MATH 2008 STUDY GUIDE 1 – Key 1.2: (2-5) Provide a counterexample for each generalization. 1) If a number is even, then it ends in a 2. 24 is even and ends in a 4 2) If a number is prime, then it is odd. 2 is an even prime number 3) If an even number is divided by an even number, its quotient is even. 12 and 4 are even, but 12/4 = 3 is odd (18-20) Find the negation of each statement. Determine whether the statements and the negations are true or false. 4) Every rectangle is a square. - false Not every rectangle is a square. – true 5) Every square is a rectangle – true Not every square is a rectangle. - false 6) A car driving 90 mph on I-75 in Georgia would be speeding. – true A car driving 90 mph on I-75 in Georgia would not be speeding. – false (21-24) Identify the hypothesis and conclusion of each statement. 7) If an even number is multiplied by an odd number, then its product is an even number. Hypothesis: an even number is multiply by an odd number Conclusion: its product is an even number 8) If there is a high demand for a product, then its price will increase. Hypothesis: there is a high demand for a product Conclusion: its price will increase (26-28) Determine the truth of statements. 9) Jack says that if the temperature drops below 50 degrees then he will not play tennis. Describe a situation where Jack would be lying. Jack is lying if the temperature drop below 50 degree and he does play tennis. 10) Describe the situation where the following statement is false: If Serena wins the second set of her tennis match, she will win the match. The statement is false if Serena wins the second set of her tennis match but loses the match. 1.3: (5-14) Use problem solving strategies 1) Tim said that he lost 5 coins, none of which was a half dollar, totaling 75 cents. What coins might Tim have lost? He lost 2 quarters, 2 dimes, and a nickel How many solutions does this problem have? One, more than one, or none. One. Three quarters would be too many (only 3 coins), and 1 quarter too few (even adding 4 dimes would give only 65¢). Therefore, there had to have been 2 quarters, and the only way to get the other 25¢ in 3 coins is 2 dimes and 1 nickel. 2) What problem solving strategy would you use to solve the following: a) Chris is 4 years older than her brother Jack. If the sum of their ages is 30, how old is each person? If you add 4 years to Jack’s age, they would be the same age. If you add 4 to the total of their ages, it would be twice Chris’s age: 34/2 = 17. Therefore, Chris is 17 and Jack is four years younger, or 13. Check: Chris is 4 years older than Jack, and 17 + 13 = 30. b) On the first day, Joe started a club and he was the first member. Each day after that, one more member joined the club than on the previous day. How many people were in the club on the 40th day? Write a table of values and find a pattern: Day # joined Total in club 1 1 1 2 2 3 3 3 6 4 4 10 5 5 15 … … … 40 40 The number of club members on the nth day is the sum 1 + 2 + 3 + … + n = n(n + 1)2. Based on this pattern, on day n = 40 there are 40(41)/2 = 820 members. 3) Solve the following problem: If 5 people are at a party, and each person shakes every other person’s hand exactly once, then how many handshakes are there? Consider the people to be A, B, C, D, and E. Draw a diagram representing the handshakes and count the lines. Keep in mind that a person does not shake hands with himself, and each pair only shakes hands once: A A B B C C D D E E There were 10 handshakes in all. 2.2: (6-8, 17-21) Create a word problem: 1) involving joining sets to go with the equation 2,000 + 356 = 2,356 If Bobby has a collection of 2000 marbles and Joe has a collection of 356 marbles, how many marbles would they have if they put their marbles together? 2) comparing two sets to go with the equation 12 – 5 = 7 If Landon has 12 toys and Becky has 5 toys, how many more toys does Landon have than Becky? 3) finding the missing addend to go with the equation 12 – 5 = 7 If Sarah has 12 grapes and eats 5 of them, how many does she have left? (9, 12) Identify the property of addition used. 4) To figure out 8 + 2 I don’t have to start at 2 and count up 8 more. I can start at 8 and count up 2 more. Commutative property: 2 + 8 = 8 + 2 5) I can never remember 8 + 7, but I know 7 + 7 = 14 and I just add 1 more to get 15. Associative property: (1 + 7) + 7 = 1 + (7 + 7) 6) (a + b) + c = (b + a) + c Commutative property: a + b = b + a 7) 0 + a = a + 0 Commutative property Be able to work problem 13 pg 86: Identify two addition and two subtraction equations modeled by each illustration. a) Addition: 3 + 1 = 4, 1 + 3 = 4; Subtraction: 4 – 1 = 3, 4 – 3 = 1 b) Addition: 15 + 10 = 25, 10 + 15 = 25; Subtraction: 25 – 10 = 15, 25 – 15 = 10 (15, 16) Rewrite the subtraction equations as addition equations: 8) 18 – 8 = n n + 8 = 18 9) y – 83 = 129 129 + 83 = y 10) r – s = t t+s=r 2.3: (4-6, 10) Create a word problem: 1) involving joining equivalent sets to go with 4 x 30 = 120 If each of the 30 students in the class has 4 pencils, how many pencils does the class have? 2) involving joining segments of equal length to go with 4 x 30 = 120 If 4 30-foot boards are laid end-to-end, how far would they stretch? 3) involving the area of a rectangle to go with 25 x 4 = 100 What is the area of a rectangle of length 25 inches and width 4 inches? (7-9) Write a multiplication equation for each question and find the solution 4) A furniture maker has four chair styles and six different upholstery fabrics for each chair. How many different chairs can be made? 4*6 = 24 (#11) Which properties of multiplication are used: 5) To remember 8 x 4 I just need to remember 4 x 8 Commutative property 6) I can never remember 8 x 6, but I just remember 8 x 5 and add one more 8 to get 48. Distributive property: 8*(5 + 1) = 8*5 + 8*1 (#12) 7) Use the distributive property to find each product: a) (20 + 5) x 3 20*3 + 5*3 = 60 + 15 = 75 b) 4 x (5 + 6) Note that the operation in the parentheses should be ADDITION. 4*5 + 4*6 = 20 + 24 = 44 c) (t + 10) x (3t + 2) (t + 10)*3t + (t + 10)*2 = 3t2 + 30t + 2t + 20 = 3t2 + 32t + 20 (13-17) Create a story problem to go with 24 / 6 = 4 for each situation: 8) separating a set into same size equivalent sets. If 24 eggs are placed into half-dozen cartons, how many cartons will it take? 9) Finding the missing factor. If Brad has $24 and tickets cost $6, how many tickets can he buy? 10) separating a set into a specific number of equivalent sets. If a set of 24 crayons is split between 6 children, how many crayons does each child get? (#19) Write each division equation as a multiplication equation 11) 18 / 6 = n 6*n = 18 12) 0 / b = c b*c = 0 13) y / 42 = 126 42*126 = y (#20) Write each multiplication equation as a division equation 14) 15 x 3 = n n/3 = 15 15) r x s = t t/s = r 16) 8 x 0 = 0 0/8 = 0 (#21) Use the division algorithm to find the quotient and remainder 17) 26 / 3 8R2 18) 292 / 21 13 R 19 19) 4 / 7 0R4 5.1: (#1) Give the integer that represents each situation and then give the opposite situation and the integer that represents it. 1) Receive a check for $10 10 Write a check for $10. -10 2) Temperature 8 degrees below 0 -8 Temperature 8 degrees above 0. 8 (2-5) Fill in the blanks The opposite of a positive integer is a negative integer. The opposite of a negative integer is a positive integer. The integer 0 is neither positive nor negative. The absolute value of a negative integer is always a positive integer. (18-25) Evaluate the following: 3) – 23 – n for n = – 9 -23 – (-9) = -23 + 9 = -14 4) n – (– n) for n = – 9 -9 – (-(-9)) = -9 – 9 = -18 5) – |– 8| = - | -8 | = -8 6) – | – 14| – (– 12) = - | -14 | - (-12) = -14 + 12 = -2 7) | –3 – |7 – 7| | = | -3 - | 7 – 7 | | = | -3 – 0 | = | -3 | = 3 (26-28, 31-34) Word problems with integers 8) The price of Allgain stock fell $5 per share on Monday and rose $9 per share on Tuesday. Write and solve an equation that shows the total change. -5 + 9 = x; x = 4 9) After finding that he had been billed incorrectly for $459, a store owner showed a profit for the week of $9,236. Solve the equation a – (– 459) = 9,236 to find the profit before the bill was dropped. a + 459 = 9236; a = 8777 10) A football team lost 9 yards because of a quarterback sack and then gained 12 yards on a pass play. Write an equation to find the result of the two plays and solve it. Use a negative integer in your equation. -9 + 12 = x; x = 3 (#29, 31) 11) Order from smallest to largest: 23, -18, 18, -15, 9, -2, -12, 17 -18, -15, -12, -2, 9, 17, 18, 23 12) Use a number line to show why – 5 > – 9. 5.2: (6-17, 24-31) Calculate the following: 1) 25 (– 15) = -375 2) |– 8| (– 13) = 8(-13) = -104 3) – | – 56| (14 + (– 9)) = -56(5) = -280 4) (– 5)(– 4)(–3) = 60 5) (– 18 / 6) + (– 14) = -3 – 14 = -17 6) – 6 (– 54 / – 9) = -6(6) = -36 7) [53 + (– 5)] / [12 – (– 4)] = (53 – 5) / (12 + 4) = 48 / 16 = 3 (#18, 19) 8) The product of an odd number of negative integers is a negative integer. 9) The product of an even number of negative integers is a positive integer. (32-36) Word Problems 10) Over a 4 month period, the total steady change of a corporation’s sales income was a – $12 million. Determine the average change of sales income per month. -$12 million / 4 = -$3 million 11) The temperature outside of an airplane drops 7 degrees Celsius for each kilometer increase in altitude. Determine the change in outside temperature if the airplane increases its altitude by 5 kilometers. -7*5 = -35 degrees