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Mathematics III—Semester B

Math III or die hard.

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

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Shmoop's Mathematics III 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.

Sometimes, less is more. Well not this time, because we've got more Mathematics III for you. It doesn't matter what you're in the mood for, we've got more of it.

In this second semester, there's more Common Core-aligned readings, more problems sets, more activities, and more...more than you can shake a stick at. You could even use two sticks and not be able to keep up. Here's the lowdown on what's going down:

  • Look out, 'cause we're coming out the gate with a triple threat—think you can handle geometry, trigonometry, and modeling all at once? We'll lay down the law with the Laws of Sine and Cosine, deal with problems involving density, and bring our geometry knowledge into the third dimension.
  • While geometry and modeling are easily spooked, trigonometry won't stay down for the count. We've found it's secret, though—with the unit circle, we can easily find any value we need from any trig function. We'll even use it to do the formerly impossible and graph these wily and wavey functions.
  • Next up, we'll visit the Build-A-Function Workshop. Take all the functions you know and love, then create Frankenstein monsters by mashing them together.
  • We hope your gears are well greased, 'cause now we're switching them out to tackle some statistics—and answer some important questions at the same time. What makes a distribution of data normal, compared the all the other types out there? What's the difference between a population and a sample? And what's the point of all this stats, anyway?
  • We'll cap things off with a bit of probability. We'll even see some old friends from earlier in the course, the Binomial Theorem and the normal distribution. But who's this mysterious Bayes that's arrived on the scene?

Mathematics III is a two-semester course, and hey, this is Semester B. If you want to back up and look at what Semester A has to offer, go here.

If you like your math a bit more animated than words on a page, then Shmoop's videos (helpfully integrated into the course) will tickle your fancy and your funny bone.

Technology Requirements

A computer and internet access are kind of required to take this online course; who'da thunk it? A scientific or graphing calculator would be helpful here, but it's not required.

Unit Breakdown

7 Mathematics III—Semester B - Geometry, Trigonometry, and Modeling

This is where we'll study the strange interplay between geometry and trigonometry. It turns out we can use trig to find the area of a triangle, and the laws of cosines and sines are just other tools in our belt for finding missing sides and angles in triangles. Then we'll enter the third dimension and do a little modelling with 3-D solids. Get ready to strike a pose. No duck face allowed.

8 Mathematics III—Semester B - Trigonometric Functions

Get ready for a face-to-face meeting with trigonometric royalty. After introducing the unit circle and a brief review of the basic trig ratios, we'll be ready to up the ante by graphing trig functions, and shifting and flipping them all over the coordinate plane. We'll finish things off with some applications of what you can actually do with your newfound trig connections.

9 Mathematics III—Semester B - Building Functions

This unit is for the pragmatist in all of us. We're going to do our best Bob the Builder impression and build some functions. Using all the functions we've learned about so far, we'll toss them in a pile and mix-and-match them to our heart's content. It may sound like a frightening, genetic engineering project gone wrong, but we promise we know what we're doing.

10 Mathematics III—Semester B - Statistics

We're doing a 180 here by moving into statistics. We're talking everything from summary stats to the normal distribution and everything in between. By the end of it all you'll never look at data (maybe even the world) the same way again. We're 100% confident.

11 Mathematics III—Semester B - Probability

After a unit on statistics, it's only natural that we move into probability. These two topics are a lot more similar than you might think. Probability is all about taking what we know about a situation and making our best guess as to what'll happen next. Sometimes we'll be wrong, but that's just how this game works. Given that we live in a world so random that a butterfly in Kazakhstan might cause a war in Timbuktu, having the math to predict events is pretty cool.

Recommended prerequisites:

  • Mathematics III—Semester A
  • Mathematics II—Semester A
  • Mathematics II—Semester B
  • Mathematics I—Semester A
  • Mathematics I—Semester B

    • Sample Lesson - Introduction

    Lesson 10.10: Random Sampling

    From Getting a Couple Together, the Director's Cut

    (Source)

    It's almost time for the summer blockbuster season, and we want to predict what the next big hit will be. Will it be the action-packed Dirigible Under Fire 2: The Legend of Curly's Gold? Maybe the romantic comedy Getting a Couple Together? The horror flick coming out next week, Oh God, Where Did All of These Tentacles Come From, has been getting good reviews, too.

    We take a poll of our friends, and they all agree that the must-see movie of the summer is Watching Paint Dry: A Documentary. We might need to get new friends because their taste is pretty awful. While they might be a sample of the movie-watching population, they aren't a good one. If we want a representative sample, we can't just sample individuals that are easy to find and measure.

    Sample Lesson - Reading

    Reading 10.10.10: Hyper Monkey Zombie Ninja Sample

    Not all samples are created equal. In fact, we only want a random sample; every other type of sample can take a hike. See, when we take a sample, we want it to be an unbiased glimpse into the population as a whole. If we're not careful to take a random sample, we might have a sample of people only named Steve, and then we'll never get them to stop talking about evolution.

    What do we mean by random sampling, though? It doesn't mean that the sampling is done arbitrarily or haphazardly. Instead, it's random in the same way that flipping a coin or rolling a die is: every outcome is equally likely to occur, and every recorded result is independent from the others. That means that everyone that might possibly be included in a sample has an equal chance to be included, and the outcome of one measurement doesn't affect the next.

    How can we possibly get a random sample? It's actually pretty tough. We can assign numbers to members of a population and draw numbers for our sample at random, or we could throw darts at a map while blindfolded. Sometimes we don't have an accurate count of the population, and other methods need to be used. In any case, we should use a random number generator to pick our numbers instead of trying to think up random numbers ourselves. People in general are actually pretty bad at that.

    Just because the sample is selected randomly doesn't mean the result is random as well or that the outcome will be 50/50. If we asked 100 random people if their favorite food was broccoli, we wouldn't expect half of them to say yes, would we? However, there are probably a few strange people in the whole world who would, and if our sample is random, they will hopefully be represented.

    If broccoli lovers are too rare, though, and our sample is too small, they might slip through our fingers by chance. That's one reason why it's important to have as large a sample as we can. Another reason is because we're lonely and would like the company.

    Recap

    Taking a random sample from the population means that everyone has an equal and independent chance of being included. This helps create an unbiased sample; random sampling is the whole reason we can estimate a population parameter using a sample. Computers and random number generators are a big help when trying to randomly select things to sample.

    Sample Lesson - Activity

    1. A magazine wants to conduct a poll to find out what the most popular pet fish type is for fish owners that are also subscribers. They decide to send out 50,000 random texts to their subscribers. Even if everyone that receives a text responds and the texts are truly randomly sent, why might this not be a representative sample of their reader's opinions?

    2. A fair 6-sided die is tossed 8 times and the results are 1, 1, 1, 4, 5, 6, 1, 1. From this information, is this a random sample and why?

    3. If we want to randomly pick the order of a set of treatments for an experiment, which method is the most random?

    4. If we wanted to randomly pick some flowers on the prairie for a research study, why is it better to assign numbers to each flower and let someone else pick out the numbers randomly?

    5. Which of the following is not a random sample?

    6. Which of the following is an example of random selection?

    7. Which of the following is a true statement about random sampling?

    8. Does random sampling mean that there is no order or method to the approach?

    9. What do you need to be able to make inferences about a population as a whole?

    10. Picking every student sitting in the front row of your math class for a survey on study habits among the students at your school is a random sample. Is this true or false?

    11. A coin is flipped 27 times and heads comes up every time. This is an an example of a random sample. Is this true or false?

    12. There are several ways Susie could go about this. One way would be to survey every tenth person who comes in throughout the day.