Week5 Thursday, August 23, 2012 7:27 PM This week we are going to learn how to isolate a variable using the four steps below 1) Distribute( Grouping symbols) 2) Combine Like Terms a) same side = b)opposite side = opposite sign 3) Addition/Subtraction 4) Multiplication/Division New Vocab for the week: Inverse Property Identity Property Property of Equality One-Step Equation Two-Step Equation Multi-Step Equation Isolate a Variable Math Equation 1st9weeks Page 1 Mon. Review Redo Thursday, August 23, 2012 7:27 PM 1st9weeks Page 2 Tue One Step Thursday, August 23, 2012 7:27 PM Isolating a variable begins this week.... There's a temptation when you are learning how to isolate a variable to see a few simple problems, understand it enough to get the correct answer and not really understand the mathematics behind it. Remember, the power of math is the use of logical arguments. If you understand the concepts underneath what you are learning you don't have to rely on tricks, and when the problems get more difficult you can use logic to support the correct procedure. With that said I had you write down 4 steps to memorize because our ultimate goal is to ISOLATE A VARIABLE: 1) Distribute( Grouping symbols) 2) Combine Like Terms a) same side = b)opposite side = opposite sign 3) Addition/Subtraction 4) Multiplication/Division Last week we covered how to combine like terms and to distribute. This week we are going to learn the last two steps. Next week we are going to finish off the brief unit by tying all these steps together. Before reading any further check out this video to prep your mind on the notes One Step equation video http://www.youtube.com/watch?v=FLbD0vgJ3Xc So remember when I wrote about the distributive property, I talked about logic. Today we are learning about the Inverse/Identity Property of Addition/Subtraction/Multiplication/Division WARNING: Your mind is going to go in reverse.....Say "Mary had a little lamb" ...now say it backwards...is your brain fuzzy...it should be, if not you may want to get that checked out Let's take a simple problem x + 31 = 53 What plus 31 will give you 52? Easy right, 22....think about what your mind had to do to figure that out. Most of us don't have that memorized, so you probably thought 1) I can subtract 31 from 53 and that will give me the answer 2) or I can count up from 31 1st9weeks Page 3 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 -5 = - 5 ---------------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.... 1st9weeks Page 4 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 - 4 = -4 --------------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 +1 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 1st9weeks Page 5 understand the logic and those that are just looking to follow a pattern. UNDERSTAND THE LOGIC There are two ways to look at a fraction, let's take 3 x=9 2 1) You can look at a fraction as x is a number multiplying with 3 and dividing by 2...so it's like two operations 2) You can look at it like it's just a number multiplying with x, one operation Every student sees it differently. Kids either fall into these two groups. One way will totally confuse you and the other way will make so much sense. If both ways confuse you come see me for help I'll show you the first way..... 1) If x is being multiplied by 3, what's the inverse of multiplying by + 3....divide by + 3 to both sides....now the next one...if x is being divided by 2 what's the inverse, multiply by 2 to both sides, here's how it looks 3 x = 9 -------2(3) (3) -----------------------(2)(1)x = 3 ( 2) ------2 ---------------------(1)x ------ = 6........does x times 1 and then divided by 1 = x (1) so x must equal 6 mathematically Now the second way, which as of recently I've had more success with 2) 3/2 is just a number multiplying with x ….then just divide by 3/2 to both sides....Remember how to divide by a fraction Copy Change Reciprocal (Flip)...It looks like this 3 x ---2 -----------3 ----2 = 9 ---------3 ---2 -----------------------------------------------------1st9weeks Page 6 -----------------------------------------------------(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 1st9weeks Page 7 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 1st9weeks Page 8 Wed Two Step Thursday, August 23, 2012 7:27 PM Now that you know how to isolate a variable when there is one operation, now we are going to look at Two-Step Equations. This is when there are two math operations with a variable. In order to understand the mathematics you must review the orders of operations a bit...PEMDAS In order to evaluate a problem you must do these operations first: P Parentheses E Exponents M/D Multiply/Divide left to right A/S Add or subtract left to right Now I'm not sure if I was around when PEMDAS was first started, but there's a couple of mistakes in the acronym. First off the P in PEMDAS stands for Parentheses. A parentheses is just a Grouping symbol. But there are other grouping symbols that lock numbers in a BFF situation. Like X+1 ------5 I don't see a parentheses but the long division bar is also a grouping symbol. It creates a BFF situation in the numerator...what's not shown is (X+1)...it's assumed you know that those two are locked in together. Another grouping symbol is the square root symbol This also creates a BFF situation also...so if I could change PEMDAS I would change it to GEMDAS G Grouping symbols E Exponents M/D Multiply/Divide left to right A/S Add or subtract left to right Let's look at a simple two step equation first: 2x + 1 1st9weeks Page 9 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 1) Multiply by 2 2) Add 1 to the answer Now let's throw an equation into the mix 2x + 1 = 11 Now we need to isolate the variable. But to undo the problem we must ****Do the inverse of every operation in reverse***** Stay with me, hang with me for a few seconds Instead of doing the two steps from above, I'm going to cancel out the numbers around x using the properties we learned yesterday. So to both sides of the equation I'm going to: 1) Subtract 1 2) Divide by 2 It looks like this 1st9weeks Page 10 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 --2 + 5 --- = 6 2 OR it can be written 1st9weeks Page 11 OR it can be written 1(x+5) ---= 6 2 Go ahead and try and plug in numbers into all three equations you will get the same thing. The reason they can be written that way is because the long division bar is just the distributive property. Notice the two new ways I just distributed the 1/2 into the parentheses.....An the original equation is just following the rule on how to add fractions, like denominators and add the numerators. You are welcome to solve any of these variables on your own using the methods we've already learned, I'm going to teach you a short cut. In the original equation the long division bar creates a BFF situation. So we must break that up by canceling out the 2. What's the inverse of dividing by +2....multiplying by +2...Now we have just a +5, what's the inverse - 5.....here's how the math works Remember the underlying math is (+1)x + 0 = 7 1st9weeks Page 12 Thur Multi-Step Thursday, August 23, 2012 7:27 PM Let's review what we've gone over this week 1) Solving a one step equation 2) Solving a two step equation Today we are going to cover multi-step...We are going to combine what you learned last week with what we've been covering this week. We have covered most of the 4 steps of isolating a variable. The highlighted steps are what you've learned: 1) Distribute( Grouping symbols) 2) Combine Like Terms a) same side = b)opposite side = opposite sign 3) Addition/Subtraction 4) Multiplication/Division Notice that the only step we've not learned is how to combine like terms on the opposite side of the =. Hopefully you've gained some confidence of each of the skills learned. Today we are going to combine those skills in solving for multi-step equations. A multi-step equation is when a student must isolate a variable by combining like terms and using the inverses. Let's take for example: 3( x + 1) - 5( x+ 2) = 7 Now we can focus on the 4 steps I've laid out since last week. First step: You must get rid of any grouping symbols, for the most part this will mean distribute, but remember that's not the only grouping symbol Second step: Combine like terms Third/Fourth step: Use the inverse property to cancel out numbers and isolate a variable If your skills are strong and you commit to these steps, this is going to be easy, if not, it's going to frustrate you a bit. Let's solve: COPY THIS 1st9weeks Page 13 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 1st9weeks Page 14 one is 5n the next one is 5(n+1) and the last is 5(n+2) Now if I'm adding these three terms it will look like this 5n + 5(n+1) + 5(n+2) and since they must add up to 90 I can drop an = 5n + 5(n+1) + 5(n+2) = 90 Now that we've figured out the underlying math, let's go ahead and go through the 4 steps 1) Distributive 2) Combine Like TERMS 3) Inverse Add/Subtract 4) Inverse Multiply/Divide DONE Now remember what the question was asking....what are the 3 consecutive multiples of 5...It's NOT 5,6,7 because the first term is 5n...so the three consecutive multiples are 25,30,35....do they add up to 90, yes they do. I know at first this looks like it's much work, but keeping it neat and organized will save you much time and wasted energy in the future. As long as you follow the 4 steps of isolating a variable, and understand the mathematics behind it, you won't have any problems, and it will be pretty systematic 1st9weeks Page 15 Fri Quiz Thursday, August 23, 2012 7:27 PM 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 1st9weeks Page 16