Modeling Decimals and Fractions
Fraction Segment Strips, Segment Strips and Grids, Solving Fraction Computations
In the last few lesson of Course I, segment strips and grids were introduced as a way of modeling fraction operations. You can visit the links in red directly above to view these introductory lessons. In Course II, we expand on some of those ideas, and make some connections between fraction and decimal computations and operations.
In order to make those connections, the red segment strip is used with those previously introduced. It is shown here along with the original six strips.
The red strip is divided in base ten, with the distance between the longest marks equal to ten cm. The distance between two of the medium marks is one cm, and between the smallest marks, one mm.We can compare the lengths of the other strips to this one to get some decimal comparisons. If the white strip equals one unit, 1.0, then the other colors measure as follows:
Approximate measurements:
Using these tools, various measurement and computation situations can be explored on cm grid paper, and segment grid paper (10/12 cm):
By using segment strips and grid paper, the students can see models for the following decimal operations:Area = 2.1 x 1.3 = 2.73 u2 (multiplication)Perimeter = 1.3 + 1.3 + 2.1 + 2.1 = 6.8 u (addition)
Difference between length and width = 2.1 - 1.3 = .8 u (subtraction)
Other situations explored in class:
These explorations help develop models for decimal operations for problems like:
The fraction operations model is reviewed with problems like:
The thrust of all of this is to develop mental models for basic operations with fractions and decimals. By visualizing the various models in their minds, many students become able to perform most basic decimal and fraction computations mentally, and to make good estimates for more complicated problems. There are three models in particular that we develop...
Common fractions can be solved by thinking of dividing a unit rectangle into parts as follows:
Both linear segment strips and base ten area pieces lead to two very useful mental computation strategies, called equal sums and equal differences.
In equal sums, two numbers are changed by adding an amount to one, and taking away an equal amount from the other. Thus, the sum remains unchanged, but the resulting numbers are far easier to compute mentally. For example:
3.57 + 2.86 can be changed to 3.43 (3.57 - .14) and 3.0 (2.86 + .14). 3.43 + 3.0 = 6.4346 + 29 becomes 45 + 30, which equals 75
256 + 398 becomes 254 + 400 which equals 654
In equal differences, two numbers are changed by adding or subtracting the same amount from each so the difference between the two remains the same, but is easier to compute mentally. Here's a couple of examples:
2.03 - 1.97 can be changed to 2.06 - 2.0 (adding .03 to each number). The difference is easy to calculate, .06.23 - 14 becomes 29 - 20 (6 added to each). the difference is 9.
3.7 - 1.8 becomes 3.9 - 2 = 1.9
These are simple examples, but illustrate the value of understanding the concepts underlying basic operations with fractions and decimals. Often, calculations can be performed 'in the head' instead of on a calculator or by using a paper and pencil algorithm.