BHMS Math Enrichment Club
2019-2020
Week 1: Ancient Number Systems
In our first meeting, students watched and discussed a slideshow (at right), which explained the early history of humans and mathematics. The earliest examples of humans using mathematical thinking are found in preserved animal bones that have purposeful groupings of scratches or tally marks. Some examples date back tens of thousands of years.
We also explored early number systems, such as the "base 60" system used in ancient Babylon, and early number systems used by Greeks, Mayans, Hebrews, and eventually the evolution of the Hindu-Arabic numerals we use today. We focused on the Roman Numeral system for our "challenge" assignment, which we will review at our next meeting. If you missed it, the file can be downloaded at right, along with my favorite Math Brainteasers. |
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Week 2: Number Systems- Base-5, Base-2 (binary), base-16 (hexadecimal)
The Hindu-Arabic numerals, which are used in America and much of the world, are a base-10 or "decimal" system. That means there are a total of ten symbols that can be used: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Notice that in a base 10 system, there is not an individual symbol to make a 10. Instead, you have to "regroup" and use a combination of other symbols.
In the same way, a base-5 system would have only five symbols (0, 1, 2, 3 and 4). To make the number "5", you have to regroup after the fourth symbol, and "five" would be written as "10" (which means "one group of five, and zero groups of ones"). Digital technology uses a "binary" base-2 system. The only symbols are 0 and 1. Computer scientists and programmers also use a system called "hexadecimal" (base-16), which has 16 symbols, and can be conveniently translated from the base-2 system, which can be difficult to manage with the long strings of 0's and 1's. If you would like to explore further and practice translating between different systems, click this link to visit an interactive site that makes this easy to do! Use it to check your answers to our (optional) assignment at right. |
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Week 3 and 4: Learning the Buzz-Fizz Math Game
Buzz-Fizz is a long-time favorite math game for many students. The game requires intense concentration, and the ability to analyze numbers "on your feet". For many of our Math Club students, the game was a new experience, and we will repeat it from time to time this year, as they gain skill.
Essentially, Buzz-Fizz is a simple counting game- students take turns announcing the next number in sequence as we stand in a circle. The trick is that we substitute the word "Buzz" for any number that is a multiple of 7, or has a 7 in it (like 17). We substitute "Fizz" for any number that is a multiple of 5, or contains a 5. For numbers like 70, in which both rules apply, the student says "Buzz-Fizz". As we increase in skill, we can easily raise the difficulty level by substituting for other numbers: "Bang" for 3's, "Boom" for 2's. (See file at right for complete directions.) |
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Week 5 and 6- Archimedes, Exponents and "The Sand Reckoner"
Archimedes was a great scientist, mathematician, engineer and astronomer. He lived about 2,300 years ago in the city of Syracuse, which is on the Mediterranean island of Sicily. Among his many accomplishments, Archimedes devised a way to work with very large numbers, similar to the way we use exponents and "scientific notation" today. To demonstrate this ability, Archimedes decided to calculate the number of grains of sand that would fill the entire universe. He described his procedure in a book (scroll) called "The Sand Reckoner" (to reckon means to count or calculate).
We used the same principles to repeat Archimedes' calculations. First, to demonstrate the basic idea, we calculated the number of tennis balls needed to fill our classroom. (A shoebox would hold 36 tennis balls, and we calculated that 1,742 shoeboxes could be stacked in our classroom with all furniture removed. Therefore our classroom will hold about 62,700 tennis balls!) Small trays were prepared containing exactly 0.1 gram of sand. Students took about 10 minutes to count and average these grains: approximately 115 grains in 0.1 grams. We multiplied up from there:
Multiplying out our grains of sand, our final answer is 3.09 x 10^86 grains. That's 309 with 84 zeroes, also known as 308 septenvigintillion! Whew! That's a lot of sand. |
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Week 7: The Tower of Hanoi, and Recognizing Number Patterns
The human brain has an incredible ability to recognize patterns in the world around us. Scientists think this probably evolved as a survival advantage- although we often recognize patterns even when there isn't one, such as imagining shapes of animals in clouds or constellations. When presented with a list of numbers, many people automatically look for a pattern in the sequence. We discussed the pattern of solutions to the Tower of Hanoi puzzle (pictured at right), and we identified a simple, mathematical sequence that could be used to predict the necessary moves for any number of given disks:
We looked at several other interesting mathematical patterns, including the famous Fibonacci Sequence. Then we looked at a video on Vi Hart's Youtube channel, in which she investigates the Fibonacci Sequence, and how nature uses it to produce perfect spiral patterns on plants and animals. |
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