Chap6b.-Mathematical-Patterns.pdf mathmath
- 1. PROBLEM SOLVING AND REASONING– Mathematical Patterns Core Idea “Mathematics is not just about numbers; much of it is problem solving and reasoning.” CHAPTER 6b.
- 2. learning objectives 1. become familiar of the problem solving heuristics and be able to identify the strategy most appropriate for the problem; 2. use problem-solving strategies to investigate and understand mathematical content; 3. acquire skill in developing and applying a variety of strategies to solve problems; and 4. verify and generalize results with respect to the original problem situation.
- 3. TERMS OF A SEQUENCE 5, 14, 27, 44, 65,… sequence term- a number in a sequence - an ordered list of numbers
- 4. TERMS OF A SEQUENCE 5 14 27 44 65 … q difference table - shows the differences between successive terms of the sequence.
- 5. Example. Predict the Next Term (Book: Page 70) Use a difference table to predict the next term in the sequence a. 2, 7, 24, 59, 118, 207, ___ b. 1, 14, 51, 124, 245, 426, ___ TERMS OF A SEQUENCE
- 7. THE FIBONACCI SEQUENCE 4 Month 3 pairs 5 5 pairs 6 8 pairs
- 9. THE FIBONACCI SEQUENCE A recursive definition for a sequence is one in which each successive term of the sequence is defined by using some of the preceding terms. 1, 1, 2, 3, 5, 8, … Let !" = 1, !& = 1, !' = !'(" + !'(& for * ≥ 3. FIBONACCI NUMBERS ./ .0 .1 .2 .3 .4
- 10. THE FIBONACCI SEQUENCE Example. Properties of Fibonacci Numbers (Book: Page 74) a. If ! is even, then "# is an odd number. (True/False) b. If "$% = 144 and "$) = 233, what is "$,? c. If "$- = 987 and "$1 = 2584, what is "$3?
- 11. NTH-TERM FORMULA FOR A SEQUENCE An nth-formula is constructed when there is a certain pattern in a sequence. It also generates the terms of a sequence. Example 2. Generate Terms of a Sequence Consider the formula !" = 3%& + % where % is the number of the term in the sequence. Generate the first 5 numbers of the sequence.
- 12. NTH-TERM FORMULA FOR A SEQUENCE Example. Find an nth-Term Formula Assume the pattern shown by the square tiles. a. What is the nth-term formula for the number of tiles in the nth figure of the sequence? b. How many tiles are in the eighth figure of the sequence? c. Which figure will consist of exactly 320 tiles?
- 13. NTH-TERM FORMULA FOR A SEQUENCE Practice. Find an nth-Term Formula Assume the pattern shown by the square tiles. a. What is the nth-term formula for the number of tiles in the !th figure of the sequence? b. How many tiles are in the thirteenth figure of the sequence? c. What figure will consist of exactly 419 tiles?