The Top 11 Keystones of Algebra 1


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1 The Top 11 Keystones of Algebra 1 The Top Eleven Keystones of Algebra 1 You should be able to 1) Simplify a radical expression. 2) Solve an equation. 3) Solve and graph an inequality on a number line. 4) Manipulate expressions with exponents. 5) Add, Subtract, Multiply, and Divide Polynomials. 6) FACTOR. 7) Graph a line. 8) Write the equation of a line. 9) Solve a system of linear equations or inequalities. 10) Interpret data and use basic probability. 11) Apply the Pythagorean Theorem.
2 Page 1  Keystone #1 Simplify a radical expression. Open Ended
3 Page 2  Keystone #1 Simplify a radical expression. Multiple Choice. 1) What is the simplified form of 48? A. 3 4 B. 9 6 C. 4 3 D ) The diagonal of a TV screen is 275 inches long. Which expression has the same value as 275? A B. 3 5 C D ) The diagram below shows a square. Each side of the square measures 4 5 cm and the area of the square is 40x square cm. What is the value of x? A. 1/2 B. 2 C. 10 D. 20 A = 40x 4 5 4) A rope is tied from one corner of a fence to the opposite corner. The distance is 98 yards. Which expression shows the simplified form of 98? A. 5 2 B. 7 2 C. 8 2 D ) For which value of x should the expression be further simplified? 21x A. x = 2 B. x = 5 C. x = 6 D. x = 10 6) An expression is given below. 3 10a If this expression is equivalent to 90, what must be the value of a? A. 3 B. 45 C. 90 D. 115
4 Page 3  Keystone #2 Solve an equation. Open Ended. 3x + 5 = 29 5(x + 7) 3 = 42 5x 3 = 7x + 5 x + 2 = 9 ½x + 7 = 3x x = 22
5 Page 4  Keystone #2 Solve an Equation. Multiple Choice. 1) What is the solution for this equation? 2x 3 = 5 2) Which equation is equivalent to 5x 2 7x + 1 = 14x? A. x = 4 or x = 4 B. x = 4 or x = 3 C. x = 1 or x = 4 D. x = 1 or x = 3 3) The total cost (c) in dollars of renting a sailboat for n days is given by the equation c = n If the total cost was $360, for how many days was the sailboat rented? A. 9x 2 = 14x B. 9x + 1 = 14x C. 9x + 2 = 14x D. 12x 1 = 14x 4) Solve: 3 x + 5 = 2x + 35 Step 1: 3x + 15 = 2x + 35 Step 2: 5x + 15 = 35 Step 3: 5x = 20 Step 4: x = 4 Which is the first incorrect step in the solution shown above? A. Step 1 C. Step 3 B. Step 2 D. Step 4 A. 2 C. 6 B. 4 D. 8 5) The cost to rent a construction crane is $750 per day plus $250 per hour for use. What is the maximum number of hours the crane can be used each day if the rental cost is not to exceed $2500 per day? 6) The lengths of the sides of a triangle are y, y + 1, and 7 centimeters. If the perimeter is 56 centimeters, what is the value of y? A. 2.5 B. 7.0 C. 3.7 D A. 24 C. 31 A. 25 D. 25
6 Page 5  Keystone #3 Solve & graph an inequality. Open Ended. 2x < 2x 3 < 21 3x + 9 < 15 or 4x 2 > 18 x + 2 > 5 4 x ½ x > 1
7 Page 6  Keystone #3 Solve & graph an inequality. Multiple Choice. 1) Which value of x is in the solution set of the inequality 4x + 2 > 10? 2) Which is a graph of the solution of the inequality 2x 1 5? A) 2 B) 2 C) 3 D) 4 3) A baseball team had $1,000 to spend on supplies. The team spent $185 on a new bat. New baseballs cost $4 each. The inequality b 1,000 can be used to determine the number of new baseballs (b) that the team can purchase. Which statement about the number of new baseballs that can be purchased is true? A. The team can purchase 204 new baseballs. B. The minimum number of new baseballs that can be purchased is 185. C. The maximum number of new baseballs that can be purchased is 185. D. The team can purchase 185 new baseballs, but this number is neither the maximum nor the minimum. 5. Which graph shows the solution to the inequality shown below? 15 7n 2 n 10 < 35 A. B. C. 4) Matthew can spend up to $60 for a carwash and gasoline at a service station. The carwash will cost $8 and the gasoline costs $3.50 per gallon. The inequality below can be solved for g, the number of gallons of gasoline Matthew can buy. 3.5g Which of the following is a true statement? A. Matthew can buy over 20 gallons of gasoline. B. Matthew can buy at most 14 gallons of gasoline. C. Matthew can buy 15 gallons of gasoline, but not 16. D. Matthew can buy 14 gallons of gasoline, but not Which graph shows the solution to the absolute value inequality shown below? A. B. C. D. 2x < 12 D.
8 Page 7  Keystone #4 Manipulate expressions with exponents. Open Ended. x 3 2y 0 2y 0 x 3 y 4 2 2x x 2 y 4 z 3xy 2 z 3 2x 1 y 4 6xy 2
9 Page 8  Keystone #4 Manipulate expressions with exponents. Multiple Choice. 4. 5x 3 10x 7 A. 2x 4 B. 1 5x 4 C. 1 2x 4 D. x Which expression has the same value as A B. 2 5 C. 2 3 D. 2 4
10 Page 9  Keystone #5 Add, Subtract, Multiply, and Divide Polynomials. Open Ended. (7x 25x  4) + (x x + 1) (2x 2  x + 3) (7x 2 + 2x  1) 4x(x 23x + 7) (x 3) 2 (2x + 5)(4x  1) 2x + 14 x 2 + 6x 7
11 Page 10  Keystone #5 Add, Subtract, Multiply, and Divide Polynomials. Multiple Choice. 5. Which polynomial expression represents the area of the figure shown? 6. A polynomial expression is shown below. x 2 A. 2x 2 + x 10 B. x x 5 2 C. 2x 2 10 D. x 2 5 2x + 5 The expression is simplified to What is the value of m? A. 2 B. 2 C. 3 D. 3
12 Page 11  Keystone #6 FACTOR. Open Ended. 4ab 2 + 6a 2 b 2 12a 2 b x 2 + 5x x 2 12x 72 x x x + 9 6x 2 x  2
13 Page 12  Keystone #6 FACTOR. Multiple Choice. 2. Which expression represents 2x2 12x in simplest form? A. 0 B. 2x C. 4x D. 2x + 2 x Find the greatest common factor (GCF) for the two polynomials shown. 6. Find the least common multiple (LCM) for the two polynomials. 300ab 2 c 500a 2 bc 3 A) 27xy B) 108xyz C) 108xy 2 z D) 2916xyz
14 Page 13  Keystone #7 Graph a line. Open Ended. y = 2 x = 3 y = 2x  4 y = ¼x + 4 2x + 3y = 6 x 2y = 6
15 Page 14  Keystone #7 Graph a line. Multiple Choice. 1. What is the yintercept of the graph 6x 3y = 24? A. 8 B Which set of slopes would belong to a pair of lines perpendicular to one another? A. C. 3 D. 24 B. C. D. 6. Identify whether the relation between the domain (time) and range (height) IS or IS NOT a function. I. II. A. I is a function, II is not a function B. I is not a function, II is a function C. Both I and II are functions D. Neither I or II are functions
16 Page 15  Keystone #8 Write the equation of a line. Open Ended. The slope of the line is 2 and the yintercept is (0, 5). The slope of the line is ¼ and it passes through the point (4, 3). Slope Intercept Form: Standard Form: The line passes through the points (2, 3) and (5, 0). Slope Intercept Form: Standard Form: The line passes through the points (1, 4) and (5, 1). Slope Intercept Form: Standard Form: The line is parallel to y = 3x + 19 and passes through the point (1, 1). Slope Intercept Form: Standard Form: The line is perpendicular to the line y = 2 x 5 and 3 passes through the point (2, 2). Slope Intercept Form: Standard Form: Slope Intercept Form: Standard Form:
17 Page 16  Keystone #8 Write the equation of a line. Multiple Choice. 1) Nyesha earns $1,000 per month plus a commission of 5% of the total dollar amount of each sales she makes. Her total monthly earnings, P, are represented by the equation P = t, where t represents the total dollar amount of her sales for month. Which equation will represent her total monthly earnings in dollars if her commission increases an additional 2%? 2) Marta is buying a car from her sister for $12,294. After making an initial payment of $1,200, she agrees to pay $258 per month for n months. Which equation can Marta use to determine the number of months, n, it will take her to finish paying for the car? A. B. C. D. 3) The table shows the 2008 United States firstclass postage rate, d(w), for packages of maximum weight, w. 4) The equation y = 5.25x models the amount of money (y) in dollars Jillian has at the end of x weeks. Which scenario is modeled by the equation? A) Jillian owes $ and is paying $5.25 each week. B) Jillian has $ and is paying $5.25 each week. C) Jillian owes $ and is receiving $5.25 each week. D) Jillian has $ and is receiving $5.25 each week. The pattern in the table continues. Which value represents the postage rate of a 13ounce package? A) $2.21 B) $2.87 C) $3.21 D) $4.21
18 Page 17  Keystone #8 Write the equation of a line. Multiple Choice. 5) Emily s family needs to rent a moving truck to move their belongings to a different house. The rental cost for Truck A is $42 per day plus $0.69 per mile and the rental cost for Truck B is $30 per day plus $0.79 per mile. If Emily s family rents the moving truck for two days, how many miles would they need to drive so that the rental cost for the two trucks would be the same? A) 60 B) 120 C) 180 D) 240 6) The scatter plot below shows the elevation of a mountainclimbing team during an 8 day trip. Which equation is closest to the line of best fit for this data? A) y = 30x B) y = 60x C) y = 490x + 60 D) y = 400x ) After Zach made a bicycle trip to Colorado, he used the equation y = 1 x to model 20 y, his altitude in feet, in terms of x, the number of feet he bicycled. Which best describes the rate of change in altitude as he traveled? A) For every 20 feet he traveled, the altitude increased 1 foot. B) For every 5000 feet he traveled, the altitude increased by 1 20 foot. C) For every 20 feet he traveled, the altitude increased 250 feet. D) For every foot he traveled, the altitude increased 20 feet.
19 Page 18  Keystone #9 Solve a system of linear equations or inequalities. Open Ended. x 3y = 255x + y = 3 x 2y = 21 3x 8y = 24 x + 2y = 6 3x 2y = 2 5x + 4y = 18 2x + 3y = 24 y + 2x = 7 2y 4 = 2x y 2x + 1 y > 3x + 4
20 Page 19  Keystone #9 Solve a system of linear equations or inequalities. Multiple Choice. 1. Karen makes $5 per hour babysitting and $12 per hour giving music lessons. One weekend, she worked a total of 18 hours and made $139. How many hours did she spend babysitting? 2. Molly bought 3 slices of cheese pizza and 4 garlic knots for $7.40. Grace bought 2 slices of cheese pizza and 5 garlic knots for $5.75. How much does it cost to buy 2 garlic knots? a) 7 hours b) 11 hours c) 18 hours d) 5 hours a) $2.00 b) $0.35 c) $0.70 d) $ Which of the following best describes the graph of this system of equations? y = 2x + 3 5y = 10x + 15 a) Two identical lines b) Two parallel lines c) Two lines intersecting in only one point d) Two lines intersecting in only two points 4. Sue and Maria purchased flowers. Sue purchased 5 roses and 4 daisies for $32. Maria purchased 1 rose and 6 daisies for $22. Which statement is true? a) A rose costs $1 more than a daisy. b) Sue spent $4 on each daisy. c) Sue spent more on daisies than she did on roses. d) Maria spent more on roses than she did on daisies. 5. Mark is raising money for a charity. He makes either 18 phone calls per day or 10 home visits per day. Mark wants to contact at least 100 people next week (7days). Which system of inequalities represents this situation if x represents number of days making phone calls and y is days of home visits? a) x + y 7 18x + 10y 100 b) x + y 7 18x + 10y 100 c) x + y 7 18x + 10y 100 d) x + y 7 18x + 10y Which system of linear inequalities has the solution set shown in the graph? a) y 1 y > 3 4 x + 3 b) x 1 y 3 4 x + 3 c) y 1 y < 3 4 x + 3 d) y 1 y < 3 4 x + 3
21 Page 20  Keystone #10 Probability, Statistics & Data Analysis. Open Ended. 1) The scatter plot below shows the population of Jackson, Mississippi, every two years since ) Given the followings set of numbers, find the mean, median, mode and range. Student Grade (%) Based on the trend line, what is the expected population of Jackson, Mississippi for the year 2020? 3.) There are 6 volleyballs, 2 basketballs and 5 footballs in a sports bin. If 3 balls are chosen at random, what is the probability that a football and then a volleyball is chosen if the football is replaced? 4.) A bag contains 6 red and 4 blue marbles. What is the probability that next two choices will be red if the first marble is NOT replaced? 5.) Given the following box and whisker plot, determine the interquartile range. 6.) The price of the 5 most popular TV s are listed below: $400, $489, $799, $ 1,249, and $1,489. If another TV is added to the list and the range was increased by $300, what could the price of the next most popular TV be?
22 Page 21  Keystone #10 Probability, Statistics & Data Analysis. Multiple Choice. 1) The scatterplot below shows data from an experiment that tested the amount of lactic acid present in aging cheese. 2.) The scatter plot below compares the math and science scores of twelve students in one classroom. What best describes the relationship between the age of the cheese and the amount of lactic acid present in the cheese, as shown above in the scatterplot? A) There is no correlation. B) There is a positive correlation. C) There is a negative correlation. D) There is a nonlinear correlation. Based on the trend shown in the line of best fit, which is closest to the expected grade of a student that scores an 86 in math? A. 82 B. 85 C. 87 D. 89 3) Which statement is true about the data set? 3, 4, 5, 6, 7, 7, 10 4) The faces of a cube are numbered from 1 to 6. If the cube is tossed once, what is the probability that a prime number or a number divisible by 2 is obtained? A) Mean = Mode B) Mean > Mode C) Mean = Median D) Mean < Median A) 6 6 B) 5 6 C) 2 3 D) 1 6
23 Page 22  Keystone #10 Probability, Statistics & Data Analysis. Multiple Choice. 5. The daily high temperatures in degrees Fahrenheit in Allentown, PA, for a period of 10 days are shown below Which statement correctly describes the data? 6. The points scored by a football team are shown in the stemandleaf plot below. A. The median value is 98. B. The interquartile range is 16. C. The lower quartile value is 76. D. The upper quartile value is 96. What was the median number of points scored by the football team? A) 24 B) 27 C) 28 D) 32 7.) A bag contains 5 red, 2 green and 3 blue marbles. What is the probability that the next two choices will be blue if the first marble is NOT replaced? 8.) What is the Interquartile range of the following box and whisker plot? A B C A. 20 B. 16 C. 6 D. 7 D. 3 10
24 Page 23  Keystone #11 Pythagorean Theorem. Open Ended. 1. A right triangular shaped garden is planned for the front of the school. If the 2 shorter sides are 7 ft and 24 ft, what is the length of the third side? 2. The length of the hypotenuse of a right triangle is 26 inches and the length of one of its legs is 10 inches. What is the length of the other leg? 3. Which equation can be used to find the length of the rope, x, holding the tent? x 4. A park is in the shape of a rectangle 20 miles long and 15 miles wide. How much shorter is your walk if you walk diagonally across the park than along two sides of it? a) x = b) x = c) x = d) x = a) 25 miles b) 50 miles c) 10 miles d) 35 miles 5. A right triangle has a hypotenuse of 50 inches and a leg of 35 inches. What is the length of the other leg? a) b) 5 51 c) 15 d) The diagonal of a square is 20 ft. What is the perimeter of the square? a) 80 b) 40 2 c) 10 2 d) 5
The Keystones of Algebra 1
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