Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Line (geometry) wikipedia , lookup
Hyperreal number wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Series (mathematics) wikipedia , lookup
Large numbers wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Collatz conjecture wikipedia , lookup
Geometry A Unit 1 Day 1 Notes 1.1 Patterns and Inductive Reasoning Patterns can be expressed visually, numerically, verbally and using formulas. I. INDUCTIVE THINKING - _____________________________________________ _____________________________________________________________________ A. CONJECTURE - _______________________________________________ _____________________________________________________________ 1) COUNTEREXAMPLE - ___________________________________ ___________________________________________________________. 2) Give a counter-example to the statement: "Even numbers are divisible by 3." _______________ 3) Give a counterexample to the statement: "If you've been to Florida, then you've been to Disneyworld". _________________________________________________________ 2. Visual Patterns/ Expressing Patterns Verbally A) 1) Sketch the next figure in the series. 2) Describe the pattern you saw that led you to your conjecture. _______________________________________________________________ ______________________________________________________________. 3. Numeric Patterns/ Expressing Patterns Verbally - In each example, give the next two numbers in the series, then describe the pattern you observed. 1, 4, 16, 64, _________, _________ . Describe the pattern you continued in words. ______________________________________________________________________ _____________________________________________________________________. -5, -2, 4, 13, _________, _________ . Describe the pattern you continued in words. ______________________________________________________________________ _____________________________________________________________________. II. More examples of multiple representations of patterns. A. Verbal/Visual/Numeric In each pattern, a specific number of toothpicks are used to create a pattern. Find the number of toothpicks in each figure and make a conjecture about the number of toothpicks needed to make the next figure. m Figure 1: Number of toothpicks: __________ Figure 2: Number of toothpicks: ___________ Figure 3: Number of toothpicks: ____________ Figure 4: Number of toothpicks: ____________ B. Numeric/Formula 1. Find the sum of the first 50 odd numbers. a. The problem above can be found by brute force, but a pattern in smaller questions might lead to an observation that cuts our work considerably. Find the sum of the first 1 odd numbers. 1 Find the sum of the first 2 odd numbers. ________ Find the sum of the first 3 odd numbers. _______ Find the sum of the first 4 odd numbers. ________ - These smaller examples suggest a formula. Sum of first n odd integers = ___________ Find the sum of the first 50 odd numbers. _______________ C. Formula/Visual/Numeric 1. Complete the chart to find and plot three points on the graph of the line y = 2x - 3. Then find two more points by continuing the visual or numeric patterns. x y 1 0 1 HW: Geometry A Unit 1 Day 1 HW