Learning Outcomes:
At the end of the lesson, the students will be able to: Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts; Write clear and logical proofs; Know how inductive reasoning differ from deductive reasoning; Solve problems involving patterns and recreational problems following Polya’s four steps; Use reasoning as a cognitive tool to arrive at conclusion or solutions to problems; Increase awareness on the importance of reasoning and problem solving; and Organize one’s methods and approaches for proving and solving problems.
This chapter will enhance our skills in reasoning and problem solving as we continue to discuss the relationships between and among
…show more content…
Why reasoning’s and problem solving useful? Why you have to study reasoning and problem solving?
Performance Task
Goal –
You are tasked to make a housing loan comparison and come up with recommendations afterwards.
Role -
You are working as one of the spokesperson of the leading firm of your province. Family Sta. Cruz comes to your office and ask you to assist them in deciding which among the housing loans offered by DBP, Bank of Commerce and PNB will allow them to save money and payoff the loan within 10 years. . Audience –
Your recommendations will be presented to your firm and to Sta. Cruz family. Situation – As a spokesperson, you will coordinate to the head of the three banks you offer to Sta. Cruz family to have a data and computations for the breakdown of their housing loans. Then, ask them for the reasons why Sta. Cruz family will choose their bank.
Performance/Product –
You will create a recommendation for each bank. The recommendation will be written in two columns. The first column contains the computation while the other column will suggest a reason why Sta. Cruz Family will choose one among the following banks: DBP, Bank of Commerce and
…show more content…
A conditional statement may sometimes be true or false. To show that a conditional statement is true, the argument should consist of hypothesis followed by a conclusion that holds at all times. If a conditional statement is false, a single counterexample is needed to make the statement true.
Example 3.6: If x^2=9 , then x = 3.
Solution:
As a counterexample, let x=-3. The hypothesis is true because 〖(2)〗^2=4. But the conclusion is false because the value of x can also be -3. Therefore, the conditional statement is false.
Example 3.7: If one-digit number is multiplied, then the product is a two-digit number.
Solution:
As a counterexample, let 2 and 4 be the one-digit number to be. However, the product of 2 and 4 is 8, which is a one-digit number. The hypothesis is true because 2 and 4 are one-digit number. But the conclusion is false because the product of 2 and 4 is not a two-digit number. Therefore, the conditional statement is false.
When we interchange the hypothesis and conclusion of a conditional statement a⇒b, then we have the converse statement b⇒a. Here are some examples:
Example 3.8:
Statement: “If Dory is a fish, then she
With that in mind, it is important to understand a couple of concepts before analyzing and determining the effectiveness of that document. Although people do not always realize it, the purchase of a home is one of the b...
Finally, I will do a financial forecast in order to figure out firms’ ability to repay its loans. I will use simple percentages-of-sales forecasting technique. I will use existing trends in my forecast to show the implications of current policies before making my own recommendations. During my forecast I will use New Era Partners loan to find out the interest rates. I will make the short-term debt as my plug.
"Home Owners Loan Corporation." Next New Deal. Roosevelt Institute, 2014. Web. 16 Mar. 2014. .
The new millennium brought with it a housing boom which had reached an unsustainable level (Pollock, 2011). Housing prices grew rapidly, and Baker (2010) noted a rise in house prices of over 70% from 1995 to 2006. For example, he noted average home prices in Los Angeles rose more than $400,000 over the period of 1995 to 2006 and approximately $519,000 in San Francisco. Prices around the country increased substantially as well (Baker, 2010). To encourage homeownership, banks promoted creative financing options (i.e. adjustable rate, interest only,...
A paradox stems from a statement that apparently contradicts itself yet might still be true. In most cases logical paradoxes are essentially known to be invalid but are used anyways to promote critical thinking. The Raven’s paradox is an example of a paradox that essentially goes against what most logical paradoxes stand for in that it tries to make a valid claim through inductive logic. Carl Hempel is known for his famous accepting of this paradox with minor adjustments by the use of the contraposition rule. In this paper, however, I argue that Hempel’s solution to the Raven’s paradox is actually unsuccessful because he fails to take into account a possible red herring that serves as evidence against his solution. Irvin John Good is responsible for the formulation of the red herring argument as he tries to prove that the observation of a black raven can potentially negate the Raven’s paradox as valid. In addition to Good’s claim, Karl Popper and his view of falsificationism also functions as evidence to reject Hempel’s solution. Using Popper’s view as a basis, Israel Scheffler and Nelson Goodman formulate the concept of selective confirmation to reject the contraposition rule used by Hempel. Based off of all of the rejections that Hempel’s solution has it can clearly be seen that the Raven’s paradox has flaws that principally lead it to it being invalid.
In this paper I intend to analyze logically this proposition, trying to focus the question of contradiction.
Then, he characterizes this rule as something that always and necessarily follows. Also, this rule must make the
Proposition is a statement that expresses a statement that is uncertain. It could be true or false.
Personal factors and choices affect the type of house someone chooses to live in. Personal tastes, stage of one’s life, family size and financial circumstances, health and career; all of these affect one’s decision in purchasing and financing
With 10 different principles of economics, one can apply in a decision to buy a house. One at the top of the list would be that of trade-offs for which an individual will encounter. Every single choice comes at a price especially when purchasing a new home. Such is the case when choosing to buy a home exhausting an individual of a sizable percentage of their sa...
Loan officers have many important duties that they have to do while on the job. Loan officers contact potential loan applicants, both people and companies, and they ask them if they are in need of a loan. After ...
Hypothesis (pg. 47) – a proposed statement of results made about a problem in a research study that is expected. An example of a hypothesis would be “If college football players play a full game of 60 minutes, then they are 70% more prone to concussions.”
Mathematics can be concrete and use reason as a way of knowing. These are learned concepts with repetitive procedure. Critical thinking is a type of reasoning that uses logic that will never deviate. The early concepts of mathematics taught in schools are thought to be concrete with fixed steps and formulas for solving problems. One only has to think about the steps previously taught and accept them to be true. The concepts can only be accepted to be true by using the skills to process and generate information and belief. The use of the skills as an “exercise” with no meaning or understanding is not critical thinking however. It is always believed that the area of a right triangle is one half the base times the height. Reasoning can be used through the drawing of a grid to prove this formula to be true. Therefore, mathematics uses critical thinking as a way of known skills to guide behavior based on intellectual commit...
Based on the loan features(interest rates & other charges),eligiblity criteria and services provide the following case study of main players for home loans is given below:
It is not only in my own writing that my awareness of math has been heightened. While reading articles for classes, on news websites, or blogs, I find myself paying more attention to the flow of the author’s argument. We’ve learned that in proof writing it is important to be clear, concise, and rigorous and the same applies to an argument within a paper. I’ve come to realize that if an author is trying to convince me of their point, then they also need to show me why their point is true or important. In this way, I’ve become more critical of an author’s argument; rather than just believing everything that they write, I more closely evaluate the progression o...