An activity chosen by my students during the 2012-2013 school year was to build a level 2 Menger Sponge with playing cards.
The relevant mathematical concepts illustrated by the activity can be found in “Expectations from Principles and Standards for School Mathematics Content Standards: Grade 4.” The three major relevant concepts included working with estimation and multiplication to determine how many cards and how many decks of cards would be needed for the level 2 Menger Sponge project, analyzing attributes of three-dimensional shapes and developing vocabulary to describe the attributes, and recognizing geometric ideas and applying them to problems in the classroom or real life.
When students decided they wanted to build a Menger Sponge, first I required them to figure out how many cards we would need. Second, particularly during the build, students would refer to a “face” and or an edge or a vertex piece. Third, students recognized that each cube needed to be comprised of “mostly” straight folds. Otherwise, the cubes would not fit together to form a level 1 sponge without falling apart. Most importantly, students collaborated at all stages of the project.
This activity was inspired by Dr. Jeanine Mosely’s work building a level 3 Menger Sponge with business cards. The first time I saw the idea was during a presentation in Atlanta, Georgia at Gathering for Gardner 10.
At the end of our project, we used more than 2700 cards.
Happy Pi Day (3/14)! Listed below are a few activities related to Pi! To begin, what do you notice about the date for this year’s Pi Day? Check the cartoon strip and read the date from either side.
Among other diversions for Pi Day, if you are looking to search Pi for any certain number sequence, such as your phone number, birthday, or any other significant sequence of numbers,this fun site searches Pi (a great opportunity to discuss irrational numbers).
Also, if you have extra marbles around, a flat surface, and patience, approximate Pi with an activity found here.
Finally, if you are not aware of Numberphile, you should be. Below is one of the greatest ideas I have ever seen for calculating Pi! Also, take very close note of the time of the video… another sign of clever!
The golden ratio has been a popular topic of conversation in recent weeks. In keeping with the trend, we will look at Penrose Tiles, and specifically, Penrose Magnets. Penrose magnets or tiles are pieces called “kites” and “darts” that can make patterns.
From the Penrose Magnets site: In a large quasicrystal, Kites outnumber Darts 1.618 to 1. This is the famous ratio known as the Golden Ratio (and Phi or Φ). Found in subatomic particles, sunflowers, seashells, and even the spiral of our Galaxy, Phi is at the very heart of nature. Phi is at the heart of our pattern, and is used to both determine the number of tiles needed and the length of the sides of the Darts and Kites.
The black hole number 4 has been previously discussed here. Below is a look at the black hole number 123 (click the image for a larger view):
*Another interesting note: In Italian, the number 3 (tre) is a black hole number.
Recently, Cliff Pickover tweeted about the number 4 being the only number in English have the same number of letters as its value. I tweeted a response that in Danish this is true for the numbers 2, 3, and 4 (to, tre, fire). These numbers are often referred to as honest numbers, which are related to black hole numbers. Also, there are honest expressions such as this one:
This honest expression was created by Michael Burke.
References:
Ecker, M. Number play, calculators, and card tricks: Mathemagical black holes. Retrieved from http://g4gardner.pbworks.com/f/mm-ecker.pdf
Gardner, M. (1976). The incredible dr. matrix. New York: Charles Scribner’s Sons.
O’Shea, O. & Dudley, U. (2007). The magic numbers of the professor. Washington, DC: The Mathematical Association of America, Inc.
Möbius hearts are nice items to make from Möbius strips. Pictured here is what a set of Möbius hearts look like when you have completed them. All you need to make them are paper strips, scissors, a glue stick, and a little focus.
You will need two strips with little slits cut into each end. Twist one of the strips “clockwise” and then glue it together at its ends. The other strip should be twisted “counter-clockwise” or “anti-clockwise,” depending on your location in the world, and glued together.
The strips should then be glued together at 90 degree angles. Next, take scissors and cut each strip through to the end. Your result, if worked properly, will be a set of functional Möbius hearts. Although this can be completed with simple steps and materials, this video is a great resource for more assistance.
For another mathematical heart, check out Wolfram|Alpha’s heart equation.
Finally, if you are not familiar with Jim Steinmeyer’s Impuzzibilities series, you should familiarize yourself with them. As a variation on his “Nine Card Problem,” remove any 9 of the hearts from a deck of cards and spell out the name of the card to arrive at it from the pack of chosen hearts.
A post at The Endeavour brought my attention to something new about a common magic square. The post was the first time I had seen the (common) magic square referred to as Jupiter’s Magic Square. Listed below is an easy way to remember how to produce this magic square (constant 34). First draw a 4×4 grid, then fill in the grid as follows, using consecutive numbers 1-4:
Next work with inserting 5-8, then 9-12:
After inserting 13-16, you will have the following:
