base number systems
The maximum quantity that a container holds is called the base. It is very easy to recognize full containers and therefore the base. The quantity within an unfilled container always yields the number of ones. This is called the one’s place. The quantity of full containers in a larger unfilled container of containers always yields the number of base units (also referred to as lines). This container of containers is the first number that is must be represented by place symbolically. This is the one-zero’s place, which is incorrectly called the ten’s place. This place is the ten’s place ONLY for the base A. The next place value is a container of squares (the one-zero-zero’s place), and the next is a container of cubes (the one-zero-zero-zero’s place). After that, the pattern repeats with a line of cubes, a square of cubes, and a cube of cubes (the sixtth power of the base system.).
A few key activities that can be performed using the Base Number System Charts (BNS Charts) and one centimeter colored cubes.
1)Distribute the same quantity of cubes on different BNS charts and discover the different numbers they are in each base. 86A = 1059 = 1268 = 1527 = 2226 = 3215 = 11124 = 100123 = 10101102
2)Distribute specific quantities (base ten(A)) to specific bases and discover how they are all the same number: an inter-base pattern! 173A is 2129, 138A is 2128, 107A is 2127, 80A is 2126, 57A is 2125, 38A is 2124, and 23A is 2123 These all subquan to 212 which means there are two squares, one line and two centimeter cubes left over. This is written algebraically as y = 2x2 + 1x + 2, where y is the quantity base A (ten) and x is the value of the base.
3)Discover the cube, square, line relationship by arranging place values of one into these shapes. For example: 1000x forms a x by x by x cubic shape. 100x forms a x by x square shape. 10x forms a linear shape x units long. Use these special cases to create and learn values in the Zeroes Tables.
4)Discover what happens to 1000x, 100x, and 10x when you subtract one.
5)Discover what happens to xxxx+1, xxx+1, and xx+1 when you add one.
NOTE: Students should always put the manipulatives back into a base ten formation when an activity is over to quickly verify that all the manipulatives are accounted for (i.e. none are on the floor). This is only necessary until the facilitator and student learn how to quickly convert between various BNS. The BNS Charts are flexible to facilitate putting the cubes back into a physical container.
The Base Number Sheets are 18” x 24” and are dimensioned to support 1 cm cubes. However, they may be scaled down to fit on standard paper for viewing.
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