MATH LESSONS is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus!

Here are the topics that She Loves Math covers, as expanded below:  Basic Math, Pre-Algebra, Beginning Algebra, Intermediate Algebra, Advanced Algebra, Pre-Calculus, Trigonometry, and Calculus.

Note: If you’d like to receive notifications when I post on my blog, please subscribe to my site: 

Note: Please let me know if you see any errors or have suggestions for this site at Thanks!

Basic Math

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 Beginning Algebra

Introduction to Algebra 

Types of Numbers and Algebraic Properties

Solving Algebraic Equations

Linear Inequalities

Word Problems in Algebra

 Intermediate Algebra


Systems of Linear Equations and Word Problems

Algebraic Functions

Scatter Plots, Correlation, and Regression

Exponents and Radicals in Algebra

Introduction to Multiplying Polynomials

Introduction to Quadratics


Solving Quadratics by Factoring and Completing the Square

Quadratic Inequalities

Quadratic Applications

Solving Absolute Value Equations and Inequalities

Solving Radical Equations and Inequalities

Imaginary (Complex) Numbers

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Advanced Algebra

Advanced Functions: Compositions, Even and Odd, Increasing and Decreasing

Inverses of Functions

Parent Functions and Transformations

Piecewise Functions

The Matrix and Solving Systems with Matrices

Introduction to Linear Programming

Rational Functions and Equations

Graphing Rational Functions, including Asymptotes

Graphing and Finding Roots of Polynomial Functions

Solving by Factoring

Exponential Functions

Logarithmic Functions

Transformations, Inverses, Compositions, and Inequalities of Exponents/Logs

Solving Inequalities

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Right Triangle Trigonometry

Angles and the Unit Circle 

Linear and Angular Speeds, Area of Sectors, and Length of Arcs

Graphs of Trig Functions

Transformations of Trig Functions

The Inverses of Trigonometric Functions

Solving Trigonometric Equations

Trigonometric Identities

Law of Sines and Cosines and Areas of Triangles

Polar Coordinates, Equations and Graphs

Trigonometry and the Complex Plane

Calculus (Differential)

Introduction to Calculus

Limits and Continuity

Definition of the Derivative

Basic Differentiation Rules: Constant, Power, Product, Quotient and Trig Function Rules

Equations of the Tangent Line, Tangent Line Approximation, and Rates of Change


The Chain Rule

Implicit Differentiation and Related Rates

Curve Sketching


Differentials, Linear Approximation and Error Propagation

Exponential and Logarithmic Differentiation

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 Calculus (Integral)

Exponential and Logarithmic  Integration

Exponential Growth

Derivatives and Integrals of Inverse Trig Functions

Applications of Integration: Area and Volume

Integration by Parts

Integration by Partial Fractions

Antiderivatives and Indefinite Integration

U-Substitution Integration

Differential Equations and Slope Fields

L’Hopital’s Rule

Riemann Sums and Area by Limit Definition

Definite Integration


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72 thoughts on “MATH LESSONS

  1. This was VERY helpful! I had a test just tomorrow in Pre-Algebra and I was practically clueless. I am beginning to understand the areas of math we are going through, though I currently have a D in my class because the 13/20 on my unit test lowered it from a C+. I was about to give up, but I knew acing my next test would help my grade a lot. I recommend this site to any girls that are struggling with math or just need a little re-learning. Wish me good luck on tomorrows test! I hope I get this one, and I will be happier than ever…I’ve always wanted to show my mom a great math test score… 🙂

    • Thanks for the sweet comment, Emme! I’m thinking good math thoughts your way, and I know you can do it!! Thanks for visiting the site and let me know if you ever have any math questions!! Lisa 🙂

    • Renee,
      Congratulations for home-schooling your precious daughter. Thanks so much for kind words and let me know if there’s anything I can do to make the site better 🙂

  2. This site is exactly what I’ve been looking for to help my mathematically talented but easily bored son. The examples and explanations are thorough enough to educate but concise enough to hold interest.

    • Diane,
      Thank you so much – you just made my day! Please let me know if there’s anything I can add to make it better 🙂

  3. I’m having a really hard time when it comes to radicals 🙁 Too bad, I can’t find something here that will help me. 🙁 maybe you should add them to your site. 🙂 Thank you! Your website is so cool

  4. I’m working on “Laplace Transform”…
    I need some real life examples which can be solve by L.T.
    can you help me ?
    please make it faster.

  5. I love the site and how much has gotten done since I last looked, but I also see that there isn’t a geometry section. Could you work on this after your other projects?

  6. A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2.If f(1)=1, and f(-2) =-2, find f(-1) and write the complete

    • You can solve this using systems. let y = -2x^3 + ax^2 + bx +c. Plug in (0, 2) to see that c = 2. Then we have y = -2x^3 + ax^2 + bx +2. Plug in (1,1) to see that 1 = a + b. Then plug in (-2,-2) to see that -10 = 2a – b. Solve that system to see that a = -3, and b = 4. So we have y = -2x^3 – 3x^2 + 4x +2. So then f(-1) = -3. Hope that helps! Lisa

    • Thanks so much for writing, and please spread the word about She Loves Math. This makes me want to keep writing 😉 Thanks again, and let me know if I can make it better, Lisa

  7. Lisa
    I stuck on trying to figure out how to parametrize
    y=(-x^2/72)+x so as to get x=24 sqrt[2] t. Once I have the x
    I can figure the y which is -16t^2 +24Sqrt[2] t.
    (sqrt is abbreviation for square root)
    Thank you for your help

    • Thanks for writing, Manuel. I think when you go from an equation with x and y to parametric equations with t, t could basically be anything. But if they wanted the y in the form y = -16t^2 + something, we could set -x^2/72 to -16t^2 (the first term of the y), and solve for x to get 24sqrt(2)t. Does that make sense? Lisa

  8. Lisa
    As you said, I could have used any parameter for x.
    I was just wondering where the author chose
    24Sqrt[2] t, from the rectangular equation y=x-x^2/72 to get
    y=-16t^2 +24Sqrt[2] t. I tried all kinds of parameters for
    y=x^2/72 and sure enough, while I get different values for y,
    Thanks again. Basically, I didn’t understand what was meant by

  9. Hello Lisa
    Please excuse if you received this comment before.
    I was wondering why you haven’t recommended the use of I believe the best calculator available, namely, Wolfram Alpha. It can be downloaded thru your app in your iPad or from the internet for free. It’s better on the iPad. I think it would help to take the drudgery out matrix`manipulations, vectors, quadratic equations, 3D graphing, et al.
    As far as I’m concerned you are the best Just thought you may be interested.
    You probably know this anyway.

    • Manuel,
      Thanks for writing! I don’t have a lot of experience with Wolfram Alpha and just downloaded the app on my ipad. It looks fantastic! I have used Mathematica in the past; I guess it was based on this.
      Thanks so much for the suggestion; that’s a great idea! They don’t use it at the high school, so I haven’t had to help any kids on it, but it’s definitely on my to-do list.

      Thanks again and spread the word about She Loves Math 🙂

  10. I am having trouble with this section, if you could get back to me asap it would be greatly appreciated.
    1)In the nineteenth century, the Austrian monk Gregor Mendel noticed while crossbreeding plants (peas in particular) that often a characteristic of the plants would disappear in the first-generation offspring but reappear in the second generation. He theorized that the first-generation plants
    contained a hidden factor (which we now call a gene) that was somehow transmitted to the
    second generation to enable the characteristic to reappear.
    As an example, suppose we denote the gene that produces the yellow seed by Y and the gene that
    produces the green seed by g. The uppercase Y indicates that yellow is the dominant gene and
    the lowercase g indicates that green is recessive. The table below shows the possible theoretical
    outcomes that can occur when we cross two first-generation plants.
    First Generation Plant
    Y g
    First Generation
    Y YY Yg
    g gY gg
    Notice from the table that of the 4 possible outcomes, 3 of the plants will be yellow (since 3 have
    the dominant Y gene) while 1 of the plants (since only 1 has two recessive genes) will be green.
    From this we can say that the theoretical probability of the second-generation plant being green
    is ¼ or 0.25.
    There is of course a difference between theoretical and experimental. The following table lists
    some of the actual results that Mendel obtained in his experiments in crossbreeding peas.
    Experimental Results
    Characteristics That Were
    Crossbred First-Generation Plants Second-Generation Plants
    Tall versus Short All tall 787 tall
    277 short
    Smooth versus Wrinkled
    Seeds All smooth seeds 5,474 smooth
    1,850 wrinkled
    Yellow versus Green Seeds All yellow seeds 6,022 yellow
    2,001 green
    Notice from the results that tall, smooth, and yellow are all dominant genes. Also notice that based on the experimental results, the probability of a second-generation plant being green is 0.2494. This agrees fairly well with the theoretical probability.
    From the tables of possible theoretical outcomes and experimental results, answer the following questions.
    1) Assume that we are crossbreeding genetically pure tall plants with genetically pure short plants. Create a table (like the one for yellow and green plants) indicating the possible combinations of tall and short peas.
    2) What is the theoretical probability that a plant will be short?
    3) What is the experimental probability that a plant will be short?
    4) How do theoretical and experimental probabilities compare?
    5) Assume that we are crossbreeding genetically pure smooth-seed plants with genetically pure wrinkled-seed plants. Create a table (like the one for yellow and green plants) indicating the possible combinations of smooth-seed plants and wrinkle-seed plants.
    6) What is the theoretical probability that a plant will have smooth seeds?
    7) What is the experimental probability that a plant will have smooth seeds?
    8) How do theoretical and experimental probabilities compare?

    • Thanks for writing! I’m sorry – I’m having trouble making sense of the tables; let me see if I can find this problem online somewhere. Sorry about that. Lisa

  11. I THINK I understand the problem statement and questions. I have re-stated the problem and answered the questions in a file on my website. I hope it helps. The document is available on Click on the “Math Docs” tab and look for the Algebra 2 file named “Statistics Example 01.”

  12. How long will the calculus site in construction?? coz i hope it will help me in my calculus project assignment. i hope it will settle faster, pleaseeeee

    • Thanks for writing! I’m SO sorry about not completing the Calculus portion yet – I hope to by the fall! Keep checking back! Lisa

  13. this problem makes me confused, can you help solving this?
    a receptacle contains x ounces of acid. A second receptacle contains x ounces of water. From the acid y ounces are removed, placed in the water, and the contents thoroughly mixed.Then, from the second receptacle, y ounces of the mixture are removed and placed in the acid. find (a)the concentration of of acid in the second receptacle, and (b)the concentration of water in the first receptacle

    • Here’s how I’d do this one – I got y/(x+y) or y/(x+y)*100 % for concentration of acid in the 2nd receptacle. Then I get (y * (x/(x+y))/(y+x-y) = y/(x+y), or y/(x+y)*100 % for the concentration of water in the first receptacle (amount of water over total amount). You can try it with real numbers. Does that make sense? Lisa

  14. I’m having trouble at this problem… hope you reply..
    Two pipes running simultaneously can fill a swimming pool in 6 hours. If both pipes run for 3 hours and the first is then shut off, it requires 4 hours more for the second to fill the pool. How long does it take each pipe running separately to fill the pool

  15. Please add log inequalities (for pre cal)
    there’s a problem that I can’t seem to figure out:

    log (12x) + log (X-1) > 2

    would the domain restriction be X>1 or would it be the domain restriction of the two log combined – log (12x^2 – 12x?

  16. Hello Lisa,

    I love your website. It would really be helpful if you could cover non-differential Calculus for us as well!

    • Thanks so much for your kind words – comments like these make me want to keep writing! I will get to non-differential Calculus eventually – hopefully this summer! Please spread the word about 🙂 Lisa

  17. This website is amazing! Even though I am a boy (and 42 years old), I just want to say I really appreciate all the hard work you have put into it. I wanted to brush up my skills a bit for a standardized test I have to take soon for nursing school, and this is by far the most comprehensive, complete, logical review I’ve seen. Exactly what I needed! Thank you.

    • David,
      Thanks so much for writing!! And kudos to you for being a nurse – such a wonderful field. I’ve thought about making the site more generic – not “she loves math” but something else – so still may. Let me know if you have any more feedback on the site! Lisa

  18. Hello Lisa,

    I looked through your website and I couldn’t really find anything that dealt with Binomial Series. I am having trouble with using the alternate definition of binomial coefficients to show that if n is a non-negative integer then (n k) = 0 whenever k > n. How does this allow me to express the binomial expansion as the infinite series (a+b)^n = (summation from k=0 to infinity) Σ (n k) (a^n-k)b^k

  19. Hi Lisa,
    Your website is awesome! I was wondering if you could tell me what I am doing wrong in this last step of this algebra word problem.
    It says that each helper can make 2 large cakes or 35 small cakes per hour. The kitchen is available for 3 hours and 20 large cakes and 700 small cakes are needed. How many helpers are required? The answer is 10 helpers are needed.
    Now using proportions I was able to determine that 10 helpers/hour are needed to make the 20 large cakes. Also 20 helpers are needed to make the 700 small cakes. So then adding the 10 helpers/hour+20 helpers/hour gives you 30 helpers/hour . But they give you 3 hours to complete the cakes so to get the answer of 10 they are dividing the 30 helpers by the 3 hours but the units do not cancel to give you helpers when you do that because its 30 helpers/hour/ 3 hours = 30 helpers/hr^2 which does not make any sense? Can you please explain how they are getting the unit just to be helper when they have the rate 30 helpers/hour and the 3 kitchen allotted hours? This question is really bugging me!

    • Here’s how I’d do this: Since 20 large cakes are needed, we would need 10 hours (2 large cakes per hour). Since 700 small cakes are needed, we would need additional 20 hours (35 small cakes per hour). So we’ll need 30 hours total. Since 3 hours are available, we would need 10 helpers. Does that make sense? Lisa

  20. Hi Lisa,
    A few days ago I was searching internet to find useful math tutorial websites.
    I found lots of them but this website is something else ! Excellent !
    Are you going to support the text parts with tutorial videos ?
    And I am looking for a website like your website that teaches physics to high school and college students from basics through advanced level. In any chance do you have any idea to introduce me a few?
    I visited some websites like KHAN ACADEMY but I did not like them so much. To me, they were not attractive enough.
    And the last thing do you have any idea how a student could be encouraged and motivated to learn pre-calculus or calculus while many of them know after graduating from college or even universities and when they get a job they will nearly never use what they had learned before!
    To be honest with you I myself know very few doctors or nurses who use calculus techniques although they need to pass the related exams while studying at the college or universities.
    Thank you.

    • Alireza,
      Thanks so much for your kind words! I’d love to do videos – I just want to finish the site (Calculus) first. And please let me know how I can make the site better.
      Thanks again!

  21. Hi Lisa,
    The Wronskian method is valid for all set of function or just for certain set of function? If just the certain set of function, can you list some of the example. Thank you.

    • Thanks so much for the compliment and the suggestion. I will ask my husband about this, since he develops iphone apps 😉 Please spread the word about 🙂 Lisa

    • Thanks so much for your kind words – this makes me want to write more and more! Not sure if I could handle a physics site, though 😉 Lisa

    • Thanks for writing! I have a few sentences talking about sets in this section, but not a lot. I’ll put it on my list to add more.

      Thanks for saying you like my site – please spread the word about! And let me know if you want me to add anything else or if you see any errors 🙂 Lisa

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