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#### Using Estimation and Mental Arithmetic

The accurate and efficient use of an algorithm rests on having a sense of the magnitude of the result. Estimation techniques enable students not only to check whether they are performing an operation correctly but also to decide whether that operation makes sense for the problem they are solving.

The base-10 structure of numerals allows certain sums, differences, products, and quotients to be computed mentally. Activities using mental arithmetic develop number sense and increase flexibility in using numbers. Mental arithmetic also simplifies other computations and estimations. For example, dividing by 0.25 is the same as multiplying by 4, which can be found by doubling twice. Whether or not students are performing a written algorithm, they can use mental arithmetic to simplify certain operations with numbers. Techniques of estimation and of mental arithmetic are particularly important when students are checking results obtained from a calculator or computer. If children are not encouraged to use the mental computational procedures they have when entering school, those procedures will erode. But when instruction emphasizes estimation and mental arithmetic, conceptual understanding and fluency with mental procedures can be enhanced. Our recommendation about estimation and computation, whether mental or written, is as follows:

Whether or not students are performing a written algorithm, they can use mental arithmetic to simplify certain operations with numbers.

• The curriculum should provide opportunities for students to develop and use techniques for mental arithmetic and estimation as a means of pro moting a deeper number sense.

#### Representing and Operating with Rational Numbers

Rational numbers provide the first number system in which all the operations of arithmetic, including division, are possible. These numbers pose a major challenge to young learners, in part because each rational number can represent so many different situations and because there are several different notational schemes for representing the same rational number, each with its own method of calculation.

An important part of learning about rational numbers is developing a clear sense of what they are. Children need to learn that rational numbers are numbers in the same way that whole numbers are numbers. For children to use rational numbers to solve problems, they need to learn that the same rational number may be represented in different ways, as a fraction, a decimal, or a percent. Fraction concepts and representations need to be related

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