# A Guide to Technical Math 1

The main goal of this class is to learn algebra by doing. The purpose of this course is to equalize the mathematics knowledge of all incoming students. Some modules may be new for some students while old and review for others. By the end of the course all students will be at the same level, able to tackle more difficult concepts and succeed in their program. For this reason, it is important to remind students that although they may find some parts boring and review, later ones will be more challenging, and components of every module should provide new insight to all students. So feel free to add additional insights and concepts if it helps capture the attention of the top students, but remember many students are also seeing basic concepts for the first time.

Make sure to update the course homepage to the Modules tab in around week 3.

As a first semester class some time should be dedicated to College orientation. The students do generally receive a lot of this information as part of an orientation week from the College, but having a professor highlight and repeat some information can make a huge difference to College success. Some key orientation details to review are:

– The Learning Support Centre for Tutoring support

– Office365 including Teams and the Office App

– The Digital Creativity Centre

– Discuss what Rough Work means to you. This term may sound obvious to you, but “What is 2+2, show your work” is confusing to understand exactly what you’re looking for

– A Canvas Orientation including:

– The Calendar tab, and ability to connect to their personal device with Google or Outlook

– The Module layout of the course

– The Profile setup page

– The Notifications setup page

## Class

Students are in class to learn, and for this course the best way to learn is by doing. Spend as much time as possible with the students doing work and asking questions, and only explain content in a lecture. Typically the course is scheduled in large 2hr blocks. Students should be spending this time with a fairly even mix of practicing examples, working on in-class activities, listening to explanations, and doing homework (in that order of importance). So if you’re ever talking for more than 10-15min to the whole class, chances are students are stuck waiting to learn.

### In-class Activities

A major part of this course is doing practice in class, and to help track and guide this there are a series of guided in-class activities designed for the course. Each week there is a dedicated in-class activity which matches the mini-lessons and examples in the notes. Content with associated activities are denoted at the bottom of the relevant page. These activities use the Live Poll features in MyOpenMath where, once the professor opens the poll, students will use their own device to enter the answer. Afterwards the proportion of students with the correct answer is displayed. If the question is not done well enough based on your assessment, or student feedback, a mini lesson re-explaining can be done, with additional examples, then a new version of the question can be regenerated and students can try again. These activities can be used to assess the students’ knowledge of the course notes either after a mini lesson and examples, or if the students feel confident enough, to skip over a concept they feel they learnt well enough previously.

Since these activities will often span multiple classes, and students may not attend every class, up to 3 activities are not counted towards their final grade. Instead, 3 alternative online discussions are run during the semester where students can engage online to share their favourite math video/channel, a math related meme, and uses of math in the news and events. This also means receiving a perfect grade in the in-class activities is not essential, but is obviously encouraged.

In addition to the the guided activities, classes are partially held in computer labs to facilitate additional exploration. Software such as GeoGebra, Worlfram|Alpha, and Desmos can be used to allow students to graph and generally explore equations and math concepts. A number of lessons are specifically designed around having the students graph equations, and use applets with sliders to explore changes in variables within the equations.

## Evaluation

Evaluation should focus on ensuring the students’ numerical mark matches, as closely as possible, their understanding of the course material. Understanding of the course material should be consistent throughout all the modules, with no module being considerably lower. Failure to understand even a single topic could have repercussions in later courses where knowledge of that concept is assumed from the beginning.

The evaluation of students’ knowledge is divided between: in-class activities where a basic understanding of the concepts is assessed; homework, where a deeper understanding is explored through many more complex and varied questions; and exams where the knowledge demonstrated in homeworks is validated.

### Homework

Homework material is designed to be investigated before class, learnt and practiced in class, and finally completed after class. The homework feature in MyOpenMath allows students to retry questions, and obtain new versions of questions for additional practice. The tries are designed to allow students to redo a question quickly once they realize their answer is wrong. This is often an “Ah Ha!” moment where they simply “… forgot to carry the two”. Each try receives a small penalty to reflect the failure to recognize the error before checking, and the issues that will come in testing and “the real world”. Should the student try too many times, additional versions (same concept but different numbers and context) of the question can also be generated. These versions will also carry a small penalty, but typically less than a try. In this way students are encouraged to obtain new versions, after getting a try wrong, and to get the question right on the first try.

### Exams

Exams are a way to validate the understanding in the Homework. For this reason it is not essential that the full breadth of course understanding is demonstrated in the exams. They can be used as a last resort for students shown significant improvement, but should remain that, a last resort. Note that students failing to validate their understanding in the exams are at risk of failing the course regardless of their overall grade.

Use of the Veyon enables faculty to view all computers in a Lab and monitor computer usage during tests. A screen recording using Kaltura can also ensure any cheating is properly documented. The goal here is not to catch cheaters though, but rather encourage good practices and naturally avoid it in the first place.

It is essential that students submit good rough work for all exam questions. After exams instead of reviewing the rough work alone, spend time with students that need it, discussing their work. Discussing an error or misunderstanding with that student can be much more enlightening than trying to decipher their rough work, although the rough work does validate their knowledge from the time of the exam. Students can message the instructor for minor errors, such as inconsequential rounding and typos, but for students should meet with the professor for larger more complex errors. This can be greatly streamlined using the Canvas appointment booking system.

In all evaluations students should be encouraged to use the “Message Instructor” option when evaluated grades do not match understanding. By using this the student’s comment, the question, and links to their evaluation are automatically forwarded to the professor for review. This allows the professor to quickly review the issue and correct the grade if necessary.

As the course builds in difficulty, constant review of older concepts is done in subsequent homework and exams. Homework and exam content should be modified if the class struggled significantly and did not demonstrate understanding of prior concepts. Course success should ultimately be based on complete course understanding across all modules and not perfection of a select few.

## Course Modules

The course consists of 5 modules. Each of these modules is based on simpler concepts students should have seen in previous schooling, but covers more depth and breadth than most have seen before. Once each concept is introduced it is then woven into and augmented in each of the following modules and courses. In some cases it is critically essential that students at least grasp the core fundamentals of the first, say Algebraic Expressions, before moving on to the second, Solving Equations.

### 1. Calculations

This module focuses on a number of basic calculation principles including the order of operations, dimension analysis, precision and accuracy. Time is also spent ensuring students are familiar with their calculator of choice and how to use various display modes, scientific and engineering notation,

### 2. Algebraic Expressions

Simplification and expanding of equations involving variables with exponents the main focus here. It is important to explain exactly what an exponent is, especially negatives and zero, and reinforce that variables are placeholders for actual numbers. This may elicit questions like “if these were numbers we wouldn’t do all this” which is correct, and shows understanding. The Midterm generally falls after this module.

### 3. Solving Equations

In this module we look manipulating equations to solve for a variable. These are mostly applications of properties from the Algebraic Expressions module. Special care is needed on implicit equations and equations with variables in the denominator. Considerable time should be spent graphing and exploring equations with software, including the use of applets that focus on key principles. Various forms of word problem applications are important here including mixture problems, related rates and variation. Some time discussing checking answers should be included here.

### 4. Right Angle Trigonometry – Concepts and fundamentals related to right angle triangles

Concepts such as inverses, radians, unit circles, and other foundational concepts are covered here. A visual understanding of how the trigonometric functions relate, and the principals behind using radians should be demonstrated.

### 5. Oblique Angle Trigonometry and Vectors – Working with non-right triangles and vector problems

The Sin and Cos laws are introduced and explored as well as polar coordinates. Significant time is also spent on solving vector problems as this concept is essential in many second semester classes.

## Points of note to enhance your teaching

If you have made it this far, you are interested in learning how to best teach this course.

The best support for students in this course is fast and reliable feedback. In addition:

- Regular lectures and office hours are essential for student support. Adding a Teams meeting to the class speeds communication with the professor and enables collaboration outside of class.
- Focus reminders as possible to only those students that need it by using the “message students who” feature in the Canvas grade book.
- Create a culture of “bug bounties”. Create an atmosphere where students can ask questions, query their results and seek help when they’re unsure. Good feedback, and a small bonus, can be nice to thank the student for their work.

## Course completion

As mentioned earlier the goal of the course should be understanding and insight into all modules with no module lacking. The final exam only validates their learning from the course and should also include content from all previous modules especially where students previously struggled. In this way the final exam can act as a guide in determining any borderline student has have sufficient skill to advance. Any student advancing beyond this course without a competent understanding of all modules will struggle or fail later, with no recourse to move back or succeed. Typically this course, and the second semester course, are offered multiple times throughout the year (unlike subsequent courses), so a failure here is an opportunity to learn fundamental concepts and will often not as significantly hold them back as future courses will.

Lastly, these students naturally form small communities and work groups. Use these to your advantage to promote group learning and strengths and magnify the effort you put in.