MATH 121: Mathematical Ideas

September 10 (Sunday)

Reflection Day

There are no classes today; it is the last day of Summer. Today makes a great day to make your New School-Year Resolutions.

  Thinking back over the past several terms, how will you change to make this term a success for you? This is such an important question that I may ask it on the first day of the term (Monday) in the form of a quiz. When you start a professional job somewhere, you will need to convey how you will help the company grow and how the company will help you grow. Good jobs will provide opportunities for growth to you. Make sure you take advantage of them.

September 11 (Monday)

Introduction to the Class

Today is the first day of the term. Our class meets in SMC A-206.

 Today’s lecture concerns learning about each other and what this class has to offer you as an adult. It should be rather easy, but come prepared to share your deepest, darkest secrets. 😺

September 13 (Wednesday)

Thinking Mathematically

Today’s lecture concerns “thinking mathematically.” What does this mean? How is it different from thinking scientifically? … or is it? These questions, and more, will be covered today. I look forward to seeing you!

• Read for today: Hawkes, Section 1.1.
• Slides for today: Section 1.1
  ↠ Remember to download and read over these slides before class.

September 15 (Friday)

Estimating and Evaluating

Today should start with a quiz. All quizzes in class cover both the past and the current. This is one way I can encourage you to properly prepare for classes. The readings and the slide decks are invaluable places for you to gain an idea of what we will be covering in class. Knowing this allows you to create a scaffold even before class. This framework gives you a place to put the information I cover… and it gives you an opportunity to formulate questions and propose them at a useful time.

 As you can tell from the slide deck for the day, we will be examining methods for approximating the answer and for checking. Note that these techniques are designed to be done in your head. While they help you think through the calculating process, they also help you work your “math muscles.”

• Read for today: Hawkes, Section 1.2.
• Slides for today: Section 1.2
  ↠ Remember to download and read over these slides before class.

September 18 (Monday)

Surprise Sick Day

Oops. The Knox Bug caught me. Today is a day I am taking off so that I do not spread this vile disease to others and harm them. Please use it to work on the Hawkes assignments.

September 20 (Wednesday)

Surprise Sick Day

The Knox Bug still has me in its grip I am also taking today off so that I do not spread it. Please use it to work on the Hawkes assignments.

September 22 (Friday)

Problem Solving

We end the week with a look at several techniques for solving problems. These problem-solving techniques apply to all disciplines, not just to mathematics. They help you work through problems, thus allowing you to write better essays. Why did I mention essays? It’s because well-written essays have a lot in common with mathematical proofs. The body of the essay step-by-step proves the thesis statement in a logical and clear manner— like a good mathematical proof.

• Read for today: Hawkes, Section 1.3.
• Slides for today: Section 1.3
  ↠ Remember to download and read over these slides before class.

September 25 (Monday)

Work Day: Chapter Assignment 1

Today is a workday for the first chapter assignment (after a quick quiz, perhaps) so that you have an opportunity to get all of your questions answered. Make sure you refer to the calendar for when Chapter Assignment 1 is due. Also, be aware of its requirements.

 Chapter Assignments

 Syllabus Information

September 27 (Wednesday)

Set Notation

We start the second learning module today. It concerns sets. Abstractly, a set is just a group of objects. The objects in a set have something in common (they are in that set for a reason). We see sets all around us, if we look.

 This lecture focuses on laying the mathematical foundation for sets and the study of sets. Eventually, we will get into mathematical operations on sets. Remember that mathematics likes to abstract the world around us, allowing us to see connections between seemingly disparate phenomena.

 Something to keep in mind.

• Read for today: Hawkes, Section 2.1.
• Slides for today: Section 2.1
  ↠ Remember to download and read over these slides before class.
• Hawkes Learning System Assignment due today: Chapter 1.

September 29 (Friday)

Subsets and Venn Diagrams

Frequently, we will need to learn about the relationship between sets. A Venn Diagram is a graphical way to do this. Recall one of Pòlya’s problem solving techniques was to draw a picture. A Venn Diagram is a picture of one or more sets. We use Venn diagrams to visualize sets and set operations.

 We also introduce the concept of a subset, which is a set that is wholly included in its parent set (superset).

• Read for today: Hawkes, Section 2.2.
• Slides for today: Section 2.2
  ↠ Remember to download and read over these slides before class.
• Chapter Assignment due today: Chapter Assignment 1.

October 2 (Monday)

Operations with Sets

We started this module by defining set terms. We then covered how to visualize sets (Venn Diagrams). Now, we look at some basic set operations. Again, as with all mathematical operations, you need to be able to move between the spoken (or written) word and the mathematical symbol and back. This is how one makes mathematics more useful.

 As we go through today’s lecture, think of examples of these set operations (especially unions and intersections). In fact, here is an opportunity for you to earn some extra credit: Before the start of today’s class, email me a Venn diagram meme.

• Read for today: Hawkes, Section 2.3.
• Slides for today: Section 2.3
  ↠ Remember to download and read over these slides before class.

October 4 (Wednesday)

Applications and Survey Analysis

We complete the second learning module today.

 This lecture examines several applications of sets. Ultimately, pay attention to the logic of survey analysis and determining the cardinality of each of the eight sections of a three-set analysis. Doing so will help you practice Pòlya’s problem solving techniques of the previous chapter.

• Read for today: Hawkes, Section 2.4.
• Slides for today: Section 2.4
  ↠ Remember to download and read over these slides before class.

October 6 (Friday)

The First Examination

This first examination covers Chapters 1 and 2. It is worth 50 points so that you have a check on how well you are studying for the course (how effectively) without destroying your grade. It may be helpful to reread the syllabus to ensure that you are working most efficiently.

October 9 (Monday)

Linear Equations and Functions

Chapter Five brings us to familiar mathematics: functions and their uses. Throughout this chapter, keep track of how mathematics is used to model the world. This will help you to see purpose behind the mathematical calculations.

 This section looks at the basics of modeling: dependent and independent variables. Beyond those two terms, pay attention to when linear functions are appropriate to use. Also, pay attention to how to manipulate symbols (onion method?) to solve equations.

• Read for today: Hawkes, Section 5.1.
• Slides for today: Section 5.1
  ↠ Remember to download and read over these slides before class.

October 11 (Wednesday)

Linear Modeling

Now that we are more familiar with linear functions, let us focus on the modeling part: linear regression. Note that we will not be doing actual linear regression. We will approximate it by picking points and calculating the slope and y-intercept from them. One interesting extension is to examine parallel and perpendicular lines.

• Read for today: Hawkes, Section 5.2.
• Slides for today: Section 5.2
  ↠ Remember to download and read over these slides before class.
• Hawkes Learning System Assignment due today: Chapter 2.

October 13 (Friday)

Modeling with Quadratics

Objects operating under the influence of gravity will follow a parabolic path. Today, we look at modeling using quadratic equations. This means we need to be able to identify parts of the parabola, determine their coordinates, and use the quadratic formula. I will also introduce so-called parametric equations, because they simplify modeling of certain types of phenomena.

• Read for today: Hawkes, Section 5.6.
• Slides for today: Section 5.6
  ↠ Remember to download and read over these slides before class.
• Chapter Assignment due today: Chapter Assignment 2.

October 16 (Monday)

Solving Exponential and Logarithmic Equations

Linear functions are useful for modeling phenomena that show a constant change. Exponential functions, on the other hand, are useful for modeling phenomena that show a constant percent change. Examples include almost anything dealing with populations, because population increase is a function of the current population.

• Read for today: Hawkes, Section 5.7.
• Slides for today: Section 5.7
  ↠ Remember to download and read over these slides before class.

October 18 (Wednesday)

Institute Day

Today is Fall Institute. There are no classes. Today offers you an opportunity to explore all that Knox offers you.

October 20 (Friday)

Modeling Activity

We will have an in-class activity designed to strengthen your understanding of Chapter Five.

October 23 (Monday)

Introduction to Probability

Our final chapter deals with probability, a topic near and dear to my heart.

 As usual, we start by introducing and using the terminology of probability. Note that a lot of this section hearkens back to Chapter Two. Thus. It is important to continually review.

• Read for today: Hawkes, Section 10.1.
• Slides for today: Section 10.1
  ↠ Remember to download and read over these slides before class.

October 25 (Wednesday)

Counting a lot

Elementary probability relies on counting elements in a set (either the set of successes or the universe). In simple cases, the counting is simple. However, this is not always the case. Here, we introduce factorials, combinations, and permutations. These are quite useful in counting quickly. They are also useful in generalizing results (as section 10.5 will show).

• Read for today: Hawkes, Section 10.2.
• Slides for today: Section 10.2
  ↠ Remember to download and read over these slides before class.
• Hawkes Learning System Assignment due today: Chapter 5.

October 27 (Friday)

Counting to Find Probabilities

Today serves to unite the previous two sections to illustrate how we can calculate probabilities in rather complicated cases.

 Once we do that, we take the next step and learn how to calculate probabilities of combinations of events (unions and intersections and complements). Again, being able to use the terminology from Chapter Two will definitely come in handy today. Continual review is important in learning.

• Read for today: Hawkes, Section 10.3, 10.4.
• Slides for today: Section 10.3, 10.4
  ↠ Remember to download and read over these slides before class.
• Chapter Assignment due today: Chapter Assignment 5.

October 30 (Monday)

Bayes’ Law Activity, Part 1

This is the first of two activities dedicated to creating and extending your understanding of Bayes’ Law… and your ability to use a spreadsheet program to perform calculations. Please bring your computer today.

November 1 (Wednesday)

Bayes’ Law Activity, Part 2

This is the second of two activities dedicated to creating and extending your understanding of Bayes’ Law… and your ability to use a spreadsheet program to perform calculations. Please bring your computer today.

November 3 (Friday)

Binomial Probability

When a probability distribution is experienced over and over again, we tend to name it and study it. The only one we will be looking at is the Binomial distribution. This distribution is found all around us, whenever we count the number of successes out of a specific number of trials. Examples include the number of students in this class who wear a hat, who are Sophomores, who wear contact lenses, etc. There are five requirements for a phenomenon to follow a Binomial distribution, so pay close attention to it.

• Read for today: Hawkes, Section 10.5.
• Slides for today: Section 10.5
  ↠ Remember to download and read over these slides before class.

November 6 (Monday)

Expected Value

The concept of an “expected value” is IMHO overhyped. The expected value of a random variable is the long-run average of its observations. That is all. It does not need to be a possible observation. It does not have to be a likely observation. It may not be a useful number at all. It is just the long-run average. But, for those times that the long-run average is useful, let us see how to calculate it.

• Read for today: Hawkes, Section 10.6.
• Slides for today: Section 10.6
  ↠ Remember to download and read over these slides before class.

November 8 (Wednesday)

In-Class Activity

This is an activity designed to pull together all of the probability material. Come prepared with your computer.

• Hawkes Learning System Assignment due today: Chapter 10.

November 10 (Friday)

Examination Two

This second examination covers Chapters 5 and 10 (and 1 and 2). It is worth 100 points.

• Chapter Assignment due today: Chapter Assignment 10.

November 13 (Monday)

Course Summation

And here we are, at the end of this course. It is rather difficult to believe that we have gone through 10 weeks together; it seems like only yesterday that we first met in the SMC And yet, here we are, at the end.

 Today, we will have a presentation on what we have done in this class. It should help you see that you have all come so far in such a short period of time.

November 14 (Tuesday)

Last Day of Classes

Today is the last day of classes for the term. Please check the syllabus for what that means.

November 15 (Wednesday)

Reading Day

Reading days were originally created by Knox College to provide students with days set aside for nothing other than bringing together all of the material learned during the term— and during the time at Knox College. Education is a process of changing yourself. It provides you with the opportunity to better see the relationships throughout the world.

 If you are only interested in short-term gains (which is better than nothing), then today is a day to cram. If you are interested in mastering the material you have learned thus far, then today is a day to reflect on what you have learned in life.

November 16 (Thursday)

Reading Day

The second reading day should have you continue your reflection of what you have learned this term— in all classes. Tie the learning together to make it a part of who you are.

November 17 (Friday)

Final Examination Period

November 18 (Saturday)

Final Examination Period

Our final examination takes place today. It starts at 1:30pm in our classroom and ends at 4:30pm. Make sure you show up early because I hand out the exam a few minutes early so I can state the important aspects of the exam. As always, you should be studying throughout the course. You should not cram for the final.

November 19 (Sunday)

Final Examination Period

November 20 (Monday)

Winter Vacation Begins

Yay! This is the start of Winter “Vacation.” What will you do to help your future?

 I hope you spend some time recuperating from the end-of-term stresses. But, I also hope you take time to do two other things. First, you should take time to reconnect with your hobbies. Second, you should take time to reflect on what went well this term, what did not go so well, and what you can learn from those two sets of events.