Mathematics II—Semester A

It's like Math I, only completely different.

Shmoop's Mathematics II course has been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.


We get it. Sequels aren't always the best. Sometimes the original is just so good, though, that it'd be a crime not to make another one. Just imagine a world without a second Dr. Doolittle or Paul Blart: Mall Cop.

We'd all be at a loss...

Mathematics I left so many questions unanswered that we just had to make another. Who really killed Pythagoras? Who gets the girl? Where's Curly's gold? Okay, the questions we'll be answering are a little less dramatic, but you get the point.

Mathematics II is the substantially more geometrically flavored sequel to Mathematics I. In fact, that's what Semester A is all about. Here's what lies ahead:

  • We'll start with an introduction to mathematical reasoning and logic. What better way to start a math course than by learning how to actually think about math?
  • From there we'll move on to similarity of various 2-D shapes, especially triangles. They seem to be getting all the attention these days. We'll even describe their centers. It turns out there's a lot of them.
  • Next up, we'll focus on right triangles and navigate our way through the trigonometry landscape. There'll be a lot of sharp corners.
  • Then, we'll take a break from triangles and move into quadrilaterals and circles.
  • We'll close out the semester by entering a new dimension. Shmoop's going 3-D, but you'll have to provide the glasses.

FYI: Mathematics II is a two semester course, and you're checking out Semester A. If you want to see what's in the next semester, click here.


We also know the written word can only go so far. Anyone who's ever tried to decipher a text message from a love interest knows this. That's why this course is loaded with videos for the visual learner in all of us. Here's a taste of what's to come:


Unit Breakdown


  1. Mathematics II—Semester A - Reasoning and Proof

    Ever wanted to be like the investigators on CSI? (Definitely not the Miami version.) If so, then this unit is for you. We'll learn everything about logic and reasoning, from conditionals and contrapositives to syllogism and detachment. And of course, we can't forget the gravy of geometry: proofs. Pass the potatoes, please.

  2. Mathematics II—Semester A - Similarity and Dilation

    While a lot of shapes may not be exact twinsies, they can be similar. Here we'll introduce the ratios and proportions that accompany similar figures and what they're good for. We'll also work with dilations, so get those pupils ready.

  3. Mathematics II—Semester A - Relationships in Triangles

    We've mainly been focusing on the soft outer shell of triangles—properties that deal with their sides and angles, and proving congruencies. That's all well and good, but the time has come for us to take a bite out of that creamy center—special line segments and centers within the triangle, just oozing with scrumptous new postulates, corollaries, and theorems for us to enjoy.

  4. Mathematics II—Semester A - Right Triangles and Trigonometry

    It shouldn't come as any surprise that right triangles are pretty important. Yeah, the Pythagorean Theorem is kind of a big deal, and knowing the properties of special right triangles will certainly make your life easier, but none of that compares to what happens after: this unit is where we'll first tackle trigonometry.

  5. Mathematics II—Semester A - Quadrilaterals

    Quadrilaterals are shapes that have four sides. Seems simple enough, right? Well, not exactly. Square and rectangles are familiar territory, but there are some real wildcards out there. Ever heard of trapezoids or rhombii? Yeah, they're pretty weird.  We'll learn about all the different properties and proofs concerning quadrilaterals, and a few other polygons might sneak their way into the quadrilateral party. We can't blame them. It really is hip to be square.

  6. Mathematics II—Semester A - Circles

    There's a lot more depth to circles than you might think. (Did you know that most of them have an extensive collection of leather bound philosophical works, and live in townhouses redolent of rich mahogany?). After a little bit about central angles, arc measures, and arc lengths, we'll learn about the equations of circles on the coordinate plane and go through a few constructions with them.

  7. Mathematics II—Semester A - 3D Geometry

    What happens when we leave the relative comfort and security of the 2-D world and go 3-D? Well, this is the unit where we'll find out. We'll take 2-D shapes and stretch them into accompanying 3-D solids and even discuss the various 2-D shapes we can make by slicing and dicing 3-D solids. Might want to keep some Band-Aids close by...

  8. Mathematics II—Semester A - Real and Imaginary Numbers

    It turns out we've been leading a pretty happy, complacent existence by only working with real numbers. There's a whole other world to explore out there—one filled with so called "imaginary numbers." Fortunately, these guys behave a lot like real numbers...they've just got an added dimension.

  9. Mathematics II—Semester A - Polynomials and Quadratics

    Polynomials are one of the most important objects in math, and that's why we've got a whole unit chock full of 'em. We'll focus most on quadratics, though, going over the various ways to trick them into revealing their roots. They're pretty beguiling creatures.

  10. Mathematics II—Semester A - Quadratic Functions

    Wait, wasn't the last unit about quadratics? It was, but here we'll be dealing with them as functions, which means we can graph them. Graphing provides us with another tool to analyze these guys, and everything we learned in the last unit will help us construct even more accurate graphs. See? We're just attacking things from a slightly different angle.

  11. Mathematics II—Semester A - Functions and Inverses

    We've been working with functions for a while now, but here we'll see them like never before. After introducing some new function types, we'll throw them all together to create some mutant Franken-functions. There's even time for comparing functions, and discussing how to represent them in multiple ways.

  12. Mathematics II—Semester A - Probability

    Here's where we wrap the course up by taking a bit of a detour into probability. There's no shortage of math to learn here, as we run you through the gauntlet of rules everyone needs to know to calculate accurate probabilities. What's the probability that you'll love this unit? Strictly speaking, somewhere between 0 and 1, but we'll assume it's closer to 1.

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