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Learning Target: I can… Write algebraic expressions from word problems Introduction to Algebra Variable: A ______________________ that stands for a ________________ ALGEBRAIC EXPRESSION Uses numbers, variables, and operational symbols Key Words for Operational Symbols: Each __________ More __________ Less ___________ Share Equally ____________ Example 1 At a ballpark, team hats are sold for $15 each. Let c = the number of hats. Write an algebraic expression for the cost of a number of team hats. 2) Jake has four more books than Sarah. Let b = the number of books Sarah has. Write an algebraic expression for the number of books Jake has. You Try! 3. Every person in Mrs. Graf’s class brings two pencils to take a test. If there are c students in the class, write an algebraic expression to find the number of pencils total. 4. Lena is exactly 3 years older than her brother, Peter. If p represents Peter’s age, write an algebraic expression to represent Lena’s age. 5. Elaina and her friends went out to dinner and decided to split the bill evenly. The cost for dinner was $56.75. If there were n number of people out to dinner, how much did each person pay? OAA Example 6) Two Step Expressions 1) To get into the county fair, Patricia must pay a $5 entrance fee and $2 per ride. Write an expression to represent the total Patricia spends at the fair. 2) A group of 10 people at the Village decides to split their bill evenly. If b represents the bill and each person says they will throw in $1 for a tip, write an expression to represents the amount each person spends. 3) Jill charges a base rate of $25 when she mows a lawn. She also charges $6 per hour she mows. If h represents the number of hours she mows, write an expression to represent the amount she makes mowing her neighbors lawn. 4) Jasmine wants to rent a bike while she is on vacation. The rental fee is $8.00 plus an additional $2.50 for every hour the bike is rented. If h represents the number of hours the bike is rented, write an expression to represent the total amount spent. 5) The length of a rectangle is 6 cm. Write an expression to represent the perimeter of the rectangle, using w to represent the width. OAA EXAMPLE Tax, Tip, Discount Expressions 1) Use p to represent the original cost of an item. Write the expression for the total amount spent if the price is discounted 20%. You Try! 2) Use c to represent the original cost of an item. Write the expression for the total amount spent if the price is discounted 25%. 3) Use c to represent the cost of an item. Write an expression to represent total spent if the item is bought with 7% tax. 4) Use b to represent the bill at Bob Evan’s. Write an expression to represent total spent if you also pay a 15% tip to the waiter. EXIT Write the expression for the following situations: 1) Total candy if there are k kids in a class and each kid brings 3 pieces. 2) Total amount of money made by a waitress who makes $3 per hour plus $25 in tips. (h represents hours) Learning Target: I can… Write and evaluate expressions To Evaluate an Algebraic Expression 1. Rewrite equation and plug in a value for each variable 2. Simplify Example 1 Evaluate 3x² + 4 + y when x = -2 and y = 5 2) Evaluate 4a² + 3a + 1 when a = -2 3) Evaluate 2a² + 3b + c when a = -3, b = 4 and c = 5 4) Write and Evaluate Remember key words for writing expressions: Each __________ More __________ Less ___________ Split Evenly ____________ Write and Evaluate 1) To get into the county fair, Patricia must pay a $5 entrance fee and $2 for each ride. Write an expression to represent the total Patricia spends at the fair. Write the expression _________________ If Patricia went on 10 rides, how much did she spend? Evaluate ___________________________ Write and Evaluate 2) Elaina and her friends went out to dinner and decided to split the bill evenly. The cost for dinner was $56.75. If there were n number of people out to dinner, how much did each person pay? Write the expression _________________ If 8 people were out to dinner, how much did each person pay? Evaluate ___________________________ Exit If x = -5, what is x² + 7? Name_________________________ RED Evaluate the following expressions for x = 4 Evaluate the following expressions for x = -3 1) x – 10 1) x – 10 Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ 2) 2) 3x 3x Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ 3) 3) -x -x Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ 4) 4) 5x + 4x 5x + 4x Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ 5) 5) 32/x 30/x Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ 6) 6) 3/x + .25 3/x + .25 Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ 7) x² – 9 7) x² – 9 Rewrite expression _____________ Rewrite expression _____________ Answer ________________ Answer ________________ Evaluate the following expressions for x = -2 and y = 5 1) x–y Rewrite expression _____________ Answer ________________ 2) 3x + 2y Rewrite expression _____________ Answer ________________ 3) -x – 3y Rewrite expression _____________ Answer ________________ 4) 5x + 4y + 10 Rewrite expression _____________ Answer ________________ 5) 32/x + 15/y Rewrite expression _____________ Answer ________________ 6) 3/x + y/4 Rewrite expression _____________ Answer ________________ 7) x² + y Rewrite expression _____________ Answer ________________ 4) 5) Sal is having a party and having people sit at a long line of tables. Each table can sit 4 people, plus one person cat sit at each end of the line of tables, as shown. If there are 12 tables, how many people can sit down? 6) If there are 20 tables, how many people can sit down? 7) Name_________________________ YELLOW 3) Sal is having a party and having people sit at a long line of tables. Each table can sit 4 people, plus one person cat sit at each end of the line of tables, as shown. 1) Evaluate the following expressions for x = 4 A. x – 10 B. 3x C. -x D. 5x + 4x E. 32/x F. 3/x + .25 G. x² – 9 If there are 12 tables, how many people can sit down? 2) If there are 20 tables, how many people can sit down? Convert the following temperatures (round to the nearest degree if necessary) 80°F = ______°C 4) The cost to ship a package is $2.79 plus an additional $0.38 per pound. Write an expression to find the total spent. 50°F = ______°C 75°F = ______°C 60°F = ______°C 100°F = ______°C How much would it cost to ship an 8 pound package? 5) 6) 8) 7) Name_________________________ GREEN 1) Meg and Tom were both running for Student Council President. Meg received 50 more than twice the amount that Tom received. If Tom received v votes, write the expression for the amount Meg received. Evaluate the expression if Tom received 30 votes. 3) Sara is jogging around her neighborhood. She is running at a rate of .1 miles per minute. Write an expression to represent her total distance after m minutes. How long will it take her to run 3 miles? 2) Jason has 50 gallons of water in his bathtub. The bathtub drains at 4 gallons per minute. Write the expression Jason can use to find the total amount of water left in the bathtub after m minutes. Evaluate the expression if Jason has let the water drain for 180 seconds. 4) Sal is having a party and having people sit at a long line of tables. Each table can sit 4 people, plus one person cat sit at each end of the line of tables, as shown. Write an expression to determine how many people can sit at x number of tables. If there are 12 tables, how many people can sit down? If there are 20 tables, how many people can sit down? If 70 people attend the party, how many tables will he need? 5) 6) 8) 7) Evaluate the following expressions for for x = -4 and y = -2 A. x–y B. 3x + 2y C. -x⁴ + y² D. 5x + 4y E. 32/x + y F. 3/x + .25y G. x² – 3y If x = 4, solve problems 1-10 1. x – 10 2. 3x 3. -x 4. 5x + 4x 5. 32/x 6. 3/x + .25 7. X² 8. x² – 9 9. √x 10.4x³ Do the rest on your own and get checked! Learning Target: I can… Solve one and two step equations Inverse Operations The inverse of each operation cancels it out. Inverses you will use: Add ~ Subtract Multiply ~ Divide Squared ~ Square Root Multiply by a fraction (ex: ¾) ~Multiply by its reciprocal (4/3) Solving Equations 1. Separate the equation 2. Start on same side of variable, but furthest from the variable 3. Inverse operations 4. Isolate the variable Examples: One step: 2x = 46 Two step: 26 + 2x = 46 31.5==57.9 9c a –x 61.3 + 36 = 106 97.5 = f/15 x² = 44.89 You try 1) 2x = -7 2) 6.5 = ½x 3) x/4 = 16 4) 6 = -¾x 5) x 6.7 OAA Example 4.5a = 17.1 a=? Solve for a. 10 + a 4 = 15 4r + 6 = 14 -11 = -5b – 6 y -7 + 8 = 19 EXIT Solve 4p + 27 = 43 Learning Target: I can… Show how to solve equations 1.75 + 1.75 + 1.75 – 1.75 + d 4 =3.2 d = 3.2. 4 d 4 = 3.2 – 1.75 d = 1.45 4 (4) d = (4)1.45 4 d = 5.8 The cost of a bag of doughnuts is $5.80. Subtract 1.75 from each side. Simplify. Multiply each side by 4. Simplify. Three Step 2x² - 13 = 19 -60 = -3a² + 15 93 = 4x² – 7 EXIT Solve for a. 3a² – 35 = 397 -x = 17 10 = 1 - x 12 – x = -3 25 – 3x = 2.5 16 – 2x = 9 5 – 3x = 11 4 – 2y = 7 Learning Target: I can…write equations to solve word problems Word Problems - Example Johnny makes $12.75 per hour at his job. If he made $459.00 this week, how many hours did he work? Identify x ______________ Equation _______________ Answer ________________ Word Problems – Example 2 Justin bought 2 apples and 1 pear for $4.00. The pear was $1.78. Write an equation and solve to find the cost of each apple. Identify x ______________ Equation _______________ Answer ________________ 1. Mark bought 4 pens and 1 notebook for $3.25. The notebook was $2.25. Write an equation and solve to find the cost of each pen. Identify x ______________ Equation _______________ Answer ________________ 2. Sara bought 2 bags of chips and a coke for $7.50. The coke was $1.50. How much was each bag of chips? Identify x ______________ Equation _______________ Answer ________________ 3. The fair costs $3 to get in and $2 for every ride. If John spent $21, how many rides did he go on? Identify x ______________ Equation _______________ Answer ________________ 4. Sara is jogging around her neighborhood. She is running at a rate of .1 miles per minute. How long will it take her to run 3 miles? Identify x ______________ Equation _______________ Answer ________________ 5. Logo Shirts, Inc. is going to charge Urbana Junior High $5 for each t-shirt they make for the school. The company will also charge the school a flat fee of $35 to set up the design. If they spent $500 total how many shirts did they buy? Identify x ______________ Equation _______________ Answer ________________ 6. Mike, who has 6 video games, has half as many games as Paul. How many does Paul have? Identify x ______________ Equation _______________ Answer ________________ 7. Meg received 90 votes for Student Council President, which was 50 more than twice the amount that Tom received. How many did Tom receive? Identify x ______________ Equation _______________ Answer ________________ 8. The 347 students who listed soccer as their favorite sport were 13 less than three times the number of students who listed basketball as their favorite sport. How many listed basketball as their favorite sport? Identify x ______________ Equation _______________ Answer ________________ 9. 331 students went on a field trip. Six buses were filled and 7 students traveled in cars. How many students were in each bus? Identify x ______________ Equation _______________ Answer ________________ You bought a magazine for $5 and four erasers. 10. You spent a total of $25. How much did each eraser cost? Identify x ______________ Equation _______________ Answer ________________ You bought a magazine for $5 and four erasers. You spent a total of $25. How much did each eraser cost? A. 5x + 25 = 20 B. 5x + 4 = 25 C. 4x + 5 = 25 D. -4x + 5 = 25 EXIT Jacqueline bought a CD for $16.95 and eight DVDs. The total cost was $52.55. Write the equation and find the cost of each DVD. Equation: ___________________ Answer:_________________ Logo Shirts, Inc. is going to charge General Sherman $6 for each t-shirt they make for the school. The company will also charge the school a flat fee of $35 to set up the design. The school spent $575 all together. Which equation represents this situation? A. 35x + 575 = 6 B. 6x + 575 = 35 C. -6x + 35 = 575 D. 6x + 35 = 575 331 students went on a field trip. Six buses were filled and 7 students traveled in cars. How many students were in each bus? a. 6x + 7 = 331 b. 7x + 6 = 331 c. 331x + 6 = 7 d. 7 + 331x = 6 Jasmine wants to rent a bike while she is on vacation. The rental fee is $8.00 plus an additional $2.50 for every hour the bike is rented. If she spent $20.50, how many hours did she rent the bike? Writing Functions 1. Look at the first pair of x,y values 2. Consider different ways to get from x to y 3. See what works for the rest of the table Function: y = 2x x y 3 6 4 8 5 10 What is the funtion? a. y = 3x b. y = x + 3 c. y = x + 4 d. y = 2x x y -5 -2 -1 2 1 4 3 6 Describe the rule of the table. a. y = 3x b. y = x + 3 c. y = x + 4 d. y = 2x x y 2 6 3 9 4 12 5 15 Describe the rule of the table. a. y = x - 4 b. y = 4x c. y = x + 4 d. y = -4x x y 2 -8 3 -12 4 -16 5 -20 When a table needs more than one operation Tricks: 1. It typically will be multiply then add or multiply then subtract 2. …so look for what you can multiply to get close to y 3. Example: Rule: y = 4x + 1 x y 1 5 2 9 3 13 4 17 When the rule needs more than one operation x y 1 2 3 8 5 14 7 20 9 ? 10 ? Rule: y = 3x – 1 What is the rule? a. y = 2x b.y = 2x + 2 c. y = 2x – 2 d.y = x + 2 x y 1 4 2 6 3 8 4 10 What is the rule? a. y = 3x b.y = 2x + 2 c. y = 3x – 2 d.y = 3x + 1 x y 2 7 4 13 6 19 8 25 x 1 2 3 4 a. b. c. d. y = 2x y=x+2 y = 2x + 2 y = x² y 1 4 9 16 Writing rules for patterns Example: Write a rule for the nth term in the sequence 2, 8, 14, 20, … 1. Make a table that pairs the term number and value 2. Find the rule: 3n-1 n value 1 2 2 8 3 14 4 n 20 Problem of the Day (Thursday) Creating Function Tables x Rule (2x + 1) y (x,y) Graph the function y = 5x - 3 x 5x - 3 y (x,y) 1 5(1) - 3 2 (1,2) 2 3 4 Graph the function y = ½x + 2 x ½x + 2 y (x,y) 1 ½(1) + 2 2.5 (1,2.5) 2 3 4 y = 6x Graphing Functions Linear Function: Makes a straight line Nonlinear Fuction: Does not make a straight line Is the function linear or nonlinear? y = 2x + 1 x 2x + 1 y (x,y) 1 2(1) + 1 3 (1,3) 2 3 4 Linear Is the function linear or nonlinear? y = x² x x² y (x,y) 1 1² 1 (1,1) 2 3 4 Nonlinear CPS Practice Problems! Linear/Nonlinear Function activity Learning Target: I can… Find the nth term of a sequence What are the first 5 terms of the sequence 2n? A. B. C. D. 2, 4, 6, 8, 10 1, 2, 3, 4, 5 2, 4, 8, 16, 32 1, 3, 5, 7, 9 What are the first 4 terms of the sequence n²? A. B. C. D. 2, 4, 6, 8 1, 2, 3, 4 1, 4, 9, 16 1, 3, 5, 7 What are the first 4 terms of the sequence n(n + 1)? A. B. C. D. 2, 4, 6, 8 2, 6, 12, 20 1, 3, 5, 7 2, 3, 4, 5 What are the first 4 terms of the sequence n(n + 4) – 6? A. B. C. D. -1, 4, 6, 8 2, 6, 12, 20 -1, 6, 15, 26 -1, 3, 4, 5 Find the first 4 terms of the sequence -3n + 10 A. B. C. D. 7, 4, 1, -2 13, 16, 19, 22 7, 9, 11, 13 9, 11, 13, 15 Patterns and Tables Write an expression for the rule for 2, 4, 6, 8, 10, . . . . Make a table that pairs the term numbers and the values in the sequence. What is the rule for the nth term in the sequence 3, 5, 7, 9… a. n + 2 b. 2n c. 2n + 1 d. 3n - 1 What is the rule for the nth term in the sequence 3, 5, 7, 9… a. n + 2 b. 2n c. 2n + 1 d. 3n - 1 What is the rule for the nth term in the sequence ½, 1, 1½, 2… a. b. c. d. n/2 2n ½+n n–½ What is the rule for the nth term in the sequence 3, 7, 11, 15… a. b. c. d. y = 3n y = 4n – 1 y=n+4 y = 2n + 1 Write a rule for the picture Write the rule. How many squares will be found in the 6th figure? Write the rule. How many people will be found in the 10th row? Write the rule. How many smileys will be earned on day 9? Learning Series Practice! Problem of the Day (Thursday) What is the rule? a. y = 4x – 1 b. y = 4x c. y = 2x + 2 d. y = 5x - 1 x 1 2 3 4 y 4 9 14 19