About This Chapter
Mathematical Reasoning and Problem-Solving - Chapter Summary and Learning Objectives
A lot of people have a hard time seeing the practical applications of math to everyday situations, but this chapter aims to help you understand just that. You'll learn how to use mathematical concepts to make arguments and solve problems. This chapter can help you understand the following:
- Logical fallacies
- Conditional propositions
- Logical equivalence
- Inductive and deductive reasoning
- Using estimation to solve problems
- Using mathematical models to solve problems
|Critical Thinking and Logic in Mathematics||Learn to make logical arguments, premises and conclusions|
|Logical Fallacies: Hasty Generalization, Circular Reasoning, False Cause and Limited Choice||Look at the logical fallacies of limited choice, hasty generalization, false cause and circular reasoning|
|Logical Fallacies: Appeals to Ignorance, Emotion or Popularity||Look at the logical fallacies of appeal to emotion, appeal to ignorance and appeal to popularity|
|Propositions, Truth Values and Truth Tables||Use truth tables to determine truth values and negations|
|Logical Math Connectors: Conjunctions and Disjunctions||Recognize conjunctions and disjunctions|
|Conditional Statements in Math||Use conditional statements to make hypotheses and propositions|
|Logical Equivalence: Converse, Inverse, Contrapositive and Counterexample||Recognize when statements are logically equivalent|
|Reasoning in Mathematics: Inductive and Deductive Reasoning||Describe how inductive and deductive reasoning apply to mathematical reasoning and problem solving|
|Reasoning in Mathematics: Connective Reasoning||Look at the concepts of connective reasoning, compound statements and simple statements|
|Polya's Four-Step Problem-Solving Process||Learn to solve problems using Polya's problem-solving method|
|Mathematical Principles for Problem Solving||Learn to give examples of the five principles of mathematical problem solving|
|The Three-Way Principle of Mathematics||Make analogies, draw graphs or diagrams and make examples to illustrate situations|
|Solving Mathematical Problems Using Estimation||Use estimation to solve mathematical problems|
|Using Mathematical Models to Solve Problems||Formulate mathematical models and solve complex problems using problem-solving techniques|
1. Critical Thinking and Logic in Mathematics
Logic has its own unique language and way of defining what is true and false. Watch this video lesson to learn how you can critically think in the language of logic while working with math.
2. Logical Fallacies: Hasty Generalization, Circular Reasoning, False Cause & Limited Choice
Watch this video lesson to see how you can identify cases where logic is not sound. Learn the characteristic traits of hasty generalization, circular reasoning, false cause, and limited choice.
3. Logical Fallacies: Appeals to Ignorance, Emotion or Popularity
Watch this video lesson to see examples of the logical fallacies of appeals to ignorance, emotion, and popularity. You will also see how to identify them.
4. Propositions, Truth Values and Truth Tables
Watch this video lesson and learn what truth values are and what a truth table looks like. Learn how to go from a proposition to its negation and how that affects the truth values and the truth tables.
5. Logical Math Connectors: Conjunctions and Disjunctions
Watch this video lesson to learn how to identify conjunctions and disjunctions. Also learn the connectors that are used with each. Learn how you can use them to make statements.
6. Conditional Statements in Math
Sometimes, what is true in the mathematical world of logic is false in the real world. Check out this lesson to learn how to identify conditional statements and how you can differentiate between what is logically true and what is true in reality.
7. Logic Laws: Converse, Inverse, Contrapositive & Counterexample
Logical statements can be useful, but only if we are able to determine their validity. In this lesson, we'll look at the various forms of a logical statement and see how they relate to each other.
8. Reasoning in Mathematics: Inductive and Deductive Reasoning
Many people think that deductive and inductive reasoning are the same thing. It is assumed these words are synonymous. They are not. This lesson reveals the reality of these two types of reasoning.
9. Reasoning in Mathematics: Connective Reasoning
Connective reasoning is reasoning that has an operation, or a way to connect two phrases. The five main logic connectives will be reviewed in this lesson.
10. Polya's Four-Step Problem-Solving Process
Problem solving can be a problem. Any problem is solved easier with an action plan. Polya's 4-Step Problem-Solving Process is discussed in this lesson to help students develop an action plan for addressing problems.
11. Mathematical Principles for Problem Solving
Solving problems is not just a simple, straightforward process. There are a few principles that can help you as you approach any problem solving scenarios. This lesson covers those principles with examples.
12. The Three-Way Principle of Mathematics
What methods are there to solve and understand mathematical problems? This lesson will review three methods to understand mathematical problems (verbal, graphical, and by example). Each will be illustrated with examples.
13. Solving Mathematical Problems Using Estimation
Estimating is a method of calculating a result that is close to, but not exactly, the correct answer to a math problem. Why would you ever need to do this? This lesson reviews estimating and answers the question as to why you would do it.
14. Using Mathematical Models to Solve Problems
Mathematical modeling simply refers to the creation of mathematical formulas to represent a real world problem in mathematical terms. This lesson reviews the creation and pitfalls of mathematical models.
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