Base 5 Grouping Models
Numeration - the representation of quantity with symbols - is the foundation to the number sense so necessary to mathematical understanding. Especially important in estimation and computation, it is at the root of many students difficulties in these areas. By examining this numeration process in the unfamiliar world of base 5 notation, the student is made to think about the concepts involved, free of misconceptions held in the familiar world of base 10.
The basic model for base five looks like this:
In this model, the unit represents 1, the strip represents 5, and the mat represents 25. For example, to represent the quantity 39 with base five pieces, a collection like this would be formed (We'll use the minimal collection - the collection that uses the fewest number of pieces - rather than using 39 units or 7 strips and 4 units):
The
base 5 notation for this collection would be 124base five
(1 mat, 2 strips, and 4 units)- the base 5 number that represents the
quantity 39. In the notation, the symbols representing the pieces are
arranged from largest to smallest.
Here are some other examples of base 5 notation, can you fill in the ???:
Speaking of larger and smaller, what would some other pieces look like? You may have noticed that, beginning with the unit, the pieces are formed of multiples of 5. This pattern can continue in both directions. Getting larger...
,,,and so on.
The pieces can also represent fractional quantities...
This pattern can continue indefinitely, as well, to very small quantities. No matter which direction we go, we're always combining five pieces or breaking up a piece into five parts. Perhaps you've already noticed a pattern in the shapes of the individual pieces...
Here are some final ideas to ponder...
In the next lesson, we'll explore some other bases, and place value.