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ALGEBRA 1 SUMMER PACKET SHERWOOD HIGH SCHOOL SUMMER 2013 General Information The purpose of this packet is to review important skills that you have learned which are necessary to succeed in Algebra 1: Calculating Decimals Calculating Fractions Evaluating Expressions Plotting points in the Coordinate Plane Graphing Equations Order of Operations Operations with Signed Numbers (Integers) Solving Equations Solving Word Problems Rounding Numbers Getting Help: When completing the problems in this packet, you should show your work and do your best to complete the problems correctly. During the summer, the answer key will be posted on the Sherwood website. Enjoy your summer! We look forward to seeing you in the fall. (If you have any questions, please contact the math office at (301) 924 – 3253.) Arithmetic with Decimals Adding Decimals Example 1: Add 4.36 and 2.89 1 1 1 Step1 4.36 +2.89 5 Step 2 4.36 +2.89 25 Line up the decimal points. Add the hundredths. Example 2: Add: 1 1 Add the tenths. Step 3 4.36 +2.89 7.25 Add the ones. Place a decimal point in the sum in line with those decimals above it. 4.3 + 12.75 + 0.093 11 Step 1 4.3 12.75 + 0.093 Step 2 4.300 12.750 + 0.093 Line up the decimal points. Place zeros in the missing digits for place holders. Step 3 4.300 12.750 + 0.093 17.143 Add the the thousandths. Add the hundredths and ones. Place a decimal point in the sum in line with those decimals above it. Subtracting Decimals Example 1: 4.72 – 2.65 4.72 - 2.65 2.07 Line up the decimal points. Subtract hundredths, tenths, and ones. Place a decimal point in the answer in line with those decimals above it. Example 2: 7.6 – 4.362 7.600 - 4.362 3.238 Line up decimal points. Place zeros in the missing digits for place holders. Subtract. Multiplying Decimals Example 1: 0.90 x 1.2 0.90 ← 2 decimal places x 1.2 ← 1 decimal place 180 0900 1.080 ← 3 decimal places Example 2: .32 x .004 128 0000 .00128 To multiply decimals, multiply as with whole numbers. The number of decimal places in the product is the total number of decimal places in the factors. .32 x .004 ← 2 decimal places ← 3 decimal places When you multiply decimals, sometimes you need to write one or more zeros in the product to equal the total number of decimal places in the factors. ← 5 decimal places Dividing decimals Example 1: 2.48 ÷ 2 1.24 2 2.48 Rewrite 2.48 ÷ 2 as shown. Place the -2 04 -4 08 -8 0 decimal point in the quotient directly above the decimal point in the dividend. Divide as with whole numbers. Add or subtract the following decimals. 1. 4.2 + 3.7 = 2. 0.09 + 3.6 = 3. 0.72 + 3.921 + 7.5 = 4. 7.41 – 5.63 = 5. 9.4 – 7.25 = 6. 17.365 – 12.19 = Multiply or divide the following decimals. 7. 1.75 x 2.6 = 8. 0.53 x 0.008 = 10. 45.6 ÷ 8 = 11. 1.3174 ÷ 2 = 9. 2.31 x 0.002 = 12. 2826.6 ÷ 42 = Arithmetic with Fractions Adding and Subtracting Fractions Example 1: 3 1 + 8 8 If the denominators are the same, then add the numerators and keep the same denominator. Then simplify the answer & reduce, if possible. 3 1 4 1 + = = 8 8 8 2 Example 2: 5 3 – 4 6 The denominators are not the same. You must rename the fractions to have the same denominator by finding the least common denominator. Step 1 Step 2 5 10 = 6 12 5 = 6 12 – 3 = 4 12 – 3 9 = 4 12 Step 3 5 10 = 6 12 – 3 9 = 4 12 = Find the least common denominator. Rename the fractions. 1 12 Subtract the numerators. Multiplying Fractions Example 3: 2 4 x 3 5 Step 1 Multiply the numerators. 2 4 8 x = 3 5 Step 2 Multiply the denominators. 2 4 x = 3 5 15 Example 2: 5 x Step 1 3 4 Rename the whole number 5 3 x 1 4 = 15 3 = 3 4 4 using a fraction. Then follow the example above. Finally simplify answer by writing it as a mixed numeral. Dividing Fractions To divide fractions, multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor). Example 1: 2/3 ÷ 4/5 2 5 10 5 x = = 3 4 12 6 Example 2: 2/3 ÷ 4 2 1 2 1 x = = 3 4 12 6 Perform the indicated operation with the following fractions. Don’t forget to simplify (reduce) your final answer to lowest terms! 13. 5 1 – = 6 6 14. 7 5 + = 8 8 15. 3 1 + = 8 4 16. 1 1 – = 2 3 17. 4 3 – = 5 4 18. 3 16 x = 8 3 19. 2 7 x = 3 8 20. 6x 4 = 3 21. 4 x 30 = 5 22. 7 5 ÷ = 3 6 23. 9 3 ÷ = 2 8 24. 3 ÷2= 4 Evaluating Expressions Example Evaluate the following expression when x = 5 Rewrite the expression substituting 5 for the x and simplify. a. b. c. d. e. 5x = -2x = x + 25 = 5x - 15 = 3x + 4 = 5(5)= 25 -2(5) = -10 5 + 25 = 30 5(5) – 15 = 25 – 15 = 10 3(5) + 4 = 19 Evaluate each expression given that: x=5 y = -4 z=6 1. 3x 5. y+4 2. 2x2 6. 5z – 6 3. 3x2 + y 7. xy + z 4. 2 (x + z) – y 8. 2x + 3y – z 9. 5x – (y + 2z) 13. 5z + (y – x) 10. xy 2 14. 2x2 + 3 11. x2 + y2 + z2 15. 4x + 2y – z 12. 2x( y + z) 16. yz 2 Graphing Points in a plane are named using 2 numbers, called a coordinate pair. The first number is called the x-coordinate. The x-coordinate is positive if the point is to the right of the origin and negative if the point is to the left of the origin. The second number is called the y-coordinate. The y-coordinate is positive if the point is above the origin and negative if the point is below the origin. The x-y plane is divided into 4 quadrants (4 sections) as described below. Quadrant 2 Quadrant 1 Quadrant 3 Quadrant 4 All points in Quadrant 1 has a positive x-coordinate and a positive y-coordinate (+ x, + y). All points in Quadrant 2 has a negative x-coordinate and a positive y-coordinate (- x, + y). All points in Quadrant 3 has a negative x-coordinate and a negative y-coordinate (- x, - y). All points in Quadrant 4 has a positive x-coordinate and a negative y-coordinate (+ x, - y). Plot each point on the graph below. Remember, coordinate pairs are labeled (x, y). Label each point on the graph with the letter given. 1. A(3, 4) 2. B(4, 0) Example: F(-6, 2) 3. C(-4, 2) 4. D(-3, -1) 5. E(0, 7) +y F -x +x Determine the coordinates for each point below: Example. ( 2 , 3 ) 6. (____, ____) 9. (____, ____) 12. (____, ____) 13. (____, ____) 10. (____, ____) 11. (____, ____) 8. (____, ____) 7. (____, ____) Complete the following tables. Then graph the data on the grid provided. Example: y = -2x - 3 X Y -3 3 -2 1 -1 -1 0 -3 Work: x = -3 y = -2(-3) – 3 = 6 – 3 = 3 Therefore (x, y) = (-3, 3) x = -2 y = -2(-2) – 3 = 4 – 3 = 1 Therefore (x, y) = (-2, 1) x = -1 y = -2(-1) – 3 = 2 – 3 = -1 Therefore (x, y) = (-1, -1) x=0 y = -2(0) – 3 = 0 – 3 = -3 Therefore (x, y) = (0, -3) 14. y = x + 2 X 0 Y 1 2 15. y = 2x X Y 0 1 2 3 16. y = -x X -3 -1 Y 1 3 17. y = 2x - 3 X Y 0 1 2 3 18. y = 1 x+1 2 X Y 0 2 4 6 19. y 3 x 1 2 X Y -2 0 2 20. y X 2 x 1 3 Y -3 0 3 Order of Operations To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifying expressions that contain multiple operations. 1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets above or below a fraction bar) 2. Simplify any term with exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. One easy way to remember the order of operations process is to remember the acronym PEMDAS or the old saying, “Please Excuse My Dear Aunt Sally.” P - Perform operations in grouping symbols E - Simplify exponents M - Perform multiplication and division in order from left to right D A - Perform addition and subtraction in order from left to right S Example 1 Example 2 2 – 32 + (6 + 3 x 2) 2 – 32 + (6 + 6) 2 – 32 + 12 2 – 9 + 12 -7 + 12 =5 -7 + 4 + (23 – 8 ÷ -4) -7 + 4 + ( 8 – 8 ÷ -4) -7 + 4 + ( 8 - -2) -7 + 4 + 10 -3 + 10 =7 Order of Operations Evaluate each expression. Remember your order of operations process (PEMDAS). 1. 6+4–2∙3= 4. (-2) ∙ 3 + 5 – 7 = 2. 15 ÷ 3 ∙ 5 – 4 = 5. 29 – 3 ∙ 9 + 4 = 3. 20 – 7 ∙ 4 = 6. 4∙9–9+7= 7. 50 – (17 + 8) = 16. (12 – 4) ÷ 8 = 8. 12 ∙ 5 + 6 ÷ 6 = 17. 18 – 42 + 7 = 9. 3(2 + 7) – 9 ∙ 7 = 18. 3 + 8 ∙ 22 – 4 = 10. 16 ÷ 2 ∙ 5 ∙ 3 ÷ 6 = 19. 12 ÷ 3 - 6 ∙ 2 – 8 ÷ 4 = 11. 10 ∙ (3 – 62) + 8 ÷ 2 = 20. 6.9 – 3.2 ∙ (10 ÷ 5) = 12. 32 ÷ [16 ÷ (8 ÷ 2)] = 21. [10 + (2 ∙ 8)] ÷ 2 = 13. 180 ÷ [2 + (12 ÷ 3)] = 22. ¼(3 ∙ 8) + 2 ∙ (-12) = 14. 5 + [30 – (8 – 1)2] = 11 - 22 23. 3[10 – (27 ÷ 9)] = 4–7 15. 5(14 – 39 ÷ 3) + 4 ∙ 1/4 = 24. [8 ∙ 2 – (3 + 9)] + [8 – 2 ∙ 3] = Operations with Signed Numbers Adding and Subtracting Signed Numbers Adding Signed Numbers Like Signs Different Signs Add the numbers & carry the sign Subtract the numbers & carry the sign of the larger number (+)+(+)=+ ( – ) + (– ) = – ( +3 ) + ( +4 ) = +7 (– 2 ) + (– 3 ) = ( – 5 ) ( + ) + (– ) = ? ( +3 ) + (–2 ) = +1 (–)+(+)=? ( –5 ) + ( + 3 ) = –2 Subtracting Signed Numbers Don’t subtract! Change the problem to addition and change the sign of the second number. Then use the addition rules. ( +9 ) – ( +12 ) = ( +9 ) + ( – 12) ( +4 ) – (–3 ) = ( +4 ) + ( +3 ) ( – 5 ) – ( +3 ) = ( – 5 ) + ( – 3 ) ( –1 ) – (– 5 ) = ( –1 ) + (+5) Simplify. 1. 9 + -4 = 7. 20 – - 6 = 2. -8 + 7 = 8. 7 – 10 = 3. -14 – 6 = 9. -6 – -7 = 4. -30 + -9 = 10. 5–9= 5. 14 – 20 = 11. -8 – 7 = 6. -2 + 11 = 12. 1 – -12 = Multiplying And Dividing Signed Numbers If the signs are the same, If the signs are different, the answer is positive the answer is is negative Like Signs Different Signs (+ ) ( + ) = + ( +3 ) ( +4 ) = +12 (+)(–)=– ( +2 ) ( – 3 ) = – 6 (– ) (– ) = + ( – 5 ) ( – 3 ) = + 15 (–)(+)=– ( –7 ) ( +1 ) = –7 (+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12 (+)/(–)=– ( +2 ) / ( – 3 ) = – 6 (+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12 (–)/(+)=– ( –7 ) / ( +1 ) = –7 Simplify. 1. (-5)(-3) = 7. -7 = -1 2. -6 = 8. (3) (-4) = 2 3. (2)(4) = 9. 8 = -4 4. -12 = 10. (-2)(7) = 11. -20 = -4 5. (-1)(-5) = -1 6. -16 = 8 12. (2)(-5) = Solving Equations To solve an equation means to find the value of the variable. We solve equations by isolating the variable using opposite operations. Opposite Operations: Addition (+) & Subtraction (–) Multiplication (x) & Division ( ) Example: Solve. 3x – 2 = 10 +2 +2 Isolate 3x by adding 2 to each side. 3x 3 = 12 3 Simplify Isolate x by dividing each side by 3. x = 4 Simplify Check your answer. 3 (4) – 2 = 10 12 – 2 = 10 10 = 10 Please remember… to do the same step on each side of the equation. Always check your work by substitution! Substitute the value in for the variable. Simplify Is the equation true? If yes, you solved it correctly! Try These: Solve each equation. 1. x+3 = 5 6. w – 4 = 10 2. c – 5 = -8 7. 3p = 9 3. -7k = 14 8. –x = -17 4. h = 5 3 9. m = 7 8 5. 4 d = 12 5 10. 3 j= 6 8 11. 2x – 5 = 11 16. 4n + 1 = 9 12. 5 j – 3 = 12 17. 2x + 11 = 9 13. -3x + 4 = - 8 18. -6x + 3 = - 9 14. f + 10 = 15 3 19. a – 4 = 2 7 15. b4 = 5 2 20. x6 = -3 5 Use substitution to determine whether the solution is correct. 21. 4x – 5 = 7 x=3 23. -2x + 5 = 13 x=4 22. 6–x=8 x=2 24. 1–x=9 x = -8 Word Problems Math Dictionary + – X ÷ = More than Increased Greater than Sum Older Less than Shorter Decreased Difference Reduced Twice Times Of Product Double Divided Each part Half Quotient One-third Same as Total Is Equal to Sum Use the math dictionary to help you translate words to algebraic symbols. Matching – Put the letter of the algebraic expression that best matches the phrase. _____ 1. _____ 2. _____ 3. ______4. _____ 5. two more than a number two less than a number half of a number twice a number two decreased by a number a. b. c. d. e. 2x x+ 2 2–x x–2 x 2 Careful! Pay attention to subtraction. The order makes a difference. Translate to an algebraic expression, then reread to check! Use an Equations to Solve Word Problems: Translate each word problem into an algebraic equation Using x for the unknown, write a “let x =” for each unknown, and then write an equation to represent the relationship in the word problem Solve the equation Substitute the value for x into the let statement(s) to answer the question asked in the word problem Check your answer by substituting the solution for the unknown in the word problem and for x in the equation to determine if the solution is correct. For Example: Suzi is 6 years older than Marianne. The sum of their ages is 24. How old is each girl? 1. What are you asked to find? Let variables represent what you are asked to find. How old is each girl? Let x = Marianne’s age Let x + 6 = Suzi’s age 2. Write an equation to represent the relationship in the problem, the sum of their ages. x + (x + 6) = 24 3. Solve the equation for the unknown. x + (x + 6) = 24 2x + 6 = 24 -6 -6 2x = 18 2 2 x = 9 Marianne = x = 9 years old Suzi = x + 6 = 9 + 6 = 15 years old Word Problem Practice Set 1. Seven less than a number is 3. What is the number? 2. Janet weighs 20 pounds more than Anna. If the sum of their weights is 250 pounds, how much does each girl weigh? 3. Three times a number is 2. What is the number? 4. The quotient of a number and 4 is 12. What is the number? 5. Sarah drove 3 hours more than Michael on their trip to Texas. If the trip took 37 hours, how many hours did Sarah and Michael each drive? 6. The are 125 more 8th graders at Rosa Parks than at Farquhar. If there is a total of 515 8th graders at the two schools, how many 8th graders are there at each school? 7. The school lunch prices are changing next year. The cost of a hot lunch will increase $0.45 from the current price. If the next year’s price is $2.60, what did a hot lunch cost this year? 8. Matt spent twice as much as Jim at the mall. If they spent $105 together, how much did each boy spend? 9. Roberto worked four hours longer than Laura studying for his final exam. If they studied for a total of 20 hours all together, how long did each student study? 10. Three-fourths of the student body attended the pep rally. If there were 1230 students at the pep rally, how many students are there in all? Rounding numbers Step 1: Underline the place value in which you want to round. Step 2: Look at the number to the right of that place value. Step 3: If the number to the right of the place value you want to round is less than 5, keep the number the same and drop all other numbers. If the number to the right of the place value you want to round is 5 or more, round up and drop the rest of the numbers. Example: Round the following numbers to the tenths place. 1. 23.1296 Tenths 23.1 2. 64.2685 64.3 the digit to the right of the tenths place, 6, indicates to round up 3. 83.9721 84 the digit to the right of the tenths place, 7, indicates to round up which makes the 9 a zero (10) and the 3 a 4. the digit to the right of the tenths place, 2, indicates to drop off Round the following numbers to the tenths place. 1. 18.6231 _____________ 6. 0.2658 ______________ 2. 25.0543 _____________ 7. 100.9158 ______________ 3. 3.9215 _____________ 8. 19.6816 ______________ 4. 36.9913 _____________ 9. 17.1483 ______________ _____________ 10. 0. 9601 ______________ 5. 15.9199