Download Algebra 1-2 - Poway Unified School District

POWAY UNIFIED SCHOOL DISTRICT
ALGEBRA 1-2
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

y
z
x
10



2.0*
xy
x y
2z2 + 3z
3 – 2(y –3)2
12 + y ÷ 4 • 2
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)
b 2c 2 d 2
bcd
Simplify:


24

2 5 2
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1

3( 6 )
54  3 6


10 ( 10  20 )

100
169

x8
Write using all positive exponents:
 –8a5b–4

102

104
A.
B.
C.
D.
If
m 4
m5
(38)2=
10-6
10-2
102
108
x  3 , what is the value
of x?
A.
B.
C.
D.
A.
B.
C.
D.
The square root of 150 is between
34
36
310
316
If x = 7, the  x =
A. 7
3 or 0
3 or 3
0 or 3
9 or 9
1
B. 
7
1
C.
7
A.
B.
C.
D.
10 and 11
11 and 12
12 and 13
13 and 14
The perimeter, P of a square may be
found by using the formula
1
 P 
4
A , where A is the area of
the square. What is the perimeter of
the square with an area of 36 square
inches?
D. 7
A. 9 inches
B. 12 inches
C. 24 inches
D. 72 inches
3.0
Students solve equations and inequalities involving absolute value.

10 |x| + 5 = 11

|2x + 1| = 5

The sum of three consecutive integers is –237. Find the integers.

Assume k is an integer and solve for k:
10 – 2|k| >4
b) {–3, –2, –1, 1, 2, 3}
c) {–3, –2, –1, 0, 1, 2,}
For these types of multiple
d) {–2, –1, 0, 1, 2}
choice questions, students
e) {–2, –1, 1, 2, 3}
should be able to work

If x is an integer, what is the solution to x  3  1?
backwards from the given
a. {3}
answers to determine the
b. {3, 2, 1, 0, 1}
solution.
c. {3}
d. {1, 0, 1, 2, 3}
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2

Assume y is an integer and solve for y.
y2 9
4.0*
a) {11, 7}
b) {7, 7}
c) {7, 11}
d) {11, 11}
 If x is an integer, which of the following is the solution set for 3|x| = 15?
a) {0, 5}
b) {, 5}
c) {5, 0, 5}
d) {0, 45}
Students simplify expressions prior to solving linear equations and inequalities in one variable,
such as 3(2x  5) + 4( x  2) = 12.
Solve:

4
x  16
5



1/3 (x – 7) = 5x
¾ (8n – 4) = –2
–3(2 + 3x) = 12

x 3x

 22
6 4



6t + 1 = 6t – 8 (no solution)
14 – (2q + 5) = –2q + 9 (identity)
5x + 3 – 2x = 7 – 4x

3
4

2x 1 2x


10 |x| + 5 = 11
|2x + 1| = 5
Which of the following is equivalent to 4(x+5)  3(x +2) = 14 ?
a. 4x + 20 3x  6 = 14
4x + 5  3x + 6 = 14
b.
4x + 5  3x + 2 = 14
c.
4x + 20  3x  2 = 14
d.
The diameter of a tree trunk varies directly with the age of the tree. A 45-year-old tree has a
trunk diameter of 18 inches. What is the age of a tree that has a trunk diameter of 20 inches?
a. 47 years
b. 50 years
c. 63 years
d. 90 years
Solve and graph on a number line inequalities in one variable.
 x + 2 ≥ –2
 5 – y < 12
 4(x + 4) > 2 – 3x
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3
Which of the following is equivalent to 9  3x > 4(2x  1)?
a) 13 <11x
13 > 11x
b)
10 > 11x
c)
6x >0
d)
Which of the following is equivalent to 3x – (2 –x) < 2x – 5?
b) 3x – 2 – x < 2x – 5
c) 4x + 2 < 2x – 5
d) 4x – 2 < 2x – 5
e) 6x + x < 2x – 5
In the inequality 2x + $10,000  $70,000, x represents the salary of an employee in a school
district. Which phrase most accurately describes the employee’s salary?
a. At least $30,000
b. At most $30,000
c. Less than $30,000
d. More than $30,000
Which of the following is equivalent to 1  2 x  3( x  2) ?
a)
b)
c)
d)
5.0*
1
1
1
1
2x > 3x  2
2x > 3x  5
2x  3x  6
2x > 3x  7
Students solve multi-step problems, including word problems, involving linear equations and
linear inequalities in one variable and provide justification for each step.
Solve:
 1/3 (x – 7) = 5x
 ¾ (8n – 4) = –2
 –3(2 + 3x) = 12

x 3x

 22
6 4
 6t + 1 = 6t – 8 (no solution)
 14 – (2q + 5) = –2q + 9 (identity)
 5x + 3 – 2x = 7 – 4x

3
4

2x 1 2x
 The length of a rectangle is 4 more than 3 times its width. If the perimeter is 56 inches, what is
the length of the rectangle?
 The two rectangles below have dimensions as shown. Which of the following expressions
represents the area of the shaded region?
a) 3w + 2
b) 3w – 2
c)w + 2
d) w – 2
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4
 After examining the equation a = 2b + c2, John stated that a will always be greater than b.
Which of the following is a counter example to John’s statement?
b)
b ≤ 0 and c = 0
c)
b and c are negative numbers
d)
b = 0 and c = any number
e)
b > 0 and c < 0
 Solve for x.
5(2x  3)  6x < 9
a) x < 1.5
b) x < 1.5
c) x < 3
d) x < 6

Which of the following is equivalent to the equation shown below?
20
4

x x5
6.0*
a) x(x5) = 80
b) 20(x  5) = 4x
c) 20x = 4(x  5)
d) 24 = x + (x  5)
Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y =
4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the
region defined by 2x +6y < 4).

Find the slope of the line that contains (4,–9) and (–9,9).

Find the slope of a line from the equation: 3x – y = 20

Write the rule for the following functions:
n
2
0
1
5
9
f(n)
4
–2
1
13
25
a)
f(n) = n + 2
b)
f(n) = 3n – 2
c)
f(n) = 2n
d)
f(n) = –2n – 2
e)
none of these
 Find the rate of change (slope) given:
minutes
cost ($)
2
3
4
8
6
13
8
18
10
23
 Graph :
3
x 1
4
 2x – y = 8
 Compute the x and y intercepts: 2x + 5y = 10
 y
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 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
2
.
3
d
6

What is the value of d?
a. 3
b. 4
c. 6
d. 9
What is the slope of the line shown in the graph below?
a.
2
b.

c.
1
2
1
2
d. 2


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)
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
Which scatter plot shows a negative correlation?

Which of the following is the graph of y 
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x  2?
2
(Version 1, 5/03)
7
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.
4
 What is the equation of a line that includes the point (9,3) and has a slope of  ?
3
3
a) y = 3x –
4
4
x + 13
3
4
c) y =  x + 15
3
b) y = 
1
4
x
3
3
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)
d) y =


8.0
**Students should be able to transform equations from standard form
slope-intercept
form (e.g., write 2x + 3y = 6 in slope intercept form).
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?
1
a) y = x – 8
3
b) y = 3x – 6
1
c) y = – x + 4
3
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 y 
A. 3
B.

1
3
C.
1
3
1
x 2?
3
D. 2
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8

What is the slope of a line perpendicular to the line y 
1
x  2?
3
a. 3
b. 
c.


1
3
1
3
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?
1
x7
4
b) y  4 x  3
c) y  4 x  3
1
d) y   x  7
4
a)
9.0*
y
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?
7 x  3 y  8

4 x  y  6
a.
b.
c.
d.
(2, 2)
(2, 2)
(2, 2)
(2, 2)
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
The solution to a system of inequalities is graphed below. What are the inequalities?
1

y   x
a. 
2
 y  2 x  3
1

y   x
c. 
2
 y  2 x  3



1

y   x
b. 
2
 y  2 x  3
1

y   x
d. 
2
 y  2 x  3
 y  3x  5

 y  2x
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:
x  y  1

4x  2y  6
Solve by substitution:
1

y  x
4

x  2y  12

Solve by elimination:
2a  3b  6

4a  3b  6

Solve using any method:
1

x   y  2
2

2x  y  4

Solve using any method:
y  x

x  y  5
**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.
* Comprises 70% of the California Standards Test
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10.0*
Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
Simplify:
 –3xy – (–9xy)
 4x – 2(x – 3) – 4(x – 1)
 5x2 + 2x – 3x + 4x2












2 2
3
x  3x  x 2
3
4
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
8a
4

a  16 a  4
3x  12 x  4

x
x
2
x  x  12
x

2
x
x4
2
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.
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

11.0
* Comprises 70% of the California Standards Test
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11
12.0*
Students simplify fractions with polynomials in the numerator and denominator by factoring
both and reducing them to the lowest terms.
Simplify:
x2  x  6

x2  4x  3
3x 2  9 x
x3
2
a  8a
a 2  64
n2  7n  12
3n2  12



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:







14.0*
x2  x  6
x2  4x  3
5
4
 2
y2 y  y6
4x2  x
15

5x
2x  2
3t  12 t  4

5t
10t 2
1
3

2
y  5y  4 5y  5
3 5
 1
4x x
3
4

x4 x
Students solve a quadratic equation by factoring or completing the square.
 Solve by factoring:
x 2  x  12  0
3 x 2  11x  4  0
2 x 2  18  0
5 x 2  13 x  6  0
x2  4 x  5
x 2  8x  ___  0 , what term would make the equation a perfect square?

In the equation

The solution to a quadratic equation is {4, 2}. What is the original equation?
a. x  6 x  8  0
2
b. x  2 x  8  0
2
c. x  2 x  8  0
2
d. x  6 x  8  0
2
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12
15.0*
**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 x 2.
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.
* Comprises 70% of the California Standards Test
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13
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.
a)
17.0
b)
c)
d)
e)
 Which function is modeled by the table?
x
f(x)
–2
5
–1
2
0
1
1
2
2
5
a) f(x) = 2x + 1
b) f(x) = x2 + 1
c) f(x) = –2x + 1
d) not a function
 If f(x) = –4x – 10, what is f(3)?
 If f(x) = 3x2 – 5x, what is f(–2)?
 Is the relation {(1,7), (3, 4), (5,0), (1, 2)} a function? Why or why not?
 Given: f(x) = 2x + 1, find f(3).
Students determine the domain of independent variables, and the range of dependent variables
defined by a graph, a set of ordered pairs, or a symbolic expression.
# of times brush
4
12 18 8
7
12 10 10 15 8
13
teeth
# of cavities
8
1
0
4
6
2
1
3
1
5
0
OR





* Comprises 70% of the California Standards Test
Identify the dependent and independent variables
Create a scatter plot
What type of correlation is there
Draw a trend line
Predict the number of cavities if you brush your
teeth 9 times
(Version 1, 5/03)
14
19.0*
Students know the quadratic formula and are familiar with its proof by completing the square.

In the equation x  8 x  b  0 , what is the value of b so that the quadratic makes a perfect
square?


Solve using the quadratic equation: x  8 x  6  0 .
What term makes the equation true?
2
2
x2 
a)
b)
c)
d)

20.0*
b
c
x  ___ 
 ___
a
a
b
a
b
2a
b2
4a 2
b2
2a 2
Complex concept
Which expression could you use to solve 2 x  5  3 x ?
2
a.
5  52  (4)(2)(3)
4
b.
3  32  (4)(2)(5)
4
c.
(3)  (3) 2  (4)(2)(3)
4
d.
(3)  (3)2  (4)(2)(5)
4
**Students should memorize and be able to use the Quadratic Formula.
Students use the quadratic formula to find the roots of a second-degree polynomial and to
solve quadratic equations.
 Solve by using the quadratic formula (round to then nearest hundredth);
3x 2  2 x  4  0
21.0*
**Important to use vocabulary that may appear on test. Students need to know that roots,
zeros, and x-intercepts all mean the same thing. Students need to know how to record the
solution as an exact number and rounded to the nearest hundredth.
Students graph quadratic functions and know that their roots are the x-intercepts.
 Graph y = 2x2 + 4x – 5 (Be sure to find and label vertex, axis of symmetry, and two points on
each side of the vertex.)
 Identify the graph of the function: y = x2 + 5

What are the x-intercepts for
a)
b)
c)
d)

Given:
y  x 2  8x  33 ?
(11, 0), (3, 0)
(11, 0), (3, 0)
(0, 11), (0, 3)
(0, 11), (0, 3)
y  x 2  8x  48 , find the x-intercepts.
* Comprises 70% of the California Standards Test
(Version 1, 5/03)
15

22.0
Given:
y  x 2  8x  33 , find the vertex of the parabola.
**Teachers need to stress the vocabulary that zeros of the function, x-intercepts, and roots
are interchangeable. Be sure students know the graph of a quadratic equation is a parabola.
Students use the quadratic formula or factoring techniques or both to determine whether the
graph of a quadratic function will intersect the x-axis in zero, one, or two points.

The solution to a quadratic problem is x 
5  7
. How many times does this parabola cross
2
the x-axis?
a. zero
b. one
c. two
d. more than three

The solution to a quadratic problem is x 
the x-axis?
a.
b.
c.
d.

5 0
. How many times does this parabola cross
2
zero
one
two
more than three
The solution to a quadratic problem is x 
5 7
. How many times does this parabola cross
2
the x-axis?
a. zero
b. one
c. two
d. more than three
 Find the intercepts:
x2  4 x  4  0
x2  2x  8  0
x2  x  6  0
**Teach discriminant.
* Comprises 70% of the California Standards Test
(Version 1, 5/03)
16