History of Numeration Systems

I just stumbled upon this great little video about Ancient Numeration Systems.  It does not go into depth on any particular system, but it wanders through the following:

  • Tally marks
  • Sumerian symbols
  • Babylonian symbols
  • Egyptian symbols
  • Roman symbols and modifications of it
  • Number systems based on the body (Zulu)
  • Commerce-based number systems (Yoruba in Nigeria)
  • Number systems involving knots and string (Persians, Incans)
  • Numerals 0-9 (invented in India)
  • Place value
  • Fractions as a solution for “fair-share” situations in culture
  • Unit Fractions (Egyptians)
  • Fractions with base-60 (Sumerians and Babylonians), still used for time measurements today
  • Abacus (Chinese)
  • Use of the “bar notation” in modern-day fractions
  • Computation by the double-half method (Russian)
  • Computation by a doubling procedure (Egyptian)
  • Computation by an abacus (Europe and Asia), the “handheld calculator of its day”
  • Introduction of Arabic Numerals in Europe
  • Importance of mental math algorithms to check for reasonableness

This would be a great introduction video to a unit that involves Numeration Systems.

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Collection of Math Games

To view the collections of Math Games, hover over the Games Menu, and go to one of the dropdown categories.

 

 

 

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Signed Numbers: Colored Counters in a “Sea of Zeros”

The “colored counter” method is an old tried-and-true method for teaching the concept of adding signed numbers.  However, to show subtraction with the colored counter method has always seemed painful to me … that is, until I altered the method slightly.

Now all problems are demonstrated within a “Sea of Zeros” and when you need to take away counters, you can simply borrow from the infinite sea.  Voila!  Here’s a short video to demonstrate addition and subtraction of integers using the “Sea of Zeros” method.  You can print some Colored Counter Paper here.

Video: Colored Counters in a Sea of Zeros

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Playing to Learn Math (new version)

I am at the Kansas City Math Technology Expo this weekend doing two talks.

Today’s talk was Playing to Learn Math? I gave this at TexMATYC in the spring, but just updated it to add some non-digital types of play that you can use in the classroom.  There are five great math games mentioned in this presentation. Direct links to these games are below:

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Prime Number Manipulatives

tcm_blog_button2For those of you who are curious what we were actually doing in class in yesterday’s post (Record with a Document Camera and a Flip), we were “fingerprinting” the factors of composite numbers with their prime compositions. The wooden “prime number tiles” are created using the backs of Scrabble tiles and the white tiles you see on the board are the student version of the prime number tiles (each of them cut out a sheet of their own prime number tile manipulatives at the beginning of class).  Just for the record, I would like to confess to destroying more than 5 Scrabble games in the last 6 months.

I tell students that the prime numbers are like the building blocks of the whole numbers, in the same way that the nucleotide bases (adenine, guanine, cytosine, and thymine) are the building blocks of DNA. It was ironic, then, that at the end of our prime factorization fingerprinting, the board looked (to me) the way an agarose gel electrophoresis looks.

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You can also use the prime number tiles to line up the prime factorizations of two numbers (leaving spaces when the prime is not in both numbers) and read the GCF as the intersection of the two lines of tiles and the LCM as the union of the two lines of tiles.

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I’ve tried explaining these concepts in writing before (circling common factors and using colors or highlighting to show the steps), but the ability to easily rearrange the prime numbers after they’ve been found makes all the difference.  Students find the prime factorization using a factor tree or short division, then place out all the tiles for each number.  At this point, they can rearrange each row into least-to-greatest order.  Then line up the primes, bumping the whole row when there is no overlap in primes between the composite numbers.

We also were able to look at more interesting problems now that students could “see” the inner workings of the factorizations better.  For example, find a pair of numbers with a GCF of 42 and an LCM of 4620 (you may not use the “trivial” answer of 42 and 4620).  By constructing a “board” of prime number tiles, starting with the factorization of the GCF and the LCM, students begin to see which primes have to be in the two numbers and which can only be in one of the two numbers.  Note that there will be more than one answer here … which is another interesting discussion to have!

I’ve never seen “prime number tiles” in any math manipulative kit … they were just something I created last semester.  I think this method of using Prime number tiles would also be helpful with explaining how to find the GCF in the factoring unit of algebra, and with explaining how to find the LCD in fractions.

Feel free to print the prime number tiles and use them in your own classes if you’d like.

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