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Mathematics II—Semester A

It's like Math I, only completely different.

  • Credit Recovery Enabled
  • Course Length: 18 weeks
  • Course Type: Basic
  • Category:
    • Math
    • High School

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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:

Technology Requirements

Seeing as this is a fully online course, a computer with internet access is going to be a must. A scientific or graphing calculator will be a big help as well, but isn't required.

Unit Breakdown

1 Mathematics II—Semester B - 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.

2 Mathematics II—Semester B - 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.

3 Mathematics II—Semester B - 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.

4 Mathematics II—Semester B - 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.

5 Mathematics II—Semester B - 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.

6 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.

7 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.

Recommended prerequisites:

  • Mathematics I—Semester A
  • Mathematics I—Semester B

    • Sample Lesson - Introduction

    Lesson 7.05: Volumes of Similar Solids

    A photo of a baby elephant leaning it's head against an older elephants leg.
    "Did you have to name me Dumbo?"

    (Source)

    Baby animals are all the rage these days, and we think we know why: proportionality!

    No, wait, we're serious. The reason a baby elephant is just so entertaining to look at is because we have a pretty good idea of what a normal, giant-sized one looks like. It's like someone took an elephant and zapped it with a hilariously tiny scale factor. The miniature version is adorable and funny because it has the same proportions as the real deal. That tiny trunk! Those stubby little legs!

    We've covered similar shapes before, so you can probably see where this is going. The baby elephant is similar, mathematically speaking, to its mom. And humans have always delighted in little versions of things.

    Three-dimensional shapes can be similar just like flat, two-dimensional ones. The ratios get a little more complicated, but it's nothing we can't handle. Now where's the nearest miniature zoo?

    Sample Lesson - Reading

    Reading 7.7.05: Mini-Mes

    When working with our old, busted flat shapes, we learned that 2D shapes are similar when they have the same proportions and angles. Basically, everything about them is the same except for their relative sizes. That's the same situation we face when we deal with similar solids in 3D.

    Similar Solids in 3D 

    What did the similar solid tell its partner? #Twinning.

    And why wouldn't it say that?! Similar solids have the same shape and angles; their sides create the same proportions; they even have the same favorite ice cream flavor. The sizes of the two solids are different, sure, but they are still proportional. How can we double check if two solids are similar?

    Excellent question. If there is a constant ratio between corresponding measurements between the solids (the scale factor), the solids are similar. When we worked with similar 2D shapes, we would compare the lengths of the sides for proportionality, right? Well, with 3D shapes we can compare the side lengths and we can compare the area of the sides, too, a small perk of working in that extra dimension.

    Just like regular 2D shapes turned out to always be similar (if they were the same shape), regular solids are always similar too. Any two spheres, for example, are similar to each other no matter how big or small they are—because spheres always have the same proportions. Similar solids include tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons.

    That's what makes similar shapes and similar solids…well…similar to each other. However, similar solids have features that similar shapes don't.

    When we write ratios of a solid to check for proportionality, we've got to be careful since we're dealing with length, width, and depth, this time around.

    Since areas are squared, the areas of two similar figures have a square relationship. But volumes are cubed, so the volumes of two similar figures have a cubic relationship. We're pretty sure there's no Facebook relationship status for that.

    What do we mean by "cubic relationship"? We mean that the volumes of two similar solids are related by their scale factor cubed. For example, if two solids are related by a scale factor of 3, the smaller one will have , or  the volume of the larger solid, not  the volume. If two solids are related by a factor of 7, then the smaller one will have or the volume of the larger solid.

    Congruent solids, by the way, are also the same idea as their 2D versions: to be congruent, two shapes have to be exactly identical in all their dimensions. We're not sure why there's such a fancy word for "literally the exact same," but there you go.

    Recap

    Comparing proportions is the name of the game when it comes to similar figures. For most shapes, we'll need to compare at least a couple proportions to see if we find the same scale factor for each. Once we know the figures are similar, we know that the ratio length:area:volume follows the pattern k:k2:k3, like some Russian nesting dolls.

    Sample Lesson - Activity

    1. Two cylinders are similar. The larger cylinder has an altitude of 6 cm, and the smaller cylinder has an altitude of 2 cm. If the volume of the smaller is 10 cm3, what is the volume of the larger cylinder?

    2. If the dimensions of a cube are multiplied by a scale factor of 2, what will the volume be multiplied by?

    3. The dimensions of a hexagonal prism with a volume of 100 cm3 are multiplied by a scale factor of 4. What will the new volume be?

    4. The dimensions of the rectangular prism below are doubled. Find the new volume.

    5. A pyramid's dimensions are enlarged by a scale factor of 3. If the new volume is 8,100 cm3, what was the original volume of the pyramid?

    6. If the dimensions of an octagonal prism are multiplied by a scale factor of 10, what will the new volume be multiplied by?

    7. By what scale factor must the smaller prism be multiplied to obtain the larger prism?

    8. By what factor is the volume of the smaller cone multiplied to obtain the larger cone?

    9. Two cones have radii in the ratio of 1:2. What is the ratio of their volumes?

    10. Two cubes have sides in the ratio of 1:3. If the larger cube has a volume of 13,500 m3, what is the volume of the smaller cube?

    11. Two square pyramids are similar with the lengths of their edges in the ratio 2:3. What is the ratio of their volumes?

    12. The original volume of a cylindrical tank is 2700 liters. A similar cylindrical tank has a volume of 21,600 liters. By what scale factor were the dimensions of the first tank multiplied to obtain the larger tank?

    13. A hexagonal prism with an altitude of 10 cm has a volume of 600 cm3. If the prism is enlarged with a scale factor of 5, what is the area of the base of the enlarged prism?

    14. Two cylinders are similar. If their volumes are 600 ml and 4800 ml, what is the ratio of their radii?

    15. The two vases below are similar. If the volume of the smaller vase is 500 ml, what is the volume of the larger vase?

    16. A manufacturer would like to increase their package volume by 30%. What is the scale factor for the package dimensions?

    17. A manufacturer would like to increase their package volume by 20%. If the original package has a length of 7.50 cm, what will be the length of the new package?