wccusd grade 8 benchmark 1 study guide · 2015. 9. 30. · wccusd grade 8 benchmark 1 study guide...

WCCUSD Grade 8 Benchmark 1 Study Guide

of 14 MCC@WCCUSD (WCCUSD) 09/29/15

1 Convert the following numbers into fractions. a) 0.58 b) 0.16

8.NS.1

1a´ Convert the following numbers into fractions. a) 0.009 b) 2.4 Choose whether each number is Rational or Irrational.

8.NS.1

Solution:

a) 0.58 = 58100

0.58 is 58 hundredths.

=2 ⋅292 ⋅50

Decompose the numerator and

=2950

Simplify.

b) Assign a variable to the value 0.16 , such as N. Multiply that value by a power of 10 according to how many numbers in N repeat. Since 2 numbers repeat, we will multiply by 100 because it has 2 zeros. N = 0.16 Assign a variable 100(N) = 100(0.16 ) Multiply by a power of 10 100N = 16.16 Simplify 100N – N = 16.16 – 0.16 Subtract N from both sides 99N = 16 Simplify 99N = 16 Divide to solve for N 99 99

N = 1699

Simplify

denominator. .

1b´

1) –5 A Rational B Irrational 2) 0 A Rational B Irrational 3) 3.67 A Rational B Irrational 4) 35.18 A Rational B Irrational 5) 3.955611632… A Rational B Irrational 6) π A Rational B Irrational



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

918

99

=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 43 (2y – 8) = 6 for y.

8.EE.7b

3a´ Solve ( ) 104352

=+r for r.

Select all of the steps that are valid in solving

the equation 34(5x − 4) =12 .

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

( )8243

−y = 6 Write the original equation

( )8243

34

−• y = 16

34• Multiply each side by the reciprocal

2y – 8 324

= Simplify

2y – 8 383•

= 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 ( )82

43

−y = 6 Write the original equation

( )8243

34

−• y = 16

34• Multiply each side by the reciprocal

2y – 8 324

= Simplify

2y – 8 = 8 Simplify 2y – 8 + 8 = 8 + 8 Inverse operations: zero pairs 2y = 16 Simplify

216

22

=y Inverse operations: division

y = 8 Simplify

3b´

A) Multiply both sides of the equation by 43.

B) Multiply both sides of the equation by 4.

C) Distribute 34

to 5x, 4, and 12.

D) Distribute 34

to 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. 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

214

22 −

=x Inverse operations: division

7−=x 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 AD and BC then compare.

8.EE.6

!

A

B

C

D

E

!

Solution: To identify the slope from a graph, locate two points and use the

runrise ratio or subtract with the slope

formula 12

12

xxyym

−

−= .

AC

BE

m =

riserun

=46

=2 ⋅22 ⋅3

=23

m =riserun

=69

=2 ⋅33⋅3

=23

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 = 32

x + 1. The equation

of Line 2 is y = − 34x − 72

. When the solution (–2, –2) is

substituted into both equations, we get a true statement.

!

y-int. (0,1)

3

2

y-int.

(0,− 72) –3

4

Graph of the system: y = 3

2x + 1

y = − 34x − 72

line 1

line 2

line 1

line 2



7 What is the solution of this system of equations?

8.EE.8b

7´ What is the solution of this system of equations?

8.EE.8b

5x + 2y = 9x = −y−3"#$

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

324

33

−=

−− y Inverse operation: division

8−=y 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 38−=x Multiply

16916165 +=+−x Inverse operations: 255 =x Simplify

525

55

=x Inverse operations:

5=x 5=x Simplify The solution of the linear system of equations is the point (5, – 8).

y = 6x −156x −8y = 36"#$



8 What is the solution of this system of equations?

4x + y =157x + 2y = 24!"#

8.EE.8b


6x + 5y =192x +3y = 5!"#

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.

4x + y =157x + 2y = 24!"#

−8x − 2y = −307x + 2y = 24"#$

– 8x – 2y = – 30 7x + 2y = 24 Add the equations – 1x = – 6

16

11

−−

=−− x Inverse Operation: division

6=x 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

218

22 −

=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: 90− 603− 2

=301= 30. 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



You Try Solutions: 1a´ Convert the following numbers into fractions.

a) 0.009 b) 2.4 Choose whether each number is Rational or Irrational.

8.NS.1

2´ Solve the equation for c: –8 = 9c – (c + 24).

8.EE.7b

1) –5 A Rational B Irrational 2) 0 A Rational B Irrational 3) 3.67 A Rational B Irrational 4) 35.18 A Rational B Irrational 5) 3.955611632… A Rational B Irrational 6) π A Rational B Irrational

1b´

Solution: a) Assign a variable to the value 0.009 , such as N. Multiply that value by a power of 10 according to how many numbers in N repeat. Since 3 numbers repeat, we will multiply by 1000 because it has 3 zeros. N = 0.009 Assign a variable 1000(N) = 1000(0.009 ) Multiply by a power of 10 1000N = 9.009 Simplify 1000N – N = 9.009 – 0.009 Subtract N from both sides 999N = 9 Simplify 999N = 9 Divide to solve for N 999 999

N = 1111

Simplify

b) 2.4 = 2 410

2.4 is 2 and 4 tenths

= 2 2 ⋅22 ⋅5

Decompose the numerator and

= 2 25

Simplify

denominator

Solution: Distributive Property/Inverse Operations

)24(98 +−=− cc )24(198 +−=− cc

−8 = 9c+ (−1)(c)+ (−1)(24) )24()1(98 −+−+=− cc

)24(88 −+=− c 24)24(8248 +−+=+− c

c816 =

88

816 c

=

c=2 2=c

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 22



3a´ Solve ( ) 104352

=+r for r.

Select all of the steps that are valid in solving

the equation 34(5x − 4) =12 .

8.EE.7b

4a´ Solve 8c + 5 = 4c – 11 for c.

∴c = –4 8.EE.7a

Solution: Inverse Operations

10)43(52

=+r

1025)43(

52

25

•=+• r

25043 =+r

2543 =+r 425443 −=−+r

213 =r

321

33

=r

7=r Solution: Decomposition

10)43(52

=+r

1025)43(

52

25

•=+• r

225543 ••

=+r

2543 =+r 42143 +=+r

213 =r 733 •=• r ∴ 7=r

A Multiply both sides of the equation by 43.

B Multiply both sides of the equation by 4.

C) Distribute 34

to 5x, 4, and 12.

D) Distribute 34

to 5x only.

3b´

Solution: Inverse Operations

11458 −=+ cc 1144548 −−=+− cccc

1154 −=+c 511554 −−=−+c

164 −=c

416

44 −

=c

4−=c Solution: Decomposition

11458 −=+ cc 114544 −=++ ccc

1154 −=+c 551154 −+−=+c

164 −=c 444 −•=• c ∴ 4−=c

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

6´ Given the graph of a system of equations below. Select all of the statements that are true about the system.

8.EE.8a

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

!

A B C D E

Solution: To identify the slope from a graph, locate two

points and use the runrise ratio.

AD :m =riserun

= 09

= 0

BC: m =riserun

= 04

= 0

The slopes are equal. The slope between any two points on the same line are equal.

!

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).




8.EE.8b


6x + 5y =192x +3y = 5!"#

8.EE.8b

y = 6x −156x −8y = 36"#$

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

4284

4242

−−

=−− x Inverse operation: division

2=x 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 1512 −=y 36812 =− y Multiply 2412812 +=− y Decomposition 248 =− y Simplify 388 −•−=•− y Decomposition 3−=y 3−=y Simplify The solution of the linear system of equations is the point (2, – 3).

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.

⎩⎨⎧

=+

=+

5321956

yxyx

⎩⎨⎧

−=−−

=+

15961956yxyx

6x + 5y = 19 – 6x – 9y = –15 Add the equations –4y = 4

44

44

−=

−

− y Inverse Operation: division

1−=y 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.



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: 100− 4010− 4

=606=10.

8.EE.5

wccusd grade 8 benchmark 1 study guide · 2015. 9. 30. · wccusd grade 8 benchmark 1 study guide...

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