Geometry - Summer 2016

Transcription

1 Geometry - Summer 2016 Introduction PLEASE READ! The purpose of providing summer work is to keep your skills fresh and strengthen your base knowledge so we can build on that foundation in Geometry. All of the topics that are covered in this packet have been learned in previous courses either in your basic math classes or in Algebra 1. There is background information in some sections, and all sections have links to websites which can provide more in-depth explanations of concepts that you may have forgotten. You are expected to complete all of the problems in each section, either on the pages provided or on additional paper. Please make sure that each problem is clearly identifiable and that your work is well-organized and neat. It is important that you take this work seriously as it is expected that you will be comfortable with all of these topics and be able to do additional problems like the problems in this packet when we begin school in September. You are expected to utilize all available resources, including the links provided here, old books, notes, additional websites, and/or assistance from your parents. But please remember, you need to make sure that you know how to do the problems included in the packet. If after using the resources outlined here, you still have questions, feel free to me for assistance over the summer (dmurphy@chas .org) or bring your questions to class with you when we return. All of the material included in this packet will be assessed early in the year, so please make sure you clearly understand all of the problems in this packet or ask for help. Be sure to read all of the directions and information in each individual section to ensure that you get the most out of the packet. General Links As mentioned above, review materials are provided in some sections, and every section has links to websites for specific topics that you might want to review. Simply click on the link within this document and it will take you to the associated web page. In addition, here are some links to the Pearson tools for an Algebra 1 book that you might find helpful. While it is not the exact book that we use at Chase, the material is the same and the site does not require a log in. Use this page to review topics from the book: Use this page to view homework tutorials on specific topics that you might want to review: suffix=0775 The more you review over the summer, the better prepared you will be for success in the fall!

2 Name Section 1: Basic Skills I. Computation with Integers Calculate each of the following without the use of a calculator. Remember that the Rules of Order ( Order of Operations ) always apply: 1. Perform any operation(s) inside grouping symbols (following Order of Operations if necessary). 2. Simplify any terms with exponents. 3. Multiply and divide in order from left to right. PEMDAS 4. Add and subtract in order from left to right. For more complex expressions show your work so we can see where you went wrong if you make a mistake = = = = = = = = = = = = Online Resources for This Section

3 II. Operations with Fractions Remember a few rules: 1. You need a common denominator when adding and subtracting fractions, but not when multiplying or dividing. 2. Dividing is the same as multiplying by the reciprocal of the second fraction. 3. When multiplying, always REDUCE FIRST - it makes things much easier than multiplying and then reducing bigger numbers. For example, it is easier to evaluate by first reducing = than to multiply first and then reduce the resulting fraction. 4. Remember that you may have to change mixed numbers to improper fractions. 5. You may leave answers as improper fractions (this is actually my preference). Perform each indicated operation. Show all of your work

4 Online Resources for This Section: tions/v/adding-fractions-with-like-denominators c/v/multiplying-fractions multiplying-fractions-and-mixed-numbers

5 III. Ratios and Proportions Reminders about ratios, unit rates, and proportions: Write each ratio or rate in simplest form: to : g.in 4 oz mi.in 4.3 h. 5.

6 Solve each proportion. Remember the rules of multiplying fractions and reduce first if possible it makes life easier. 6. * = 7. = + 8. =, 9., = 10.. = = 12. A canary s heart beats 130 times in 12 seconds. How many times does its heart beat in 50 seconds? Online Resources for This Section:

7 IV. Percents and Percent Applications Write each decimal as a percent: = % = % = % Write each percentage as a decimal: 4. 8% = % = % = Solve each percent problem. 7. What is 25% of 50? 8. What percent of 58 is 37? % of what number is 90? is what percent of 40? is 75% of what number? % of 58 is what? Online Resources for This Section:

8 Section 2: Algebra Skills I. Expressions Write an algebraic expression for each phrase. 1. a number t times more than the product of 8 and n 3. a number z plus more than the product of 9 and a 5. 5 less than the quotient of a number p and 4 Simplify each expression ( ) ( ) ( 6) ( 2 3 ) Evaluate each expression for the given values of the variables m + 6n 2 n 3 ; m = 2 and n = (3x) 2 (x 3 y 2 ); x = 3; y = b + 4b 2 a 3 ; a = 3 and b = (3f) 2 (f g) 3 ; f = 2; g = Order the numbers from least to greatest. 9 10, 11, Order the numbers from least to greatest. 6 5,, Estimate 83 to the nearest integer.

9 19. Estimate 150 to the nearest integer. 20. Which property is illustrated by 5 + n = n + 5? 21. Which property is illustrated by (x + y) + z = x + (y + z)? Online resources for this section:

10 II. Simplifying Expressions 1. 6n 2 5n ( x) 4. 4(p ( 5)) 5. 2ab 2 7ab (2t (4 7)) 2 7. ( 6 12x) 9. Write an equation for the sentence: the difference of 12w and 9 is Write an equation for the sentence: the sum of 8y and 7 is Suppose you used the Distributive Property to get the expression 12x + 3y 9. With what expression could you have started? Online resources for this section:

11 III. Solving Equations Solve each equation. Check your answer. 1. 9g + 12 = ( 2) 1 3 z+ = 3. n = x 12 = 4x ( 8) 16 5 x = x 6. x 4 5 = m = = 1 5 y 9. p 6 = p x + 4 = 5(x + 1) 3(x + 2) 11. Jackie earns $172 per week at her part time job. She is saving this money to buy a used car that costs $2000. At this rate, how many weeks will it take her to earn enough money to buy the car? Define a variable and write an equation for each situation. Then solve. 12. The length of a rectangle is twice the width. An equation that models the perimeter of the rectangle is 2w + 4w = 36 where w is the width of the rectangle in ft. What are the length and the width of the rectangle? 13. Two consecutive odd integers can be modeled by n and n + 2, where n is an odd integer. The sum of two consecutive odd integers is 80. What are the integers? Online Resources for this Section:

12 IV. Inequalities Is each number a solution of the given inequality? 1. 4y a. 3 b. 1 c x + 2 > 5 a. 3 b. 4 c Write an inequality to model each situation. 3. The high temperature will be at least 75 F today. 4. The class can contain at most 28 students. Write an inequality for each graph Solve each inequality. 7. r q y < 3n < < 5c + 7 < b > 42 or 4b > x > y f 5f x < x 8 < 2x d < (x 4) < (5y + 13) 5.7 < x + 2(3x 11) (x 11) 4x < 72

13 Write a compound inequality that each graph could represent Solve each equation. Check your solutions x + 13 = = w x = y = Which of the following inequalities could be represented by the graph? I. n 1 < 2 II. 4x < 12 or x < 2 III. 1 < 3h + 4 < 13 A. I only B. I and II C. II and III D. I and III Online Resources for This Section: alities-on-a-number-line alities_on_a_number_line alities-using-addition-and-subtraction alities-using-multiplication-and-division g-inequalities

14 V. Functions Sketch a graph to represent the situation. Label each section. 1. The temperature of the water decreases over the first few hours in the refrigerator. 2. The sales of the company have increased steadily over the years. 3. The temperature changed as Shelly preheated the oven, cooked the bread, and turned the oven off. For each table, determine whether the relationship is a function. Then represent the relationship using words, an equation, and a graph Each set of ordered pairs represents a function. Write a rule that represents the function. 6. (0, 0), (1, 3), (2, 6), (3, 9), (4, 12) 7. (0, 8), (1, 7), (2, 6), (3, 5), (4, 4) 8. (0, 7), (1, 2), (2, 3), (3, 8), (4, 13) 9. (0, 8), (1, 6), (2, 4), (3, 2), (4, 0) 10. 2, 1 16, 1, 1 4, (0, 1), (1, 4), (2, 16) 11. 2, 10, 1, 4, (0, 2), (1, 4), (2, 10) 9 3

15 Write a function rule that represents each sentence more than two-thirds of a is b less than the product of a number y and 2 is z times the sum of a number t and 5 is s. Identify the domain and range of each relation. Use a mapping diagram to determine whether the relation is a function. 15. {( 3, 6), (0, 2), (1, 0), (2, 3)} 16. {( 1, 4), (0, 0), (1, 4), (2, 8)} Find the range of each function for the given domain. 17. f (x) = 2x + 1; { 2, 0, 2, 4, 6} 18. f (x) = x 3 + 1; { 2, 1, 0, 1, 2} 19. f (x) = 12x 10; { 3, 1, 0, 1, 3} 20. f (x) = x 2 7; { 2, 1, 0, 3, 4} Online Resources for This Section:

16 VI. Linear Equations Find the slope of the line that passes through each pair of points. 1. (-3, -4), (2, 1) 2. (0, 0), (8, 9) 3. ( 3, 1), ( 1, 5) 4. 3,5 4, 5,2 4 In the space below, draw a coordinate plane and graph each equation 1 5. x + 2y = 6 6. y = x 3 7. y 2 = 2(x 3) 2 Write each equation in slope-intercept form. 8. 6x + 9y = x = 3y 12

17 Find the x- and y-intercepts of the graph of each equation x + 12y = x + 3y = 24 Write an equation in point-slope form for the line that has the given slope m and that passes through the given point m = ;(0, 2) 13. m = 2; (0, 1) 4 Write an equation in slope-intercept form for the line that passes through the given points. 14. (2, 3), (1, 5) 15. (5, 2), ( 16, 4) Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the given line ( 3,5); y = x ( 7, 3); x = 4 2 Write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the given line. 19. (5, 1); y = 4x (4, 2); y = 3

18 Online Resources for This Section: Graphing Linear Equations Using Slope-Intercept Form Graphing Linear Equations Using Point-Slope Form Graphing Linear Equations Using Standard Form Converting Between Forms

19 VII. Systems of Equations Solve each system using substitution. 1. 3x 5y = 1 2. x + 2y = x + 3y = 9 x y = 1 2x 3y = 12 3x + 4y = 5 y x y 4. 7x = 2y x + = = x 4y = 3x y = 5 2x y = 4 3 Solve each system using elimination. 7. x + y = x + 3y = 9 9. x + y = 7 x y = 6 2x 2y = 4 3x 2y = x 8y = x + 0.3y = x 7y + 10 = 0 8x 7y = 7 0.7x 0.2y = 0.8 y 2x 3 = 0 Online Resources for This Section:

20 VIII. System Word Problems Write a system of equations to model each situation. Solve by any method. Ten years from now, A will be twice as old as B. Five years ago, A was three times as old as B. What are the present ages of A and B? The ratio of incomes of two persons is 9:7. The difference in their weekly incomes is $200. What are their weekly incomes? A change purse contains a total of 100 nickels and dimes. The total value of the coins is $7. How many coins of each type does the purse contain? Online Resources for This Section:

21 IX. Exponents Simplify each expression. Use positive exponents. 1. a 4 b 7 c 0 2. ( )( ) 3. p q q r (m 3 n 5 m 1 ) 3 5. x 4 y u 5 v 4 ( u 3 v 2 ) 3 x 3 y If z =, which expression has the greatest value? 2 A. z 6 z 4 B. (z 2 z 5 ) 2 C. (z 3 ) 5 D. (z 2 z 4 ) 3 Determine whether each number is in scientific notation. If it is not, write it in scientific notation Solve each problem using scientific notation. 12. At the end of 1993, there were 109 nuclear power plants operating in the United States. These plants generated a total of 6,520,000,000,000,000 Btu (British thermal unit) of electric power in How much energy was generated per plant? 13. A red blood cell is m in diameter. There are about 20,000,000,000,000 red blood cells in a 125-lb person. If all of the red blood cells were lined up end to end, how long would the line be? Online Resources for This Section

22 X. Polynomials Find the degree of each monomial. 1. 6xy 2. 3b 2 c m 7 n Simplify each sum or difference. 4. 6r 3 + 7r u 2 v 19u 2 v 6. (5g 2g) + (2g 2 + 6g) 6. The perimeter of a pentagon is 20t + 7. Four sides have the following lengths: 6t, 2t, 4t 5, and 5t + 1. What is the length of the fifth side? Simplify each product. 8. 3x(x + 6) 9. z 2 (z 9) 10. 2x(4x 2 7x + 6) Factor each polynomial x n 3 40n n b 2 c bc 5 Simplify each product. 15. (x 2)(3x 4) 16. (3x + 2)(x + 7) 17. (4x 1)(2x + 5) Simplify each product. 18. (x + 6) (2s + 7) (3x 8) 2 Complete. 21. k 2 + 9k + 18 = (k + 3)(k + ) 22. x 2 11x + 28 = (x 4)(x ) Simplify each product. 23. (v + 7)(v 7) 24. (5s t) (3p q)(3p 2 10q)

23 Find an expression for the area of each shaded region The area of a rectangular coffee table is given by the trinomial t 2 + 7t 8. The table s length is t + 8. What is the table s width? Online Resources for This Section: y-a-polynomial ion-and-subtraction-of-polynomials g-and-subtracting-polynomials-1 g-and-subtracting-polynomials-3 /multiplying-binomials /square-a-binomial /special-products-of-binomials /multiplying-polynomials

24 XI. Facoring Polynomials Factor each expression. 1. r r g 2 8g m 2 + 2m d 2 13d y y w 2 75w n 3 24n 2 + n p 4 + 6p 3 8p 2 4p 9. 8h h + 16 Online Resources for This Section: oring_polynomials_1 oring_polynomials_by_grouping_1 l2-t1-we1-factoring-special-products-1

25 XII. The Quadratic Formula Use the quadratic formula to solve each equation. 1. 7c 2 + 8c + 1 = w 2 28w = j 2 3j = x 2 6x + 4 = n 2 6n = d 2 + 2d + 9 = a 2 + 4a 6 = p p = d 2 8d + 3 = 0 Which method (factoring or the Quadratic Formula) would you choose to solve each equation? Justify your reasoning. 10. h 2 + 4h + 7 = a 2 4a 12 = y 2 11y 14 = p 2 7p 4 = x = f 2 2f 35 = 0 Online Resources for This Section: oring-special-products mple%201:%20solving%20a%20quadratic%20equation%20by%20factoring mple%202:%20solving%20a%20quadratic%20equation%20by%20factoring ng_quadratics_by_factoring ing-a-quadratic-by-factoring ormula_tutorial/v/using-the-quadratic-formula