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Inverse Property. Identity Property. Property of Equality. One-Step Equation. Two-Step Equation. Multi-Step Equation. Isolate a Variable. Math Equation.
Typology: Exercises
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This week we are going to learn how to isolate a variable using the four steps below
Thursday, August 23, 2012 7:27 PM
Thursday, August 23, 2012 7:27 PM
Both are correct procedures. Today we are going to give you the why's and the how's for any problem. In the world of mathematics solving for that mysterious number is called Isolating a Variable... And notice the difference from last week when we looked at math expressions, now we have an = sign. This is called a mathematical equation. In mathematics language the equal sign is a very powerful symbol because it means both sides must be the same...not sort of the same...exactly the same, but x + 31 = 53 doesn't look like 53 = 53, which is exactly the same This is where the Inverse Property comes in, the Inverse Property states: When a number is combined with its inverse, it is equal to its identity. Identity means itself. Let's look at Addition and Subtraction 1st ,think of a number.....how about 20 20 stands alone.....what can I add with 20 to get 20 20 +? = 20 There's only one number, that number is zero....So any number adding with zero gives you that number...that may seem silly, but that is the logic we are going to use to isolate a variable today..... So let's take x + 5 = 9 What can I do to find the identify of X? Well using the logic I just used x + 0 = x, right? An equation is like a scale, if I take something off of a balanced scaled I must do the same to the other side in order to balance it again. This is called the Property of Equality. Students get freaked out by this because they aren't used to changing equations. They are used to solving them. Well in math you have the power to do ANYTHING to an equation. Later on we are going to use the property of equality to completely change the look of an equation to get what we want. In order to use the property of equality you must understand what the inverse property is.... If x + 0 =x, what can I do to the left side of the equation to cancel out the 5? The inverse or the opposite of a +5 is - 5 this creates a 0. But what I do to one side I must do to the other side...it looks like this x + 5 = 9
x + 0 = 4 using the logic we used earlier. X must equal 4....we don't ever write the 0, but it's important that you see why you are cancelling out numbers. WARNING: Learn your sign rules for addition and subtraction...Sing the SIGN SONG...in 5,4,3,2,1....
What about: x + 4 = - 10 What is the inverse of +4, it's - 4, that will create a x + 0 , It looks like this x + 4 = - 10
x + 0 = Uh Oh....do you know your sign rules for addition and subtraction? Same sign add and keep the sign, even if it's negative...... x + 0 = - 14 or x = - 14.....Review question, what coefficient is in front of x? + 1 right....that leads us to our next property Inverse of Multiplication/Division So let's practice some logic again. Think of a number....how did I know you were going to pick 8.... 8 *? = 8 ; 8 times what will give me it's identity? There's only 1 number, and that number is + 8 * 1 = 8 mathematical fact the same holds true for division.... 8/? = 8 ; only 1 will make this true 8/1=8 mathematical fact So if I have 4x = 32 the logic is "x multiplied with +1 will give me x, so what's the inverse of multiplying by +4" Common mistake!! At this point student will say divide by - 4 because they just learned the Addition Inverse property, let's think about that: does 4 / - 4 = +1 No, never.... So the inverse of multiplying by +4 is divide by +4, it looks like this 4x = 32
4 4 1 * x = 8 ; x must equal 8 because the logic of math Fractions Every year students have a difficult time with fractions. But this is where I can see the students that understand the logic and those that are just looking to follow a pattern. UNDERSTAND THE LOGIC
(1)x = 6 x must be 6 Now let's look at how it's going to look on your quiz. Knowing math is nothing unless you can put it in real life terms: First define what you don't have, or what you are looking for. In this example we want to know how many people speak Mandarin, let's call that a variable M Next define the numbers you do have it looks like this M = Mandarin total 487= English Speaking Now let's look at the relationship in the problem: 487 was 512 fewer or less that how many spoke Mandarin... I know it makes sense to add 487 and 512. But let's try and set up a one step equation. You could also say that M - 512 = 487 now use what we learned today to solve.....how about this problem: Again define what you are looking for and assign a letter. We don't know how much Jason weighs J= Jason's weight Now define the numbers 144= how much Ben weighs Now examine the relationship: Ben weighs 3 times more than Jason
So it makes sense that 3J = 144 Now practice your algebra Algebra takes practice....this is when you must drill through repetition. The homework looks like a long assignment, but all you are really doing is addition, subtraction, multiplication and division....it should not take more than 20 mins.... I'm available at 615am for morning help
Notice there is no equation symbol yet, because you need to look at GEMDAS. If I were to evaluate this expression using an input of 3, what would my answer be: It would be 7 correct, didn't you multiply by 2 and then add 1. You used the order of operations. WARNING BRAIN FREEZE COMING SOON..... Well in order to isolate a variable you need to work in reverse of what you did to solve the problem...OUCH, WHAT, BUT???!! You're kidding right? I wish I were kidding. Now as we get into this you may get frustrated and start thinking "When am I ever going to use this" What you are learning is how to think in reverse, and how to follow a procedure...That style of thinking is extremely important when you are going to get out into the big bad world of real life problem solving. So will you be isolating a variable everyday of your adult life, probably not. Will you be solving problems and probably have to reverse engineer some situations, absolutely! Back to the problem 2x + 1 Let's review the steps to solve it
It looks like this In the background, what I didn't show is that you created an equation that looks like this (1)x + 0 = 5
(1) Doesn't anything times +1 / divided by +1 and then added with 0 get you back to the original #….YES math proves it correct 100% of the time!! Here's a problem that trips up students all the time: x + 5 -------- = 6 2 Remember that long division bar is a grouping symbol. One of the tricky parts about math is that there are different ways to write the same equation. Equations that don't look, but are exactly the same thing. Just a heads up, the above equation can be written two different ways. We'll learn more about this later, but it's good to keep in the back of your head that it can be re-written x + 5 --- --- = 6 2 2 OR it can be written
Let's review what we've gone over this week
Thursday, August 23, 2012 7:27 PM
The numbers or symbols that are in different colors are your actual thoughts and actions. Notice we are pulling together all the skills you've learned so far. SIGN song, Multiplication rules, Combine like terms, Inverse and Identity to solve this equation. Let's look at a word problem: This is a great question because you've really got to analyze what you are looking for, and you've got to understand numbers. First what are we looking for: three consecutive multiples of 5 Let's start with understanding what a consecutive number is. Let's define a number any number. let's call it n. How do you define three consecutive numbers in a row. You could say that if the first one is n n....the next one must be n + 1 and the next one must be n + 2 Let's say I evaluate by plugging in 1, wouldn't that give me 1,2,3...three consecutive numbers...But there's something different about these three numbers they must be multiples of 5. I could say the first one is
This quiz will not have many review questions from functions. It will be about what was learned this week and last week Review Simplifying Combine Like Terms Distributing One Step Two Step Multi Step---Extra Credit since we just learned that You will be quizzed on setting up an equation, writing it out and solving for a variable. The activities we did in class
Thursday, August 23, 2012 7:27 PM
