My pre-algebra class hit the topic of equations just as the NFL season moved into the playoffs. The result was this series of class notes called “The Game of Algebra.”
We used the Singapore Math NEM 1 textbook, which is full of example problems and quality exercises. These notes simply introduce or review the main concepts and vocabulary in a less-textbooky way.
I hope you find them useful.
The Algebraic Term
Every game has its own special equipment. Football uses a ball and a playing field that are different from those of any other game. Chess has its special board, pieces, and pawns.
In algebra, one of the most important pieces of equipment is…
- The algebraic term (pdf 75KB)
Algebra is a game, playing with ideas. Like football or chess or any other game, algebra can be fun, or it can be torture.
A game is fun if:
- You are in shape.
- You know the rules.
- You have a well-matched opponent who makes the game a challenge, so that winning is sweet.
A game is torture if:
- You are out of condition.
- You don’t know how to play.
- Your opponent is much too weak, so the game is tediously boring.
- Your opponent is much too strong, so the game is impossible.
For the rest of this semester, we are going to focus on learning some basic rules of the game of algebra.
Here is rule #1: Substitution (pdf 60KB)
The Distributive Law
You have done this for years in arithmetic, whenever you multiplied large numbers:
Now you need to practice doing it with algebraic terms. You need to do it over and over again, until it becomes automatic. This is one of the foundational moves of algebra — so make your foundation firm!
- The Distributive Law (pdf 80KB)
The Balance Rule
An algebraic equation is like a balance scale. If the two sides of the scale are in balance, then the weights in the two pans must be the same. In an equation, the equal sign means that the numerical value of the expression on the left hand side is the same as that on the right hand side.
- You can add the same amount of weight to both pans (or subtract the same amount from both pans), and your scale will still be in balance.
- Similarly, you can do any mathematical operation you wish to your algebraic equation, as long as you do the same thing to both sides of the equal sign.
That is… The Balance Rule (pdf 60KB)
5 Steps to Solve Almost Any Equation
Time for a workout! The next few lessons will give you practice in solving single-variable (mostly linear) equations. Remember what you have learned, and follow these steps for…
- Solving equations (pdf 54KB)
- Simplify each side using the Distributive Law.
- Collect the variables on whichever side of the equal sign has the greater coefficient. Remember the Balance Rule.
- Collect the constants on the opposite side. Don’t forget to balance your equation.
- Divide both sides by the coefficient of the variable.
- If the variable is squared, take the square root of both sides. Don’t forget the negative root!
To Solve a Formula, Use All The Rules
Now you are ready to play with multiple variables…
- Using a formula (pdf 42KB)
Do you remember all the rules and vocabulary we have studied so far?
- Algebra quiz (pdf 30KB)
Are You Ready for Some Football?
You play the game of algebra by solving problems. A good word problem will challenge you to use everything you have learned. Here are 5 steps that will help you fight your way through the toughest problems in your pre-algebra book:
- Algebra story problems (pdf 72KB)
- What do you want to find? Give it a name and write it down!
- What do you know? Write this down, too, so it isn’t cluttering up your mind.
- Do what you can. Write each fact given in the problem as an algebra equation, and do any calculations you can.
- If you get stumped, remember your algebra rules. Can you substitute one thing for another? Can you use the Distributive Law or the Balance Rule to simplify your equations?
- Go back and check the original question. Did you answer what the problem asked? Does your answer make sense?
For Further Study
If you need more help with algebra, here are some great online resources:
- Pre-Algebra: A Complete Course
- SparkNotes Math Study Guides
- Interactivate Mathematics
- S.O.S. Mathematics—Algebra
- College Algebra Tutorials
Photo by StuSeeger.