Poway Unified School District
Algebra 2A-2B
Standards and Exemplars
Spring, 2003
1.1 / Student use properties of numbers to demonstrate whether assertions are true or false.True or False:
· a(b + c) = ab + ac
· np = pn
· a + b = b + a
· (a + b) + c = a + (b + c)
· a*b = ba
· 0 + a = a
· a * 1 = a
Evaluate if x = –2, y = 3, z = –1:
·
· x – z
· 3x2 + 2y
· xy – z
·
· 2z2 + 3z
· 3 – 2(y –3)2
· 12 + y ÷ 4 • 2
2.0* / Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
Simplify:
· (–3)4
· –34
· 2x3 – 4x3
· 2a3(3a4)
· (–5x)4
· (2w2x6y)3
· 4x2y5(4x – 3xy2 + 2y3)
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/ Simplify:
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Write using all positive exponents:
· –8a5b–4
·
A. 10-6
B. 10-2
C. 102
D. 108 / (38)2=
A. 34
B. 36
C. 310
D. 316 / The square root of 150 is between
A. 10 and 11
B. 11 and 12
C. 12 and 13
D. 13 and 14
If , what is the value of x?
A. -3 or 0
B. -3 or 3
C. 0 or 3
D. -9 or 9 / If x = -7, the - x =
A. -7
B.
C.
D. 7 / The perimeter, P of a square may be found by using the formula , where A is the area of the square. What is the perimeter of the square with an area of 36 square inches?
A. 9 inches
B. 12 inches
C. 24 inches
D. 72 inches
6.0* / Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x +6y < 4).
· Find the slope of the line that contains (4,–9) and (–9,9).
· Find the slope of a line from the equation: 3x – y = 20
· Write the rule for the following functions:
a) f(n) = n + 2
b) f(n) = 3n – 2
c) f(n) = 2n
d) f(n) = –2n – 2
e) none of these
· Find the rate of change (slope) given:
minutes cost ($)
2 3
4 8
6 13
8 18
10 23
· Graph :
·
· 2x – y = 8
· Compute the x and y intercepts: 2x + 5y = 10
· Graph and write the equation of a line in slope-intercept form that:
· Passes through the point (3,4) and has a slope of 2
· Passes through (–1, – 4) and (2,3)
· Graph the linear inequalities on a coordinate plane.
· x + 6y > –6
· 2x – y ≤ 3
· y > –2x + 1
· x ≥ 5
· The slope of the line shown below is .
· What is the y-intercept of the line 2x -3y = 12?
a. (0, -4)
b. (0, -3)
c. (2, 0)
d. (6, 0)
· What are the coordinates of the x-intercept of the line 3x + 4y = 12?
a. (0, 3)
b. (3, 0)
c. (0, 4)
d. (4, 0)
· What is the slope of the line shown in the graph below?
· Which scatter plot shows a negative correlation?
· Which of the following is the graph of
7.0* / Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations using the point-slope formula.
· What is the equation of a line that includes the point (9,3) and has a slope of ?
a) y = 3x –
b) y = + 13
· Is (3,–4) a point on the line 2x + 3y = –4? (yes/no)
· Which of the following points lies on the line 4x + 5y = 20?
a. (0, 4)
b. (0, 5)
c. (4, 5)
d. (5, 4)
**Students should be able to transform equations from standard form slope-intercept form (e.g., write 2x + 3y = 6 in slope intercept form).
8.0 / Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
· Give the slope intercept form of the equation of the line that is perpendicular to 4x–9y = –9 and passes through (– 4,7).
· Which equation models a line parallel to y = 3x – 6?
a) y = – 8
b) y = 3x – 6
c) y = – + 4
d) y = 3x + 1
· Give the slope-intercept form of the equation of the line that is parallel to 4x - 9y = -9 and passes through (-4, 7).
a. 4x -9y = -79
b. 4x + 9y = 47
c. 9x - 4y = -64
d. 9x + 4y = -8
· What is the slope of a line parallel to the line ?
A. -3
B.
· What is the slope of a line perpendicular to the line
a. -3
b.
c.
d. 2
· Which of the following statements describes parallel lines?
a) Same y-intercept but different slopes
b) Same slope but different y-intercepts
c) Opposite slopes but same y-intercept
d) Opposite slopes but same x-intercept
· Which of the following could be the equation of a line parallel to the line y = 4x - 7?
a)
b)
c)
d)
9.0* / Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
· Millenium High and Ridgemont High have decided to offer their students a sports event discount card. At Millenium High, each student can buy a discount card for $10.00 and then pay $1.00 to get into each game. At Ridgemont High, each student can buy a discount card for $15.00 and then pay $.75 to get into each game. How many games would a student need to attend in order for the total cost at Ridgemont High to be the better deal?
· Norm bought 5 oranges and 3 bananas for $2.70. Maria bought 3 oranges and 7 bananas for $3.70. How much are one orange and one banana?
· The length of a rectangle is 4 more than 3 times its width. If the perimeter is 56 inches, what is the length of the rectangle?
· After examining the equation a = 2b + c2, John stated that a will always be greater than b. Which of the following is a counter example to John’s statement?
a. b ≤ 0 and c = 0
b. b and c are negative numbers
c. b = 0 and c = any number
d. b > 0 and c < 0
· What is the solution to the system of equations shown below?
a. (-2, -2)
b. (-2, 2)
c. (2, -2)
d. (2, 2)
· The solution to a system of inequalities is graphed below. What are the inequalities?
a. b.
c. d.
· What is the solution of the system of equations shown above?
a) (1, -2)
b) (1, 2)
c) (5, 10)
d) (-5, -10)
· Solve by graphing:
· Solve by substitution:
· Solve by elimination:
· Solve using any method:
· Solve using any method:
**Students must be able to interpret the solution to a system of equations graphically—the intersection is defined by a point, the same line, or parallel lines.
10.0* / Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques.
Simplify:
· –3xy – (–9xy)
· 4x – 2(x – 3) – 4(x – 1)
· 5x2 + 2x – 3x + 4x2
·
· 3x2 + 2x – (5x2 + x – 6)
· –2x(4x4 + 2y)
· (5p + 2)(3p – 7)
· (4x – 7y)2
· (5x2 + 6)(5x2 – 6)
· 2z2 + 3z
· 3 – 2(y –3)2
· 12 + y ÷ 4 • 2
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· Create an algebraic expression from the following statement:
“4 less than eight times a certain number”
· The sum of 3 consecutive integers is -237. Find the integers.
**Students must be able to complete word problems involving consecutive integers, age, length and width of a rectangle, and any one-variable word problem.
11.0 / Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
Factor completely:
· 5x3 – 10x2
· x2 – 81
· 9x2 – 64
· x2 + 8x + 16
· x2 – 15x + 54
· x4 – 2x3 – 3x2
· 5x3 – 20x
· 2x2 – 5x – 6
· 3x2 + 5x + 2
**Teachers need to teach:
· When A=1
· Difference of Squares
· Binomial squared
· GCF
· When “A” is prime
12.0* / Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
Simplify:
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13.0* / Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
Simplify:
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Solve:
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14.0* / Students solve a quadratic equation by factoring or completing the square.
· Solve by factoring:
· In the equation , what term would make the equation a perfect square?
· The solution to a quadratic equation is {4, -2}. What is the original equation?
a.
b.
c.
d.
**Students should know how to solve a simple completing the square problem. Also, focus on solving quadratics by factoring using a prime number as the coefficient of x2.
15.0* / Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
Work problems:
· Mr. Jacobs can correct 150 quizzes in 50 minutes. His student aide can correct 150 quizzes in 75 minutes. Working together, how many minutes will it take them to correct 150 quizzes?
a) 30
b) 60
c) 63
d) 125
· Stephanie is reading a 456-page book. During the past 7 days she has read 168 pages. If she continues reading at the same rate, how many more days will it take her to complete the book?
a. 12
b. 14
c. 19
d. 24
· A new copier can make 72 copies in 2 minutes. When an older photocopier is working, the two photocopiers can make 72 copies in 1.5 minutes. How long does it take the older photocopier working alone to make 72 copies?
· Company A can install chairs in a theatre in 10 hours. Company B can install them in 15 hours. The owner of the theatre wants the chairs installed in leas than one (8 hours) day. If the companies work together, can they install the chairs in less than one day?
· It takes painter A 3 hours to paint a certain area of a house. It takes painter B 5 hours to do the same work. How long would it take them working together, to do the painting job?
Rate problems:
· Livingston is 25 miles east of Bozeman, Montana. Lisa left Bozeman at 2:00 p.m., driving east on I-90 at 65 mi/h. Jerome left Livingston at 2:00 p.m., driving west of I-90 at 55 mi/h. At what time will Lisa pass Jerome? How far will Lisa be from Bozeman when she passes Jerome?
· Two city buses leave their station at the same time, one heading east and the other heading west. The east bound bus travels 35 mph, and the westbound bus travels 45 mph. In how many hours will they be 60 miles apart?
· A bicyclist travels 20 miles per hour faster than a walker. The cyclist traveled 25 miles in the time it took the walker to walk 5 miles. Find their speeds.
· A train leaves a station and travels east at 72 km/h. Three hours later a second train leaves on a parallel track and travels east 120 km/h. When will it overtake the first train?
Percent of mixture problems:
· One solution is 80% acid and another one is 30% acid. How much of each solution is needed to make a 200 L solution that is 62% acid?
· A solution containing 30% insecticide is to be mixed with a solution containing 50% insecticide to make 200 L of a solution containing 42% insecticide. How much of each solution should be used?
** Introduce as time allows. At this time, it is not clear how much this is being emphasized on the state assessments or to what depth.
16.0
18.0 / Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.
· Determine if this relation is a function: {(2,5), (5,6), (2,7)}
· Find the range of the function y = 3x2 + 5x + 1 when the domain is {–1, 0, 1}
· A function f is defined by the set of ordered pairs {(1,–2), (2,–4), (3,–6), (4,–8)}
· Which of the following statements about the function f is true?
a) The domain of f is the set {–2, –4, –6, –8}.
b) The range of f is the set {1, 2, 3, 4}.
c) f(2) = –2
d) f(4) = –8
· State whether or not each is a function and justify your answer.
· Which function is modeled by the table?
x f(x)
–2 5