Algebra—Approaching Problems

Learning Objective(s)

·         Review the properties of real numbers and the order of operations.

·         Develop a strategic approach to solving algebraic problems.

Introduction

Algebra is a powerful tool for solving problems and gaining knowledge. But like every skill, it takes time to master it. As you proceed, don't be surprised or discouraged to encounter ideas and situations that confuse you. After all, the world is big and complicated, so it's going to take more than simple math to figure it out.

Luckily, there's a step-by-step way to approach any situation in algebra. When you come upon a problem, just take a deep breath, follow the steps below, and you'll find your way.

Steps to Solutions

Sometimes, especially as you get more comfortable with algebra, you'll see a problem and know exactly what to do. Other times, you'll run headfirst into a problem that takes you by surprise, and you'll wonder how to even begin. Whether the path to a solution is obvious or not, there's a reliable series of steps that will help take you in the right direction:

 Problem-solving strategy   1) Understand the problem 2) Look for familiar patterns 3) Break the problem up 4) Visualize 5) Try different techniques 6) Don’t give up!

1) Understand the problem:  Figure out both where you are and where you need to go before you start working. Look carefully at the problem to be sure you know what the question is. Identify the known and the unknown quantities. Start thinking about the techniques that could be useful, and the kinds of answers you expect to find.

2) Look for familiar patterns:  Use your experience to guide you. Have you seen similar problems? Is there anything about the form of an equation or the shape of a graph or the phrasing of a question that seems familiar? If you can compare formulas or terms or shapes to problems you've solved before, it's a good bet that the strategy you used to solve them last time will help again now.

3) Break up the problem:  A big complicated problem is often just several small and simple questions stuck together. See if there are pieces you can pull out to work on separately. Maybe you don't know how to find all the variables all at once, but you might be able to figure them out one at a time. Once you solve the easy parts, those answers can help you get the rest of the way. Divide and conquer!

4) Visualize:  A picture is worth a thousand words. Sometimes a verbal description or an equation don't really speak to you. A quick sketch or a careful graph can sometimes make relationships clear in a way that words or numbers don't. A chart, table, or diagram might spark a fresh understanding of the situation. When you create a visual representation of a problem, you give yourself a whole new way to look for answers.

5) Try different techniques:  If at first you don't succeed, try, try again, isn't just a cliché; it's also a good idea. There will be times when the first approach you try doesn't get anywhere. Don't give up, try another way. Think about why one technique didn't work—did it give you an answer, but not the one you needed? Maybe you broke up the problem without realizing it, and this answer will lead to other answers and eventually to the right answer. Did you get stuck partway through because you didn't have enough information? Perhaps you need to go back and make sure you understand the problem.

6) Don’t give up!  The truth is out there. If you keep trying, and vary your approach, you will get it. Take a second to step back and look it over with fresh eyes. Try going back over your notes. Eventually, you’ll find what works!

Recalling Mathematical Properties

If you’ve ever played a video game or packed a suitcase for a long trip, you know that the order and placement of objects makes a difference in the outcome. Some things can be moved around to wherever they are convenient, while others can only go a certain way or they won’t fit.

The arrangement of numbers and variables in algebraic expressions works like that too. Some situations allow us to switch values and operations around. Others require that numbers can only be handled one way or the process will go wrong. The ability to play with the parts of an equation (or not) are described by mathematical properties.

The same properties that you learned about when you studied arithmetic also work in algebra. If it's been a while since you've used them, review the properties below to refresh your memory.

Associative Property

The Associative Property describes how the association, or grouping, of numbers and variables can be changed in expressions and equations.

The Associative Property of Addition holds that numbers in an addition sequence can be added in any order, and the value of the expression will not change:

For all real numbers a, b, and c, (a + b) + c = a + (b + c)

The Associative Property of Multiplication states that numbers in a multiplication sequence can be multiplied in any order, and the value of the expression will not change:

For all real numbers a, b, and c, (ab)c = a(bc)

However, the Associative Property does not hold for subtraction or division. In these cases, the order in which operations are performed will affect an expression’s value.

Commutative Property

The Commutative Property describes how numbers and variables can move, or commute, inside expressions and equations.

The Commutative Property of Addition says that when two values are added together, changing their location does not affect their sum:

For all real values a and b, a + b = b + a

The Commutative Property of Multiplication tells us that numbers in a multiplication sequence can be moved as well, without affecting the value of the expression:

For all real values a and b, ab = ba

However, the Commutative Property does not hold for subtraction or division. In these cases, the order in which operations are performed will change an expression’s value.

Distributive Property

The Distributive Property allows us to distribute, or spread out, the multiplication process in an expression or equation:

For all real numbers a, b, and c, The Distributive Property is one of the most important tools in algebra. It provides a way to rearrange expressions full of variables, parentheses, and operations, and make complex problems easier to solve.

Properties of Equality

Equality, as represented by the equals symbol, tells us that the two sides of an algebraic expression have the same value. Equality has several properties that are useful in solving algebraic equations. These properties apply to all 4 mathematical operations. They let us add to, subtract from, multiply, and divide expressions while still preserving the equality of an equation.

The Addition Property of Equality allows us to add the same amount to both sides of an equation:

For real numbers a, b, and c, if a = b, then a + c = b + c

The Multiplication Property of Equality allows us to multiply both sides of an equation by the same amount:

For real numbers a, b, and c, if a = b, then ac = bc

The Subtraction Property of Equality allows us to subtract the same amount from both sides of an equation:

For real numbers a, b, and c, if a = b, then a - c = b - c

The Division Property of Equality allows us to divide both sides of an equation by the same amount:

For real numbers a, b, and c, if a = b and c is not 0, then Identity Property

The Identity Property describes numbers that can be added or multiplied to other numbers without changing the values of those numbers. Identities allow a term to maintain its identity even after addition or multiplication. Perform an operation on a number with an identity, and the result is the original number.

The Addition Property of Identity states that any number plus zero equals that number:

For all real values of a, a + 0 = a

Because adding zero doesn’t affect the value, or identity, of a number, 0 is called the additive identity.

The Multiplication Property of Identity states that any number times 1 equals that number:

For all real values of a, a 1 = a

Because multiplication by 1 doesn’t affect the value of a number or variable, 1 is known as the multiplicative identity.

Inverse Property

The Inverse Property involves addition to get to 0 and multiplication to get to 1. The word inverse means to turn inside out or upside down—in mathematics an inverse essentially turns a number around on itself.

The Inverse Property of Addition says that any number plus the negative of that number  equals zero:

For every number a, a + (-a) = 0

Because we find the additive inverse of a number just by repeating the number with the opposite sign, the additive inverse is sometimes called the opposite.

The Inverse Property of Multiplication says that any number multiplied by 1 over that number equals 1:

For every number a, The multiplicative inverse of any number or variable is just the number flipped upside down. This value is also called the reciprocal.

A typical algebraic equation includes multiple operations. Where do you even begin when you're faced with a statement bursting with addition and subtraction, multiplication and division, exponents, radicals, parentheses, and brackets? The Order of Operations has the answer! It specifies the order in which to perform the operations in an expression or equation. Just like the properties of real numbers apply in both arithmetic and algebra, so does the order operations.

If you're a little rusty on the Order of Operations, here’s a reminder:

 The Order of Operations specifies the order in which to perform the operations in an expression or equation.   There are 4 steps that must be carried out in order:   ·         First, perform all calculations that are inside parentheses.   ·         Second, resolve all the exponents in the expression.   ·         Third, do all the multiplication and division, in sequence, from left to right.   ·         Fourth, perform all addition and subtraction, again from the left side of the expression to the right.

The order of operations is often called PEMDAS, from the first letter of the operations. This trick helps us remember the correct sequence to follow.

 PEMDAS   Parentheses Exponents Multiplication & Division Addition & Subtraction

PEMDAS tells us exactly what steps to follow when evaluating expressions. It's just like giving everyone identical directions to a meeting place—if we all make the same turns in the same order, we should all end up in the same spot. And if we all go through the same sequence of operations when we solve or simplify an equation, we should all end up with the same answer.

Summary

Algebra problems are easier to understand and to solve when you establish a strategy for working with them. First, remember the arithmetic! The same properties of real numbers and the order of operations that worked when you only had numbers to think about also apply to equations with variables.

Second, don't be intimidated! Algebra problems have answers, and you can find them. Take the time to look at a problem carefully to make sure that you understand what you need to do. Look for familiar patterns or clues on what to do. Break a problem into smaller pieces that are easier to solve. Visualize the situation by making sketches or other graphics. If you don't get an answer the first time through, back up and try a different method. Above all, don't give up. Mistakes and wrong turns are just steps on your way to the right answer.