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Math 8 Assessment Review Absolutely Positively Everything You Need To Know New York State says there are certain math skills and concepts they expect you to know. These skills fall under the headings: Algebra Geometry Measurement Number Sense and Operations How To Use This Program This program is designed to give you a good idea of what you already know and what you need to learn a little better. Each slide has a topic (written in yellow) you are expected to know and some sample questions. If you get all of the questions on the slide right, click to the next slide. If you miss any questions, click on the link and you will go to a tutorial that will help you learn what you need to know. Number Sense Let’s see if you know your integer rules. 1) 5 + (-7) -2 7) -2 x 9 -18 2) -3 + (-1) -4 8) -2 x -9 18 3) -4 + 9 5 9) 5 x -4 -20 4) 5 – (-2) 7 10) -10 x (-5) 50 5) 2 – 5 -3 11) 15 / (-5) -3 6) -7 – 3 -10 12) -15 / (-5) 3 Use the order of operations to simplify expressions. Do you know it? Simplify each: 1)18 – 5 + 3 16 2)50 - 4·9 14 1) If a = 5 and b = 3, evaluate ab2 45 2) If a = 4, b = -3 and c = 5, evaluate a(b + c) – (a – b) 1 Be able to express large and small numbers in scientific notation. Do you know it? Express each in scientific notation: 1)872,000,000 8.72 x 108 2)0.00000104 1.04 x 10-6 Apply the Laws of Exponents for Multiplication and Division. Do you know it? 1) x2·x3= x5 2) (3x4) (5x5) = 15x9 3) v8/v2 = v6 4) 12a6/3a2 = 4a4 Evaluate Expressions with Integral Exponents. Do you know it? 1) 34 81 2) 50 1 3) 102 100 4) 3-2 1/9 5) 199 1 Read, write and identify percents less than 1 and greater than 100. Do you know it? 1) Express 125% as a decimal in simplest form. 1.25 2) Convert 0.0003 to a percent. 0.03% 3) Describe a situation where a percent can be greater than 100%. Invest money and get back more than you had. 4) Convert 3/8 to a percent. 37.5% Apply percents to tax. Do you know it? Suffolk County has an 8.5% sales tax. If Jenna buys some cleats for $50 …………………. a)How much tax does she have to pay? $4.25 b) How much does she give the cashier? $50 + $4.25 = $54.25 Find the percent of increase or decrease. Do you know it? Todd buys a painting for $25 and then sells it on eBay for $30. What is the percent of increase on the value of he painting? 20% increase Use percents to find the sale price of an item. Do you know it? Overstock is having a Christmas sale. Originally a pair of earrings cost $78 and are on sale for 45% off. What is the new price? $42.90 Calculate markup on an item and determine the price the store sells an item for (retail.) Do you know it? Fred’s Flyfishing Emporium marks up every item he sells by 120%. If the store buys a DVD for $8, how much does he sell it for? $17.60 Calculate simple interest. Do you know it? In order to buy a jeep, Jimmy is thinking of taking out a loan. He would need to borrow $5,000 for 2 years at an interest rate of 9%. a)How much interest would Jimmy have to pay? $900 b)What is the total Jimmy would have to pay back? $5,000 + $900 = $5,900 Estimate the percent of a quantity in an application. Do you know it? 1)Two out of nine kittens are black. Estimate what percent are black. 20% 2) Students in a school in Tennessee voted on their favorite country singer. 52% of them voted for Tim McGraw. If 250 students were surveyed, how many voted for Tim McGraw? 125 Use estimation to analyze how reasonable an answer is. Do you know it? Tara buys 12 different magic markers when she shops at Staples. Each marker costs $1.49. She thinks the total cost of the markers will be $15. Is she right? No. The cost is closer to $18. (1.5 x 12) = 18 Add and subtract monomials. Do you know it? 1)5x + 2x = 7x 3) 3c – 7c = -4c 2) 3x2 + 7x2 = 10x2 4) ab – 4ab= -3ab Solve multistep equations by combining like terms, distributing or moving variables. Do you know it? 1)4w + 3 – 2w = 17 3) 5(2x – 3) = 25 w=7 x=4 2) 7y – 3 = 4y - 15 y = -4 Graph a table of values or pattern from an equation. Do you know it? Using a table of values, graph y = 3x – 4. x Y y 0 -4 1 -1 (0,0) x 2 2 3 5 Create algebraic tables using charts/tables, graphs, equations and expressions. Do you know it? Tara reads 3 pages per minute. One Saturday she begins reading on page 17 and reads for 5 minutes. Create a table of values to show what page (p) she is on after any minute (m). m p m p 0 17 1 20 2 23 3 26 4 29 5 32 Build a pattern to develop the sum of the interior angles of a polygon. Do you know it? How many degrees in the interior angles of a ….. Triangle 180 Quadrilateral 360 Pentagon 540 Hexagon 720 In general? (Number of sides – 2) x 180 Write an equation to represent a function from a table of values. Do you know it? Write a function rule that explains the relationship between x and y in each table. x y x Y 0 3 -1 3 1 5 1 5 2 7 3 7 3 9 5 9 Y=2x+3 Rule: ________ Y=x+4 Rule: ________ Translate verbal sentences into algebraic expressions. Do you know it? Write each as an algebraic expression: 1) A number, n, increased by five. n+5 2) Seven less than a number, n. n-7 3) The product of three and a number, n. 3n 4) Two more than four times a number, n. 4n + 2 5) The sum of a number squared and one. n2+ 1 Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship. Do you know it? Jamal bought a plant that was 3 cm tall. Each week it grew 2 cm. Graph the height (h) on any week (w). h w h w h 0 3 1 5 2 7 3 9 2 1 0 w Add and subtract polynomials. Do you know it? 1)Find the sum of 3x – 2 and x – 5. 4x - 7 2) Add: (5a + b) + (2a – 2b) 7a - b 2 2 3) Add: 3x – 2x + 5 – 2x – 7x – 5 x2 – 9x 4) Subtract: (5d – 3f) – (2d – f) 3d – 2f Multiply a binomial by a monomial or binomial. Do you know it? Simplify each expression: 1) 3(x – 7) 3x - 21 2) -2(2d – 9) -4d + 18 3) x(x + 2) x2 + 2x 4) (x + 3) (x + 5) x2 + 8x + 15 5) (x – 5) (x + 2) x2 – 3x - 10 Factor algebraic expressions using the Greatest Common Factor (GCF). Do you know it? Factor each: 1) 3x + 12 3(x + 4) 2) 5h – 20 5(h – 4) 3) x2 + 5x x(x + 5) 4) x2 – 8x + 12 (x – 6) (x – 2) Solve multistep inequalities and graph the answer on a number line. Do you know it? Solve the following in equalities and graph each on a number line: 1) 3x – 7 < 5 x<4 4 2) 4 – 2x < 10 x>3 3 Solve multistep linear inequalities by combining like terms, distributing or moving variables from one side to the other. Do you know it? Solve each inequality for the variable: 1)3d – 5d + 4 > 12 2) 4x + 6 < x – 3 d < -4 x < -3 Geometry Identify the hypotenuse, right angle and legs of a right triangle. Do you know it? In the picture of the right triangle below, identify the legs and the hypotenuse. hypotenuse leg leg Be able to find a missing leg or hypotenuse using the Pythagorean Theorem. Do you know it? Find n in each diagram: 13 4 n 5 n 3 n = 12 n=5 Identify pairs of vertical angles as congruent. Do you know it? 5n - 1 3n + 27 Find n n = 14 Identify pairs of supplementary and complementary angles. Do you know it? Label each diagram as complementary or supplementary. supplementary complementary Calculate the missing angle in a complementary or supplementary pair. Do you know it? Find the value of n in each diagram: 67 n 71 n n = 109 n = 23 Determine the angle relationships when two parallel lines are cut by a transversal. Do you know it? List all of the angles that are 1 2 equal to each 3 4 other. <2, <3, <5, <8 6 5 8 7 equal eachother. <1, <4, <6, <7 equal eachother. Calculate the missing angle measurements when two line are cut by a transversal. Do you know it? m<a = 115 a b c d m<b = 65 m<c = 65 m<d = 115 115 e m<e = 65 g f m<f = 115 m<g = 65 Identify different transformations in the plane, using proper function notation. Do you know it? 1) 2) rotation reflection 3) translation Identify horizontal and vertical line symmetry as well as point symmetry. Do you know it? Which of the following letters have horizontal, vertical and/or point symmetry? C horizontal Y vertical S point J no symmetry Be able to draw a line reflection. Do you know it? Reflect the line segment over the x-axis. y (0,0) x Be able to draw a rotation of 90, 180 and 270 degrees. Do you know it? Rotate the rectangles (one for each) below 90 degrees, 180 degrees and 270 degrees. 90 180 270 Be able to draw and compute the result of a translation. Do you know it? Translate the line segment two units to the right and down four. Measurement Calculate distance using a map scale. Do you know it? The scale of a map is 1 inch = 80 miles. On the map, the distance from Patchogue to Callicoon is 2.5 inches. How far is it from Patchogue to Callicoon? 200 miles Calculate unit price using proportions. Do you know it? A box of Corn Flakes cereal sells for $3.78. The volume of the box is 18 ounces. What is the unit price for an ounce of Corn Flakes? 21 cents or $0.21 per ounce Compare unit prices for least and most expensive. Do you know it? Costco sells two different size packages of chicken. One package sells 3.5 pounds for $8.19 and the other sells 5 pounds for $10.79. Which one is a better buy and how much do you save per pound? The 5 lb bag is cheaper by $0.18 per pound. Set up and use proportions to solve word problems. Do you know it? Emily babysits in order to make some money. On Saturday, she gets paid $12 for 2.5 hours of babysitting. On Monday, she gets paid $30. How long did Emily babysit? 6.25 hours Determine if two figures are congruent or similar. Do you know it? 1)What does congruent mean? same size & shape 2)What does similar mean? same shape sides in proportion Is the pair of triangles below similar or congruent (assume all corresponding angles are equal.) similar Use a proportion to solve for a missing side in similar figures. Do you know it? Find x. 12 9 6 x x=8 Absolutely, positively everything you need to know! Adding Integers There is one question you need to ask yourself when adding integers: Are the signs the same? Are the signs the same? Yes No Subtract. Keep Add. Keep sign. Sign of bigger #. Adding Integers With The Same Sign Remember: Add and keep the same sign! 1) 3 + (+5) 8 2) -3 + (-5) -8 3) 7 + 4 11 4) -7 + -4 -11 5) -3 + (-100) -103 6) -2 + -2 -4 7) -12 + (-3) -15 Adding Integers with Different Signs If the signs are different, subtract and keep the sign of the bigger number. 1) 5 + (-3) 2 2) 5 + (-4) 1 3) 5 + -5 0 4) 5 + (-6) -1 5) -3 + 4 1 6) -9 + 7 -2 7) 8 + -5 3 Mix & Match – How Many Can You Get? 1) -3 + 5 2 6) -2 + -3 -5 2) 7+8 15 7) 9 + -9 0 3) 1 + -5 -4 8) -7 + (-7) -14 4) -10 + -5 -15 9) -4 + 1 + 8 5 5) -8 + -2 -10 10) -2 + (-1) + (-4) -7 You understand when you can get 9 or more right ! Subtracting Integers If you let it, subtracting integers can get very confusing. If you are having trouble, let’s try a new way and hope we don’t get confused. DON’T SUBTRACT!!!! That’s right – let’s add instead – after all, you already know how to add! The hardest part of integers is when all of the different rules start confusing us. We already know the rules for adding, so let’s just add. The only thing we have to remember is that the sign in front tells us whether the number we are adding is positive or negative. Here’s what I mean: 4 – 2 can also mean (+4) plus (-2). So just read it that way. 4 + (-2) = 2. Let’s try another: 5 – 7 should be read as +5 + -7. And we already know that is 2. Practice. Practice. Practice. Question Conversion Answer 3–7 3 plus -7 -4 -5 – 2 -5 plus -2 -7 -8 – 2 -8 plus -2 -10 -8 -4 – 4 -4 plus -4 9-5 9 plus -5 OR just 9 - 5 4 The Last Detail of Subtraction In case you had not noticed, the last set of problems lacked the type of question that looked like 5 – (-3). Let’s just write this as 5 “-” (-3). In order to do this type, we need to remember that when you negate or minus ( “-” ) a negative, you make a positive. So 5 – (-3) is 5 “+” 3. Question Conversion Answer 4 – (-5) 4+5 9 2 – (-3) 2+3 5 -5 – (-2) -5 + 2 -3 -4 - 10 -4 plus -10 -14 Subtraction Practice Question Conversion Answer 1) 7 – (-3) 7+3 10 2) 5–8 5 plus -8 -3 3) -7–2 -7 plus - 2 -9 4) 10 – (-5) 10 + 5 15 5) 3-7 3 plus - 7 -4 What Do We Know So Far? 1) When you add and the signs are the same, you add same sign ______ and keep the ____________. 2) When you add and the signs are different, you subtract sign of the bigger number _________ and keep the _____________________. 3) addition DON’T Subtract!!!! Treat the problem like _________ positive and the sign in front tells you if it is ___________ or negative ___________. If you see two negatives in a row (like + 5 – (-2) ), change the “-” “-” to a ___. Addition and Subtraction Mixed Bag 1) 8 + -4 4 6) -3 – 9 -12 2) 12 – (-5) 17 7) -10 – (-2) -8 3) -2 – 1 -3 8) 5 + -4 1 4) -3 + -4 -7 9) 7 + (-7) 0 5) -4 + -1 + -3 -8 10) -4 – (-1) + -7 -10 Multiplication & Division There are several ways to explain multiplying and dividing with signed numbers, but let’s try a less common method. Here’s the deal: just multiply (or divide) the numbers and only when you are doing so BY a negative do you change the sign. It’s easy! Question Thought Process Answer -3 x 5 3 times 5 is 15 and keep the neg. -15 -3 x -5 -3 x 5 is -15, BUT -5 means change sign +15 8 x -4 8 x 4 = 32,BUT -4 means change sign -32 -5 x -9 -5 x 9 = 45, BUT -9 means change sign -45 Multiplication and Division Practice 1) -5 x – 2 10 5) (-10) / 2 -5 2) -18 / 9 -2 6) -1 x -3 3 3) 12 / -4 -3 7) -10 x -8 80 4) 5 x -5 -25 8) -20 / 5 -4 In #s 1, 3, 4, 6 and 7, you multiplied or divided BY a negative, so the sign changed. In all of the others, you multiplied or divided BY a positive, so it stayed the same ! Completely Mixed Practice 1) 7 + (-5) 2 6) -3 x -8 24 2) -3 – (-4) 1 7) 8 – 10 -2 3) -22 / 11 -2 8) -9 + (-2) -11 4) -6 x -5 30 9) 10 – (-4) 14 5) -4 + 3 -1 10) -25 / -5 5 Challenging Questions 1) Explain why when you add two negatives you get a negative but when you multiply them, you get a positive. When adding, you keep the same sign but when multiplying by a negative you change the sign. 2) When you see a problem that says 5 – (-3) with those two minus signs in a row, what do you have to remember to do? Change the “-” “-” to a “+”. 3) addition There is no subtraction. It is really ___________ and the sign in the front tells you if it is positive or negative. Absolutely, positively everything you need to know! The Order of Operations The order or operations tells us what to do and when to do it when trying to evaluate an expression. Some people remember PEMDAS and others PLEASE EXCUSE MY DEAR AUNT SALLY. Either way, it all means the order is: Parenthesis (aka Packages) Exponents Multiplication / Division (left to right, they are equal) Add / Subtract (left to right, they are equal) What Operation Would You Do First? 1) 9–2+9 subtract 2) 17 + 4 x 5 multiply 3) 10 (5 – 3) subtract 4) 10 – 14 ÷ 2 divide 5) 83 + 5 x 100 exponent Step by Step Order of Operations Go slow and answer this question step by step. As you click on the mouse, each step will appear. Evaluate: 25 – 3 x 22 OK. First you have to do 22 = 4 So it becomes 25 – 3 x 4 Next you have to do 3 x 4 = 12 Finally, it is 25 - 12 13 Let’s Do Another Step BY Step Evaluate 5(7 – 3) – 8 + 1 First things first …… do the package 7 – 3 = 4 So we have 5(4) – 8 + 1 I sure hope you realize multiplication comes next. 5 x 4 = 20 OK. Now 20 – 8 + 1 Don’t get fooled!!! Addition and subtraction are equal. 20 – 8= 12 Finally, 12 + 1 13 Perfect Practice Makes Perfect 1) 5(7 – 9) 3) 2(4 + 3)2 5 (-2) 2(7)2 -10 2 · 49 98 2) 22 – 7 + 10 ÷2 4) 102 – 43 22 – 7 + 5 100 – 64 15 + 5 36 20 Algebra & The Order of Operations Use the values a = 10, b = -5, c = 4, d = 2 and e = -2: 1) ab2 2) e3 + ab 3) ac + be 10 · (-5)2 (-2)3 + 10(-5) 10 · 4 + (-5)(-2) 10 · 25 -8 + 10(-5) 40 + 10 250 -8 + (-50) 50 -58 Summary The Order of Operations is: 1) Parenthesis (aka Packages) 2) Exponents 3) Multiplication / Division 4) Addition / Subtraction *** DANGER!!! Be very careful with the last two steps. Multiplication and Division are equal to each other. If they are the only steps remaining, just go left to right. The same holds true for addition and subtraction. Absolutely, positively everything you need to know! Equation Basics There are 4 concepts we must remember at all times with basic equations: 1) The strategy is to get letters on one side and numbers on the other. 2) First identify what is happening to the variable. 3) Do the opposite, undo or do the inverse operation. 4) Be sure to do the same thing to both sides!!!!! Basic Two Step Equation Solve for x: 3x – 1 = 14 +1 things are happening: subtracting and Two +1 multiplying. First attack the subtraction. 3x = 15 3 3 Now attack the multiplication. x = 5 More Practice: n/2 + 5 = 3 5h + 4 = 9 12 = g/3 + 9 -5 -5 - 4 -4 -9 -9 n/2 = -2 5h =5 3 = g/3 n = -4 h =1 9 = g More Two Step & A Multi Choice Trick Before we can possibly understand the harder equations, we have to master these. Solve for x: 4x + 7 = 3 10 = x/9 – 1 12 = 3c - 12 x = -1 99 = x 8=c In a multiple choice question, if you have a hard time solving the equation, you can substitute and check the answers. Solve for x: 7x – 5 = -33 a) -5 b) 5 7(-4) – 5 = c) 8 d) -4 -28 – 5 = -33 check! Two Step Inequalities The biggest difference between an equation and an inequality is that an equation has only one answer and an in equality has an infinite number of answers. For the most part, they are exactly the same. Here’s a comparison: 2x + 1 = 21 2x + 1 < 21 -1 -1 -1 -1 2x = 20 2x < 20 x = 10 x < 10 Step by step is identical. The only difference is the symbol – for now. Let’s practice some more . More Inequalities 5f + 7 ≤ 22 f/3 – 2 > 8 -3d – 1 < 11 f≤3 f > 30 d > -4 + right!!!! That’s 1 +1 Let’s try a multiple choice: -3d < 12 d > -4 NOT less Which is a solution of 2x + 7 < -13 -3 than! Why? -3 a) -12 Whenever you x SEE IT !!!!! You If we solve, we get x<-10. ÷ an inequality ordivided BY a neg. b) -11 Only (d) is less than -10. BY a negative, sign! SWITCH the c) -10 you have to You can also check each d < change the -4 d) -9 answer. symbol. Watch the steps! Click now. Graphing Inequalities The < and > symbols are nothing more than the “less” and “greater” ends of the number line. As long as the variable is on the left, the graph on a number line goes the way the symbol points. Let’s graph each: x>2 x≥2 2 2 What is the difference between the two graphs? The ≥ has the circle filled in. Graph these: x<2 x≤2 2 2 More Graphing What happens if you solve an in equality and the answer looks like this: 7 < x The letter is on the right……. that’s not the way w want it. All you need to do is put it in the left and MAKE SURE the inequality points to the same thing. 7 < x is the same exact thing as x > 7 Solve and graph each of the following: -4x + 11 < -1 -10 > 3x - 1 x>3 -3 > x x < -3 3 -3 Solving Multistep Equations There are two primary types of equations we will have to deal with here. One has the variables on the same side and the other has them on opposite sides. SAME SIDE OPPOSITE SIDE 3n + 7 + 4n = 21 5b – 3 = 3b + 17 -3b -3b 7n + 7 = 21 2b – 3 = 17 -7 -7 +3 +3 n = 2 b = 10 Remember this KEY SAYING: Same Side, Same Operation. Opposite Side Opposite Operation. What Would Your First Step be? Would you do the same operation and combine like terms (same side) or use the opposite operation (opposite side). 3x + 7x – 3 = 11 5d + 7 = d - 9 same operation opposite operation combine terms 2n – 9 = n + 11 2x + 7x – x = 42 opposite operation same operation combine terms Perfect Practice Makes Perfect Let’s try these four. Don’t forget about it! Same side same operation opposite operation _________________. Opposite side ________________. 3w + 2 + w – 5 = 5 6f + 1 = 28 – 3f 4w – 3 = 5 9f = 27 w =2 f=3 2x + 3x + 7x – 10 = 134 5x + 2x – 3 = 4x - 15 12x – 10 = 134 7x – 3 = 4x – 15 x = 12 3x = -12 x = -4 Equations Using Distribution If you see parenthesis (or a package) in an equation, you will probably have to distribute. 3(x – 5) = 9 2(x + 4) = 7x – 12 3(x – 2) + x – 3 = 11 3x – 15 = 9 2x + 8 = 7x – 12 3x – 6 + x – 3 = 11 + 15 +15 -2x -2x 4x – 9 = 11 3x = 24 8 = 5x – 12 x =5 x = 8 +12 +12 20 = 5x 4=x Perfect Practice Makes Perfect 1) 3n + 8 – n + 2 = 40 3) 3(h – 7) – 2 = 7 n = 15 h = 10 2) 8q – 3 = 5q + 18 4) x + x + 10 + 2x + 20 = 110 q=7 x = 20 The Get It - Got Its scale An equation is like a __________. As long as you do the same thing _____________ to both _______ it stays balanced. sides multiply divide When solving an inequality, if you _________ or ________ negative by a _________, you have to switch the sign. With multiple step equations, if you see parenthesis, you distribute can expect to _____________. If the variables or same numbers are on the same side, use the _____ operation ___________, but if they are on opposite sides, use the ____________ ____________. opposite operation Absolutely, positively everything you need to know! Adding and Subtracting Monomials To be good at adding and subtracting monomials, you need to know two things: #1: How to add and subtract integers #2: That whenever you add or subtract anything on the face of the earth, the type of thing stays the same. 2 “puppy dogs” + 3 “puppy dogs” = 5 “puppy dogs” 2 “kitty cats” + 3 “kitty cats” = 5 “kitty cats” But you can’t add 2 puppies + 3 kittens!!!!! $2 + $3 = $5 2x + 3x = 5x 2ab + 3ab = 5ab What you start with ($, x, ab or puppy) is what you end with!!! Identifying Polynomial Types What is a polynomial? The word is large enough to look intimidating, but it is actually made up of two simple parts. Poly means many (polygon, polytheism) and nomial means term (x, 3y, 4ab and x2 are all terms.) So a polynomial has many terms. 3w – 5y 2x2 – 8x + 11y 2p – q + 1 – w are all polynomials. We can be a bit more specific. 2w – 5y is a binomial because it has two terms and 3a – 2b + c is a trinomial because it has three terms. 5x is a monomial. Perfect Practice Makes Perfect Simplify each expression: 1) $4 + $3 3) 3w – 2w 5) 4ab + 3ab $7 w 7ab 2) 4d + 3d 4) 8h – h 6) 4xy - xy 7d 7h 3xy There are two critical things you absolutely MUST remember: 1) Only like things can be added. 2) You get what you start with. More Perfect Practice Simplify each expression: 1) 3c – 7c 3) -4w + 3w 5) ab + ab -4c -w 2ab 2) 3x2 – x2 4) -q + 8q – 2q 6) 3a2b + 2a2b 2x 2 5q 5a2b Remember: Things must be the same. Things stay the same. Adding & Subtracting Polynomials This is pretty much the same – but we have to really emphasize that only like things can be put together! We can use Poppa’s Bucket Principle to clean up the mess. Everything has a bucket and only the same things go in a bucket! 3a + 5b – 2c + a – 2b – 3c 3a 5b -2c Find the a -2b -3c sum in each +4a +3b -5c bucket. Final Answer: 4a + 3b – 5c (like the buckets say) Perfect Practice Makes Perfect Simplify each by combining like terms: 1) 2a – 3b + a – 2b 4) 2q2 – 3q + 5 – q2 + q – 3 3a – 5b q2 – 2q + 2 2) 3x + 5 + 5x – 7 5) a – 3b – 5c – a + 2b – c 8x - 2 -b – 6c 3) n2 – 3n + n2 + 2n 6) 3x – 2y + 3y - x 2n2 - n 2x + y Perfect Practice Makes Perfect Simplify each by combining like terms: 1) 2a – 3b + 5a – b 4) 4q2 – q + 4 – q2 + q – 2 3a – 4b 3q2 + 2 2) 4x + 5 + 5x – 3 5) a – 7b – 3c – 2a + 4b – c 9x + 2 -a -3b – 4c 3) n2 – 3n + 3n2 + n 6) 5x – 3y + 3y - x 4n2 - 2n 4x The Distributive Property The distributive property distributes or gives out a term to the others. Examples: 5(3x – 7) = 15x – 35 2(x2 + 5x – 1) = 2x2 + 10x – 2 Notice we multiplied the 5 by 3x to get 15x and the 5 by -7 to get -35. The “2” is similarly given to the trinomial in the second example. Now try these: 1) 3(7x + 2) 3) 5(3x2 – 2x + 9) 21x + 6 15x2 – 10x + 45 2) 2(5a – 3b + c) 4) -7(2x – 5) 10a – 6b + 2c -14x + 35 Double Distributing Frequently in algebra we are required to multiply a binomial by a binomial. For instance (x + 3) (x + 5). The key is to distribute one on t he binomials at a time. (x + 3) (x + 5) = x2 + 5x + 3x + 15 Distributing the blue “x” gives the blue part of the answer and the yellow “3” gives the yellow part. The only thing left to do is put the two like terms together. Click now! You can see that the 5x and 3x are like terms. Combine them to get 8x. So the final answer is x2 + 8x + 15 Now try this one: (x + 10)(x + 4) (first the “x”) x2 + 4x (then the 10) +10x+40 x2 + 4x + 10x + 40 = x2 + 14x + 40 Double Distribute Practice Find the product of each pair of binomials. 1) (x – 3)(x + 5) 4) (x + 10)(x – 1) x2 + 5x – 3x – 15 x2 – x + 10x – 10 x2 + 2x - 15 x2 + 9x - 10 2) (a + 8)(a – 4) 5) (c – 2)(c – 7) a2 -4a + 8a – 32 c2 – 9c + 14 a2 + 4a - 32 3) (m + 1)(m + 1) 6) (2x – 3)(x + 4) m2 + m + m + 1 2x2 + 8x - 12 m2 + 2m + 1 Factoring Using the GCF GCF is short for Greatest Common Factor. While distributing multiplies, factoring divides or pulls out the GCF. For instance: 5x + 10 = 5(x + 2) 5 is the GCF so we divide it out and write the quotient in the package. The GREAT thing about factoring is that all we have to do to know if we are right is distribute our answer and see if we get the question! Try these: 10x + 70 4x – 12 x2 + 5x 10(x + 7) 4(x – 3) x(x + 5) Perfect Practice Makes Perfect Factor each of the following: 1) 3c + 21 4) 5a + 5b – 15c 3(c + 7) 5(a + b – 3c) 2) x2 – 3x 5) c2 + 9c x(x – 3) c(c + 9) 3) 8b – 80 6) 2q - 38 8(b – 10) 2(q – 19) If you are ever worried that you are wrong, just distribute back and see if you get what you started with. Factoring Trinomials The last thing we need to talk about is factoring trinomials. The state says that this topic can be on the test, so we should know how to do it. The thing is that the topic has not yet been on the test, and it takes quite a while to learn. So I am going to suggest that if you want to learn this so that you can factor any trinomial, go to extra help. But it is quick enough and easy enough to answer a question in multiple choice format – just double distribute the choices! Example: Factor x2 – 7x + 12 The correct answer is c, just double distribute a) (x+3)(x+4) b) (x+6)(x+2) each of the choices to c) (x-3)(x-4) d) (x-6)(x-2) see why. Perfect Practice Makes Perfect 1) What are the factors of x2 + 9x + 20? a) (x + 10) (x + 2) b) (x – 10) (x – 2) c c) (x + 5) (x + 4) d) (x – 5) (x – 4) 2) What are the factors of x2 – x – 30? a) (x – 6) (x – 5) b) (x – 6) (x + 5) b c) (x + 6) (x – 5) d) (x – 10) (x – 3) Key Points to Ponder Remember when combining like terms, everything is addition. The sign in front just tells you if the term is positive or negative. Distributing is multiplication. Factoring is division. When factoring, all you need to do is distribute back and see if you get what you started with. When double distributing, always look for the two terms in the middle to combine. Absolutely, positively everything you need to know! What You Need to Know There are a handful of very important exponent concepts you need to know. Tops among the concepts are: 1) Scientific Notation 2) Evaluating Integral Exponents 3) The Laws of Exponents Scientific Notation Scientific notation is most commonly used as a shortcut for expressing and working with large and small numbers. In general, the format for a number expressed in scientific notation is (Number between 1 & 10) X 10# places decimal moves What is wrong with each of these? 1) 235,000 = 235 x 103 Numbers is not between 1 & 10 2) 5,000,000 = 5 x 56 The base is not 10, it is 5 Express Each in Scientific Notation 1) 42,000,000,000 4.2 x 1010; Some common mistakes are ………. 42 x 109 (42 is not between 1 & 10) .42 x 109 (.42 is not between 1 & 10) 2) 705,000 7.05 x 105; Some common mistakes are ……….. 705 x 103 (stopped moving the decimal as 0’s stopped) 70.5 x 104 (70.5 is not between 1 & 10) Scientific Notation Done Right 1) Convert 52,000,000,000 to scientific notation. 10 places to move We need to get the number into proper form. So …. In order to get 52,000,000,000 to proper form I need to move the decimal (and count) until I get 5.2. The answer is 5.2 x 1010 2) Convert 0.00000073 to scientific notation. 7 places to move Now we need to move the decimal to the right until we get a number between 1 & 10. Count the places! The exponent is negative because the number is small (0.0000……) The answer is 7.3 x 10-7 Perfect Practice Makes Perfect 1) 507,000 3) 11,000,000 5.07 x 105 1.1 x 107 2) 0.0000004 4) 0.00000003108 4 x 10-7 3.108 x 10-8 5) The sun is 93,000,000 miles from the earth. Express this number in scientific notation. 9.3 x 107 Evaluating Integral Exponents We already know that an exponent means how many times you use the base as a factor. For instance: 23 = 2 x 2 x 2 = 8 and 15 = 1 x 1 x 1 x 1 x 1 =1 It gets a little tricky when the exponent is negative or zero. First point to remember: Anything (except 0) to the zero = 1. Evaluate each: 50 = 1 x0 = 1 ab0 if a = 3 and b=9. 3 · 90 3x1=3 Negative Exponents The hardest exponent to compute is a negative exponent! Just looking at 5-2 makes you think the answer ought to have a negative sign, like say ……. -25. It doesn’t. A negative exponent has absolutely, positively nothing to do with a negative sign!!!!!! A negative exponent means to use the reciprocal or put the exponent in the denominator. For example: 3-2 = 1 / 32 = 1/9. Notice the neg. exp. means “one over” the positive exponent. 7-2 = 1 / 72 = 1/49. Perfect Practice Makes Perfect Evaluate each: 1) 5-3 1/125 4) 10-3 1/1000 2) 3-2 1/9 5) 7-1 1/7 3) 2-5 1/32 6) 9-2 1/81 The Laws of Exponents This is another topic students can find confusing. If you stay focused, I think you will find that the rules make sense. xaxb = xa+b What we are seeing here is that the base stays the same when multiplying and you add the exponents. xa ÷ xb = xa – b What we are seeing here is that when divide, the base stays the same and you subtract the exponents. For instance: 8 x5x3 = x x8 ÷ x2 = x6 the exponent is 6 not 4 because you subtract. Some Practice & Clarification 1) x3x7= x10 2) 3233 35 3) c6÷c3 c3 I hope they seemed easy for you . Just remember that the base ALWAYS stays the same and you add or subtract the exponent depending on the operation. Sometimes students get confused if you put numbers in front (called coefficients.) The thing you have to remember is that numbers are numbers and that they behave like numbers – not like exponents!!!!! Examples: (3w2)(5w) = 15w3 The 5 and 3 are numbers not exp’s! 5x3=15 10x8÷5x2 = 2x6 The 10 and 5 are numbers not exp’s 10/5 = 2 Perfect Practice Makes Perfect 1) 12c5÷4c 3c4 4) (7v3)(-3v6) -21v9 2) (3x)(4x2) 12x3 5) 20a9b4÷(5a3b) 4a6b3 3) 100x2÷10x 10x 6) (-4xy2)(-5xy) 20x2y3 Remember to keep the bases the same, treat exponents like exponents and coefficients (numbers) like numbers! What You Need to Know In scientific notation, you must have a number between 1 & 10 times 10# places you move the decimal. Anything to the zero power is 1 A negative exponent means 1 over the positive exponent. With respect to Exponent Laws, treat exponents like exponents bases __________, keep the _______ the same and treat the coefficients numbers (also called _____________) like numbers. Absolutely, positively everything you need to know! The Perpetually Problematic Percents Percents drive some kids NUTS! One big reason for that is that you don’t solve all of the problems the same way. First the bad news: Any student who just wants to be told what to do instead of having to think with percents is going to have a hard time. Now the good news: Any student willing to remember and work with a couple of basic ideas can master percents. Key ideas: 1) Percent means “out of 100.” 2) The percent is the “part” out of 100. 3) Fractions, decimals and percents are all the same thing (like hello, hola and bon jour). Percent Means “out of 100” Simply put, percent means out of 100. So if 55 out of 100 dentists prefer Trident gum, 55% prefer it. If 99 out of 100 students think Math is the best subject, 99% think Math is the best subject. But what percent think Math is the best subject if only 2 out of 5 think it is the best? Do you know what a lot of students write? 2%!!!!! Can you believe that? The question said 2 out of 5 – not 2 out of 100! To find the percent we must find out how many out of 100. Converting Between Decimals and Percents Converting between decimals and percents is one of he easiest math skills. Remember, percent means out of a hundred. Also remember that two decimal places is the equivalent of “100”. % to Decimal Decimal to % Move decimal 2 places left 2 places right Percents are bigger so we move to the right Examples: 50% = .50 (or just .5) .721 = 72.1% 115% = 1.15 (2 places only) .005 = 0.5% (2 places) Converting to a Percent There are two main ways to convert fractions like 2 out of 5 (2/5) to a percent. since 5x20 = 2/5 means 2 ÷ 5 which 1) 2 = ? 100, 2x20=40 is .4000………. To 5 100 … convert to a percent, 40/100 = 40% move the decimal two places right or 40%. Convert each to a percent: 1) 7/10 2) 3 out of 25 3) 3/8 70% 12% 37.5% Convert Percent to a Fraction To convert a percent to a fraction, just write it “out of 100” and reduce if necessary. 25% = 25 out of 100 = 25/100 reduced = ¼. Now you try. 45% = 45 out of 100 = 45/100 reduce to 9/20 90% = 90 out of 100 = 90/100 reduce to 9/10 37% = 37 out of 100 = 37/100 this is reduced. What if we get one like 37.5%? Then we need to do 37.5/100 but this looks funky. Convert to 375/1000 to get rid of the decimal (move numerator & denominator one place each. You try 31.7%. 31.7 out of 100 = 317/1000 Perfect Practice Makes Perfect Convert each to a percent: 1) 0.57 57% 2) 12/100 12% 3) 12/50 24% 4) 1.39 139% 5) 5/8 62.5% 6) 0.001 0.1% Convert each to a decimal: 7) 77% 0.77 8) 29% 0.29 9) 0.2% 0.002 10) 145% 1.45 11) 100% 1 12) 25.5% 0.255 Converting to Fractions We already discussed how, in order to go from a percent to a fraction, we should “put it “over” or “out of” 100 and reduce. To go from a decimal to a fraction is just as easy. Just read it properly and reduce ……. Remember, one decimal place is tenths, two are hundredths, three thousandths and so on. For example: 1) 0.5 read properly is 5 “tenths” = 5/10 = ½. 2) 0.34 read properly is 34 “hundredths” = 34/100= 17/50 3) 1.233 read properly is 1 and 233 “thousandths” = 1 233/1000 Perfect Practice Makes Perfect Convert to a fraction in simplest form: 1) 0.1 2) 0.79 3) 0.433 1/10 79/100 433/1000 4) 0.38 5) 75% 6) 0.009 19/50 3/4 9/1000 7) 110% 8) 0.747 9) 0.5% 1 1/10 747/1000 1/200 Fractions, Decimals, Percents Summary Besides remembering that fractions, decimals and percents are just different ways of saying the same thing, we need to remember how to convert amongst them. two Decimals and percents move the decimal ________ places right left to the ________ for a percent and _________ for a decimal. Decimals to fractions you just read it properly and ______ reduce divide _______. Fractions to decimals you just ________. over Percents to fractions, just put it ______ 100. The Percent Proportion Most students use the percent proportion in percent problems, so we will might as well, too. The percent proportion is: Percent = part (is) 100 Whole (of) Your job as a student is to determine what number goes in which place. Now, the 100 always stays where it is. The word is usually tells the part and of usually tells the whole. A lot of people think underlining can be a big help in these word problems. DON’T try to solve the question right away – try to find the parts. Problem One Jenny takes a science test that has 25 questions. She gets 21 of them right. What percent did Jenny get right? First things first ……. Let’s underline the important stuff! Two numbers are underlined, 21 and 25. Which one represents the whole number of questions and which one the part? % = part Now the part = 21 and whole = 25, so 100 whole N = 21 cross multiply 25n = 2100 100 25 25 25 n = 84 Problem Two Remember, find the important stuff and underline!!!!! In order to pay her way through college, Tammy works and earns $800 a month. If 65% of this money goes to tuition, how much money does Tammy spend each month on tuition? The whole amount Tammy earns is $800 % = part (is) 100 whole (of) 65 = n 100 n = 52,000 100 800 n = 520 The Hardest Part of Some Percent Problems Why do some percent problems give students a hard time? I think one of the biggest reasons is that many percent problems, and for that matter many math questions in general, require only one process to get the answer. Typically, all of he following percent related questions require TWO STEPS!!!!!! 1) Sales tax 2) Sale price 3) Tips (gratuities) 4) Mark up Problem Three: Sales Tax Probably the easiest way to understand a sales tax problem is first find the sales tax amount and add it to the cost of the item; after all, isn’t that how it works in a store? Rob buys a fishing rod for $80. The tax rate is 8.5%. a) How much tax does Rob pay? 8.5 = n (tax is part) 100 n = 680 100 80 (whole cost) n = $6.80 b) How much does Rob give the cashier? Rob gives the cashier $80 + $6.80 = $86.80 Another Sales Tax Question John buys two bats for $150 each. If the sales tax rate is 8%, how much does he have to pay the cashier? Maybe it will help to keep underlining……. First off, John spends $300 (two bats) 8 = n 100 300 100 n = 2400 n = 24 John has to give the cashier $300 + $24 = $324. Problem Four: Sale Price This should be easy to remember. Everybody shops during a sale because items cost less and less means to subtract! Find the amount you save and subtract it. Darla wanted to buy Alfalfa some flowers. They usually cost $20, but this week only are on sale for 40% off. How much does Darla have to spend on the flowers? % = part 100n = 800 100 whole n=8 40 = n 100 20 Darla pays $20 - $8 = $12. Another Sale Price Question Sue is looking to buy a new rug for the dining room. A rug she likes ordinarily costs $1800 but is on sale for 25% off. Sue has $1500 to spend. Can she afford the rug? % = part 100 n = 45,000 100 whole n = 450 25 = n The new price is $1800 - $450 100 1800 which is $1350. Yes, Sue can afford the rug! Tips and Gratuities A normal tip for a meal at a restaurant is 15% of the bill. If Bob and Betty went to dinner and the meal cost $45.00, how much should the leave the waitress as a tip? Round your answer to the nearest whole dollar. How much should they give the waitress? 15 = part (tip is part) 100 n = 675 100 whole bill n = $6.75 15 = n Rounded to the nearest whole dollar, the tip is $7. They should 100 45 give the waitress $45 + $7 = $52. Mark Up Problems: More Two Step Questions Mark up is how stores make money. Stores buy the items they sell for much cheaper than a customer pays for them. The amount the increase the price is called “mark up,” and because it increases the price you have to add. Remember, many of these questions are tricky because you often need to do two steps. Rob sells fishing supplies. He marks up all of the items he sells 120%. If he pays $40 for a fishing reel, how much does he charge? % = part 100 n = 4800 Now for Step #2: So the cost 100 whole price n = 48.00 Rob sells the 120 = n So the mark rod for is $40 + up is $48.00. $48 = $88 100 40 Another Mark Up Question Bob owns a pet supply store. With fish tanks, he marks the price up 40%. If Bob pays $150 for a fish tank, how much does he sell it for? % = part (mark up) 100 n = 6000 100 whole cost n = 60 40 = n This is the mark up!!! 100 150 The tank sells for $150 + $60 = $210. A Commission Question Mrs. Lindquist is a real estate agent. Recently, she sold a piece of land for $510,000. The commission (amount of money the real estate agent makes) she gets is 1.55%. How much money does Mrs. Lindquist make? % = Part 100 n = 790,500 100 Whole cost n = $7,905 1.55 = n Mrs. Lindquist makes 100 510,000 $7,905 on the sale of the land. Percent of Increase and Decrease Percent of increase and decrease problems still depend on your being able to determine the part and the whole. The thing is that, with respect to increase and decrease problems, the amount of change is always the part and the whole is always the starting amount. % = part % = amount of change 100 whole 100 original amount Last year, the SOA wrestling team won 8 matches. This year they won 10 matches. What is the percent of increase? % = amnt change n = 2 (10 – 8) 8n = 200 100 orig. amount 100 8 n = 25 The percent of increase is 25% Another Percent of Increase / Decrease Mr. Smith figures it is about time he retires and gets to go fishing whenever he wants. Ahhhhh, what a life! He is looking over his financial information and determines that he makes $95,000 a year now, but when he retires, he will only make $75,000. To the nearest whole percent, what is Mr. Smith’s percent of decrease? % = amount of change 75,000n = 2,000,000 100 starting amount n = 26.6666667% n = 20,000 To the nearest whole percent, Mr. Smith has a 100 75,000 27% decrease in income. Simple Interest Simple interest is really easy and if you don’t know it, by the time you are done with these questions you’ll agree it really is easy. The formula for simple interest is: I = prt I is interest. p is principle (amount of money). r is interest rate as a DECIMAL. t is time in YEARS. How much interest does Johairy get if she puts $5,000 in an account that earns 7% interest for 5 years? I = prt I = (5000)(.07)(5) I = $1,750 Two More Simple Interest Questions Cathy invests $200,000 in a money market fund at 4.5% interest. If she leaves it in the account for 4 years…… A) How much interest does she earn? I = prt I = (200,000)(.045)(4) I = $36,000 B) How much is in the account? The amount in the account is $200,000 + $36,000 = $236,000. Percent Summary Percents are not easy! Some of the hardest percent questions are those that involve mark up, discount, sales, sales tax and tips. The thing that makes them two steps tricky is that they often have __________________. One strategy that often helps is being sure to ___________ underline the important parts of each question. Finally, be sure to write and then substitute into the percent proportion. The percent proportion says % = part (is) 100 whole (of) Absolutely, positively everything you need to know! Measurement on the State Test On the 8th Grade Assessment Test, New York State is not limited to measuring with a ruler. The state wants to see much more. Can you use some pieces of measured information to compute others. These situations include but are not limited to: 1) distances on a map. 2) similar figures. 3) unit price. 4) convert units like inches and feet as well as Celsius and Fahrenheit temperatures. Measurement The scale of a map is 1 inch = 80 miles. On the map, the distance from Patchogue to Callicoon is 2.5 inches. How far is it from Patchogue to Callicoon? First things first – 1 inch = 2.5 inches write a proportion. 80 miles n miles inches = inches n = 80 x 2.5 miles miles n = 200 miles. The scale of a map is 1 inch = 60 miles. On the map, the distance from Patchogue to Monticello is 2.5 inches. How far is it from Patchogue to Callicoon? First things first – 1 inch = 2.5 inches write a proportion. 60 miles n miles inches = inches n = 60 x 2.5 miles miles n = 150 miles. Perfect Practice Makes Perfect The scale of a map is 1 inch = 15 miles. If the map shows that it is 1.8 inches across Brookhaven, how many miles wide is Brookhaven? 1 = 1.8 inches = inches 15 n miles miles n = 27 miles A box of Corn Flakes cereal sells for $3.78. The volume of the box is 18 ounces. What is the unit price for an ounce of Corn Flakes? unit means “one” so the question is s aying what is the price for one ounce. $3.78 ÷ 18 ounces = $0.21 per ounce More Practice John Haag was a locally famous fly tier on Long Island. He used to buy enormous quantities of feathers. He would buy 5 pounds of feathers for $240.00. What was the unit price (per ounce) that John paid for feathers? dollars = dollars ounces ounces 80 n = 240 (16 oz)(5lbs) = 80 n = 3 $240 = n 80 oz 1 oz Costco sells two different size packages of chicken. One package sells 3.5 pounds for $8.19 and the other sells 5 pounds for $10.79. Which one is a better buy and how much do you save per pound? $8.19 ÷ 3.5 = $2.34 $10.79 ÷ 5 = $2.158 = $2.16 $2.34 - $2.16 = $0.34 The 5 lb bag is cheaper by $0.18 per pound. Emily babysits in order to make some money. On Saturday, she gets paid $12 for 2.5 hours of babysitting. On Monday, she gets paid $30. How long did Emily babysit? dollars = dollars 12n = 75 hours hours n = 6.25 $12 = $30 Emily works 6.25 hrs. 2.5 n Perfect Proportion Practice Mr. Lindquist goes fishing for rainbow trout. In his first 2.5 hours of fishing, he catches 10 beautiful rainbow trout. At this pace, how long will it take Mr. Lindquist to catch 18 trout? time = time 10 n = 45 trout trout n = 4.5 2.5 = n It takes Mr. Lindquist 10 18 4.5 hours Perfect Practice Makes Perfect One dozen tomatoes cost $3.15. How much do 30 tomatoes cost? dollars = dollars tomatoes tomatoes $3.15 = n 12 tom. 30 tom. 12 n = $94.50 n = $7.875 n = $7.88 Determine if two figures are congruent or similar. Do you know it? 1)What does congruent mean? same size & shape 2)What does similar mean? same shape sides in proportion Is the pair of triangles below similar or congruent (assume all corresponding angles are equal.) similar Use a proportion to solve for a missing side in similar figures. Do you know it? Find x. 12 9 6 x li’l = li’l 9 x = 72 big big x=8 9 = 6 12 x Perfect Practice Makes Perfect 1) Two triangles are similar. Find the value of x. 20 n n = 10 4 10 2) A photograph is enlarged. The original picture had a length of 5 and a width of 3. If the new picture has a length of 7.5, what is the width? length = length 5 = 7.5 5 n = 22.5 width width 3 n n = 4.5 What You Need To Know There are two main types of questions that fall under the heading of “measurement.” One involves unit price and the other involves proportions. divide When finding unit price, always _________ by the quantity. When using a proportion, first write the proportion in words substitute ______, then ___________ the values. Finally, _______ cross __________ to get the answer ! multiply Absolutely, positively everything you need to know! The Three Angles of a Triangle 180 The three angles of a triangle add up to _____. Find n in each diagram: 57 n-6 90 n n n 90 + 57 = 147 180 – 147 = 33 n + n + n – 6 = 180 3n – 6 = 180 n = 62 Word Problems & Triangles 1) The three angles of a triangle can be represented by x, 2x – 5 and 2x + 20. Find the value of x. x + 2x – 5 + 2x + 20 = 180 5x + 15 = 180 x = 33 2) The three angles of a triangle are n, n and n + 90. Is the triangle acute, right or obtuse. n + n + n + 90 = 180 The triangle is obtuse 3n + 90 = 180 because one of the angles is 30 + 90 = 120 degrees. n = 30 The Pythagorean Theorem The Pythagorean Theorem says a2 + b2 = c2. You can use this formula as long as you always remember that c is the hypotenuse. Another way to do the Pythagorean theorem is leg2 + leg2 = hyp2. Find n in each diagram: n2 + 62 = 102 10 n2 + 36 = 100 n 12 6 - 36 - 36 n 5 n2 = 64 52 + 122 = n2 N = 8 25 + 144 = n2 13 = n Word Problem A ladder is leaning against the side of a building. The ladder is 15 ft long and the base of the ladder is at a point that is 9 feet from the house. How high up the house is the ladder? n2 + 92 = 152 n2 + 81 = 225 n 15 n2 = 144 n = 12 9 Parallel Lines Cut By a Transversal One of the most famous situations in geometry is when 2 parallel lines are cut by a transversal. There are three things you must know: 1) All acute angles are =. 2) All obtuse angles are =. 3) Any acute angle + any obtuse angle = 180. Parallel Lines & Angles Name all of the acute angles in the diagram. <1, <4, <5, <8 1 2 What do you know about these 3 4 angles? They are all equal. 5 6 Name all of the obtuse angles in 7 8 the diagram. <2, <3, <6, <7 What do you know about these angles? They are all equal. If a question pairs an acute angle with an supplementary obtuse angle, they are _________. Parallel Lines & Angles Name all of the angles in the diagram that are congruent to 1 2 <1. Once you select them, click 3 4 and they will appear light blue. 5 6 7 8 Name all of the angles in the diagram that are congruent to <7. Once you select, click and they will <2 and <8 are appear red. supplementary ______________. Parallel Lines & Angles Which angle is the alternate interior angle to <4? <5 1 2 3 4 Which angle is the alternate 5 6 interior angle to <3? <6 7 8 Which angle is the consecutive interior angle to <5? <3 Which angle corresponds to <4? <8 Parallel Lines & Angles Which angle is the alternate interior angle to <5? <4 1 2 3 4 Which angle is the alternate 5 6 interior angle to <6? <3 7 8 Which angle corresponds to <1? <5 Which angle corresponds to <2? <6 Parallel Lines & Angles If m<5 = 70, find the measures of all the other 1 2 angles: 3 4 m<1 = 70 5 6 m<2 = 110 7 8 m<3 = 110 m<4 = 70 m<6 = 110 m<7 = 110 m<8 = 70 Parallel Lines & Angles If m<5 = 75, find the measures of all the other 1 2 angles: 3 4 m<1 = 75 5 6 m<2 = 105 7 8 m<3 = 105 m<4 = 75 m<6 = 105 m<7 = 105 m<8 = 75 Parallel Lines & Angles If m<4 = 3x – 18 and m<5 = x + 24, find the m<4. 1 2 3 4 Equal or Supplementary? Find x. Substitute 5 6 <4 & <5 are both acute, so 7 8 they are equal. 3x – 18 = x + 24 -x -x 2x – 18 = 24 m<4 = 3(21) – 18 + 18 +18 = 63 – 18 = 45 2x = 42 x = 21 Parallel Lines & Angles If m<4 = 2x + 20 and 1 2 m<6 = 3x + 10, find x. 3 4 <4 is acute and <6 is obtuse, so the two are supplementary! 5 6 7 8 2x + 20 + 3x + 10 = 180 (since they are on the same side, like terms go together) 5x + 30 = 180 - 30 -30 5x = 150 x = 30 Identifying Shapes: recall the name & click Number of Sides 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 8 Octagon Angle Types There are four angle types you need to know. Name them and their definitions. Acute Between 0 and 90 Obtuse Between 90 and 180 Right 90 degrees Straight 180 degrees Label Each Angle Type Acute Right Straight Obtuse A Few More Questions What is the largest acute angle that you could double and still get an acute angle? 44. If you tried 45 and doubled it you get 90 – a rt <. What is the smallest acute angle you could triple and get an obtuse angle? Well, we need to find the smallest number that tripled is bigger than 90. Since 30 x 3 = 90, the angle we are looking for is 31………. 31 x 3 = 93. A Few Questions ………. 1) An angle that measures 54 degrees is _________. a) Acute b) Obtuse c) Straight d) Right 2) A 180 degree angle is a __________ angle. a) Acute b) Obtuse c) Straight d) Right 3) Perpendicular lines make what kind of angles? Right Angles!!!!! Problems to Ponder Two congruent acute angles add up to a right angle. How many degrees in each angle? 45 degrees. 45 + 45 = 90 Is it possible for a quadrilateral to have no acute angles? Sure ! A rectangle has 4 right angles. Triangle Types: Choose ALL possible choices from the words below: Scalene Equilateral Acute Right Isosceles Obtuse 1) This triangle has all sides equal. Equilateral 2) This triangle has no sides equal and all angles less than 90 degrees. Scalene. Acute. 3) This triangle has two equal sides. Isosceles 4) This triangle has an angle greater than 90 and less than 180 degrees. Obtuse 5) The angles of this triangle are 90, 45 and 45. Right. Isosceles. How many degrees in the 3 <s of a triangle? _______ 180 How many degrees in the missing angle below? 58 + 52 = 110 180 – 110 = 70 58 52 70 Find the value of n This is isosceles, so it has two congruent angles as well as two 48 congruent sides. What does the base angle on the left equal? n – like the one on the right Therefore …………….. n 2n + 48 = 180. Solve this! 2n + 48 = 180 - 48 -48 2n = 132 n = 66 ! The angles of a triangle measure n, 2n and 3n. 1) Find n. 2) Is the triangle best classified as acute, right or obtuse? 3 angles add up to 180 degrees! n + 2n + 3n = 180 6n = 180 30 n = 30 n = 30 2n = 60 90 60 3n = 90 Find the value of n The vertex angle of an isosceles triangle measures 80. Find the 80 measure of each of the base angles. The three angles sum to 180, so N + N + 80 = 180 n n 2N + 80 = 180 - 80 -80 2N = 100 N = 50 Tricky Triangle Questions Johnny says an equilateral triangle is not isosceles. Is Johnny right or wrong? Johnny is wrong. An isosceles triangle has two sides equal. Two of the sides of the equilateral triangle ARE equal. It does not matter that the third sides is, too. Is it possible for a triangle to have two right angles? No. Numerical explanation: two of the angles would already sum to 180 (90 + 90). There would be no room for a third angle. Geometric explanation: Two “sides” of the “triangle” would never meet. 90 90 Angle & Triangle Questions 1) An acute triangle has three acute angles. Does an obtuse triangle have 3 obtuse angles? Why or why not? No! The three angles of a triangle sum to 180. All obtuse angles are greater than 90, so three of them would sum to 270 or more. 2) True or false and why: An obtuse triangle can never be isosceles? False! One example is a triangle with angles of 100, 40 and 40. Also, see the diagram below. The Quadrilateral Family Papa Parallelogram Opposite Sides Parallel & Equal Opposite Angles Equal Angles Sum to 360 Randy Rectangle Rebecca Rhombus Parallelogram + 4 Rt <s Parallelogram + Congruent Diagonals 4 Equal Sides Squiggy Square The Tricky Trapezoid What makes a trapezoid tricky? 110 110 It is kind of like a parallelogram – but it only 70 70 has one pair of parallel sides! A parallelogram has two pairs. How is a trapezoid like a parallelogram? The interior angles add up to 360. Check out the angles……….. Quadrilateral Questions Always. Sometimes. Never. A rectangle is a parallelogram. Always – the opposite sides of a rectangle are congruent. A rhombus is a square. Sometimes. But a rhombus does not have to have a right angle and a square does. A trapezoid is a parallelogram. Never! A trapezoid only has one pair of parallel sides. Quadrilateral Questions Always. Sometimes. Never. A parallelogram has 4 congruent sides. Sometimes. When it does, it is a special type of rhombus parallelogram called a rhombus. The opposite angles of a square are each 100 degrees. Never. They are each 90 degrees. The sum of the angles of a parallelogram is 360. Absolutely! Think of the 4 rt angles of a square. 90 x 4 = 360 90 Quadrilateral Question Two consecutive sides of a rhombus are 3b – 5 and b + 11. All Sides of a Find b. Find the length of a side. rhombus are equal! Strategies: 3b – 5 = b + 11 3b - 5 1) Diagram -b -b b + 11 2) Equation 2b – 5 = 11 3) Solve +5 +5 2b = 16 Side: 3x8 – 5 = 19 b = 8 Quadrilateral Question Two opposite sides of a parallelogram measure 4(x – 5) and 28. Find the value of x. Opposite sides of a 4(x – 5) parallelogram are equal! 28 4(x – 5) = 28 DISTRIBUTE! 4x – 20 = 28 + 20 +20 4x = 48 x = 12 Quadrilateral Word Problems The measure of an angle of a rectangle can be expressed as 2x – 10. Find x. Strategies: 1) Diagram 2) Equation 2x + 10 3) Solve The angles of a rectangle are right angles…… =90 2x + 10 = 90 - 10 -10 2x = 80 x = 40 Match the Name to the Pic 1) Rectangle only 2) Parallelogram 3) Square 4) Rhombus Key Terms & Key Pictures 1) Define Complementary. Two <s add up to 90 2) Define Supplementary Two <s add up to 180 What is the trick for remembering that complementary means 90 and supplementary means 180? C comes before S, 90 before 180 3) What is the complement of 40? 50 4) What is the supplement of 100? 80 Find the complement and supplement of an angle of 40 degrees. Complement = 50; Supplement = 140. What is the difference between the supplement and complement of 40? 90 Will the difference between the complement and supplement of any given angle always be 90? Why or why not? Yes! Think about it …… Complementary adds up to 90 and supplementary to 180. The difference between those two numbers is 90 . Complementary or Supplementary? Which picture is which and find the value of n. Supplementary 2n + 68 = 180; n = 56 2n 68 Complementary 63 8; 3n + 66 = 90 3n - 3 Pairs of Angles Find the value of n in each diagram. 7n + 8 2n + 43 3n - 17 2n + 12 These angles are called These angles are called ________ and they are vertical supplementary ___________ and they _______. equal sum to ______. 180 7n + 8 = 2n + 43 3n – 17 + 2n + 12 = 180 -2n -2n 5n – 5 = 180 5n + 8 = 43 +5 + 5 - 8 - 8 5n = 185 n = 37 5n = 35 n=7 Counterexamples in Geometry Ellen says that every pair of supplementary angles must have one acute and one obtuse angle. Is Ellen right or wrong? If you think she is right, why? If you think she is wrong, give a counterexample. At first thought, we might think Ellen is right because the sum of two acute angles is less than 180 and the sum of two obtuse angles is greater than 180. So we think that we need one of each (example: 80 + 100). But Ellen is wrong! Two right angles are also supplementary and are neither acute nor obtuse. Perimeter and Area Perimeter measures the outside _____________ of a figure. 10 in 32 in Area measures the inside _____________ of a figure. adds multiplies Which operation? Perimeter ______. Area __________. Find the perimeter and area of the rectangle above. Perimeter Area Add all sides. 84 inches Area = length x width 10 in+32 in+10 in+ 32in 10 in x 32 in = 320 sq. in. Circles: Circumference & Area Circumference is like perimeter – but only applies to a circle. Area doesn’t change – it is all about how much is inside. What is the trick to remember the formulas? Cherry Pie is delicious. Apple pies are, too. C = πd A = πr2 Parts of a Circle For our purposes, there are two main parts of a circle we need to know ………….. diameter and radius. What is the difference? The diameter goes all the diameter way across. The radius only goes halfway. radius If the diameter if a circle is 18 inches, what is the radius? The radius only goes halfway across so it must be half of 18 inches …… 9 inches. Find the Circumference and Area of the circle. C=πd (d is diameter) 10 m C = 3.1416 x 20 m C = 62.832 m A = π r2 (r is radius) A = 3.1416 x 10 m x 10 m A = 314.16 m2 Surface Area of a Rectangular Prism How many faces does this prism have? 6 I’ll name one side. You name its opposite partner. FRONT LEFT TOP BACK RIGHT BOTTOM Finding The Surface Area Step 1: Find the front’s area. TOP 5 in x 8 in = 40 in2 x 2 = 80 in2 Why double the 40? The Back! 5 in FRONT 2) Find the top area and double it. 8 in x 4 in = 32 in2 x 2 = 64 in2 8 inches 4 in 3) Find the area of the right and double it because RIGHT of the left “partner” or equal face. 4 in x 5 in = 20 in2 x 2 = 40 in2 80 in2 + 64 in2 + 40 in2 = 184 in2 Surface Area of a Cylinder What shapes do you get when you unfold a cylinder? Two congruent circles on the top and bottom and a rectangle. If the width of the rectangle on the side is the height, how do you find the length? Radius = 10; Height = 25 It is the circumference of the circles. Computing the Surface Area The reference sheet gives the following formula for the surface area of a cylinder: SA = 2πrh + 2πr2 Substitute and crank out the antza . SA = 2(3.1416)(10)(25) + 2(3.1416)(102) = 1570.8 + 628.32 = 2199.12 Shaded Areas Find the area of the shaded region. 8 in 6 in 5 in Area of green rectangle: 8 in 8 in A = l x w = 8in x 25in=200in2 25 in Area of the triangle: A = ½ b h = ½ (6 in) (8 in) = 24 in2 Area of the parallelogram = b x h = (5 in) (8 in) = 40 in2 Area of the shaded region is 200 in2 – (24 in2 + 40 in2) 136 in2 Find the Area of the Shaded Region Area Square – Area Circle Area of the square = side x side 5 inches Since the radius is 5, side is 10 in 10 in x 10 in = 100 in2 Area of the circle = πr2 A = (3.1416) x (5 in) x (5 in) A = 78.54 in2 100 in2 – 78.54 in2 = 21.46 in 2 Key Points to Remember 180 The three angles of a triangle add up to ______. When two parallel lines are cut by a transversal, = 1) all acute angles are ___. = 2) all obtuse angles are ___. 3) any acute angle + any obtuse angle =180 ___. = Vertical angles are always ___. Complementary means two angles that add up to ____ and 90 180 supplementary means adds up to ____.

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