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6.1 ConceptsBy middle grades children should understand that numbers can be represented in various ways, so that they can see that 1/4, 25%, and 0.25 are all different names for the same number. NCTM Standards 2000, page 33Section 6.1 contains diagrams which illustrate why the first three decimal places for 1/3 are .333, and relates these diagrams to the long division algorithm for dividing 1 by 3. Form similar diagrams for 1/6 to show why its first three decimal places are .166, and relate your diagram to the long division algorithm for dividing 1 by 6. Write a few sentences to show how the diagrams could be continued to obtain further digits in the decimal for 1/6. 6.1 Concepts Students’ understanding and ability to reason will grow as they represent fractions and decimals with physical materials and on number lines . . . . NCTM Standards 2000, page 33.For any two decimals on a number line, no matter how small their difference, there is always an infinite number of decimals between them. Write a convincing argument with examples to illustrate this property. 6.1 Teaching Without a solid conceptual foundation, students often think of decimal numbers incorrectly; they may, for example, think that 3.75 is larger than 3.8 because 75 is more than 8 (Resnick et al. 1989). NCTM Standards 2000, page 216Describe what you feel would provide a “solid conceptual foundation” to enable school students to understand inequality of decimals with different numbers of decimal places. Explain with diagrams how you would help school students understand why 3.75 is less than 3.8. 6.1 Teaching Suppose an elementary school student asked you why writing zeros to the right of a whole number increases the value of the number, but when zeros are written to the right of a decimal point the value of the number does not increase. Write a response that would make sense to this student. |