A fun (and funny) fact: Algebra students need to know properties inside-out and backward and forward in order to succeed in the later parts of the course.
It’s a fact and a pun, really, when you know what the Algebra properties are and how to use them.
Algebra introduces new concepts while building on concepts previously introduced in other courses. Remembering Algebraic Properties (also called Algebraic Laws) is one of those building-block type of concepts.
Students get a little experience with properties and identities in pre-Algebra.
Most Algebra teachers make that assumption when taking the deep dive into Communitive, Associative, and Distributive properties, typically early in the course.
Seemingly easy topics, but they can trip up any student, if not careful. Instead of just memorizing the properties, here are easy ways to learn it and keep it all straight.
First, let’s figure out what’s the point. (Another math joke for you)
Why Do Students Need to Know Algebraic Properties?
Knowing your Algebraic Properties is going to help you solve equations and simplify expressions.
Most of it is just putting a name to what students already know.
Typically, when students learn how to solve equations, they aren’t consciously learning the properties. They just know that they can subtract five from both sides to get x by itself, for example.
But they need to know that what they’re doing has a proper name. Then, they need to know why it works.
By the end of the unit, students should be able to look at a completely worked-out problem and specifically identify the properties used to come to the answer.
We want to develop students that don’t just know how to calculate. There are calculators for that.
We want to develop students that know how to explain what’s going on, to explain the details of doing the math.
If a student can explain it, the student can perform it with higher accuracy and precision.
That plays a big role when dealing with more abstract thinking.
Get identifying properties in Algebra right and the student is much better off in Geometry class when working with proofs.
The Algebra content is on a simpler introductory level, so it’s good practice.
The Commutative Property is Driving The Point
Look at the root word and you can figure out what this property is about. Remember the Commutative Property by thinking about commuting to work or just driving from one place to another.
The Commutative Property means you can move your problem around and still get the same answer. It would still be equal. Remember, this can only be done on one side of the equation.
For example, the two expressions below are the same because of the Commutative Property.
| 5 x 3 = 15 | 3 x 5 = 15 |
Easy, right? It just makes sense. Now, you know why it makes sense and you can use it for more complex equations.
The Commutative Property works with multiplication and addition. And with those two operations only.
Knowing when to use this is a real lifesaver during multiple-choice tests. That’s because students may correctly solve the problem, but their final answer looks a little different than the given choices.
He or she may think their answer is incorrect. And it’s that uncertainty that often causes some students to get their problem wrong. They get the right answer, then second-guess themselves because of a lack of confidence in their math skills.
The Commutative Property reminds the student that answers can look slightly different but still be correct.
Associative Property: Not Quite the Circle of Trust
We have groups of friends and then we have associates. And that’s OK, according to the Associative Property, which tells us numbers can be grouped in parentheses in any combination and still be the same.
The root word, again, helps to remember the Associative Property.
It doesn’t matter if you have terms (or friends, in our example) in the parentheses (or in your group) or not. You will come up with the same answer.
Properties take out a lot of the guesswork for beginner Algebra students, like we talked about above with the Commutative Property.
Knowing the properties also help students work faster.
Now, math definitely isn’t a race, especially at the beginner level.
But as problems get more complex, it helps to lean on math laws that you don’t have to question the validity.
Parentheses easily intimidate students, I find. So remembering to spot the Associative Property definitely helps.
And again, the Associative Property works only with the addition and multiplication operations.
Distributive Property: Everyone gets a piece of the pie
The Distributive Property tells us that we are to multiply everything outside the parentheses with everything in the parentheses.
Remember the Distributive Property by thinking about a teacher passing out test papers. Everyone in class will get a test to take, whether it’s difficult or not.
There’s a little more complexity here. But follow the property step-by-step and you can get it right.
This property gets the most use since students use it a lot in the Quadratics Unit of the course.
A common mistake here is not distributing to all the terms in the parentheses, I’ve noticed. These students say they don’t notice or just forget.
Typically, the parentheses drops off on the following line in the Distributive Property.
Properties and Identities
Identities are a type of Algebraic property. Identities reflect and relate to each other. They tell you what something is.
For example,
1 + 1 is the identity for 2
32 is the identity for 9
The Additive Identity says that anything plus zero will be itself.
The Multiplicative Identity says any number times 1 is going to equal itself.
Think about it this way: Identities are the most defining characteristic and it doesn’t change.
Besides Additive and Multiplicative, there are a lot of more identities as you go further up in math. But those two are the main ones used in Algebra.
So Easy that Everyone Can Do It. Or Can They?
The difficulty in Algebra properties comes from this:
Knowing how and when to use them.
Teachers usually just teach the properties and move on. So, there’s usually not a lot of time spent putting the knowledge into practice.
More practice trains the student’s eye to recognize certain telltale signs.
Values and variables that shift around from one line to the next without much else changing is characteristic of the Commutative Property.
If the numbers or variables don’t move from their original position, but you see a change inside the pair of parentheses, then you know it’s the Associative Property.
Look for the pattern of the outside value being multiplied to the values inside the parentheses to identify the Distributive Property.
In Conclusion
Remembering Algebraic properties is like learning basketball. Kids just want to have fun. Their goal is just to learn to shoot the ball in the hoop before knowing the terms Jumper, Lay-Up, Half-Court shot, etc.
The Commutative Property, Associative Property, and Distributive Property are just putting a definition to what a student usually is already doing.
With more practical application and these helpful hints, your student can learn correctly remember the Algebra properties, and set a great foundation for future math courses.
