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Tuesday, November 13, 2007

Place Value and Algebraic Thinking

Surprisingly, an important foundation to algebraic thinking is the ability to simply read a number correctly. Teachers often blow off reading numbers, but when mathematical expressions are correctly read, the math speaks. It almost screams, “This is how you solve me!” Even before students encounter mathematical expressions,simply reading a number can clarify the difference between mere digits, 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.” The worst habit is mixing digits and numbers, or putting "and" in the wrong place. The worst way to read the example is "three hundred and ninety-two point six seven," yet math teachers model the poor reading of numbers everyday. 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 (loose ones), 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 10^2, 10^1, 10^0 or 7^2, 7^1, 7^0 (or whatever, depending on the base).

Students often practice writing in expanded notation without ever grasping the 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 5x^2 + 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 10^2) + (6 x 10^1) + (3 x 10^0). If x = 7, then the expanded notation is (5 x 7^2) + (6 x 7^1) + (3 x 7^0). 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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