A picture's worth a thousand words.
If you've been honing your Pictionary skills ever since you could pick up a pencil, then geometry has your name written all over it.
Geometry isn't your typical math course. Sure, there's math involved, but it's more about drawings and doodles than numbers and calculations. (Score!) Then again, it's also about using logic and reasoning to back up your arguments. Unfortunately, "Because I said so!" only goes so far. (Boo!)
With the help of hundreds of readings, activities, examples, and problems, our Common Core-aligned Semester A includes
- learning how to work with geometric tools from the undefined notions (how mysterious!) to a compass and straightedge.
- figuring out formulas and terms so you can get the answers you want/need.
- mastering logic and proofs to give us the tools we need to work with other figures (like…more points, lines, and planes).
- discussing triangular congruence and similarity, and even helping them find their centers. (By the time we're through with them, they'll be all SAS and no sass.)
P.S. Geometry is a two-semester course. You're looking at Semester A, but you can check out Semester B here.
Unit 1. Intro to Geometry
We'll start off by introducing you to geometry and giving you a brief rundown of the reoccurring concepts you'll see throughout the course. We'll learn about some of its fundamental building blocks, like points, lines, and angles, and even venture into the third dimension and the Cartesian plane. Complimentary peanuts will be provided.
Unit 2. Reasoning and Proof
Ever wanted to be like the detectives or lawyers on Law & Order? 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 meat and potatoes of geometry: proofs. Pass the gravy, please.
Unit 3. Transformations
In this unit, we'll be the masters of motion, transforming images to wherever we want, facing whichever way we want, and even in whatever size we want. We can even decide whether we want to perform these transformations on or off the coordinate plane! Don't let the power get to your head, though.
Unit 4. Parallel and Perpendicular Lines
Soon after learning about the properties of parallel lines, we'll shake things up by throwing transversals into the mix. Make sure you hold onto your congruent angles because these angles and their theorems are the main course of this unit. We'll perform constructions, prove theorems, and even talk about the slopes of these lines on the coordinate plane. And naturally, leaving out perpendicular lines just wouldn't be right.
Unit 5. Congruent Triangles
It's the same ol' same ol'. Well, we think that sameness—or congruence—can be good. (Oreos haven't changed in over 100 years, and they're still delicious!) In this unit, we will learn how to prove congruence in triangles and how to use that congruence to solve problems. We won't help you find a better place to stash your Oreos, though. You'll have to do that on your own.
Unit 6. Relationships Within Triangles
Up to now, most of our lessons have been focused on the tasty outer shell of triangles—properties that deal with their sides and angles, and proving congruencies. That's all good and fine, but the time has come for us to dive into the creamy center—special line segments and centers within the triangle, just dripping with delicious new postulates, corollaries, and theorems for us to enjoy.
Unit 7. Similarity
You probably thought you had escaped triangles, but we aren't finished with them yet. We're going to cover some new triangle theorems. At this point, these will practically seem like second nature. We'll also learn about the scale factor, which is the ratio by which the polygon is reduced or expanded. Who knew so much information was bundled up in just three sides?
Sample Lesson - Introduction
Lesson 3: Developing Definitions of Reflections
People have a love-hate relationship with mirrors. When we first roll out of bed, look in the mirror, and see that hideous monster staring back at us, we aren't their biggest fan. Apart from that, we seem to love them. Why would we spend so much time taking selfies if we didn't?
They help us get ready in the morning, tell us if we have food in our teeth (gross), and help us keep track of those suspicious-looking moles. If Disney movies were true, we could also get our mirrors to talk to us and show us our futures. Wouldn't that be cool?
The deceiving thing about mirrors is that they don't show us exactly the same thing. Instead of an exact replica, we get what's called a mirror image.
News flash: a mirror image is a reflection, and reflections are what this lesson is all about. We will be learning about how to do reflections of an image—just like what we see when we look in a mirror. Though you might want to run a comb through that hairdo of yours.
Sample Lesson - Reading
Reading 3.3: Reflections
When we look in a mirror, we see ourselves as everyone else sees us, except flipped. What we are seeing there is a reflection. It shows an image that's the same as the preimage in terms of size and shape, but flipped over a line. (When will it show who we are inside?)
What this means is that we take all the points of the preimage and make sure that the image's corresponding points are the same distance away from the line—but on the other side of it. That's the nitty gritty of how reflections work: the distances from the points of the figure to the line are conserved.
Since every point in the preimage is being mapped to a corresponding point on the opposite side of the line, we end up with a figure that has the same size and shape as the original, even though it isn't necessarily superimposable. (That's just a fancy word for "they look the same when we put one on top of the other.") This is true no matter what shape we start with: rectangles, parallelograms, trapezoids, other regular polygons, whatever you want.
A reflection is an isometry that results from flipping an image over a line. We take a center line and replicate every single vertex of the shape on the opposite side of the line, conserving each point's distance from the line. We end up with a figure that has the same size and shape as the original, though it might not be in the same position anymore. This is true for any shape, including rectangles, parallelograms, trapezoids, and other regular polygons. This even works if the line of reflection passes through the figure.
Sample Lesson - Activity
Activity 3.3c: Problem Set
- Credit Recovery Enabled
- Course Length: 19 weeks
- Course Number: 310
- Grade Levels: 9, 10, 11
- Course Type: Basic
Algebra I—Semester A
Algebra I—Semester B
Just what the heck is a Shmoop Online Course?
Common Core Standards
The following standards are covered in this course:CCSS.Math.Content.8.G.C.9