Pre-Algebra II—Semester B
Scatter plots, not scatter brains.
People often ask, "Which came first, the chicken or the egg?" as though it were some kind of super stumper. It's really not, though; you kind of need an egg before you can get a chicken. Hatching from eggs is something they are kind of famous for.
It's pretty much the same thing with Pre-Algebra and Algebra. Before we can move on to the big bad world of Algebra, we need to master the tools of Pre-Algebra and get them under our belt. Don't worry—it's a pretty big belt. In this Common Core-aligned course, we'll use loads of drills, examples, and projects to cover
- systems of linear equations and inequalities
- functions and relations
- triangles, triangles, and maybe some more triangles
- the basics of three-dimensional geometry
- statistics and probability
Are you ready for the egg of truth to be turned into an omelet of understanding?
P.S. Pre-Algebra II is a two-semester course. You're looking at Semester B, but you can check out Semester A here.
Unit 8. Systems of Linear Equations and Inequalities
Get out your thinking cap and your scheming scarf, because we're going to start working with two of the most common algebraic tools that people use to plan and strategize: systems of linear equations and linear inequalities. You can use them to make a business plan, strategize your rise to power, and orchestrate your eventual conquering of the world. Pretty useful, right?
Unit 9. Functions and Relations
We've only recently been introduced to functions, but we're already becoming fast friends. They've been telling us all about themselves: what they look like in algebraic or graph form, what kinds of x and y values they like, their favorite Fall Out Boy song, and what food allergies they have. You know; all the little details that friends should know.
Unit 10. Triangles and the Pythagorean Theorem
Triangles are really simple, right? How many ways can there be of putting three sides and angles together? Well, the shapes themselves might be simple and predictable, but there's still a lot to say about them. We're going to study these three-legged creatures in their natural habitat, and even use Pythagoras's theorem to prove that some of them are more right than others.
Unit 11. 3D Geometry
We'll be spending this unit getting to know cylinders, cones, and spheres inside and out: literally. We're going to talk about their insides and outsides (i.e., their volumes and surface areas). There will be formulas galore, so it will help to pull out the calculator for this one. We'll end things with a hands-on project to get psyched up about circular solids. Whoo!
Unit 12. Statistics
Are two pieces of data related in some way? We're going to try to find out in this unit, using plenty of two-way frequency tables and scatterplots. We'll guide you through nonlinear associations, linear models, and both numerical and categorical data. By the end of it all, you'll be BFFs with lines of best fit. Get the scrapbook ready.
Unit 13. Probability
It's time to start unlocking the secrets to predicting the future. (Think less divination and more blackjack.) We'll make tables and diagrams to show the probabilities of different events, calculate our chances of getting a winning poker hand or lottery ticket, and dabble in the black art of simulation. All to bring probability to life. Mwahaha!
Sample Lesson - Introduction
Lesson 3: Domain and Range
Humans are fascinated by transformation; just look at our classic comic book heroes. A hapless nerd gets bitten by a spider or blasted by gamma radiation and—poof!—he's transformed into a heavy hitter with a cool (yet tragic) backstory. Most supervillains also have some major transformation in their personal histories. A little lab accident here, a little acid in the face there, and another well-meaning physicist or politician turns to a life of crime.
Even before they're old enough to hold a comic book right-side up, kids love to watch something turn into something else. Did you ever push Play-Doh through that gadget that turns it into a pile of green spaghetti? Maybe you were preparing for a career as a mad (culinary) scientist. Or maybe it's just that children are deeply rooted to life's bottom line: inputs and outputs. (Changed any diapers lately? Then you know exactly what we're talking about.)
Like toddlers (and evil geniuses), functions are all about turning things into other things. Sometimes, we'll even replace y with f(x) when dealing with functions, but all that f(x) really means is, "We're gonna push x through this here contraption, and you'd best believe that something else will be coming out the other end." Let's just hope it's as pungent as what comes out of babies' other end.
Since inputs and outputs are at the heart of functions, we're going to spend some time playing with them. If you're lucky, maybe we'll let you use our Squeezy Freezy function to transform lemonade into delicious lemonade slushies.
Sample Lesson - Reading
Reading 9.3: Master of Your Domain (and Range)
Every field has its jargon, so mathematicians have special names for the inputs and outputs of functions. The set of numbers that can be plugged into a function is called the domain, and the set of numbers that can come out is known as the range. If this makes more sense to you in the form of a cheesy song parody, watch this video.
Domains and ranges are not the same for all functions. In the very, very near future, you will probably be given a function and asked to find its domain and range. So let's start preparing ourselves for that moment right about…now!
Determining a Function's Domain
So why would functions even have different domains? If x is the independent variable, can't it be anything it wants to be? In many functions, yes it can. But certain nonlinear functions put x in a corner, where its options are more limited. Watch out for:
- Functions that have x in the denominator. Remember the old fraction motto, "Real heroes don't let denominators equal zero"? It's still true in the world of functions. Any time you see a function that has x in the denominator, ask yourself what values of x would cause that denominator to equal zero. Then call for the bouncer and kick those values out of Club Domain.
- Functions that have x inside a square root. We can only take the square of positive numbers, so any values of x that make the contents of the square root negative are not allowed in the domain of the function.
Determining a Function's Range
When determining range, the key question is "What y values are impossible for this function to have?" In the case of the nonlinear function y = x2, there's no way that y can have a negative value. We just can't square a real number and get a negative answer, no matter what kind of awful day it's had. When in doubt, graph it out; we'll see that every point on the line has a positive y value:
Look at that frown-turned-upside-down! The graph of the function shows that when the domain of y = x2 is the set of all real numbers, then its range is the set of all real, positive numbers.
You may be wondering why we specify "all real numbers" when talking about seemingly unlimited domains and ranges. There are functions that are equipped to handle imaginary or complex numbers, but we won't see any on this trip. (You have to charter a special coordinate plane for that.) Still, when talking about domain and range, we like to be clear that the functions we're dealing with aren't into any freaky stuff. This is a family-friendly website.
- A function's allowed inputs (x values) are known as its domain.
- The allowed outputs of a function (y values) are known as its range.
- When finding the domain of a function, watch out for x within a denominator or a square root. Those domains will exclude at least one x value.
- Graphing the function often reveals limits of the domain and range.
Sample Lesson - Activity
Activity 9.3: Problem Set
- Credit Recovery Enabled
- Course Length: 18 weeks
- Grade Levels: 7, 8
- Course Type: Basic
Pre-Algebra I—Semester A
Pre-Algebra I—Semester B
Pre-Algebra II—Semester A
Just what the heck is a Shmoop Online Course?
Common Core Standards
The following standards are covered in this course:CCSS.Math.Content.7.G.A.2