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Q&A #1479 |
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You have quite an extensive list! I will try to see what I can do to provide some kind of answers. 1. The first thing that comes to my mind here is really some kind of science course. I co-teach an integrated math & science course with a science colleague and she has a wonderful activity on "WHY Significant Digits" where she has rulers with different kinds of markings. Ruler one has 0 and 10, when students measure they are allowed to estimate one digit beyond the indicated markings. So on this ruler they can measure to a single digit. Instead of accuracy, though, science folks refer to precision of measurement so this would minimize error in measurement situations. A ruler marked with 0,1,2,3,4,5,6,7,8,9, and 10, could be used to find a precision within one decimal place. 2. This one seems a little contradictory! In order to compare and contrast patterns, one needs some kind of numbers (data). There are formulas taught in some kind of advanced algebra or pre-calculus course that teach how to find terms and sums of arithmetic or geometric patterns. Arithmetic patterns are patterns that have a common difference, like 2, 5, 8, 11, ... or 100, 90, 80, 70 .... Geometric patterns are patterns that have a common ratio like 2,4,8,16,... or 1,5,25,125,... One is able to predict in each case if one is provided with a few of the numbers in the overall pattern. A staircase height would be arithmetic, because the height would increase by a common difference each time. Bacteria growth in which the bacteria halved each hour, would tend to be geometric because the growth would involve a common ratio (multiplying factor). 3 & 4. This one sounds like methods that are used to find roots that are irrational in nature, possibly a pre-calculus or calculus class. This process would be similar to the content of your #4. Often successive approximations are used in the iterative and recursive process. Iteration and recursion involve taking the answer and using it to replace in the formula or process to find the next, then that answer is used to find the next, and so on! A simple example would be to take the square root of a number, then take the square root of that answer, and the square root of that answer. Graphing calculators allow for this to be done, by taking the square root of a number, then doing the same with sq.rt.(answer) and then pressing <ENTER> over and over. Eventually, the calculator will round to one. This occurs because of the idea of limits as they apply to this pattern. sq.rt.(x) can be written as x^(1/2). Moving outward, the next term is x^(1/4)... you get x^(1/2),x^(1/4),x^(1/8).... consider just the powers. As the powers continue to grow they approach zero. So what we have is an approximated expression for x^0 which in turn approximates ONE on the calculator. -Claudia, for the Teacher2Teacher service
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