Numerical and Algebraic Expressions and Equations (The Algebra Unit)
Exponent Review
We begin our Algebra journey by talking about the order of operations......However,.....we cannot talk about the order of operations without first discussing the operations and meanings of the operations first. Therefore, we start with one of the more "mystical" operations in math, exponents. I say that exponents are mystical because they are probably the least understood of all of the operations in the order of operations.
The video below provides a brief review of exponents.
Negative Exponents
Now that we are reminded of how to view and use exponents. We will spend some time looking at 0, 1, and negative exponents.
The two videos below were taken from my Algebra website. They are NOT curricular content for 7th grade, however, some students may be eager and ready for them. I may ask you to watch one in class, or you may watch them on your own if you would like a challenge.
Scientific Notation
Scientific notation is simply a different way of writing numbers. That's it! We use scientific notation specifically to write very very large numbers, or very very small numbers. For example, we might use scientific notation to rewrite the number 234,000,000,000,000,000. Why? Because let's say you needed to use the number 234,000,000,000,000,000 in a formula or in your scientific research. How convenient is it to keep writing 234,000,000,000,000,000 in your notes or lab report? Or, lets say you wanted to multiply:
234,000,000,000,000,000 x 456,000,000,000,000,000,000
There are a lot of place values in the two numbers above. So many, in fact, that your calculator does not have enough space to display the product, and it would take a lot of time and mental anguish on your part to multiply these the 'old fashioned way' using the multiplication algorithm. So you might say, "isn't there an easier way to find this product?!?!". Yes. Scientific notation.
The video below is not my video, but it is pretty close to what we discuss when explaining scientific notation.
It is important to note that since scientific notation is a type of number, technically speaking anything you can do with an order decimal or fraction can also be done with scientific notation numbers. In other words, multiplying, dividing, adding, subtracting, exponents, equations, fractions.......The possibilities are endless! Lucky for you as 7th graders we are only talking about how to write numbers in scientific notation. We will save the operations for next year. :)
However, if you are up for it, I will gladly share a few worksheets and activities that investigate operations with scientific notation. If you are willing, I am able!
Order of Operations Review
Now that we are more comfortable with exponents we can review the order of operations. There are two main focuses of this review, one is to clarify that the order of operations is not a strict "order", but more of a suggested order. There are times when the order does not matter, or when one operation might be preferable rather than following the order (As long as what you are doing is mathematically correct!).
Secondly, we will be discussing how the order is not PEMDAS (as some of us may have seen), but rather GEMDAS. The G stands for all grouping symbols (including parentheses). Grouping symbols can include things like brackets, absolute value, radicals, fractions (numerators and denominators), etc...
Like Terms and Adding/Subtracting Like Terms
After reviewing the order of operations and how to algebraically show our work when simplifying expressions, we can throw in some variables. Before we simplify variable expressions we first need to learn about the phrase 'like terms'. The video below shows how we determine what 'like terms' are, and how to add or subtract like terms:
Distributive Property
Solving Equations Introduction
Algebra is the ability to take something that is unknown and solve it by comparing it to something that IS known. For example, in the expression x + 2, x can be any value. If I asked you, "what is x" it would be difficult if not impossible to tell me the exact value. However, if I compared x + 2 to something else you could use this comparison to find the exact value for x. In other words, what if I said x + 2 = 10? What does x equal?
When solving an equation, you are essentially finding the value that will make the equation true. For example, in the equation x + 1 = 6, the number 5 is the solution because it makes the equation true. Watch the video below for an introduction to the steps I wants to see when solving equations:
Solving Two- Step Equations
Two step equations are equations that require two steps to solve them. Generally speaking, the act of solving two-step equations is fairly simple, however, we must first decide which operation to do to my expressions. Watch the video below for an explanation on how I would like you to approach two-step equations:
Solving Difficult Multi-Step Equations
Solving and Writing Inequalities
Naturally, if we are going to learn how to solve equations then we are also going to solve inequalities too. Solving inequalities is actually virtually identical to solving equations (with a couple of tricks). Typically, however, we tend to get more tripped up on the writing of inequalities (this is normal!). Watch the video below for a brief introduction to both writing and solving: