Tuesday, November 13, 2007

Place Value and Algebraic Thinking

Students need to learn to read numbers correctly. Reading numbers is often glossed over, but when mathematical expression are correctly read, the math speaks. It almost screams, “This is how you solve me!” Simply reading a number can clarify the difference between the digits or characters, and the number itself. For example, 392.67 as digits is “three nine two point six seven”, and as a number, “three hundred ninety-two and sixty-seven hundredths.” Because the placement of the word “and” marks the boundary between the wholes and the part of a whole, students need to develop facility with the use of “and”. Another important by-product of the chocolate factory activity is a keener sense of rounding. They learn that rounding a number is a matter of distinguishing place value.

After work with cases, boxes, and leftovers, children can readily accept the renaming of the place value columns as “10 x 10,” “10,” and “1,” or even, for example, “7 x 7,” “7,” and “1” depending on the base. Then it is an easy precursor to exponents to rename the columns as 102, 101, 100 or 72, 71, 70 or whatever, depending on the base.

Students often practice writing in expanded notation without ever grasping real significance of what they are doing. In algebra, many polynomial expressions are really bases in disguise. For example, the base 10 tally and the base 7 tally were both 563. Algebraically this could be expressed as 5x2 + 6x + 3, where x is the base. The algebraic expression is nothing more than expanded notation. If x is 10, then the expanded notation is (5 x 102) + (6 x 10) + (3 x 1). If x = 7, then the expanded notation is (5 x 72) + (6 x 7) + (3 x 1). Rewriting numbers as polynomial expressions often makes calculations in different bases much easier, and The Chocolate Factory activity enhances such algebraic understanding.

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