1. Regents Review #5

2. Set Notation Universal Set = {10,11,12,13,14,15,16,17,18,19,20,21} A = {14, 21} B = {12,15,18,21} A and B are subsets of the Universal Set A B A intersection B “In Common” {21} A B A union B “Combination of both Sets” {12,14,15,18,21} B’ or B c The Complement of B “Not B” {10,11,13,14,16,17,19,20}

3. Interval Notation [ means to include ( means do not include For Example: [12,16) means…. All Real Numbers 12 through 16, including 12 but not 16 Ex: Which interval notation represents the following inequality? -4 < x 6 (1)[-4,6](2) (-4,6)(3) [-4,6)(4) (-4,6] (4) (-4,6]

4. Properties 1) Commutative (addition) 2) Commutative (multiplication) 3) Associative (addition) 4) Associative (multiplication) 5) Distributive A + B = B + A (A)(B) = (B)(A) (A + B) + C = A + (B + C) (A x B) x C = A x (B x C) A(B + C) = AB + AC or A(B – C) = AB – AC

5. Properties 6)Identity (addition) 7) Identity (multiplication) 8)Inverse (addition) (additive inverse) 9)Inverse (multiplication) (multiplicative inverse) A + 0 = A (A)(1) = A A + (-A) = 0 A x (1/A) = 1

6. Properties 10) Closure Given a set of numbers, if an operation is performed and the result is part of that set, then the set of numbers is closed under that operation. Example: Is the set of whole numbers closed under addition? Yes, any whole number added to another whole number will always result in a whole number Example: Is the set of whole numbers closed under subtraction? No, if the difference is taken between 3 and 4, the result is an integer 3 – 4 = -1, therefore the set is not closed under subtraction

7. Properties Name The Property 1) 2) (q)(y + z) = (y + z)(q) 3) a x 1 = a 4) (9 + 8) + 11 = 9 + (8 + 11) 5) r + 0 = r 6) m(p + q) = mp + mq 7) c + (-c) = 0 8) For which operations are the set of integers closed? Inverse Property of Multiplication Commutative Property of Multiplication Identity Property of Multiplication Associative Property of Addition Identity Property of Addition Distributive Property Inverse Property of Addition Add, Sub, Mult

8. Ratios, Rates, Proportions A paper copy machine takes 4 minutes and 20 seconds to duplicate 520 pages. Find the rate at which the machine duplicates in pages per second. 4 minutes = 4(60) = 240 seconds 240 seconds + 20 seconds = 260 seconds 2 pages per second 2 pgs/sec

9. Ratios, Rates, Proportions Two inches of rain fell in an hour and a half. At this rate, how many inches of rain, to the nearest hundredth, will fall in 10 minutes? Set up a proportion to find the number of inches it will rain in 10 minutes. 0.22 inches of rain will fall in ten minutes

10. Ratios, Rates, Proportions Mike takes 3 minutes to read an article of 312 words. How many minutes will it take him to read an article of 884 words at the same rate? Set up a Proportion 312x = 2652 x = 8.5 8 ½ minutes 8 minutes and 30 seconds

11. Ratios, Rates, Proportions The measures of two supplementary angles are in the ratio of 4:11. What is the measure of the larger angle? Let 4x = the measure of the smaller angle Let 11x = the measure of the larger angle 4x + 11x = 180 15x = 180 x = 12 The smaller angle = 48 degrees The larger angle = 132 degrees The larger angle measures 132 ○

12. Ratios, Rates, Proportions Converting Units of Measurement You can convert units of measurement by setting up a proportion EX: Convert 20 feet to yards 1 yard = 3 feet 3 x = 20 x = 6.66666…. 6 2/3 yards

13. Ratios, Rates, Proportions Convert 25 yards to inches 1 yard = 3 feet 1 foot = 12 inches 75 = x 75 feet 900 = x 900 inches

14. Proportions Anita’s wages vary directly as the number of hours she works. If she can earn $29.80 for working 4 hours, how much will she earn when she works 30 hours? This problem can be solved using a proportion 4x = 894 x = 223.5 Anita will earn $223.50

15. Percents Ronald bought a DVD that cost $18.99. His total payment including sales tax was $20.51. Find the sales tax rate to the whole percent? Tax = % (Tax Rate) x Original Price Tax = 20.51 – 18.99 Tax = $1.52 1.52 = N% (18.99) N% = 1.52/18.99 N% = 0.080042…. 8% tax rate

16. Percents The accompanying circle graph shows expenditures for state and local governments. 1.If the total spending is $50,000, how much money was spent on highways? 2.Approximately how many times the amount of spending on highways is spent on education? 3.Approximately what fraction of the total expenditures are spent on highways and public welfare combined? 1. N =.07(50,000) N = $3,500 2.N =.34(50,000) N = $17,000 17,000/3500 = 4.857…. About 5 times as much 3.7% + 14% = 21% 21/100 is about 20/100 About is spent on highways and pw

17. Trigonometry Make sure calculator is in degrees! Go to Mode and highlight degree (ENTER) Finding a Side Use a Proportion Sine, Cosine or Tangent Finding an Angle Use the inverse Trig Functions (2 nd Key on Calc) Sin -1,Cos -1 or Tan -1 20 units A ladder leans against a building. The foot of the ladder is 6 feet from the building. The ladder reaches a height of 14 feet on the building. To the nearest foot, find the measure of the angle that the ladder makes with the ground. 67 ◦

18. Statistics Quantitative Data is data that is numerical, meaning it can be counted or measured Ex: height of a flagpole, weight of a backpack Qualitative Data is data that is not numerical, meaning it can be observed Ex: type of toppings on a pizza, favorite ice cream flavor Univariate refers to single variable data Ex: the number of pets each student owns Bivariate refers to two variable data Ex: A person’s shoe size compared to their height

19. Statistics Correlation vs. Causation A correlation is a relationship between two variables Ex: Not every correlation shows causation (one variable causes the other) Which graph above shows causation? Money Spent on potted plants Number of potted plants purchased Number of swimsuits sold Number of winter coats sold The first graph because the more money you spend on plotted plants, the more plants you will have The second graph shows a correlation but the number of swimsuits sold does not cause the number of winter coats sold. The temperature outside and seasons cause the sales of the items.

20. Statistics A population is a group that you want information about. A sample is part of a population that is used to make estimates about the population In a random sample, each member has an equal chance of being selected, and the sample is representative of the entire population A biased sample favors one or more parts of the population over others Ex: You want to conduct a survey to find out what type of music people listen to Determine which of these scenarios is biased A.You ask every fifth person leaving a Taylor Swift concert about the type of music they listen to B.You ask every fifth person leaving the local mall about the type of music they listen to

21. THE REGENTS IS WEDNESDAY, JUNE 12 th STUDY! STUDY! STUDY! END OF REGENTS REVIEWS Now it’s your turn to review on your own! Using the information presented today and your review packet, complete the practice problems in the packet.