Intro to Mathematical Reasoning and Proof
Term: Yearlong 2020–21, September 8–May 28
Target Grade Levels: Grades 9–12
Schedule: 2x / week, 60–75 min.
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Section 1: Coming Soon
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Unit 1: An Introduction to Proof and Mathematical Thinking
The first part of the course provides an introduction to mathematical reasoning, logic, and proof techniques at the high school level. Students will learn about mathematical logic and what it means to prove mathematical statements. Topics include truth tables, the meaning of implication, proof techniques such as proof by contradiction, and mathematical induction. They will learn how to use rigorous mathematical reasoning to prove mathematical statements. Students will also learn the basics of the theory of sets and functions and what it means to say that two sets have the same number of elements.
Unit 2: Counting Techniques and Probability
In the second part, building on the concepts about sets, functions, and proof from part 1, students will learn how to derive powerful methods and techniques for counting and about the basics of probability theory and statistics.
Here are some examples of practical questions in counting that students will be able to solve:
- How many license plates with 7 digits composed of either capital letters or numerals 0-9 are possible such that exactly 3 distinct letters are used?
- Suppose a bag of 20 marbles contains 12 red marbles and 8 blue marbles. In how many ways can 5 marbles be drawn such that exactly 3 blue marbles are chosen?
- In how many ways can a group of 21 students be split into 3 teams with 7 students on each team?
One of the fundamental notions of probability is that if the outcomes of an experiment are equally likely, then the probability of an event is the number of ways that the event can happen divided by the total number of possible outcomes. In this way, counting techniques are useful in computing probabilities. Students will learn some of the fundamental discrete probability distributions and about statistical concepts such as mean, variance, standard deviation, and percentiles.
Prerequisites: It is recommended that students complete Algebra II or equivalent before taking this course, and a minimum of Algebra I or equivalent is required.
Syllabus: View course syllabus here.
- An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions, 1st Edition, by Peter J. Eccles
- Probability: For the Enthusiastic Beginner, 1st Edition, by David J. Morin
- Paper for homework and scratch-work
*Required materials are not included in the purchase of the course.
Dr. John Dever earned a PhD in mathematics from Georgia Institute of Technology in 2018. Before that he earned both bachelor’s and master’s degrees in mathematics from the University of Mississippi. For the 2018-2019 school year he was a Visiting Assistant Professor at Bowling Green State University in Ohio. He has had over eight years of experience teaching a wide variety of mathematics courses at the college level. He has also taught middle school and high school mathematics for three years as a volunteer teacher and tutor at an Orthodox Christian school. He enjoys cultivating interest and curiosity in mathematics among students. He prioritizes student participation and discussion in class as means of helping students to build confidence and see the interconnections of the mathematical ideas under discussion. He hopes that students will begin to view mathematics as both a creative activity, in which they may be active participants, as well as a means of practical problem solving. firstname.lastname@example.org
Computer: You will need a stable, reliable computer, running with a processor with a speed of 1 GHz or better on one of the following operating systems: Mac OS X with Mac OS 10.7 or later; Windows 8, 7, Vista (with SP1 or later), or XP (with SP3 or later). We do not recommend using an iPad or other tablet for joining classes. An inexpensive laptop or netbook would be much better solutions, as they enable you to plug an Ethernet cable directly into your computer. Please note that Chromebooks are allowed but not preferred, as they do not support certain features of the Zoom video conference software such as breakout sessions and annotation, which may be used by our teachers for class activities.
High-Speed Internet Connection: You will also need access to high-speed Internet, preferably accessible via Ethernet cable right into your computer. Using Wi-Fi may work, but will not guarantee you the optimal use of your bandwidth. The faster your Internet, the better. We recommend using a connection with a download/upload speed of 5/1 Mbps or better. You can test your Internet connection here.
Headset: We recommend using a headset rather than a built-in microphone and speakers. Using a headset reduces the level of background noise heard by the entire class. Headset Recommendations: USB | 3.5mm
Zoom: We use a web conferencing software called Zoom for our classes, which enables students and teachers to gather from around the globe face to face in real time. Zoom is free to download and easy to use. To download Zoom:
- Visit zoom.us/download.
- Click to download the first option listed, Zoom Client for Meetings.
- Open and run the installer on your computer.
- In August, students will be provided with instructions and a link for joining their particular class.
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This registration will be finalized when the student's placement assessment has been returned by the course instructor with placement confirmation.