Taking the Mystery out of Finding 'x'
One of the common fears among homeschool parents is how to teach their older child algebra. For many of them, the instructions “solve for x” resurrects their own feelings of inadequacy in mathematics. So they seek a parent who did well in algebra and beseech her to do a group study. I think the anxiety comes from the way algebra is presented in most American classrooms, which leads to a perception that it is a complicated system of letters and numbers that only the really smart students can understand. Contrary to popular opinion, I submit that algebra is easy to learn, and its roots are actually taught in the third grade.
Third grade? You got to be joking! No, I am not. How many of you remember the problem box plus two equals 5 from elementary school? The box represented a missing number, and you asked the question: what number plus 2 gives you 5? Since you knew your addition facts, you put 3 in the square. Believe it or not, it was a user friendly form of algebra. The algebraic form x + 2 = 5 is the exact same problem. It also asks the question: what number plus 2 gives you 5? In algebra, the square is replaced by an x to represent the unknown number. Another difference is how you present the answer. Instead of putting the 3 in the box, in algebra, you write x = 3.
Although x is a very popular letter to represent an unknown number in algebra, it is nothing special. Any letter can be used. For example, 8 + a = 16 has the same form of a question: 8 plus what number gives you 16? Again it is a basic addition fact, and you would write the algebraically correct answer a = 8.
As you can see in our first two examples, the order of the letter and the number is not important. Math teachers give this principle a fancy name, but we will avoid it for right now. Making students memorize the names of a bunch of mathematical laws is what perpetuates the myth that algebra is mysterious and hard. Yes, there are rules, but they can be explained in simpler terms without the technical language. Unfortunately your student will eventually run into these laws, and hopefully by then, you will understand the material well enough to figure out what they mean.
The next concept could be explained with a simpler example, but I think the larger numbers make a stronger point. The problem is x + 98 = 302. Unfortunately this does not refer back to a memorized math fact. So how would you solve the question: what number plus 98 gives you 302? If you said subtract 98 from 302, or 302 - 98, you earn a gold star.
Now remember we are dealing with an equation, which means both sides represent the same amount. You are allowed to take something from one side, but to keep everything equal, you have to take the same amount from the other side. In our example, it would look like this x + 98 - 98 = 302 - 98. We have the problem 302 - 98 which we need to find our answer on the right, but to maintain the equation we also subtract 98 from the left side. We can figure out that 302 - 98 = 204, but how do we handle x + 98 - 98? Well, we work with what we already know. (Pardon the alliteration.) We know that 98 - 98 = 0. So we can rewrite the equation as x + 0 = 204.
From this point I could take you straight to the answer, but I want to demonstrate another algebra principle first. From arithmetic, you know any number plus 0 is that number, for example, 1 + 0 = 1; 2 + 0 = 2; 101 + 0 = 101. Remember that x or any other letter used in algebra represents a number, and as a result, behaves just like a number. So if 3 + 0 = 3, we can also say x + 0 = x. So our equation reduces to x = 204, our algebraically correct form of an answer!
Okay, we covered addition, so lets look at multiplication. In elementary school, multiplication problems looked like this: 3 x 4. Since x is often used as an unknown number, the traditional multiplication sign 'x' might cause confusion. So in algebra we use a different symbol { • }, so 3 x 4 = 3 • 4. As in addition, the unknown quantity can be placed anywhere in a algebraic multiplication problem. So we can have 5 • y = 10. In words, it can be stated: 5 times what number equals 10? To answer this, we turn this time to our multiplication facts. As a result, the answer is y = 2.
When we have a number multiplied by a letter, the multiplication sign is not usually shown. 3 • y is simplified to 3y. So anytime in algebra when you see, for example, 2z, it is always understood to mean “2 times z.” So what if you have 4x = 28? Again think of it as: 4 times what number gives us 28? Recalling once again our multiplication facts, we can determine that x = 7.
Now we are ready to put it all together. How do you solve 9x + 3 = 48? Look back at the problem x + 98 = 302. We solved it by subtracting 98 from both sides. Using the same trick, we will subtract 3, like this: 9x + 3 - 3 = 48 - 3. Once we simplify everything, look what we have–9x = 45. Hey! This looks like a problem we can solve! Because 9 times what number equals 45? What else, but x = 5.
I know this discussion just scratched the surface of what a student encounters in a complete algebra course. By removing all of the technical terms and laws, I hope it was easier to see how algebra really works. Next I would encourage you to pick up your child’s textbook and look at a section, such as “Integers,” and try to put it in a non-threatening form. For example, integers is just a fancy word for both negative and positive numbers. Then look around and find where you use negative and positive numbers in everyday life. (Hint: Look at a thermometer.) Continue to do this until you have brought the entire book down to terms that you can understand. Finally go out there and teach your student!
©2009 Homeschooling Today magazine, Nehemiah Four, LLC