Are you one of the many parents who think they can’t help their child with math homework, especially algebra? This article will give you a big picture understanding of the importance of algebra and some tips for solving algebra problems. We will also provide some tools you can use to help your student with their algebra homework.
What is Algebra?
In my years as a math teacher and then as a parent, one comment I heard over and over from other adults was “I liked math until I got to algebra.” Many parents feel like they can’t help their child with their math homework once that child is in an algebra class, because their own understanding of algebra is shaky at best. What is it about algebra that seems to cause such distress? And what is algebra anyway?
The word “algebra” come from the Arabic “al-jabr” which means “putting together broken parts”. The type of algebra usually studied in high school, elementary algebra, is an extension of arithmetic where some quantities are unknown and must be found. More advanced algebra, also called abstract algebra, creates abstractions of our familiar numbers and then solves problems in those abstract systems.
What Are Variables and Why Does Algebra Have Them?
The follow-up to the common complaint I heard, “I liked math until I got to algebra,” is usually along the lines of “when they started mixing letters in with the numbers.” Why are there letters in my math? What are they used for?
To understand why letters are used in math, we need to understand the concept of a variable. A variable is simply a quantity that can change. For example, the symbol “3” always means the number three. But if we want to represent a number that is unknown, or that could be different values under different conditions, we would use a variable. The most common variables we see, especially in algebra, are represented by lower-case letters, such as $x.$
However, students learn the concept of an unknown quantity much earlier than algebra. You may remember seeing worksheets from your child’s early elementary school days with problems such as $$5 + \_\_\_ = 8.$$
Guess what — this is an algebra problem! A young child will be able to figure out what number added to five equals eight, and fill in the blank correctly with 3. Problems of this type introduce the concept of algebra without introducing variables and without needing to teach complex algorithms for solving an equation. Those topics are covered heavily in elementary algebra.
Algebra Word Problems
Coming in a close second to the most common complaint I hear from other adults about their own math experience has to do with those dreaded word problems! Well, I am sorry to say that word problems are one of the most important reasons to study math. No one outside of a classroom will ever hand you a math problem already formed and ask you to find the answer. No, in your daily life, you usually have to figure out both the problem and the solution.
Suppose you decide to join a gym. They offer you two choices: pay a \$99 joining fee and then \$10 a month, or pay no joining fee and then \$25 a month. Which is the better deal?
Obviously the second option is better if you only intend to use the membership for a few months. But is it really the better deal? After how many months would the first option have been the cheaper one? We can find the answer with algebra.
The key to solving word problems is to be very clear about what you are trying to find. Then to be very clear about what you already know. That advice bears repeating: first understand what you are trying to find. Read the problem again, and clearly articulate what is unknown. Second, clearly lay out what information you do know, and think about how to use the known information to find the unknown amount.
Word Problem Example
In this case, we are trying to find the number of months that makes the two payment options equal. What we don’t know, the unknown quantity, we are going to represent with a variable, and since we are looking for a number of months, let’s use the letter $m.$
Next we need to lay out what we know. The first option is a one-time fee of \$99, then \$10 per month. If you have a hard time expressing this as a mathematical expression, figure out what the amount would be for a few concrete examples. For one month, it would be $99 + 10.$
For two months, it would be $99 + 10 * 2.$ Once you see the pattern, you can extend it to $m$ months, $99 + 10m.$ (Hint: “per” usually means multiplication.)
The second option has no one-time fee, it is just \$25 per month. That expression, again using $m$ as the number of months, is $25m.$
We are looking for the value of $m$ that makes these two expressions equal. In other words, we are trying to solve the equation $$99+10m=25m.$$
The answer, if you are curious, is $m=6.6.$ In other words, if you use the membership for 6 or fewer months, the second option is the better deal. But if you use the membership for 7 or more months, the first option is actually cheaper.
How Can I Help My Child Learn Algebra?
Hopefully you want more than to help with your child’s math homework. Hopefully, you really want your child to learn the math. So let’s talk about how you can help your child learn algebra.
First, understand the big picture of algebra. Up to this point, your child has learned and practiced (and practiced and practiced) arithmetic. Hopefully, they are adept at the arithmetic operations (addition, subtraction, multiplication, and division) as well as understanding fractions and decimals. In addition, they have applied their knowledge to concrete examples such as measurement and time.
Beginning in algebra, your child will continue to use all those concepts, but just keep expanding their applications of them. A typical algebra course will cover variable expressions, solving equations, and representing solutions to equations in different ways.
After a first year in algebra, most students will take a year of geometry, and then a second algebra class. That second year of algebra will apply algebraic concepts to geometric figures and serves as a basis for many calculus concepts.
But back to that first year in algebra. Beyond understanding the big picture of algebra, what are some more tangible things you can do to help your child? You may find that a book can help. For example, the book Help Your Kids With Math covers basic arithmetic through algebra and geometry and is highly rated on Amazon.
Another highly-rated resource is the book No-Nonsense Algebra. Each lesson includes an introduction, helpful hints, a step-by-step example, as well as exercises and solutions.
Math Homework Help Apps
Maybe instead of a book, you are looking for an app to provide some additional practice for your student, and maybe even for you. The free app Algebra Touchis a basic equation solver that allows the student to drag components of the equation around in order to solve it.
Another great app is Photomath. It lets you use the camera function on your phone to scan a math problem. Then shows you step-by-step how to solve it. This is a free app, but does have in-app purchases.
Free Online Algebra Resources
There are resources ranging from complete algebra courses to extra sample problems available online. For example, MathPlanet has an algebra course broken into twelve chapters, each consisting of several lessons. It’s a great option if you are struggling with a particular topic and looking for an alternative explanation.
Another site, CoolMath, has math-based games in addition to lessons on algebra and other math subjects. If your child is struggling with algebra homework, an extra five or ten minutes on one of these sites may give a different perspective. These sites can be the extra push they need to really “get” their homework.
A Taste of Advanced Algebra
After seeing the big picture about elementary algebra, I’d like to end by giving you a small taste of where all this algebra can go. Of course, elementary algebra is important in its own right. Many situations in daily life can be solved by applying algebra, which is why most high schools require it. But what would come after elementary algebra?
The next step in your student’s advanced algebra might be a college course called Linear Algebra. Linear Algebra is all about vectors and matrices, which have applications in physics and other sciences. For some interesting reading on this topic, check out our article on Matrices and Vectors.
After Linear Algebra, many students take Abstract Algebra. Abstract Algebra expands upon our familiar number system by understanding the properties of the numbers. Then applying those properties to other sets.
Abstract Algebra Example
For a fairly simple abstract algebra example, think about the familiar set of integers, in other words the counting numbers 1, 2, 3,…, together with 0 and the opposites of the counting numbers $-1, -2, -3,\ldots$ This set of numbers has certain properties. The sum of two integers is an integer. There is an additive identity (zero) that leaves unchanged any number it is added to. Each integer has an additive inverse, that is a number that can be added to it to get zero. (These are just the negatives of the numbers, in other words $-3 + 3 = 0$ and so 3 and $-3$ are additive inverses).
Now what if we only consider the numbers on a clock face, and define addition the way we would add on a clock face. For example, $3 + 2 = 5$ as usual, but $11 + 2 = 1.$ (Start at 11 on a clock and go forward 2, where do you end up?) It turns out, this set of numbers and addition defined this way has some of the same properties as the integers. Any two clock face numbers added together gives another number on the clock face. There is an identity, that is a number that when added to any number leaves that number unchanged. Strangely, on the clock face, the identity (zero), is 12. Start at 3 on the face of the clock and go forward 12, you end up back at 3. Therefore 12 is the identity.
Finally, each number on the clock face has an additive inverse, or in other words a number that when added to it gives 12 as the answer. It should be easy to see that 10 and 2 are additive inverses. It may surprise you to see that 6 is its own additive inverse. That doesn’t happen with the regular integers! So we have taken a familiar set of numbers, the integers, abstracted from it a set of numbers with some of the same properties, and found that this set of numbers has some additional and surprising properties. That is Abstract Algebra.