Mr. L’s Math

Pi is probably the most well-known mathematical constant, and it even has its own special day - March 14 (3.14)! Throughout the world people recognize the significance of pi and its contribution to mathematics in particular and to humanity as a whole.

Each year my students celebrate Pi in several ways. We, of course, have a classroom event where students bring food - anything that’s round! They also memorize pi to 30 decimal places. We have “Pi Day Wordies”, words beginning with the letters “pi” - see link below. And many students create works of pi art - the focus of this post. Here is the link to some of the art my students have created - A Slice of Pi Day Art

Related Post:

40 Ways to Say Hi to Pi - Pi Day Wordies that Honor Pi on Its Birthday, March 14, 2008

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Graphic from Wikimedia Commons, a repository of media licensed under the GNU Free Documentation License.

Every year I ask students to memorize 30 decimal digits of pi. When people ask why, I offer several reasons based on improving student performance and ability. Here are some thoughts on the subject.

It helps students focus on a single task and do it well.

Students need to be able to work on a single thing and become better at it over time. It gives them the experience of setting a goal and then achieving it. When an entire class has the same goal of learning the digits of pi, it has a strong effect on motivation and gives students the feeling that they can do it.

It helps improve your memory.

Students find ways of remembering the digits of pi by using analogies of remembering phone numbers. Knowing 30 digits of pi is like knowing three phone numbers (including area codes). Repetition is necessary for students to gain improvement, and this is a necessary component of success in future life. Many students willingly commit hours on a daily basis to sports and other activities, and devoting time to their minds will pay long-term dividends.

It exercises your brain and keeps it in shape.

People exercise their bodies to stay physically in shape, so it makes sense to treat your brain in the same fashion. Daily practice of memorizing digits of pi allows students to improve their brain’s functioning ability and creativity. The use of mental aerobics such as this activity will make students’ brains stronger.

Here’s a Pi Memorization Poem to help with the task:

Now I will a rhyme construct (3.14159)

By chosen words the young instruct (265358)

Cunningly devised endeavors, (979)

Con it and remember ever. (32384)

Widths of circle here you see. (626433)

Sketched out in strange obscurity. (83279)

(use the number of letters of each word for the digits of pi)

Related Post:

40 Ways to “Say Hi to Pi”: Pi Day Wordies - Honoring Pi on Its Birthday, March 14, 2008
A Slice of Pi Day Art - A Tribute to the Mathematician’s Favorite Day, March 14 (3.14)

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After students become familiar with pentominoes, they are ready to use them to explore some algebra. Teachers may present this activity either using the white board, or preferably, with the overhead projector and some pre-made transparent pentominoes (colored for easy location) when projected. Using a transparency grid (half-inch or one centimeter grids work well) and pentominoes cut to the same size grid make a great visual aid that can be used over and over for demo purposes. Since defining ordered pairs always begins with the origin, it’s nice to also begin pentomino graphing with one shape located with a vertex at the origin and sides aligned with the axes. Using the simplest shape, the I, record the coordinates of the four vertex points. This reinforces the concepts that ordered pairs located on the x-axis are always of the form (x, 0) and points located on the y-axis are of the form (0, y). Some simple ideas to extend this first task involve asking students: 1. How would the coordinates change if the I shape were rotated 90 degrees clockwise about the origin (keeping one vertex fixed at the origin)? 180 degrees clockwise about the origin? 90 degrees counter-clockwise about the origin? 2. How would the coordinates change if the I shape were translated 5 units to the right? 3 units to the left? 4 units down? 6 units up? These questions/explorations are important to ask/do since they build up visualization and prediction skills. One of the most essential things we can do for students is to give them immediate feedback on their progress; this allows them to gain confidence they are moving along the right track. Next, select another pentomino and place it so that it lies entirely within the first quadrant. This is good pedagogy since kids will be working entirely with positive numbers. After gaining confidence they can move along to quadrants 2, 3, and 4. After working entirely within these quadrants, you may want to place a pentomino so that it lies partially in two quadrants.

Subtraction of integers is another basic skill that can be incorporated into this activity. Asking the length of a pentomino’s side is equivalent to subtracting the integer coordinates.

Related Posts:

Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon Golomb

Pentomino Puzzles; Spatial Sense, Geometrical Visualization, and Reasoning Skills

Pentomino Tessellations: Tiling the Plane Helps Build Problem-Solving Skills

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Pentominoes are wonderful creatures. They are fun to create, they can be colored by students, and there are a multitude of questions that can be posed, and many activities to pursue to stimulate and strengthen students’ minds.

This activity asks students to use a single pentomino to tessellate the plane. In other words, can you use only one shape as a floor tile and fit it together to cover the entire floor of a room? Some of the pentominoes can be tessellated quite easily, like the letter I, while others, like the letter F, may take awhile to fit together into a repeatable pattern.

This activity is best used after students have explored and discovered all 12 pentomino shapes. Depending on the age of the students, use 1cm, 1/2″, or 3/4″ grid paper to record results. Coloring the shapes gives a nice touch to the work. Once students find a way to tessellate a given pentomino, consider challenging them to find a different way to arrange the given shape to cover the plane. Students who are good at spatial visualization are better problem solvers. Presenting material in a visual way is always a good approach since it gives the class a common reference point when reviewing.

Here is a link to some tessellations my students created. This can be viewed as a slide show: Pentomino Classroom Tessellations

Related Posts:

Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon Golomb

Pentomino Puzzles; Spatial Sense, Geometrical Visualization, and Reasoning Skills

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Yesterday’s post talked about polyominoes; pentominoes, using 5 squares, are the polyomino that has gotten the most popularity due to their versatility in generating puzzles. Martin Gardner popularized them in 1957 in his math column in Scientific American magazine.

Pentomino puzzle pieces are simple to make; using card stock (or an old manila folder), create pentomino pieces having an edge length of anywhere between 1/2″ to 1″ depending on the ages of the children in your class. Using colored paper lends a nice touch to the pieces.

The Centre for Innovation in Mathematics Teaching contains a very fine collection of pentomino resources and puzzles. The graphic above is from their site. The 10 pages of puzzles on the site are organized into several groups of related puzzles of varying degrees of difficulty.

These puzzles build spatial sense and geometrical visualization and allow students to create conceptual models. Research says that problem solving abilities increase when students gain spatial reasoning skills, and doing so in such a creative way brings joy to the students and the classroom.

Related Post:

Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon Golomb

Give Pentomino puzzles a try in your classroom; they’re sure to be a hit with your students!

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Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon Golomb is the definitive book on the subject by its creator who brought these wonderful objects to the mathematical world in 1953. Polyominoes are formed by groups of squares joined along their edges. Single squares are called monominoes. Two squares form the familiar dominoes. Then we have, in order, Trominoes (3), Tetrominoes (4), Pentominoes (5), Hexominoes (6), and so on. The graphic above shows the 12 distinct pentominoes.

Some of the simpler problem investigations teachers can ask involve finding out how many distinct tetrominoes and pentominoes there are, not counting rotations/reflections as different. Other interesting questions involve classifying polyominoes according to the type of symmetry they have.

The amount of significant mathematics developed concerning polyomininoes is simply amazing, and a large number of websites feature these rectangular wonders. There are some types of Sudoku that use polyomino-shaped regions on the grid. The game Tetris is based on seven tetrominoes (some are reflections). The graphic below shows some of the ways pentominoes can be formed into rectangles.

Links to further information:

Wikipedia article on Solomon Golomb

Google Books article containing preview pages, website links, and places to purchase

Give polyominoes a try in your classes and see why they have been so popular over the years!

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Sudoku is popular among math teachers since the puzzle requires only logic to solve, and logic/math reasoning is part of any mathematics class, often appearing in state standards. Most of the sudoku puzzles appearing in newspapers, like crossword puzzles, are of the 9×9 variety. These are normally too large for teachers to use when introducing sukoku to their students. Enter the 6×6 puzzle - some of the reasons kids and teachers like these puzzles:

* No language or math skills needed

* Easy to learn and easy to build skill

* Fun and satisfying when completed

* Good for new users and “old hands”

These are quite a bit simpler and thus take less time to complete. All six numerals, 1-6 must appear in each column, row, and 2×3 box.

A nice website to find a huge number of ready-made 6×6 puzzles is Sudoku Place. The author, Mark Danburg-Wyld, has these puzzles sorted into three categories: those with 20, 16, and 12 clues, and this determines the level of difficulty.

For those wishing to go to the next level with students or for themselves, Sudoku on Wikipedia has a nice article with techniques to use for solving these puzzles and a large number of links.

Related Post: For Firefox Users: Do It Yourself Sudoku Solver - a Great Visual Showing Progress Toward the Solution

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Adobe Acrobat Reader is the software of choice for documents that I make available for download at Mr. L’s Math since it’s a universal web standard. What follows is the “quick & dirty” guide to getting the most out of the files available for download at mrlsmath.com.

1. Adobe pdf files look exactly like original documents and preserve text, drawings, and color graphics, regardless of the software I used to create them. This is important since you want materials to look their best. Most of the books that my partner, Brad Fulton, and I write are created in MS Word and it takes quite awhile to get the look we want. Presentation files are either created with MS PowerPoint or Open Office Impress. No matter what the source file is, the pdf file created will look exactly the same at your site.

2. The files are searchable - you can use the Find command on your web browser for full-text search features to locate words in documents. In Internet Explorer or Firefox, the web browsers with the most users, Ctrl-F will bring up a search window. If you saw something and can’t remember where it was, use this search feature. You can also access the Find feature from the Edit command in the toolbar.

3. The files are also accessible - Adobe pdf documents have the ability to help make information accessible to people with disabilities. Since there are many versions of the Reader, I can’t give detailed directions for some items; just take a good look at your screen when viewing a pdf document. The View options give quite a lot of choices. There is the Full Screen View option which is nice for viewing Presentation-type documents; the pdf file will then act much like a PowerPoint presentation. You can also increase/decrease the percentage of zooming of the document to fit your taste or those of other participants.

4. You can select and copy text from a pdf document to create a document of your own. Combined with the Search (Ctrl-F) tool, this is a great way to make a document from online information. You can also Save a Copy of the pdf file for future use or for emailing to a friend as an attachment.

5. And you can print the pdf file, in whole or in part. Most of the files available at Mr. L’s Math are designed with the teacher in mind and have Overhead Transparencies for classroom use. You can print the activity files for reference, and print certain pages for transparencies. If you have a color inkjet, you can print some of these in color as a bonus.

The web and its tools make it easy to copy content, and it’s always a good idea to remind your students about copyright information. Give credit by citing sources, and make use of public domain information whenever possible. I show Wikipedia for encyclopedia content and Wikimedia Commons for images in class as example of great information that is freely available for their use.

If you wish to upgrade your version of Reader, here is the download site: Adobe Acrobat Reader. Adobe maintains a large site with many options - take a look around and enjoy the visit.

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Students will be so amazed at the patterns they find in the hundreds chart they will think you are a “mathemagician“. Once hooked, they will use algebra to understand the magic.

A favorite activity of my partner, Brad Fulton, and mine whenever asked to do a workshop for teachers, this chart can be found in primary grades to teach skip counting, multiples, factors, prime numbers, and pattern recognition.

We show you how to incorporate simple arrangements of numbers and transparent chips to arouse the curiosity and imagination of students with number sense that leads naturally to algebraic proofs that all students can see and understand.

Numerically, in the second graphic the sum of the blue chips equals the sum of the violet chips. This arrangement of chips leads to the same sums no matter where this arrangement is placed. This leads algebraically, to a simple proof. If x is allowed to equal the number 2, then x+1 equals 3, x+10 equals 12, and x+11 equals 13. Adding the blue chips and the violet chips results in the same sum of 2x+11. Since this works every time, you have a simple algebra proof. Other proofs are outlined in the activity. You can probably find more after playing with the chart.

The six-page downloadable pdf activity will print well at your site, so you can make Activity Masters and Overhead Transparency Masters for all your classes. Pick up a few transparent colored chips and you have materials for lesson plans any day of the year. Send a comment and let me know how you do with your classes!

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Most students are familiar with the “dirt equation“: D = R x T. But applying this simple equation to a nontraditional problem leaves most kids (and adults) wondering what happened.
Take the case of average speed - if you take a round trip going in one direction at 20 m.p.h. and returning at 30 m.p.h., the large majority of people will figure that the average speed is 25 m.p.h.
But this is incorrect - the actual average rate is 24 m.p.h. The calculation is quite simple, but it takes logical, step-by-step thinking; this is the specialty of mathematics. Let’s take a concrete case. Suppose there are two towns, A and B, 60 miles apart.

Then the trip from A to B at 20 m.p.h. takes 3 hours, but the return trip at 30 m.p.h. takes only 2 hours. So since our definition of average speed is Total Distance ÷ Total Time, we get 120 miles ÷ 5 hours, or 24 m.p.h.

This disturbs students the first time they see it. This is actually a good thing because you will then have their attention when explaining the seeming logical fallacy. In simple terms, what happens is that the traveler spends more time moving at the slower speed than at the faster speed, thus causing the average rate to be closer to the slower speed.

When explaining this concept in class I have had to go over this scenario several times with differing rates and distances until the kids get the idea. In each case, the average rate of speed is less than simply taking the average of the two rates.

This makes a wonderful project for class. I’ve included two for you to look at. At conferences and workshops teachers report that the student projects I share are one of the highlights - they like to see what students are capable of doing, and they like doing the projects back at their school with their own kids. I’ve trained my students with some simple concepts to do projects on the computer. After they see the basic ideas of formatting their work, their creative juices take over and the results are really quite nice. I hope you and your students enjoy them; if you have a projector in class you can display them for students to see and discuss as a source of curriculum.

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