Algebra is one of the most complex subjects of study. It is also the basis for many more difficult levels of study, like engineering, physics, business and even culinary arts. As a math tutor of mainly Algebra students, I have learned that most students have not mastered specific basics of Pre-Algebra and therefore are left stuck on more complex Algebra problems. Always remember that math builds on itself, so what you learn in the past will be used for learning something on the next level.
If you are trying to pass Algebra, here are a few Pre-Algebra concepts you need to master.
#1: Prime Factorization
At the very base, there are two types of numbers: primes and composites. Primes are numbers that can only be divided by 1 and itself (ie. 2, 5, 11, 19, etc). But composites are numbers that can be divided by 1, itself and several other numbers (ie. 4, 6, 25, 32, etc.)
What is Prime Factorization? This is a process that divides a number all the way down to it’s prime numbers. We need to know the difference between prime and composite so that we can know if we are done dividing to all the primes.
You may have seen some people use “factor trees” to model Prime Factorization, which is a super helpful visual. It’s important to master this concept to prepare yourself for finding Greatest Common Factors and simplifying radicals. Interestingly enough, both of those made it to this list as well.
#2 and 3: Finding the LCM and GCF
The LCMs (Least Common Multiples) and GCFs (Greatest Common Factors) aren't hard to find at all. We learn in elementary school that multiples are just bigger expansions of a number using multiplication, and that factors are the smaller numbers that a number breaks down to using division. With LCMs we just need to find the least of them that two numbers have in common, and with GCFs we just need to find the greatest of them that two numbers have in common. That's all.
#4: Operations with Integers
There's a difference between whole numbers and integers. Whole numbers start at zero (0) and go to positive infinity (ie. 0, 1, 2, 3, ....). Integers include whole numbers and their opposites, or their negatives. Take a look at the number line below. These are all integers. The arrows represent that the numbers go from negative infinity to positive infinity.
Integers are the numbers we see the most when performing arithmetic and Algebra. Most of us hope that we'll get positive integers in our math problems and as our answers. Sometimes when we get negatives (like -6) we think the answer is wrong. This is not always true. If you've applied the correct properties and rules and got a negative, then the answer is correct.
As far as integers go, you need to know how to add, subtract, multiply and divide them. Seems simple right? Well as a math tutor I can tell you these simple tasks are what get people all twisted up in the middle of a problem. So practice applying operations to integers on a regular basis to eliminate this barrier of solving Algebra problems.
#5: Working with Fractions
Some of my students are so afraid of fractions that I have to assure them that they actually don't bite lol. I have to take them back to the sectioned pizza pie example to calm them down. Breathe.... Breathe....
Listen, you CAN'T avoid fractions. They will pop up out of nowhere and demand that you simplify, convert and apply operations to them. They want you to do weird things to them, like add the top (numerator) but not the bottom (denominator), and divide using KCF..... huh??
Fractions are tricky and we must prepare ourselves to deal with them or else they'll deal with us.
#6 and 7: Exponents and Radicals (Square, Cubed or nth Roots)
A lot of students don't know that exponents and radicals are actually opposite operations, just like addition and subtraction. Opposite operations mean that one reverses, or "undoes", the other's work. Realizing this creates a huge "Aha!" moment for most Algebra students.
A few other facts:
Try to remember that a number (base) raised to a power (exponent) DOES NOT mean for you to multiply those two numbers. It means that the base is being multiplied by itself for the amount of times indicated by the exponent.
Ex. 4^3 = 4 x 4 x 4
When you take the square root (or third, or nth root) of a perfect square, the radical sign disappears...poof! The radical is an operation.
sqrt(9) = 3, and that's it
To simplify radicals, you must know how to find factors of numbers (refer to #1 in this list). Remember, math builds on itself!
#8: The Order of Operations
Some use PEMDAS. Some use BODMAS. With whichever acronym for the Order of Operations you use simplify an expression, you must understand it and follow it.
A few things that students don't know are:
When addition and subtraction are the last operations left in an expression, you ignore the fact that addition comes first in the Order of Operations and just perform the operations from left to right. The same goes for multiplication and division.
When you see a radical (square root) in the expression, this is performed at the same time an exponent would be performed in the Order of Operations.
PEMDAS: Parentheses, Exponents (and Square Roots/Radicals), Multiplication, Division, Addition, Subtraction
It's catchphrase: Please Excuse My Dear Aunt Sally.
BODMAS: Brackets, Orders (or pOwers), Division, Multiplication, Addition, Subtraction
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