Bits and Pieces unit goals:
Bits and Pieces I was created to help students:- Build an understanding of fractions, decimals, and percents and the relationships between and among these concepts and their representations
- Develop ways to model situations involving fractions, decimals, and percents
- Understand and use equivalent fractions to reason about situations
- Compare and order fractions
- Move flexibly between fraction, decimal, and percent representations
- Use 0, 1/2, 1 and 1-1/2 as benchmarks to help estimate the size of a number or sum
- Develop and use benchmarks that relate different forms of representations of rational numbers (for example, 50% is the same as 1/2 and 0.5)
- Use physical models and drawings to help reason about a situation
- Look for patterns and describe how to continue the pattern
- Use context to help reason about a situation
- Use estimation to understand a situation
The Mathematics in Bits and Pieces I
In this unit, students will meet several interpretations and models of fractions. These have been carefully chosen so that the move between problems will add to a deepening knowledge and comfort with fractions.
The major interpretations on which this unit focuses are
- fractions as parts of a whole
- fractions as measures or quantities
- fractions as indicated division
- fractions as decimals
- fractions as percents
Other interpretations—such as fractions as operators ("stretchers" or "shrinkers") and fractions as rates, ratios, or parts of a proportion—are postponed until later grades.
This interpretation of rational numbers is applied in situations that are continuous and in situations that consider discrete objects. The important characteristic is that this interpretation depends on partitioning an object or a set into equal-size parts and making a comparison of some of the parts to the whole object or set. For example, if there are 27 students in the class and 13 are girls, the part of the whole that is girls can be represented as In the following diagram, two parts are shaded.
The shaded portion can be represented as 2/3. The 3 tells into how many equal-size parts the whole has been divided, and the 2 tells how many of the equal-size parts have been shaded.
In the part-whole interpretation of fractions, the difficulties for students center on the following:
- determining what the whole is
- subdividing the whole into equal-size parts-not equal shape, but equal size
- recognizing how many parts are needed to represent the situation
- forming the fraction by placing the parts needed over the number of parts into which the whole has been divided
In this interpretation, a fraction is thought of as a number. For example, a fraction can be a measurement that is in between two whole measures. Students meet this every day in such references as 2-1/2 feet or 11.5 million people. Understanding this interpretation is important for students' mathematical development, and it leads to comparison of fractions and operations on fractions.
To move with flexibility between fraction and decimal representations of rational numbers, students need to understand how fractions can be thought of as indicated divisions. Sharing is a natural context in which to help students see how this interpretation is related to whole-number division. If students see that sharing 36 apples among 6 people calls for division (36 ÷ 6 = 6 apples each), then they can move to an understanding that sharing 3 apples among 8 people calls for dividing 3 by 8 to find out how many each person receives.
A byproduct of the division interpretation of fractions is the relationship between a fraction and decimal representation of the same quantity For the fraction 2/5, for example, we can find the decimal representation by dividing 2 by 5. Given the modern tools of calculators and computers, decimal representations are even more important today than in the past. Students need time to develop comfort and ease in moving between fractions and decimals, and they need to understand decimals in two ways:
- as special fractions with denominators of 10 and powers of 10
- as a natural extension of the place-value system for representing quantities less than 1
Rather than treating fractions, decimals, and percents as separate topics, this unit seeks to build the connections between them. Students will see that the ideas and concepts are related and that the differences are in the symbols used to represent those ideas. Ten percent, 10%, is simply another way to represent 0.10 or 0.1, which is another way to represent 10/100 or 1/10. Percents are introduced as special names for parts of 100.
The models of rational numbers used throughout this unit were chosen because they connect directly to the interpretations of rational numbers that the unit raises. The models on which this unit focuses are
- fractions-strip models
- number-line models
- grid-area models
- partition models
Students are introduced to fractions in a situation that uses a fraction strip as a model. Fraction strips can be created by dividing a strip of paper into equal-size parts by folding. This is a fraction strip for halves:
The collection of fraction strips are used to move to a number-line model of rational numbers. The number-line model helps make the connection to fractions as numbers or quantities. This is a number line for 0 to 2 with a few fractional quantities marked:
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Because 100 and powers of 10 are so useful in understanding decimals and percents, grid-area models are introduced and developed in this unit. This grid shows a shaded area of 12%. Students also use a more general model of fraction situations that is based on partitioning an area, such as a circle, into equal-size parts. The circle shows a shaded portion of 3/10. |
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