Free step-by-step math lessons with guided notes to help you truly understand math.
Welcome to Understand The Math, where you'll find clear explanations, worked examples, and free guided notes designed to help you build confidence and succeed in math.
Whether you're studying algebra, trigonometry, precalculus, calculus, linear algebra, or differential equations, you'll find lessons that focus on building real understanding, not just memorizing formulas.
Created by Dr. Cheryl Hile, a former Penn State mathematics professor with more than 25 years of university teaching experience and a Ph.D. in Engineering Science and Applied Mathematics.
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Understand The Math
Subspaces can be one of the trickiest topics in an introductory Linear Algebra course.
This quick reference summarizes the definition, the three subspace conditions, and several common examples.
📌 Save it for later when you're studying!
For a complete explanation of vector spaces and subspaces, watch the lesson here: https://youtu.be/Niz4RbwxOfw
3 days ago | [YT] | 5
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Understand The Math
Calculus tip! The chain rule is all about taking the derivative of the outside function and multiplying by the derivative of the inside function. Here's a quick reference with formulas and examples.
Watch the full lesson here: https://youtu.be/mnfX_i2LK1A
1 week ago | [YT] | 2
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Understand The Math
Second-order differential equations don't have to be overwhelming.
This graphic summarizes the three possible cases based on the discriminant of the characteristic equation and the corresponding general solution for each.
📌 Save this for your next quiz or exam.
If you'd like to see these ideas explained step by step with examples, here's the full lesson: https://youtu.be/9H6IfrGLJpE
2 weeks ago | [YT] | 2
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Understand The Math
Sinusoidal functions don't have to be confusing.
This graphic highlights the four key features you need to identify from
y = A sin(B(x − C)) + D
✔️ Amplitude
✔️ Period
✔️ Phase Shift
✔️ Midline
📌 Save this for your next quiz or exam.
Want to learn how to graph sinusoidal functions step by step? Watch the full lesson here: https://youtu.be/62bqzy9qtB0
3 weeks ago | [YT] | 2
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One-to-One vs. Onto Transformations
These are two of the most important—and often most challenging—concepts in linear algebra.
Here's a quick visual cheat sheet that summarizes the key differences. Save it for when you're studying! 📚
3 weeks ago | [YT] | 6
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🎉 The redesigned Understand The Math website is live!
It's now easier to find the math course you need, with free videos, guided notes, and study resources organized by course.
4 weeks ago | [YT] | 3
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Getting ready for a math exam soon?
Here’s a simple reminder that can make a big difference: how you study matters just as much as what you study.
If you missed it, I recently shared a video that walks through 5 practical strategies to help you feel more organized, focused, and confident on exam day — no matter which math class you’re taking.
If you want a clear plan for reviewing notes, organizing formulas, and practicing problems effectively, this video can help.
🎥 Watch here: https://youtu.be/XWi9SPXhaaw
Which part of math exam prep do you find most helpful—reviewing notes, memorizing formulas, or working practice problems?
7 months ago | [YT] | 1
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Starting your week with Calculus 1 review?
Here’s a set of resources many students use to feel more confident before finals.
This review series walks through exam-style practice problems with clear, step-by-step solutions. Each video focuses on one topic at a time, making it easy to review limits, derivatives, integrals, and applications throughout the week.
Exam Review Playlist:
www.youtube.com/playlist?list...
Free Practice Problem PDF:
www.understandthemath.com/calculus1-exam-review
7 months ago | [YT] | 0
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Understand The Math
Cosine is positive in Quadrant II.
See the explanation in this example: https://youtu.be/lFpbswAA3a0
7 months ago | [YT] | 1
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Reviewing Calculus 1 Example Problems? 📘
This playlist features step-by-step solutions to key Calculus 1 problems, organized by topic. Each video focuses on one example at a time so you can review core concepts and strengthen your problem-solving skills alongside your main lessons.
Topics covered include:
• Limits and continuity
• Derivative rules and applications
• Curve sketching, related rates, and optimization
• Integrals and substitution
• Applications of integrals such as area, volume, and average value
Example Problems Playlist:
www.youtube.com/playlist?list...
Free Guided Notes 📝
www.understandthemath.com/calculus-1
7 months ago | [YT] | 0
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