Pre-Algebra I—Semester A
"I graph, therefore I am."
Since elementary school, you've always been sort of a math whiz. Fractions were freakishly easy, times tables took no time at all, and geometry was practically art class. Honestly, you don't even need your calculator anymore; it's really just a paperweight at this point.
Okay, we get it. You're smart. Quit bragging.
If you're looking to step up your middle school math game, we've got just the ticket. In this Common Core-aligned math course, we'll take you through readings, examples, and problem sets that cover
- rational numbers of all sorts, including those of the negative variety. (They're less pessimistic than they sound, we promise.)
- breaking down whole numbers into fractions and decimals and performing arithmetic with 'em.
- factors, prime numbers, and other helpful number theory tools.
- ratios, proportions, rates, and percents. And expressing the same number about eighty different ways.
- graphing, finding distances, and reflecting across the coordinate plane.
- similarity, scale factors, and dilations of figures.
So if you thought that middle school math was Dullsville, USA, then think again. And maybe get a better GPS while you're at it.
P.S. Pre-Algebra I is a two-semester course. You're looking at Semester A, but you can check out Semester B here.
Unit 1. Integers and Rational Numbers
We're slapping on some scuba gear and diving down deep to find the negative numbers of the rational number line. We'll see integers, fractions, all sorts of decimals—all in positive and negative forms. While we're down here below zero, maybe we'll get lucky and get to visit the famous octopus's garden.
Unit 2. Fractions, Decimals, and Division
Working with positive whole numbers is old hat for us, but the world doesn't always make life easy. We need to be able to work with positive and negative fractions and decimals. That's not even counting when things take a turn for the worse and we need to start dividing all this stuff together. This unit will whip us into shape so that we'll be ready to take on the world at a moment's notice.
Unit 3. Number Theory
Compared to Quantum Theory, Relativity Theory, and the Lost Sock Theory (for washing machines), number theory is pretty concrete. Probably because we have so many concrete strategies to deal with it; things like prime factorization, factor trees, and the GCF and LCM. If those sound like a bunch of random words thrown together, don't worry. You'll see some concrete examples soon enough.
Unit 4. Ratios, Proportions, and Percents
Numbers don't just sit there, isolated from the rest of the world. They tell about…stuff. And things! Can't forget about the things. If you want some more details than that, then we can bring in ratios, proportions, and percents, which are all about how two things relate to each other. And we don't mean in a family tree kind of way.
Unit 5. Coordinate Graphs
Even if you have a perfect sense of direction, a coordinate plane will still help you navigate the wild world of numbers. We can use it to plot points, find distances on the plane, and reflect things across both its axes. It's like a map, compass, and GPS all in one, and you only need paper and pencil to make it work. Handy!
Unit 6. Scale Factors and Dilations
Well, we have these coordinate planes that we can put points and shapes on. We also know about ratios and proportions, which tell us how two different things are related to each other. What happens if we smash them together? We'll see that similar figures, scale factors, and dilations will all fall out of the aftermath. We suggest bringing an umbrella.
Sample Lesson - Introduction
Lesson 3: Rules of Divisibility and Factor Trees
Rapunzel, Rapunzel, let down your hair. Though, it would be way easier if you just took our advice and installed an elevator in that massive tower of yours. We know a guy who can build it on a low-interest installment plan.
Technically, we never said we were going to be knights in shining armor. But yes, this armor is pretty heavy, not to mention shiny, and we really don't feel like trudging all the way up this time. Maybe we've been barking up the wrong tower—er, tree—all along, and it's time we started calling on someone else instead.
Someone like a non-prime number. Because rather than towers, non-primes live in trees. And all those factorable branches make for easy climbing. Plus, numbers aren't really all that bad, are they? Or at least they're better than snooty princesses.
Sample Lesson - Reading
Reading 3.3: Rules of Divisibility and Factor Trees
While it's easy to write out the factors (and prime factors) of a number using multiplication signs and parentheses, we can also represent factors on branched diagrams called factor trees, where each branch on the tree represents a factor. This means that prime numbers, like 2, make for some pretty dinky factoring trees, because they only have one factor pair.
More like a factoring sapling, if you ask us.
Things are more interesting (meaning branch-ier) when we get to non-prime numbers. That's because we can continue to break up non-prime factors into smaller and smaller factor pairs, meaning even more branches. For example, 24 has some nice looking foliage.
Prime Factorization with Factor Trees
This brings us to the grand motto of prime factorization: "Factor until thou canst not factor anymore, until thou hast reached the prime numbers, the factoring equivalent of dead ends."
Yes, we use Shakespeare-speech like that when we're really trying to make a point. Or when we're trying to convince Rapunzel to come down for a visit. What can we say? Dates like that kind of stuff.
A factoring tree's "dead end" is when all the branches end at prime numbers; at this point, the tree can't grow any more branches of factoring pairs, because primes can't be factored any further.
As a trick, while you are diagramming prime factorization on a factor tree, go ahead and circle the prime factors to mark your "dead ends," and cross out the non-prime factors that you factor even further. Then, once all your branches end in circled primes, you know you're done with prime factorization. Like this:
To start a factoring tree, we first have to know at least one factoring pair (preferably one other than 1 and the original number), so that we can draw our first branches.
If we don't think of a factoring pair off the top of our heads, it can be hard to tell what we're working with: a factorable number or a prime. Maybe our thinking caps are malfunctioning.
Or maybe we just need some handy rules to get us started. These shortcuts come from the fact that all factorable numbers are divisible by their factors. That means that when one number divided by another yields a whole number, then we say it is divisible by that number.
For example, we know that 2 and 6 are a factor pair of 12, and we can see that 12 ÷ 6 = 2. Ta da. Divide by a factor, get a factor. It's a pretty fair deal.
That means if we can recognize that the number at the very top of our tree is divisible by another number other than itself, then presto, we've got at least one factor, and we know our number isn't prime. Use the divisibility rules below as shortcuts to help recognize non-primes, and what they are divisible by:
|2||A number is divisible by 2 if it is even.||4 is divisible by 2 because it's even.|
|3||A number is divisible by 3 if the sum of the digits is divisible by 3.||342 is divisible by 3 because 3+4+2=9, and 9 is divisible by 3.|
|4||A number is divisible by 4 if the last two digits are divisible by 4.||124 is divisible by 4 because 24 is divisible by 4.|
|5||A number is divisible by 5 if it ends in 0 or 5||1535 and 170 are divisible by 5 because they end in a 5 or a 0.|
|6||A number is divisible by 6 if it is divisible by both 2 and 3. That means if it is even and the sum of the digits is divisible by 3.||342 is divisible by 6 because it is even and divisible by 3 (since 3 + 4+ 2 =9, which is divisible by 3).|
|7||A number is divisible by 7 if two times the last digit subtracted from the remaining digits is equal to zero, or is also divisible by 7. (You can rinse and repeat with this one.)||126 is divisible by 7 because the last digit (6) multiplied by 2 is 12, and if we subtract this from the first two digits (12), we'll get 0.|
|8||A number is divisible by 8 if the last 3 digits are divisible by 8.||1800 is divisible by 8 because 800 ÷ 8 =100|
|9||A number is divisible by 9 if the sum of the digits is divisible by 9.||981 is divisible by 9 because 9 + 8 + 1 = 18, and 1 + 8 = 9, which is divisible by 9.|
|10||A number is divisible by 10 if it ends in 0.||50 is divisible by 10 because it ends in a 0.|
We can use a branched diagram called a factor tree to break large numbers up into their factors.
A good tip for doing prime factorization using factor trees is to cross out the numbers you've factored. This is helpful because when you are done, you'll only see the numbers that aren't crossed out. Those are your prime factors.
The divisibility rules are a set of shortcuts we can use to decide whether a number has factors, and give us a starting point for a factor tree.
Sample Lesson - Activity
Activity 3.3a: Problem Set
Sample Lesson - Activity
Activity 3.3b: Birthday Factor Tree
Like most parties, birthday parties are always more fun when they involve a little math. If you disagree, then maybe you've been going to the wrong sorts of parties.
Actually, maybe we're the ones who have been going to the wrong sorts of parties. That could explain why so few people showed up to our last birthday bash.
In any case, while we've got your attention, we'd like to make the case that birthdays can be turned into math. Literally. By taking your birthday and factoring it out into a prime factorization tree.
Who needs birthday stones when you've got a unique set of birthday primes? And don't forget to decorate your tree and add a little pizzazz. It is your birthday, after all.
Step 1: Gather all the required materials: construction paper, pencils, and maybe an eraser. Once you have them, follow the directions below to create your snazzy birthday prime factorization tree.
Step 2: Using a combination of your month and birth year (for instance May 2002 would be 52002), make a factor tree to get this number down to its prime factors.
Step 3: If the number in Step 2 happens to be prime, try reversing the year and the month like this: 20025.
Step 4: State the prime factorization for this number, and decorate your tree with all the bells and whistles; that means be sure to cross out non-prime factors on the tree, and circle the primes.
Step 5: Double-check your work by multiplying all the prime factors you found together to see if you get the original number you started with at the top of the tree. Then upload it and congratulate yourself, because this can serve as next year's party invitations.
- Credit Recovery Enabled
- Course Length: 18 weeks
- Grade Levels: 6, 7
- Course Type: Basic
- Middle School
Just what the heck is a Shmoop Online Course?
Common Core Standards
The following Common Core Standards are covered in this course:CCSS.Math.Content.6.EE.A.2