By Hsu L., Randolph T., Wang K.

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Thus, the sum of the first n counting numbers would be expressed as 1 ϩ 2 ϩ 3 ϩ . . ϩ (n Ϫ 2) ϩ (n Ϫ 1) ϩ n. The sum of these numbers can be found by noticing that the first number 1 added to the last number n is n ϩ 1, which is the same as (n Ϫ 1) ϩ 2 and (n Ϫ 2) ϩ 3. Adding up all such pairs can be done by adding up all of the numbers twice. ” Compared to less successful problem solvers, good problem solvers generated problems that were more mathematical, and their problems were more mathematically complex (Silver & Cai, 1996).

Step 3 Carry Out the Plan Arrange the pieces of paper in the shape of an equilateral triangle and check sums. Keep rearranging until three sums of 12 are found. Second Approach: Systematic Guess and Test Step 2 Devise a Plan Rather than randomly moving the numbers around, begin by placing the smallest numbers—namely, 1, 2, 3—in the corners. If that does not work, try increasing the numbers to 1, 2, 4, and so on. Step 3 1 11 11 Carry Out the Plan With 1, 2, 3 in the corners, the side sums are too small; similarly with 1, 2, 4.

Do a simulation. 17. Use a model. 18. Use dimensional analysis. 19. Identify subgoals. 20. Use coordinates. 21. Use symmetry. The first six strategies are discussed in this chapter; the others are introduced in subsequent chapters. qxd 9/13/07 5:30 PM Page 5 S T U D E N T PA G E S N A P S H O T From Mathematics, Grade 2 Pupil Edition, p. 233. Copyright 2005 by Scott Foresman-Addison Wesley. qxd 9/13/07 6 5:30 PM Chapter 1 Page 6 Introduction to Problem Solving Step 3 • • • Implement the strategy or strategies that you have chosen until the problem is solved or until a new course of action is suggested.

### Denoising array-based comparative genomic hybridization data using wavelets by Hsu L., Randolph T., Wang K.

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