this teaching and learning tool builds creative logical thinking and patterns of selfreliance. 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 3712 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!