In the summer of 2023, I presented at the Summer Math Summit (a Virtual Conference for Math Teachers). My topic was "Tips and Strategies for Teaching Precalculus."

I have over 20 years of experience teaching Precalculus and wanted to do a presentation specifically targeting upper-level math course teachers.

Rather see my video presentation on this topic? Check it out here:

## #1 Choose your topics wisely

Determining what to teach in your Precalculus course is a calculated choice. You must consider:

**The district or state standards:**be sure you are meeting the mandatory benchmarks**Dual enrollment requirements:**align your course with the college or university awarding the concurrent credit.**The school culture:**is it an honors-level course? Are all students college-bound?**The students' previous course:**did they take Algebra 2? College Algebra? Algebra 2 in summer school?**The students' future:**Are they 11th or 12th graders? Will they take Calculus next year? AP Calculus?

Check out my blog post, What to Teach in Precalculus, for a list of the topics I cover in my Precalculus class.

## #2 Try to Find Real-World Applications

Most of us are not fans of the dreaded question, "When will we ever use this?" Finding ways to share real-world applications can enrich the learning experience.

**Exponential growth/decay:**

- appreciation and depreciation Exponential Functions Project
- drugs metabolizing in the body
- population growth

**Law of Sines/Cosines:**

- construction, building furniture, landscaping
- surveying land

**Graphs of Sine/Cosine:**

- sunrise/sunset times Trig Graphs Project
- tidal heights

## #3 The Abstract and the Aesthetic

Not all Precalculus topics lend themselves readily to practical examples. Some are about symbol manipulation or learning the lingua franca of mathematics (such as the Factor Theorem for polynomials or Properties of Logarithms). But there's beauty in the abstract, from the elegance of

**polar graphs**to the puzzle-like nature of**Pascal's triangle**. These themes can captivate students through their sheer mathematical beauty.- Check out my Polar Graphs Digital Art Activity

## #4 Use Desmos.com

Even if your school is not one-to-one, you can use Desmos to demonstrate particular topics and reinforce ideas. Below are some of my favorite Desmos pages and activities.- Transformations of Functions Card Sort
- Polynomial Equation Challenges by Desmos
- Graphs of Trig Functions
- Ambiguous Case of Law of Sines
- Precalculus Collection by Desmos
- Polar Coordinates
- Polar Graphs: Butterfly

## #5 Free calculators for your students

*I understand that Texas Instruments has no plans to make a*

**SmartPhone app**for the TI-84, but there are a couple of good knockoffs:**Numworks**is a young but growing company that makes graphing calculators. They are more affordable than the Texas Instruments calculators, and some say they are more user-friendly. I have one, but I do not use it much because I have been using the 83 or 84 since I was in high school (which was a very long time ago). If you can prove you are a math teacher, Numworks will send you a free graphing calculator. Learn more at www.numworks.com/educators/offers/

## #6 Learn tips and tricks for TI-84

*This section is best seen in the video presentation starting around 1:14.*

If your students use the TI-84 calculator, it would be beneficial to know its ins and outs to make their lives easier.

The TI-84 converts angles from degrees to degrees, minutes, and seconds. This command is under the "angle" menu (**2nd** + **apps**).

On your home screen on your TI-84, try this: 52.37> DMS

The newer TI-84 operating systems allow you to type in a stacked fraction (**alpha** + **y=**). The TI-83 does NOT have this capability.

Speaking of a stacked fraction, if you put a decimal point after any number in your function, the table values will be returned as decimals rather than fractions. For example, type 𝑦 = 𝑥/(𝑥+2.) instead of 𝑦=𝑥/(𝑥+2)

Change your table settings so that you can evaluate a function for any value for x (without scrolling!). Go to **2nd** + **window** and change **Indpnt** to **Ask**. Leave the other settings alone. Now go to the table and type any value you want for x.

A great way to see this in action is with polar graphs.

Change your **Mode **to RADIAN and POLAR, then press the **y=** button and type 𝑟 = 2 + 3 cos 𝜃.

Go to the table, and in the x-column, type 𝜋/6, and voila!

## #7 Have a non-calculator test

Have a non-calculator portion on your assessments, especially if they will take AP Calculus the following year because there is a non-calculator portion of the AP calculus exam.

*Suggestion*: Copy the calculator portion on a different-colored paper. First, give the students the non-calculator portion. After they turn in the non-calculator portion, I give them the calculator portion, which is on a different-colored paper.

This is useful because when you glance around the room, it's pretty evident that students who have colored paper are allowed to have their calculators out. This is a quick way to track who is on the calculator portion and who is not.

## #8 Make connections

A lot of what's covered in Precalculus is about functions.

Every time your unit covers a new type of function (polynomial, rational, exponential, logarithmic, trigonometric), bring back these ideas of functions:

- definition (Vertical Line Test)
- inverse function
- domain/range
- increasing/decreasing
- end behavior
- odd/even

With regard to odd and even functions, if it's an Honors Precalculus class, I would definitely cover them. For an on-level Precalculus class, I don't think much emphasis needs to be put on them beyond the initial definition of odd and even. In other words, there is no need to discuss which trig functions are odd and which are even.

## #9 Practice Practice Practice

Build in time for students to collaborate with their peers and practice the topics. The shared struggle and triumph over tricky problems are powerful pedagogical tools.

Here are some links to get ideas for in-class group practice.

## #10 Do formative assessments often

Formative exams are not just about grading—they're learning moments. Even without scores, these assessments can motivate students to engage more deeply with the material. The trick is to make it meaningful without weighing heavily on their grades.

I like to give Open-Note Accountability Quizzes (ONAQs—"oh nacks"). As the name suggests, they are open-note. I require students to work independently because I want to put some pressure on them.

They are typically one page long and take about 15 minutes. Even the students with extended time accommodations only had 15 minutes.

Sometimes, I grade the ONAQs, and sometimes, I do not. If I do grade them, I will make them have very little effect on their overall grade. For example, sometimes, they count as twice a homework grade, half of a quiz grade, or 1/4 of a quiz grade. The students take it more seriously when they know it's going in the gradebook.

You can read my thoughts on formative assessments in this blog post: How to Give a Formative Assessment in Secondary Math.

## #11 Use the right tech tools

Beyond calculators, there are software tools that can amplify your instruction. Here are a few of my favorites:

- Math Illustrations software: They have a free 30-day trial. It works on both Mac and PC. It's great if you teach geometry. You can create diagrams for presentations, assessments, or worksheets. It's also great for creating non-right triangles for the Law of Sines and Law of Cosines lessons. I love that I can force an angle to be a specific degree measure rather than trying to make the triangle look accurate in PowerPoint.
- Autograph graphing software: It's FREE! It only works on PCs. This is my go-to software for creating graphs of functions I need for a presentation or worksheet. It is especially useful for calculus when doing 3D volumes of revolution. Note: I do not use Desmos for images that I want to paste into a document or presentation. Desmos is a calculator; it was not designed to make images for printing.
- GraphFree.com is a FREE website for making nice graphs to be printed or added to your presentations. It's not as powerful as Autograph in terms of the options for scale and tick marks, but it is very good.
- Planbook.com is a brilliant website for keeping your lesson plans organized. I've been using it since 2012 and have never looked back. It is so easy to
- copy plans from year to year
- Share plans with colleagues
- share parts of your lesson plans with your students
- adjust plans when there's a snow day, teacher absence, most students absent, etc
- lock lesson plans to a specific date (e.g., state testing, parent conferences, etc.) so that other plans will adjust around that locked lesson.

## #12 Teaching the Unit Circle

The unit circle is a cornerstone of trigonometry. In my many years of experience, doing the Paper Plate Activity is the best way to help students internalize this fundamental concept in trig.

## #13 Table of Trig Values

Have your students fill out a table of trigonometric values to observe patterns and relationships inherent to the unit circle. For example, students will see how many angles have a secant of 2 or a cosecant of 2sqrt(3)/3. A solid understanding of the six trig functions for all the angles on the Unit Circle fortifies students' understanding. It prepares them for more complex concepts, such as graphs of Trigonometric Functions and Polar Coordinates.

Table of Trig Values FREE Resource

## #14 Have props

My favorite prop, my only prop, is my giant Unit Circle poster.

I had mine made at a local print shop. I used PowerPoint to make the Unit Circle with all the information I wanted. I saved it as a PDF, put it on a thumb drive, and asked the print shop to make the poster large enough to fit through their largest laminating machine. The whole thing cost like $50.

Smiling in front of my giant Unit Circle poster hung using magnets on my whiteboard. |

I keep the poster up during most classes but take it down for assessments. I do require that my students memorize the Unit Circle. Though some teachers frown upon memorization, what I mean when I say "memorize the Unit Circle" is that students must be able to recreate it. The Paper Plate Activity helps them remember the radian measurements, and then I talk about using the values in Quadrant I to get the values in the other three quadrants.

*Anecdote: A year ago, I was teaching a lesson on polar coordinates for an interview. Students got to the point where they had to evaluate the sine of 𝜋/4, and they couldn't do it! I mentioned it to the teacher, and he said, "Yeah, I don't make them memorize the Unit Circle."*

Another time it is helpful to have the Unit Circle on the board is when discussing the Ambiguous Case of the Law of Sines. You can point out to students that the sine of 30 degrees and the sine of 150 degrees is the same value, and 30 and 150 are supplementary. What about the sine of 60 degrees and the sine of 120 degrees? What about the sine of 10 degrees and the sine of 170 degrees?

I highly recommend that you make your own Unit Circle poster. A colleague of mine bought one from a website. It is a nice-sized poster, but it has too much information on it (e.g., formulas for sum and difference identities). Because it has so much information, it is not big enough to be seen from the back of the room.

## #15 To Memorize or Not to Memorize

Even though my previous tip was about memorizing the Unit Circle, I'm not a fan of having students memorize formulas.

Below are lists of formulas that I require my students to memorize or not memorize.

If students are going to take AP Calculus next year, they should know that there is no formula sheet provided.

__NOT Memorize__- Sum and Difference Identities
- Double Angle Identities
- Half Angle Identities
- Polar to Rectangular
- Complex Numbers in Polar Form (multiply, divide, De Moivre's)
- Sequences & Series

__Memorize__- Unit Circle
- Reciprocal Identities
- Quotient Identities
- Pythagorean Identities - memorize sin^2(x) + cos^2(x) = 1and be able to generate the other two identities
- Cofunction Identities (Fun fact: "co" comes from "complementary." That's why it's called "co-sine." sin(30°)=cos(60°).)

## #16 Ask for help

You're going to run into cases, especially at the beginning of your time teaching a particular course, when your students ask you a question that you cannot answer. Be honest with them. Tell them that you don't know and that you will find out. And then follow up! If you try to fiddle your way through it and lie to them, they will see through that.

It's okay for your students to see you make a mistake. When you make a mistake in front of them, you are giving them permission to make mistakes, too, and it allows you to model how to handle mistakes.

If you are the only teacher at your school teaching these higher-level math courses, you can ask for help on social media!

I have a Facebook group with about 3000 high school math teachers. Everyone in the group is very helpful and friendly (because I kick them out if they're not!) If you have a content question, no shame; we've all been there. If you're looking for an idea for how to teach a topic or if you should even teach it at all, come to the group and ask!