7th Grade Math—Semester A
Keeping proportions in proportion.
Fractions and decimals might seem pretty rational, but what happens when we start replacing them with letters? Before you know it, you'll wind up with a math problem that looks more like a Shakespearean play than a number line. Good thing Shmoop's 7th Grade Math course is here to turn those incomprehensible variables into alphabet soup: easily digestible, not to mention delicious.
After a solid review of rational numbers in all their forms, we'll start working with a little algebra. Once we get our feet wet with variables and expressions, we'll deep-dive straight into one- and two-variable equations. Then comes another wave of numbers—ratios, proportions, and percents—and finally, we'll finish up our math expedition with scale factors and dilations. (Hope you brought your scuba gear!)
With all the readings, projects, and problem sets you could ever want, this Common Core-ified course gives you the 7th grade lowdown with lesson plans on:
- performing operations with fractions, decimals, and negative numbers.
- simplifying and evaluating expressions.
- solving both one- and two-variable equations and inequalities.
- representing and working with ratios, proportions, and percents.
- understanding scale factors and drawing dilations of shapes.
P.S. 7th Grade Math is a two-semester course. You're looking at Semester A, but you can check out Semester B here.
Unit 1. Rational Numbers
Numbers can take all kinds of forms, and we have to be ready for any that sneak up on us. We'll review how to defend ourselves against fractions and decimals, and practice the martial art of taking down integers. (Tip: aim for their negative signs.) With number lines, operations, and rational numbers, this unit will warm us up and prepare us for combat with any number.
Unit 2. Linear Expressions
From variables and PEMDAS to word problems, this unit will teach you how to express yourself more suavely in all kinds of situations—mathematical and otherwise. Unfortunately, there's no lesson on snappy comebacks. All of our best zingers show up three days late, too.
Unit 3. Solving Equations and Inequalities in One Variable
We'll start off covering the basic rules of solving equations. That means treating both sides of the equation exactly the same. No playing favorites! From there, we'll move onto inequalities, the hippie relatives of equations. They don't care about single values as much; as long as answers fall within a certain range, they'll go with the flow. Pretty groovy, huh?
Unit 4. Equations and Inequalities in Two Variables
We'll learn all about how two-variable equations can describe all sorts of different relationships—except your love-hate relationship with Lost. Those feelings are sort of inexplicable.
Unit 5. Ratios, Proportions, and Percents
Ratios, proportions, and percents were designed to work with groups of objects or to make comparing things directly easier. We can use them to do everything from comparing the prices of regular and jumbo packs of batteries to figuring out our savings per battery. (Four whole cents? Score!)
Unit 6. Scale Factors and Dilations
Now that we're well-versed in ratios and proportions, we'll apply these babies to two-dimensional shapes. Ever heard of dilations and similarity? We'll learn how to recognize and prove that two shapes really are related, kind of like they do on Maury—only with less swearing and more geometry.
Sample Lesson - Introduction
Lesson 4: Graphs of Proportional Relationships
Here's where we combine everything we've learned about fractions and ratios with the actual meat (or tofu, if you prefer) of algebra. Like soldering pieces of silver together to make a beautiful ring or fusing two atoms of hydrogen together to power our whole solar system, we're about to join two great things to make an even greater third thing. Brace yourself.
To be fair, we only start the fusing process here. It'll take a little while until we've got a full-fledged solar system on our hands. But as we continue exploring these different topics, we can expect to see a lot of topics mix and mingle more than middle-aged singles.
Sample Lesson - Reading
Reading 5.4: Graphs of Proportional Relationships
Proportional relationships are more than just two ratios with an equal sign in between them. They describe a relationship between two different values and we can get more values that follow this relationship by multiplying or dividing both values by the same number.
So for example, Phil can eat 46 Oreos in two minutes. What we have here is a proportional relationship because we're comparing two different units (Oreos and minutes). We also know that Phil can eat 23 Oreos in one minute because we can divide both 46 Oreos and two minutes by 2.
Since Phil can continue eating Oreos forever, there are an infinite number of ways to write out this proportional relationship. We can say that he eats 46 Oreos in two minutes, but saying that he eats 92 Oreos in four minutes is the same thing, isn't it?
Both ratios tell us the same thing about Phil's eating habits. (Namely, that they're awful.) In fact, as long as all the ratios have the same unit rate, they're all basically saying the same thing.
So the question is how can we represent these infinite ratios in a clear-cut way? Let's set up a table with the values we already know.
|Minutes Passed||Oreos Eaten|
While we can't graph this relationship on a number line, this is exactly why we have a two-dimensional coordinate plane. We're pretty sure you've seen these guys around before, but we'll introduce you just in case you were both too shy to say hello.
Coordinate Plane, but you can call him by his rapper name, Cory-P. All you need to know is that he's great for graphing pairs of numbers—one represented by the horizontal axis, and one represented by the vertical axis. You'll learn more about each other soon enough, so there's no need to rush anything.
If we assign the horizontal axis to represent minutes and the vertical axis to be Oreos, we can plot these three ratios on the plane.
These three points only represent the values in our table, but we know that we could just as easily plot how many Oreos Phil can eat in three or eight or 107 minutes, assuming he keeps going at a constant pace. But since an infinite number of those values exist, we can just draw a line connecting these points.
That line represents all the possible values that fall under this proportional relationship. Looking at the horizontal axis will tell us how many minutes have passed, while the vertical axis will tell us how many Oreos he's eaten in that amount of time.
If we look closely, we'll notice two important pieces of information that apply to all graphs of proportional relationships. First, our line goes through point (0, 0). That's because at 0 minutes (the very moment we click the stopwatch to time him), Phil hasn't eaten any Oreos yet. Don't worry, we checked.
It's also important to know that the slope of our line is equal to the unit rate. To verify this, all we need to do is look at how many Oreos Phil has eaten after one minute. We know he ate 23 Oreos, which is what our graph tells us too. That's the same as the unit rate we calculated earlier. Coincidence? We think not.
Proportional relationships have an infinite number of values that satisfy them. They can be represented as tables of values or as lines on the coordinate plane.
It's usually easiest to make a table of values for the proportional relationship first. To do this, multiply or divide both numbers in the given ratio by the same number. An easy way to check this is to make sure all ratios simplify to the same value (the unit rate).
Once you've got a solid table with at least three or four values, you can assign each unit to one of the axes (vertical or horizontal), and plot your values as points on the coordinate plane. Make sure these points all line up with one another because you'll draw a line through them. (Who saw that one coming?) The slope of the line should equal the unit rate and the line should always pass through the point (0, 0) on the graph.
Sample Lesson - Activity
Activity 5.4c: Problem Set
- Credit Recovery Enabled
- Course Length: 18 weeks
- Grade Levels: 7
- Course Type: Basic
6th Grade Math—Semester A
6th Grade Math—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.7.EE.A.1