wccusd grade 8 benchmark 1 study guide
TRANSCRIPT
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 1 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
1 Convert the following number into a fraction.
0.58
8.NS.1
1´ Convert the following number into a fraction.
0.24
8.NS.1
Solution:
0.58 = 58
100 0.58 is 58 hundredths.
502
292
Decompose the numerator and
50
29 Simplify.
denominator.
.
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 2 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
2 Solve the equation for x: 7x – (6 – 2x) = 12.
x = 2
8.EE.7b
2´ Solve the equation for c: –8 = 9c – (c + 24).
8.EE.7b
Solution: Distributive Property/Inverse Operations
7x – (6 – 2x) = 12 Write original equation
7x – 1(6 – 2x) = 12 Show distributing with “1”
12)]2(6)[1(7 xx Change subtraction to adding (-)
12)2)(1()6)(1(7 xx Distributive Property
122)6(7 xx Simplify
12)6(27 xx Commute like terms
12)6(9 x Combine like terms
6126)6(9 x Inverse operations: zero pairs
9x = 18 Simplify
9
18
9
9
x Inverse operations: division
x = 2 Simplify
Solution: Decomposition
7x – (6 – 2x) = 12 Write original equation
12)2(67 xx Distribute subtraction
126)2(7 xx Commute like terms
9x – 6 = 12 Combine like terms
9x – 6 = 12 + 6 – 6 Add in zero pairs
9x – 6 = 12 + 6 – 6 Simplify
9x = 18 Simplify
299 x Decompose multiplication
x = 2
Solution: Bar Model
x = 2
7x
6 – 2x 12 7x
-2x 18
7x + 2x -2x
-2x 18 9x
18
2 2 2 2
x x x x x
2
x x x x
2 2 2 2
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 3 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
3 Solve 4
3(2y – 8) = 6 for y.
8.EE.7b
3a´ Solve 10435
2r for r.
Select all of the steps that are valid in solving
the equation 12)45(4
3x .
8.EE.7b
Although you can use the distributive property to solve this
type of equation, it is easier to multiply each side of the
equation by the reciprocal of the fraction.
Solution: Decomposition
824
3y = 6 Write the original equation
824
3
3
4 y =
1
6
3
4 Multiply each side by the reciprocal
2y – 8 3
24 Simplify
2y – 8 3
83 Decompose the fraction
2y – 8 = 8 Simplify
2y – 8 = 8 – 8 + 8 Add in zero pairs
2y = 16 Simplify
822 y Decompose multiplication
y = 8 Simplify
Solution: Inverse Operations
824
3y = 6 Write the original equation
824
3
3
4 y =
1
6
3
4 Multiply each side by the reciprocal
2y – 8 3
24 Simplify
2y – 8 = 8 Simplify
2y – 8 + 8 = 8 + 8 Inverse operations: zero pairs
2y = 16 Simplify
2
16
2
2
y Inverse operations: division
y = 8 Simplify
3b´
A) Multiply both sides of the equation by 4
3.
B) Multiply both sides of the equation by 4.
C) Distribute 3
4to 5x, 4, and 12.
D) Distribute 3
4to 5x only.
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 4 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
4 Solve 10x + 18 = 8x + 4 for x.
x = –7
8.EE.7a
4a´ Solve 8c + 5 = 4c – 11 for c.
Select the number of solutions for the
following equations.
8.EE.7a
To solve equations with variables on both sides, collect the
variable terms on one side of the equation and the constant terms
on the other side of the equation. If, while solving the equation,
you are left with a statement without variables that is false, such
as 5 = 0, then the equation has no solutions. If you are left with
one that is always true, such as 5 = 5, it is an identity and has
infinitely many solutions.
Solution: Inverse Operations
10x + 18 = 8x + 4 Write original equation
10x – 8x + 18 = 8x – 8x + 4 Inverse operations: zero pairs
2x + 18 = 4 Combine like terms
2x + 18 – 18 = 4 – 18 Inverse operations: zero pairs
142 x Simplify
2
14
2
2
x Inverse operations: division
7x Simplify
Solution: Bar Model
10x 18
8x 4
8x 2x 18
8x 4
2x 14 4
-14 14 4
2x
-14
x x
-7 -7
4b´
1) 4u = 37 + 4u _____
A One Solution
2) 7x = 5(x – 12) _____
3) 5 + 2x – 9 = 7x – 4 – 5x _____
B No Solutions
4) 5(9 – x) = 4(x + 18) _____
5) 4(r + 1) = 6 – 2(1 – 2r) _____
C Infinitely Many
6) 3(x – 4) – x = 2(x – 6) _____ Solutions
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 5 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
5 Using the graph below, find the slopes of
AC and BE then compare.
8.EE.6
5´ Using the graph below, find the slopes of
BCAD and then compare.
8.EE.6
A
B
C
D
E
Solution: To identify the slope from a graph, locate two
points and use the run
rise ratio or subtract with the slope
formula 12
12
xx
yym
.
AC
BE
3
2
32
22
6
4
run
risem
3
2
33
32
9
6
run
risem
The slopes are equal. The slope between any two points on
the same line are equal.
A
B
C
D
E
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 6 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
6 Find the system of equations for the graph
below. Identify the solution(s).
8.EE.8a
6´ Given the graph of a system of equations
below. Select all of the statements that are
true about the system.
8.EE.8a
A) There are no solutions.
B) The system graphed is y = x + 2
y = 3x + 4
C) There is one solution at (0, 2).
D) There are an infinite number of solutions.
E) The solution is at (1, –1).
Solution: The solution to a linear system is the point(s) that is
a solution(s) to both linear equations. This means that any
point that is a solution will be a point that lies on both lines
(or the point of intersection). These lines intersect at the point
(–2, –2), therefore this point is the solution.
Line 1 has a y-intercept of 1 and a slope of 3
2. Using the
slope-intercept form its equation is y = 3
2x + 1. The equation
of Line 2 is2
7
4
3 xy . When the solution (–2, –2) is
substituted into both equations, we get a true statement.
y-int.
(0,1)
3
2
y-int.
)2
7,0(
–3
4
Graph of the system:
y = 3
2x + 1
2
7
4
3 xy
line 1
line 2
line 1
line 2
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 7 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
7 What is the solution of this system of
equations?
3
925
yx
yx
8.EE.8b
7´ What is the solution of this system of
equations?
8.EE.8b
Equation 1
Equation 2
Solution: Since one variable (x) is already solved for in the
second equation, we can use the substitution method to solve
this system.
Substitute “ – y – 3” for x in the other equation and solve
for y.
925 yx Write Equation 1
92)3(5 yy Substitute “ 3 y ” for x
92)3(5)(5 yy Distributive Property
92155 yy Multiply
91525 yy Commute terms
9153 y Combine like terms
15915153 y Inverse operation: zero pairs
243 y Simplify
3
24
3
3
y Inverse operation: division
8y Simplify
Substitute (– 8) for y in either equation and solve for x.
925 yx 3 yx Write the equation
9)8(25 x 3)8( x Substitute – 8 for y
9165 x 38x Multiply
16916165 x Inverse operations:
255 x Simplify
5
25
5
5
x Inverse operations:
5x 5x Simplify
The solution of the linear system of equations is the point (5, – 8).
3686
156
yx
xy
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 8 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
8 What is the solution of this system of
equations?
2427
154
yx
yx
8.EE.8b
8´ What is the solution of this system of
equations?
532
1956
yx
yx
8.EE.8b
Solution: Label the equations.
4x + y = 15 Equation 1
7x + 2y = 24 Equation 2
Multiply Equation 1 by (– 2) so that the coefficients of y are
opposites.
2427
154
yx
yx
2427
3028
yx
yx
– 8x – 2y = – 30
7x + 2y = 24 Add the equations
– 1x = – 6
1
6
1
1
x Inverse Operation: division
6x Simplify
Substitute 6 for x in either of the original equations and solve for y.
4x + y = 15 7x + 2y = 24 Write the equation
4(6) + y = 15 7(6) + 2y = 24 Substitute 6 for x
24 + y = 15 42 + 2y = 24 Multiply
24 – 24 + y = 15 – 24 42 – 42 + 2y = 24 – 42 Inverse Operations:
zero pairs
y = – 9 2y = – 18 Simplify
2
18
2
2
y Inverse Operations:
y = – 9 Simplify
The solution to this system is (6, – 9).
Solution: Bar Models
Substitute 6 for x The solution is
to solve for y: (6, – 9).
4x y
15
7x 2y
24
3x + 4x y + y
24
4x + y 3x + y
15 9
x 3x + y
6 9 9
4x
3x + y
y
15
24 y
24 –9
(-2)
Substitute 6 for x
and solve for y:
The solution
is (6, –9).
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 9 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
9 Compare the table and equation below to
determine which represents a greater speed.
Include a description of each that discusses
unit rates in your explanation.
Table
Time
(hours)
Distance
(miles)
2
60
3
90
5
150
Equation:
The equation for the distance y in miles as a
function of the time x in hours is:
y = 25x
Explanation: The table represents a greater speed
because they would travel 30 miles per hour. This
is found by finding the rate of change from 2
points from the data: .301
30
23
6090
The
equation shows the speed at 25 miles per hour.
Therefore the table shows a faster speed by 5
miles per hour.
8.EE.5
9´ Compare the table and equation below to
determine which represents a greater speed.
Include a description of each that discusses
unit rates in your explanation.
Table
Time
(minutes)
Distance
(feet)
4
40
7
70
10
100
Equation:
The equation for the distance y in feet as a
function of the time x in hours is:
y = 20x
8.EE.5
End of Study Guide
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 10 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
You Try Solutions:
1´ Convert the following number into a fraction.
0.24
8.NS.1
2´ Solve the equation for c: –8 = 9c – (c + 24).
8.EE.7b
Solution:
0.24 = 100
24 0.24 is 24 hundredths
5522
3222
Decompose into prime factors
25
6 Simplify
Solution: Distributive Property/Inverse Operations
)24(98 cc
)24(198 cc
)24)(1())(1(98 cc
)24()1(98 cc
)24(88 c
24)24(8248 c
c816
8
8
8
16 c
c2
2c
Solution: Bar Model
c = 2
9c
-8 c + 24
9c
c 16
c
c
8c
16
c c c c c c c c
2 2 2 2 2 2 2 2
2
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 11 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
3a´ Solve 10435
2r for r.
Select all of the steps that are valid in solving
the equation 12)45(4
3x .
8.EE.7b
4a´ Solve 8c + 5 = 4c – 11 for c.
c = –4 8.EE.7a
Solution: Inverse Operations
10)43(5
2r
102
5)43(
5
2
2
5 r
2
5043 r
2543 r
425443 r
213 r
3
21
3
3
r
7r
Solution: Decomposition
10)43(5
2r
102
5)43(
5
2
2
5 r
2
25543
r
2543 r
42143 r
213 r
733 r
7r
A Multiply both sides of the equation by 4
3.
B Multiply both sides of the equation by 4.
C) Distribute 3
4to 5x, 4, and 12.
D) Distribute 3
4to 5x only.
3b´
Solution: Inverse Operations
11458 cc
1144548 cccc
1154 c
511554 c
164 c
4
16
4
4
c
4c
Solution: Decomposition
11458 cc
114544 ccc
1154 c
551154 c
164 c
444 c
4c
Solution: Bar Model
c = – 4
8c + 5
4c - 11
-11-5
4c 4c
4c
+5
+5
4c
-16
c c c c
-4 -4 -4 -4
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 12 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
4b´ Select the number of solutions for the
following equations.
8.EE.7a
5´ Using the graph below, find the slopes of
AD and BC then compare.
8.EE.6
1) 4u = 37 + 4u __B__
A One Solution
2) 7x = 5(x – 12)__A__
3) 5 + 2x – 9 = 7x – 4 – 5x __C__
B No Solutions
4) 5(9 – x) = 4(x + 18) __B__
5) 4(r + 1) = 6 – 2(1 – 2r) __C__
C Infinitely Many
6) 3(x – 4) – x = 2(x – 6) __C__ Solutions
Solution: To identify the slope from a graph, locate two
points and use the run
rise ratio.
2
3
6
:
run
risemAD
2
1
2
:
run
risemBC
The slopes are equal. The slope between any two points on
the same line are equal
A
B
C
D
E
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 13 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
6´ Given the graph of a system of equations
below. Select all of the statements that are
true about the system.
8.EE.8a
7´ What is the solution of this system of
equations?
8.EE.8b
3686
156
yx
xy
Solution: Since one variable (y) is already solved for in the first
equation, we can use the substitution method to solve this system.
Substitute “6x – 15” for y in the other equation and solve for x.
3686 yx Write Equation 2
36)156(86 xx Substitute “ 156 x ” for
36)15)(8()6(86 xx Distributive Property
36120486 xx Multiply
3612042 x Combine like terms
1203612012042 x Inverse operation: zero pairs
8442 x Simplify
42
84
42
42
x Inverse operation: division
2x Simplify
Substitute 2 for x in either equation and solve for y.
156 xy 3686 yx Write the equation
15)2(6 y 368)2(6 y Substitute 2 for x
1512y 36812 y Multiply
2412812 y Decomposition
248 y Simplify
388 y Decomposition
3y 3y Simplify
The solution of the linear system of equations is the point
(2, – 3).
A) There are no solutions.
B The system graphed is y = x + 2
y = 3x + 4
C) There is one solution at (0, 2).
D) There are an infinite number of solutions.
E) The solution is at (1, –1).
WCCUSD Grade 8 Benchmark 1 Study Guide
Page 14 of 14 MCC@WCCUSD (WCCUSD) 10/19/16
8´ What is the solution of this system of
equations?
532
1956
yx
yx
8.EE.8b
9´ Compare the table and equation below to
determine which represents a greater speed.
Include a description of each that discusses
unit rates in your explanation.
Table
Time
(minutes)
Distance
(feet)
4
40
7
70
10
100
Equation:
The equation for the distance y in feet as a
function of the time x in hours is:
y = 20x
Explanation: The equation represents a greater
speed because they would travel 20 feet per
minute. This speed in the table is only 10 feet per
minute. This is found by finding the rate of
change from any 2 points in the data:
.106
60
410
40100
8.EE.5
Solution: Label the equations.
6x + 5y = 19 Equation 1
2x + 3y = 5 Equation 2
Multiply Equation 2 by – 3 so that the coefficients of y are
opposites.
532
1956
yx
yx
1596
1956
yx
yx
6x + 5y = 19
– 6x – 9y = –15 Add the equations
–4y = 4
4
4
4
4
y Inverse Operation: division
1y Simplify
Substitute –1 for y in either of the equations and solve for x.
6x + 5y = 19 2x + 3y = 5 Write the equation
6x + 5(–1) = 19 2x + 3(–1) = 5 Substitute –1 for y
6x – 5 = 19 2x – 3 = 5 Multiply
6x – 5 + 5 = 19 + 5 2x – 3 + 3 = 5 + 3 Inverse Operations:
6x = 24 2x = 8 Simplify
466 x 422 x Decomposition
x = 4 x = 4 Simplify
The solution to this system is (4, – 1).
Solution: Bar Model
(-3)
6x 5y
19
2x 3y
5
2x + 4x 3y + 2y
19
4x + 2y 2x + 3y
14 5
2x + y 2x + y 2x + 3y
7 7 5 2x + y 2x + y 2x + y 2y
7 7 7 –2
y y
–1 –1
2x 3y
5
x x –3
4 4 –3
The solution to this
system is (4, –1).
Substitute
–1 for y.