Download GED102 Weekly Guide Notebook week 4 and more Lecture notes Mathematics in PDF only on Docsity!
Task List
Week 4 lessons are all about problem solving, reasoning and applications of mathematics. The lessons aims to develop the skills and the requisite knowledge for more practical approaches to real-life problems with emphasis on logical reasoning and well thought-out solutions. Keep track of your progress in this lesson by checking the box corresponding to each task. _/ 1. Read/Watch Module 2 Introduction _/ 2. Read/Watch Module 2 lesson 1
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Lesson 1. Inductive and Deductive Reasoning
A. Explain the inductive approach of mathematical inference. Inductive reasoning is simply starting with from specific observations or specific cases to create broad generalizations. To discuss it in details, it starts by looking for patterns in the observation. From the patterns, a hypothesis is generalized in order to obtain a theory. However, we must understand the inferences using inductive approach do not render absolute truth, it is only a conjecture in which it can be proven as valid or invalid. It can be proven valid thru deductive process and then it becomes a theorem but if proven false, then it is a contradiction
Highlights
Answer Week HW 2.
Lesson 2. Tower of Hanoi Problem
A. State the Tower of Hanoi Problem. The tower of Hanoi is a mathematical puzzle invented by E. Lucas in 1883. It consists of three rods and a given number of disks. Initially, all the disks are placed on one rod, one over the other in ascending order of size similar to a cone shaped tower( disks are arranged from largest on the bottom and smallest on top). The objective is to move the entire stack of disks from the original location to another area. It asks for a minimum number of moves required to stack it from the stack of one rack to another. There are three rules: (1) only one disk can be moved at a time, (2) no larger disk may be placed on top of a smaller disk and (3) disk can only be moved if it is the uppermost disks on a stack.
Highlights
B. At least how many moves are needed to solve the Tower of Hanoi problem with 4 chips, 5 chips and 6 chips? For us to solve the number of moves needed to solve the Tower of Hanoi problem with 4, 5 and 6 disks, we will start with 1 chip/disk. Moving 1 disk takes only 1 move, moving 2 disks needs 3 moves and moving 3 disks take 7 moves. As you can see, there is a pattern already on the number of moves.
of disk # of moves
1 1 2 3 3 7 We can see the 2 chips adding 2 moves, and 3 chips adding four moves. So it is just merely adding the previous number of moves + the multiples of two. Thus,
of disk # of moves
4 15 5 31 6 63 C. Give a generalized solution to the Tower of Hanoi problem. With research and observation of pattern. I was able to obtain two generalized solution or equation to solve the tower of Hanoi problem. FIRST SOLUTION This pattern uses data from the previous step to help in finding the next one
- 1 chip, 1 move to transfer 1 chip from area the original position to another position
- 2 chips, 2M + 1 = 2(1) +1 = 3 moves
- 3 chips, 2M + 1 = 2(3) +1 = 7 moves
- 4 chips, 2M + 1 = 2(7) +1 = 15 moves
- 5 chips, 2M + 1 = 2(15) +1 = 31 moves
- 6 chips, 2M + 1 = 2(31) +1 = 63 moves The solution would be 2M + 1 where M is the number of moves from the previous number of chips. SECOND SOLUTION
B. Enumerate and describe the 4 phases of Polya’s method of solving a problem. Polya’s method of solving problem has 4 steps or phases. This is based from his book: “How to solve it”:
- Understand the problem The first step is to read and interpret the given context carefully. It is the best to list down all given data and components. If there is a need to draw a figure, then draw it. Determine all the unknowns and condition so that we can proceed to the next step.
- Devising/ creating a plan Find the connection between the data and the unknown. There are multiple ways to create a plan, it can either be an equation, graph, chart, diagram or etc. You can look for theorems that could be useful.
- Carry out the plan Follow the plan created and solve the problem. Check carefully each step and make sure the step is correct.
- Check Examine the result obtained. We can use deductive reasoning to determine if the plan created is a success or there is no error with the solutions and answers. Answer HW 2.
