If you’re homeschooling high school math, you’ll most likely come to the point where your student will have the option of studying precalculus.
If your student is planning on studying STEM in college it’s usually recommended that they take precalculus, as it can lay a solid foundation for Calculus 1 and prepare them for the more challenging math they’ll need to learn later.
However, because it includes of a variety of more advanced concepts in high school algebra and geometry, precalculus can sometimes be a little intimidating for those learning at home.
A good precalculus curriculum, one that outlines and explains concepts clearly to the student, can therefore be invaluable when studying this somewhat tricky subject.
A good curriculum, however, is also dependent on the student, their background, their ability and their needs. Since every student is different, finding the right curriculum can be tough.
To help out, we created this guide to help homeschooling parents and students find a precalculus curriculum that can work for them and help students thrive.
Precalculus Curriculum: What Students Learn
Precalculus, broadly speaking, is a course offered in high school that is designed to prepare students for studying calculus.
Although not technically required, calculus can be quite important to those planning on studying science, technology, engineering, mathematics (STEM) at the college level.
Prerequisites for Precalculus
Designed to prepare students for calculus, precalculus tends to include more advanced concepts in algebra and trigonometry.
As a result, students will generally need to be familiar with the concepts presented in Algebra I & II and Geometry before starting the course.
In particular, students should have a decent understanding of:
- Basic trigonometry (sine, cosine, tangents) with some familiarity of important trigonometric identities
- Quadratic equations and inequalities
- Linear inequalities
- Geometric formulas (area, volume, perimeter)
- Basic graphing models including circles, lines, y=x, y=x², y=x³, y=|x| etc, as well as understanding reflection and scaling
- and basic function notation
Common Core Precalculus Curriculum Standards
Precalculus courses can vary quite a bit from curriculum to curriculum when it comes to their scope and sequence.
More rigorous or AP-focused courses can include some more detailed calculus content or content aimed at college level courses, such as spending more time on Euclidean vectors, matrices, power functions, conic sections and more.
In general, though, a precalculus course that meets Common Core State Standards should include at least the following topics:
|Linear Equations||Solving Equations Graphically||Equations For Parallel And Perpendicular Lines|
|Application Problems For Inequalities||Quadratic And Factorable Inequalities||Linear And Compound Linear Inequalities|
|Multivariable Inequalities||Linear Programming||Graphing Linear Equations|
|Operations With Functions||Graphs For 12 Basic Functions||Graphing Piecewise Functions|
|Graphs And Transformations||Graphs Of Nonlinear Inequalities||Inverse Functions And Relations|
|Families Of Graphs||Direct, Inverse, And Joint Variation||Polynomial, Rational, And Exponential Functions|
|Solving Quadratic Equations Algebraically||Graphs Of Rational Functions||Remainder And Factor Theorems|
|Real Zeros Of A Polynomial Function||Exponential And Logistic Functions||Natural Base E|
|Logarithmic Functions And Graphs||Conics (Various)||Trigonometric Functions|
|Circular Functions||Angles And Radian Measure||Amplitude And Period Of Graphs|
|Sine/Cosine Functions||Trigonometric Inverses And Graphs||Inverse Functions|
|Area||Law Of Sines||Law Of Cosines|
|Identity And Inverse Matrices||Determinants||Reflections, Rotations, Translations, Dilations Of Matrices|
|Modeling Motion With Matrices||Matrices And Solving Linear Equations||Vector Operations|
|Complex Numbers And Their Operations||Distance And Midpoint||Geometric Vectors|
|Algebraic Vectors||Magnitude And Direction Of Vectors||Transformation Of Vectors With Matrices|
|Dot Products||Polar Coordinates||Probability|
|Combinations And Permutations||Normal Distributions||Making Decisions With Probability|
|Data Types||Shapes Of Distributions||Qualitative Vs Quantitative Displays Of Data|
What Makes A Good Precalculus Curriculum?
While there can be a lot of variation between the many precalculus curricula out there, the most effective ones tend to have few things in common.
Thorough and understandable explanations
There are a lot of abstract and complex topics discussed in precalculus, as well as a good deal of graphing involved.
These can be a little tricky for students who are seeing them for the first time and who may not necessarily be used to working in multiple dimensions or working graphically in math.
A good precalculus program should be able to explain these concepts fully and clearly to the student, getting the point across in a way that teens can understand so that the student can practice on their own with a minimum of error, confusion and frustration.
At the same time, being essentially a prep course for higher level math, it’s important that these programs not dumb down the material.
It’s important that any precalculus program not shy away from important terminology or theory, as they will most likely be used in later courses.
Preparation for Calculus
Obviously, a precalculus program should prepare students adequately for the next level of high school math – i.e. calculus.
By the end of a precalculus course, a student should be familiar with:
- Polynomial expressions and their operations,
- Graphing and graphing transformations – graph circles, lines, piecewise-defined functions and more
- Properties, algebra and graphing functions
- Algebra of functions
- And more
Before picking a curriculum, it’s therefore important for parents to look at the overall scope and sequence of the course to make sure that it touches upon the most important concepts at the very least.
Course Rigor and Pace
Precalculus curricula can approach concepts with more or less rigor, that is making the course more or less challenging for students overall.
Rigorous programs may include more challenging problems, explore theory and concepts in more depth, use a quicker pace and/or may introduce more advanced calculus topics and concepts earlier than other programs.
The more rigorous the program, the more of an intellectual challenge it can be for the student and, all things being equal, the better prepared a student will be for higher level classes.
That said, more rigor isn’t always the best option for all students.
Not every student who takes precalculus is a math wizard or interested in exploring a career in quantitative sciences.
They may be taking the course to keep their options open or to improve their prospects for college admissions by making their transcript more competitive.
Some students can benefit from a slower pace and an easier approach to the material, spending more time reviewing the foundational principles that they’ll need to succeed later on.
For these students, an overly challenging curriculum may simply serve to frustrate them or, worse, convince them that math is just not for them and prevent them from continuing their math education.
Digital vs Traditional Curriculum
As with many other courses today, those looking for a good at home precalculus curriculum have the option of studying with an online program or one that’s more traditional and textbook-based.
Each has their own particular strengths and weaknesses, and its important to consider these as they relate to the student and the way they learn best.
Online courses tend to have a variety of technologies built into them that can help students better study precalculus on their own when compared to traditional textbooks.
Many include multimedia instruction, involving video, interactive questions and more, that can create a more engaging and varied learning experience.
Further, they may even be able to draw on various online resources to explain things in ways that are more clear and understandable to the average student.
Online courses may also include adaptive learning technology, which alters the difficulty of practice so that the program can adjust itself to the student’s actual skill level and performance, making learning more personalized and keeping things challenging without becoming frustrating.
These digital options typically also include progress tracking, which can measure student progress and identify any gaps in skill that may exist or develop, which in turn can make learning more targeted and effective.
Finally, online courses often include the full suite of K-12 math for one subscription price, which makes it a lot easier for students studying a course like precalculus compared to an individual, subject specific textbook or curriculum. At any time, students can go back and brush up on specific fundamental concepts, such as in algebra and trig, if they run into trouble.
On the other hand, online math programs are usually subscription-based and more expensive than a traditional curriculum overall, charging a monthly fee rather than being a one time cost.
Similarly, they tend to licensed to one user per year, making it harder to reuse them with younger siblings, a distinct advantage that textbooks have.
Digital courses also increase the time students spend in front of a screen, which isn’t always so healthy, and tend to require more dedication and discipline from the student (or oversight from the parent).
Being online, there is a far greater temptation and opportunity to click away and do something more fun than their assigned math problems.
Summary of Online Programs vs Traditional Textbook-based Programs
|Online Precalculus Program||Traditional Textbook Program|
|Screen time||Intenstive||Low to None|
|Immediate access to review previous math content||✔||❌|
|Progress tracking and similar analytics tools||✔||❌|
|Adaptive questions and other learning technology||✔||❌|
|Easy to get distracted||✔||❌|
|Reusable with future students||❌||✔|
For more information on choosing a math curriculum
For more in-depth information on how you can pick a math curriculum that’s right for your homeschool, including what to look for and how to ensure a good fit between student and curriculum, check out our Guide To Picking The Best Homeschool Math Curriculum.
Precalculus Curricula That We Recommend
Note: All prices are current as of writing, all prices in USD.
CTCMath offers a complete digital precalculus course from down-under with a little bit more rigor and lots of practice
|Price: From $29.97 per month||Rigor: ⭐⭐⭐⭐|
|Conceptual/Procedural: Procedural Math||Progress Tracking: ✔|
|Common core: ❌||Instruction: Video|
Based in Australia, CTCMath is browser-based, complete online precalculus curriculum that’s a little bit more challenging than a typical curriculum, but otherwise still quite approachable for most students.
Taught through a variety of short videos, CTCMath covers all topics of precalculus pretty comprehensively.
A mastery method math program, each topic in precalculus is separated into its own unit and taught in a series of individual lessons, allowing CTCMath to explain each topic carefully and thoroughly and only moving on once a student has reached a level of proficiency and understanding.
The video lessons themselves are taught by Patrick Murray, a veteran math teacher with over 30 years of experience in teaching math. Murray explains concepts quite clearly and uses a variety of visual charts and graphs to make the subject’s more abstract concepts a little more concrete and understandable.
Following each video lesson, students are then given a variety of questions to test their knowledge of the material, which they’ll have to work out on on scrap paper before inputting the answer into the program.
There are also interesting and challenging weekly revisions and worksheets they can work on to solidify knowledge and work at their own pace.
Overall, CTCMath’s questions and revisions are a little more challenging than other programs and encourage students to focus on exploring and understanding precalculus concepts, rather than just answering simpler computational questions.
That said, while a little more rigorous than an average program, we don’t think that there is anything in CTCMath that will really frustrate an average student.
Interestingly, CTCMath leans on its digital nature to offer in-program diagnostic testing that can be used to figure out if a student has any skill gaps, as well as to help measure and track progress as they move along the course.
Parents should know that CTCMath is obviously not common core standards-aligned (coming from Australia) and that it’s not the most flashy or intrinsically interesting learning environment – the videos are pretty simple all things told, and the visual aids are pretty basic on screen drawings and graphs.
Overall, however, with its comprehensive coverage of precalculus topics, clear explanations and tons of practice questions, if you’re looking for an approachable online precalculus curriculum with a little challenge we feel that CTCMath might be right for you.
For more information, read our review of CTCMath
Check out the CTCMath website
With clear explanations and an easier pace, Unlock Math is an approachable precalculus course for those who get nervous around numbers
|Price: From $24.92 per month||Rigor: ⭐⭐⭐|
|Conceptual/Procedural: Conceptual Math||Progress Tracking: ✔|
|Common core: ❌||Instruction: Video|
UnLock Math is an online math program that can make precalculus a little more approachable and easier to understand as a subject, especially for those who aren’t the biggest fans of math.
Lessons are primarily taught through videos.
These are hosted by the founder, a Canadian educator and long time teacher, who presents material in a clear, concise and pretty positive way, perfect for those who are uncertain about their math skills or who suffer from math anxiety.
Overall, the lessons keep things fairly casual and stress-free, using humor, visuals and lots of enthusiasm, while still covering the material in depth and without dumbing the material down in any way.
Following the lessons, students answer a variety of practice questions to test their understanding.
UnLock Math contains a variety of digital tools that allow students to do their work totally within the program, which is fairly novel for an online math program.
Problems are presented one at a time, which makes the process far less intimidating compared to a page full of questions, and they’re adaptive, so they become more or less difficult depending on how well a student is progressing and answering them.
A mastery system, students only progress to the next lesson once they’ve achieved a specific level of proficiency in a topic (adjustable by a parent or teacher account).
As a digital curriculum, UnLock Math also contains some features to improve the learning experience.
It has a progress tracking that is accessible by both student and parent, allowing students to better understand their own progress and skill development rather than just relaying this information to the parent.
The program also rather cleverly includes a live chat and support line, which is available during weekday business hours.
Staffed by educators, this chat service can act as live on-demand help for math problems and concepts if a student finds themselves a little lost while studying.
Parents and students should be aware, however, that while the program covers all the necessary precalculus topics, we don’t feel it does so with quite as much rigor or conceptual depth as some other programs out there.
In other words, UnLock Math keeps learning approachable and on-grade but is not necessarily the most advanced.
Similarly the program does not cover high school calculus, so if a student does enjoy how UnLock Math teaches there isn’t really a more advanced course to move into.
Overall, however, if a student is a little anxious when it comes to learning precalculus, UnLock Math’s approachable, sequential and easy to understand lessons, adaptive technology and careful progress tracking can go a long way in helping them succeed with a minimum of tears.
For more information, read our review of Unlock Math
Check out the UnLock Math website
Art of Problem Solving
If you’re looking for a rigorous and enriched precalculus course to take them to the next level, check out Art of Problem Solving’s textbook
|Price: About $57.00||Rigor: ⭐⭐⭐⭐⭐|
|Conceptual/Procedural: Strong conceptual ||Progress Tracking: ❌|
|Common core: ❌||Instruction: Textbooks and problem sets|
Created by a former math olympiad winner and used by numerous winners math competition champions over the years, Art of Problem Solving (AoPS) is well-known for its rigorous, concepts-driven approach that is squarely aimed at advanced students seeking enrichment in math.
And its precalculus course is no different.
It is quite an extensive program, teaching the required precalculus topics and including a good deal of material that’s not usually included in a high school precalculus course, such as complex variables and their effect on 2D geometry,
Going into far more depth than other programs, AoPS’ precalculus textbook teaches through short, clear and to the point explanations, with thought-provoking demonstrations and examples that are then followed by a myriad of exercises to put concepts into action.
The precalculus textbook offers a wide variety of challenging problem sets with a heavier emphasis on logic and proofs than other precalc programs out there. Its problems are designed for students to work through carefully and critically, and they reward students for finding alternative methods and creative approaches to finding an answer.
In fact, some of its more advanced problems have been directly pulled from past national math competitions and olympiads, which should give some idea of the program’s rigor. Compared to other math curricula out there, AoPS is more of a problem-solving oriented program and in general there is a lot less handholding, with students expected to discover things for themselves.
That said, Art of Problem solving does offer a good deal in the way of possible strategies and methodologies that students can use, as well as discussing concepts far in advance of most other programs.
Although by no means a walk in the park, Art of Problem Solving can provide students with a far deeper and more thorough understanding of the material.
It not only can do an excellent job at preparing students for Calculus, AP exams, college or even prestigious math contests, but can serve to hone their quantitative problem solving skills and develop more creative and flexible mathematical thinking that will better serve them in the future.
Overall, of you have an outstanding student in math and are looking for a precalculus curriculum that can take things a bit further, Art of Problem Solving’s precalculus textbook might be just what you’re looking for.
For more information, read our review of Art of Problem Solving Textbooks
Check out AoPS Precalculus
A slower paced curriculum that explains concepts simply and focuses on sharpening those precalculus skills, Saxon is great for those seeking a back to basics approach to math
|Price: From $105.75 for a homeschool kit||Rigor: ⭐⭐⭐⭐|
|Conceptual/Procedural: Procedural Math||Progress Tracking: ❌|
|Common core: ✔ (non-common core textbooks available)||Instruction: Workbooks, textbooks, problem sets|
One of the more well known math programs out there, Saxon Math offers a pretty well-regarded precalculus curriculum, which it refers to as Advanced Math.
Taught through a combination of textbooks and workbooks, Saxon takes a back to basics approach to teaching math.
That is, it focuses more on the application of math, that is solving math problems accurately and efficiently, that is how to do math, rather than spending a lot of time on developing a conceptual understanding, i.e. the why.
Practically speaking, this means students can expect to spend more time learning and applying various math facts, rules and techniques in precalculus, product to sum formulas and law of sines/cosines for example, that will help them quickly solve various precalculus problems.
In terms of program rigor, with lots of computational practice questions and a quicker, spiral approach to presenting math concepts, Saxon Math can be a little more rigorous than a standard school curriculum.
That said, in terms of teaching, Saxon Advanced Math does a fairly good job at explaining precalculus concepts to students, with clear explanations and plenty of visuals and graphs, and is well known for providing a lot of review and practice in each lesson to help reinforce learning.
There are also a variety of in-activities and explorations that help connect math to the real world, providing students with a better idea of the practical application of precalculus concepts.
In fact, part of the Saxon method’s claim to fame is its process of incremental review, where older concepts are reviewed alongside newer concepts. This not only helps strengthen the retention of knowledge, but helps make math skills more automatic and quicker, as well, which is particularly come exam time.
Overall, while they may not get as much of the in-depth exploration of the whys of precalculus, if your student wants a program that can help them learn to do the math quickly and efficiently, and provide a lot of practice problems to help review and solidify concepts, Saxon Math may be the right choice right for them.
For more information, read our review of Saxon Math
Check out the Saxon Advanced Math Homeschool Kit on Rainbow Resource
Because it touches on more advanced concepts in high school algebra and geometry, precalculus can sometimes be a little intimidating to homeschoolers, especially for parents whose own math skills are a little rusty.
A math program that can explain precalculus concepts simply, comprehensively and fits the way a student learns best can be a very helpful tool when it comes to this sometimes tricky math subject.
While there are a lot of precalculus curricula out there, with a little care and some investigating, parents can easily find a curriculum that’s right for their student, which can often make the difference between math success and a tearful slog.
About the Author
David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.