VIEWS: 25 PAGES: 72 POSTED ON: 5/8/2010
TAKS Tutorial Algebra Objectives 1 – 4 Part 2 Remember… This tutorial and all the ones presented previously can be found on Mrs. Nelson’s website. There is a link to her site on the Spring High website. If you missed Objectives 1 – 4 part 1, you definitely want to go to the website to go through the lesson. Today, we have a great deal to cover! Topics include: Simplifying expressions Setting up & solving linear equations Setting up & solving linear systems Setting up & solving linear inequalities Writing linear equations from information Slope and intercepts Let’s start with expressions. This problem was on the 2006 test. This expression certainly can be simplified the ―traditional‖ way: - 21x 12x 6x - 13 24x 2 2 Distribute : 3x 18 x 13 2 Combine like terms : The only problem with the “traditional” way is that some of you do not correctly distribute the negative/subtraction signs! So let’s look at an alternate way to do this problem with a calculator. It takes longer, but if you are someone who makes simple mistakes…this is for you! The only variable in the expression is x. Store a value into your calculator for x—any number except 1, 0, or -1. Select and press your number, press STO, press X,T,Θ,n , and ENTER Now, enter your expression, just as you see it, into your calculator… -3x(7x-4)+6x-(13-24x^2) Be sure to double check that you typed the expression correctly! Now, press 2ND MATH to get TEST We are looking for equal expressions so select = You should see the = sign Now, press ENTER. If this appear after your answer choice is correct and two expressions are equal, theexpression you will see a ―1‖ as an (NO, do not press Now, oneanswer! We is the the answer. If these two Looks like option F correct by one, enter were ENTER—the calculator firstlucky the first time. just as it expressions are not equal in answer choice You will not simplify this appears. Double check to might check the other value, you will see ―0‖ as an expression for you!) options to see that you make sureget “0”. you typed it in answer. correctly. That was a lot to grasp in one problem. Let’s try another one and practice with the calculator. (2003) Type in the (2 option Get TEST first answer Type variable used is Try again withnd Math)B. The in the expression exactly Remember, choice. as it appears. You do NOTahave to ―1‖ ―n‖. Select and select = number Double for ―n‖ and is if it useeverything! Press to is check retypeequal; ―0‖ if itstore not. nd ENTER correctness! 2it in the calculator. That was a lot to grasp in one problem. Let’s try another one and practice with the calculator. (2003) That againthe answertoB. like Lookswasn’t the correct go Use the backspace Try with option option D. Don’t nd must beequal sign. Either to the NOT have to answer choice, either. 2 You do Check assume!the first it out! delete everything! the answer Enter, backspace toPress retype =, nd ENTER option it. 2 and type in choice or type over C. We will now move on to solving equations… This problem was on the 2006 test We are looking for C when they have given us F. We will now move on to solving equations… This problem was on the 2006 test We can solve this equation the ―traditional‖ way—using the ―undo‖ process. 9 104 C 32 Subtract 32 5 9 72 C Multiply by 5 5 360 9C Divide by 9 40 C Alternate method We can solve this equation by using the table feature of the graphing calculator. Enter the equation. Go to the table. Scroll down the table until you find 104 in the y-column Alternate method You we could use Or need to be the to see in the able graph and window features of CALC where the two lines the graphing intersect. That calculator place looks way Enter the equation off to the right. in y1 and 104 in y2. Adjust the window Adjust the try the again. Let’s window — at 50. xmaxYou need ymax to be higher than 104 Graph Alternate method Press 2nd TRACE so that you get CALC. Now, select Intersect. Move the cursor to be close to the point of intersection. Enter again for the second curve? And guess? This problem was NOT multiple choice. You have to bubble in your answer correctly! Be careful!!! After going through all 4 0 that work to get the correct answer, you don’t want the problem to be scored as wrong because you didn’t bubble in the answer properly! This problem was on the 2003 test. Let’s approach this one differently, since it is multiple choice. We could transform the equation so we can use the calculator, but too many people Option F is not it. mess that up—especially with the Neither is option G subtraction sign! Option H is the one! Since we were given answer choices for y, we will use the calculator to substitute and simplify until we find the y-value that gives us 18 for an answer. We already know x is 3. Here’s an inequality problem from the 2003 test where they ask you to graph, but they don’t look at the graph! We’ll solve this problem the same as the last one. In other words, don’t be fooled by how complicated a problem looks! Study the problem. See what you know that can be used to make the problem easy to do! Let me again mention—the state graders are not going to look in the test booklets for your graph! We are NOT going to use the grid to do this problem! We are going to substitute these coordinates into the expression. We want an answer that is less than 12! Option D gives us 12. We want an answer that A is more than 12 is less than 12. And so is B Sure enough, C Looks like C is the answer, is the answer. but let’s check to be sure. TAKS will not always give you an expression, equation or inequality. You may just have to read a word problem and come up with one of your own. The next several problems fit in this category. 2006 First, you want to find the profit on ONE tool set. If it sells for $19.95 and is made for $4.37, the profit is the difference in the two prices. 19.95 – 4.37 Eliminate B & C Now, the profit is made on every tool set sold, which is s. We would multiply s to the profit (19.95 – 4.37) on each tool set. 2006 Let’s reason. A total of 80 backpacks were sold. That means, if 1 $35 backpack was sold, 79 of the $50 backpacks were sold. If 2 $35 were sold, then In each case, I subtracted the 78 $50 were sold. number of $35 backpacks, x, If 10 $35 were sold, from the total of 80. then 70 $50 were sold. That is 80 – x . 2006 Since x is the number of $35 backpacks, I would need to multiply 35 and x together to get the amount of money made by those purchases. Since (80 – x) is the number of $50 backpacks sold, I would need to multiply 50 and (80 – x). 2006 ―Total‖ implies addition. Adding b and c should be 220. b + c = 220 2006 As for the money…Every brownie sold for $.75 so $.75 needs to be multiplied to b, the number of brownies. Please note that $.50, the cost of each cookie is multiplied to c. In some cases, you will be expected to use the information given in a problem situation to set up the procedure and to solve for the requested information. How you do the problem is up to you and will not be checked. That you come up with the correct answer is the important piece this time. 2006 Perimeter is the sum of the lengths of the 3 sides: 5x + 10 Anytime a figure is used, DRAW & LABEL it in the 2x + 5 2x + 5 test booklet! We have no idea how long the base is so 5x + 10 = 95 x that is ―x‖. 5x = 85 Each leg is 5 x = 17 more than double the base: 2x + 5 That is not the only way you can do this problem. You can just use the answers and figure it out. Let’s look at option A. If the base is 17, then based on what the problem says, the legs are each 2(17) + 5 = 39 The perimeter means to add the two legs and the base: 17 + 39 + 39 = 95 95 is the perimeter we want, so A is the answer. 2006 This part means $.11 times ―x‖ ―x‖ is the number of minutes Lisa can talk. Notice: all graphs have a start at zero. That is because Lisa cannot talk for less than 0 minutes, which is no talking at all. The connection fee is charged, no matter what. 2006 This part means $.11 times ―x‖ ―Not more than‖ means we have an inequality—this value is the upper limit. Lisa wants to spend $5 or less. .50 + .11x ≤ 5 .11x ≤ 4.50 x ≤ 40.9090… Again, on this problem we could use the answer choices to work toward the solution. All of the graphs have 0 as a lower limit of minutes. That makes the 0 a ―non-issue‖. We need to look at the upper limit of the graphs and see which one keeps us at $5 or less. Let’s start with option D since it has the largest upper limit. Again, on this problem we could use the answer choices to work toward the solution. If minutes is 50, then .50 + .11(50) = 6. $6 is over Lisa’s limit. If minutes is 45, then .50 + .11(45) = 5.45 This X value is also over Lisa’s limit X If minutes is 40, then .50 + .11(40) = 4.9 $4.90 is under Lisa’s $5 limit, so option B is the correct one. 2003 Anna must get the same amount of money, m, for selling the plates as it takes to make the plates before she can ―x‖ represents the number of plates make any profit. Anna must sell Her costs: m = 750 + 10x Her revenue: m = 25x As we discussed earlier, you can solve by graphing the two equations and finding the intersection. Or you can use the table and find where the two y values are the same. Or you can use substitution and solve algebraically. 2003 750 + 10x = 25x 750 = 15x 50 = x 2003 And, of course, you always have the option of working with the answer choices to see which one gives you equal values for her costs and how much she makes selling the plates. 20 plates is a NO 50 plates looks NO 30 plates is also like a winner. Now, it is time to move on to the mechanics of lines. We need to talk about Slope Y-intercepts X-intercepts Equations of lines through points If we do not get everything covered before the end of the session, please visit Mrs. Nelson’s website to complete the power point lesson! About slope… The first thing you need to note is that the slope is X negative. That means the line should be going downward. Eliminate the ones going up. Now, look for the X point (4, -3) on the remaining 2 graphs. The point must be ON the line. Rate of change is slope Graph the line and see what it looks like. The line is horizontal. It is ―running‖ but it is NOT ―rising‖. It has 0 rate of change because the y-coordinates of every point on the line are exactly the same. You want to remember: Horizontal lines have 0 slope Vertical lines have undefined slope Diagonal lines have some fraction for a slope If the diagonal line goes up as it goes right, the slope is positive If the diagonal line goes down as it goes right, the slope is negative The steeper the line (the closer it is to the y-axis), the bigger the slope The flatter the line (the closer it is to the x-axis), the closer the slope is to zero (which would be horizontal). 2006 You are expected to know that the x-intercept is where the graph ―touches‖ the x-axis. You are also expected to know that the y-coordinate is 0 for an x-intercept. Option 1: put 0 in 2/3(0) = -6x + 12 for y and find x. 0 = -6x + 12 6x = 12 x=2 2006 You are expected to know that the x-intercept is where the graph ―touches‖ the x-axis. This equation is not in the correct form for our calculator. So, divide by 2/3—Not you! Tell the calculator to Option 2: graph the do it. Just be sure to use line and see where parentheses! it crosses the x- axis. y1= (-6x+12)/(2/3) x = 2 at this point 2003 Find the point (5, -1) on the grid. You need to use the equation x – 3y = 6 to find the y-intercept. There are a few ways to do that! Remember, the state graders are NOT looking at your graphs in your booklet. This grid is for you! 2003 The x-coordinate of the y-intercept is always zero. Substitute 0 in for x and find y. 0 – 3y = 6 -3y = 6 y = -2 You could transform the equation x – 3y = 6 into slope-intercept form to find the y-intercept. -3y = 6 – x y = -2 + 1/3 x You could put the equation into a form that the calculator will graph and find the y- intercept visually. x = -2 -3y = 6 – x y = (6 – x)/-3 here 2003 Eliminate A and D X since the b-value, the y-intercept, is NOT -2. Make a point where the y- intercept is -2. Now, count the slope from (0,-2) up 1 right 5 X to (5,-1). The slope is 1/5. 2006 Use your formula chart and draw the line between R and S. You can see that the y- intercept cannot be 4.5, so eliminate those two choices. up 12 3 Now, count the slope 1.5 between R and S. right 8 2 2004 There are numerous ways to do this problem. You could type each answer choice in your calculator, go to the table, and look for BOTH points. BOTH points are here—this is it! 2004 Option 2 You are given graph paper at the end of the math section. Make yourself a set of axes and graph the two points. The y-intercept is at 3.5 7/2 = 3.5 Draw the line between them. Count the slope: down 2 2 1 right 4 4 2 2004 Option 3 You could use the slope formula on the formula chart to calculate the slope. Then, you could use slope- intercept form or point-slope form, which is also on the formula chart to calculate the y-intercept. y mx b 1 2 (3) b use the slope and (3,2) as(x, y) 4-2 2 1 2 4 3 2b get rid of the fraction by multiplyin g everything by 2 -1 - 3 4 2 7 2b add 3 to both sides 7 b divide both sides by 2 2 Now, let’s talk “linear systems” 2003 X X We – 3y needthe given Neither 0 2x just = of the y by equations is in itself. Using parentheses -3y = 0 – 2x calculator the will allow us form. We opportunity – 2x)/-3 into y = (0 for the need to get them that form ―calculate‖ calculator tofirst. put this function in y1 instead of us. X That does is too The slope in FNOT mean compared to steep as they have to this graph. be in slope-intercept form. The line going through (0, 0) must go upward. Now, let’s make sure… X X x + 2y = -7 2y = -7 – x y = (-7 – x)/2 put this function in y2 X Sure enough, we have a match. Now, let’s make sure… X X You were not asked, but let’s find the point of intersection for the two lines—the solution to the system. X Yes, this point is (-3, -2) 2006 Count the slope, follow the pattern, and extend each line. The point of intersection, the solution, is (10,2). Practice Problems