WCCUSD Grade 8 Benchmark 1 Study Guide

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.

.



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



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.


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.



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.


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



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



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



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



8 What is the solution of this system of

equations?

2427

154

yx

yx

8.EE.8b


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).



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




Table

Time

(minutes)

Distance

(feet)

4

40

7

70

10

100

Equation:

The equation for the distance y in feet as a


y = 20x

8.EE.5

End of Study Guide



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



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


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


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´


11458 cc

1144548 cccc

1154 c

511554 c

164 c

4

16

4

4

c

4c


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



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



6´ Given the graph of a system of equations

below. Select all of the statements that are

true about the system.

8.EE.8a


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).




equations?

532

1956

yx

yx

8.EE.8b

9´ Compare the table and equation below to




Table

Time

(minutes)

Distance

(feet)

4

40

7

70

10

100

Equation:

The equation for the distance y in feet as a


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.



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.

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WCCUSD Grade 8 Benchmark 1 Study Guide