sagamore middle school math 8 quarter 1 · quarter 1 name:_ # 1 math 8 unit 1- integers...

0

Sagamore Middle School

Math 8

Quarter 1

Name:_____________________ #____

1

Math 8

Unit 1- Integers

2019-2020

Date Lesson Topic Homework

First Day Return Signed Contract

1 Introduction to Integers Lesson 1- page 4

2 Operations with Integers Lesson 2 – page 9

3

Order of Operations Lesson 3 – page 12

4

Translating Algebraic Expressions

Quiz (Integers)

Lesson 4 – page 15

5

Evaluate Algebraic Expressions

(Celsius/Fahrenheit)


6

Combine Like Terms (Perimeter

Problems)


7 Distributive Property (Area Problems) Lesson 7 – page 24

Review Study!

Test Good Luck!

2

Unit 1- Lesson 1

Aim: I can identify the determine the different number sets that make up the Real Number System

Warm Up: What are the different types of numbers that make up the Real Number System?

Number set Definition Examples Non-Examples

Natural numbers

Whole numbers

Integers

Rational numbers

Irrational numbers

3

Guided Practice:

Problem Set:

Name the Integer

1) 5 degrees above 0 _____

2) a loss of 2 yards _____

3) a withdrawal of $25 _____

4) a deposit of $25 _____

5) 3 inches of rainfall _____

Put in order from least to greatest

6) 3, -8, 0, 11, -7 ____________________________

7) -11, 12, 9, -8, -1 ____________________________

Inequality Symbols

< Less than

≤ Less than or equal to

> Greater than

≥ Greater than or equal to

Use < or > to make a true statement

8) -2 ____-8 9) 0 ____-6 10) -5____-2 11) -8____-10

4

Unit 1 - Lesson 1 Homework

Write the number that describes each situation:

1) loss of 5 kg _____ 2) gain of 3 kg _____ 3) rise of 1,500 ft in elevation _____

4) 15 feet below sea level _____ 5) 12 inches of snowfall ____

Put in order from least to greatest

6) 10, -5, 1, 12, 0 _______________________ 7) 55, -25, -11, 52, -74 ______________________

Compare the integers using < or >

8) 7 ____ 4 9) 6 ____ -13 10) -6 ____ -3 11) 7 ____ -3

12) -2 ____ 11 13) -14 ____ -10 14) -4 ____ -9 15) -8 ____ 0

16) What type of numbers are used to represent temperatures below zero?

17) Circle all the numbers that are integers and explain why you chose them.

25 1½ -18 0 - 3

4 4

1

3

24

4 -17

37

5

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

18) List the first five non-negative integers.

19) List the first five positive integers.

20) 8 ∙3

2 21) 5 ∙

2

5 22) 8 ∙

7

8 23) 6 ∙

2

3

5

Unit 1 - Lesson 2

Aim: I can add, subtract, multiply and divide integers

Warm Up: You buy a belt for $10 and some socks. Each pair of socks costs $3. The total shipping cost is $3.

What is the total cost if you buy 11 pairs of socks?

A) $36 B) $43 C) $46 D) $50

Guided Practice:

Adding Integers

Signs Same Signs Different

1) _____________________________________

2) _____________________________________

1) _____________________________________

2) _____________________________________

1) -2 + -3 2) -2 + 8 3) -6 + 1 4) -4 + 4

5) 6 + (-3) 6) 10 + -5 + 3

Subtracting Integers

1) _______________________________________________________________________

2) _______________________________________________________________________

1) 6 – 10 2) -3 – 8 3) 4 - -6 4) -2 – (-8)

5) -2 - -2 6) 7 – (-3)

6

Multiplying and Dividing Integers

Odd number of negative signs Answer – Negative

Even number of negative signs

Answer – Positive

Steps for Multiplying and Dividing

1) ___________________________________________________________

2) ___________________________________________________________

1) (-6)(-2 ) 2) (5)(-3) 3) (-5)(0) 4) (6)(-2)(3)(-1)(-4)

5) (-2)(1)(-3)(4) 6) (-1)³ 7) (-1)246 8) (-1)485

9) -25

5 10) -36 ÷ -9 11)

0

8 12)

8

0

Zero Rules

When 0 is in the numerator of a fraction the answer is _________________.

When 0 is in the denominator of a fraction the answer is _______________.

*** Remember ….. When 0 is Underneath the answer is U____________

7

Using a Calculator

How to input fractions into the calculator

You must use the 𝑎𝑏

𝑐 button!

Use (−) if you need to make a number negative.

• Simple fractions such as 𝟏

𝟐 are entered as:

1 𝒂𝒃

𝒄 2

• Mixed numbers such as −𝟏𝟏

𝟐 are entered as:

(−) 1 𝒂𝒃

𝒄 1 𝒂

𝒃

𝒄 2

Examples:

1) 𝟏

𝟒+𝟐

𝟑=

Enter the following: 1 𝒂𝒃

𝒄 4 + 2 𝒂

𝒃

𝒄 3 = _________

2) 𝟏𝟑

𝟒 − 𝟐

𝟏

𝟑=


𝒄 3 𝒂

𝒃

𝒄 4 - 2 𝒂

𝒃

𝒄 1 𝒂

𝒃

𝒄 3 = _________

3) 𝟔𝟑

𝟒 ÷ −

𝟏

𝟐=


𝒄 3 𝒂

𝒃

𝒄 4 ÷ (−) 1 𝒂

𝒃

𝒄 2 = _________

4) 𝟓𝟏

𝟐×𝟐

𝟑

𝟒=


𝒄 1 𝒂

𝒃

𝒄 2 × 2 𝒂

𝒃

𝒄 3 𝒂

𝒃

𝒄 4 = _________

8

Problem Set:

Adding Integers

1) 8 + -7 2) – 8 + 2 3) -1 + -7 4) -3 + (-3)

5) 6 + (-6) 6) -2 + - 5 + 3

Subtracting Integers

1) -12 - 7 + 4 2) -5 – 6 + 2 – 11 3) 5 - 12 4) 8 – (-9)

5) -7 – (-10) 6) -4 - -4 7) -8 + 12 – 9 + 4 8) 5 – 6 + 2 – 11

9) 8 – 9 + 3 – 7

Multiply and Divide

1) (6)(-7) 2) (-3)(-9) 3) (-9)(0) 4) (-4.2)(0.3)

5) -100 ÷ -10 6) 35÷ -7 7) -12 ÷ 0 8) 0 ÷ 17

9) 6 −1

6 10) −3

5

9 ÷ 3 11) −2

2

3 × 4

1

10 12)

4

5 + 0

9


Simplify:

Multiply and Divide

1) (-3)(9) 2) (-5)(-1) 3) (-9)(0) 4) (-7)(-3)

5) (-24)(-5/8) 6) (-2/3)(-18) 7) (7)(-1)(0)(-8) 8) (-4)(-23)

9) -52

5 10) -18 ÷ -9 11)

0

7 12)

3

0

Using a Calculator

13) 3

5−2

3 14)

1

3÷ (−

1

3) 15)

5

9 ×

7

15 16)

1

5+ (−

1

5)

17) 7

6−5

6 18) 0 ÷

4

9 19) -2 ×

3

7 20) −5

5

8×−4

2

10

Add and Subtract

21) -5 – 6 22) (-5)+(-6) 23) (-2)-(8) 24) -2 + 8

25) 12 - 4 26) -4 + 12 27) -9 – 3 28) -9 ∙ -3

Word Problems

29) An elevator began on the fourth floor. It went up 6 floors, dropped 3 floors, and then went up another two

floors. What floor did the elevator stop on?

30) On Monday afternoon the temperature was 6°. That night it dropped 8°. What was the temperature on

Tuesday morning?

31) In the morning, Mrs. Boxer deposited $135 to her bank account. She withdrew $235 in the afternoon.

What number describes her account’s net change?

10

Unit 1 - Lesson 3

Aim: I can simplify numerical expressions

Warm Up:

Does the order in which these operations are done make a difference?

6 + 3 x 4 6 + 3 x 4

9 x 4 = 36 6 + 12 = 18 Which is correct, and why?

Guided Practice:

1) 5 + 2 x 6 2) 10 ÷ 5 x 2 3) 7(1 + 2) – 5 ÷ 5 4) 3 · 4 + 8

15 – 2 · 5

5) -4 + (-5)(3) 6) -9 + (-4)(-2) 7) )3612(58 8) )28(5242

11

Problem Set:

9) 7)4(2)2(3 2 10) 33

16)5(42

11) )212(734 2 12) 4)9(

3

2

13) 2(-3)2 14) 2 - 32 15) 2 - (-3)2 16) (-7)(4) – 8 17) 0.34 + 2.4(3)

18) -32 19) (-3)2 20) -33 21) (-3)3 22) (-3)4

23) 2 + 7 ∙ 4 – 15 ÷ 3 + 7 24) 20 ÷ 4 + 3 ∙ 6 – 12 ÷ 4 25) 8(4) + 9 ÷ 3 – 1 ∙ 5

12


1) 8 + 4 ∙ 2 2) 16 – 32 ÷ 4 3) 5 + 7(2) 4) 3(8 – 4) + 6

5) 7 ∙ 8– 4 ÷ 2 + 5 ∙ 6 6) 14 – 16 ÷ 8 + 9(5) 7) 5(9)+18÷3

2+3(5) 8) 9 – 14 ÷ 2 + 3 ∙ 4

9) (4

5)(35) – 14 ÷ 7 10) (7 – 5)6 + 4 11) (0.9)(0.2) + (0.6)(0.4) 12) 10 – 3(5 – 2)

13) 735)43(2 14) 54215 3 15) 24 ÷ 6 + 33 – 5(2) + (36 ÷ 4)

16) How do we simplify a numerical expression?

17) Which operation would be performed first when simplifying 2 × (3 – 8 + 5) ÷ 12?

18) Which set of numbers would be most appropriate to use to represent the number of household online-

devices a family has in their home?

A) Integers B) whole numbers C) rational numbers

13

Unit 1- Lesson 4

Aim: I can translate verbal expressions to algebraic expressions

Warm Up:

1) -4 + (-5)(3) 2) Which answer is greater? (3x)² or 3x² when x = 2

3) What is the difference between an expression and an equation?

___________________________________________________________________________________

___________________________________________________________________________________

Guided Practice:

x + 5

x plus 5

the sum of x and 5

x increased by 5

5 more than x

5 added to x

x – 5

x minus 5

the difference of x and 5

x decreased by 5

5 less than x

5 subtracted from x

5x

5 times x

the product of 5 and x

5 multiplied by x

5

x

5 divided by x

The quotient of 5 and x

Remember….

Translate and Simplify

1) The product of -2 and -6 2) 7 subtracted from -10 3) 8 less than 10

More Than / Less Than

___________________

Subtracted From

___________________

14

Write an algebraic expression for each

1) 3 times a number plus 6 2) 4 less than a number times 2

3) x divided by 8 4) 12 subtracted by x

Math each phrase with the correct algebraic expression

___ 5) n increased by 11 A) n – 19

___ 6) 11 less than n B) n + 11

___ 7) n subtracted by 19 C) n + 19

___ 8) n less than 11 D) n – 11

___ 9) n increased by 19 E) 19 – n

F) 11 - n

Problem Set: Write each as an algebraic expression (use x for the variable if it is not given) (*Standard form*)

10) m increased by 8 11) 4 less than c 12) the sum of b and 14

13) twice Don’s age increased by 8 14) 40 more than Meg’s bowling score

15) Abe’s savings decreased by $540 16) Bill’s batting average increased by 12

17) Bert’s cab company charges $1.00 plus and additional $3.00 per mile for a ride. Write an expression to

represent the total charge for m, miles.

18) Madeline’s cab company charges $3.00 plus and additional $2.00 per mile for a ride. Write an expression

to represent the total charge for m, miles.

19) To rent a paddleboat in Belmont State Park, the cost is $7.00 plus $3.50 per hour. Write an expression to

present the cost of the rental for h, hours.

20) An amusement park charges $5 for admission and $2 for each ride. Write an expression for the total cost

of admission and r rides.

15


1) w decreased by 4 2) the sum of m and 3 3) 9 less than x 4) 7 less than three times x

5) 24 less than q 6) 5 decreased by 3 times a number 7) 12 more than twice m

8) 8 less than a number divided by 5 9) $60 more than 6 times Ben’s salary

10) The sum of 25 and 3 times Joe’s age 11) 3° more than the current temperature

12) 13 less than 6 times a number 13) 7 more than twice the length of a rectangle

14) A hot-air balloon is at a height of 2,250 feet. It descends 150 feet each minute. Write an expression for the

balloon’s height at m minutes.

15) A hiking club charges $60 to join and $12 for each hiking trip. Write an expression to model the total cost

of h hiking trips.

16) A florist divides 60 roses into equal bunches of f, flowers. Write an algebraic expressions for the number

of bunches the florist can make.

17) An object’s weight on Mars is 0.38 times the object’s weight on Earth. If e represent the weight of an

object on the earth, write an algebraic expression for an object’s weight on Mars.

#18-20 Translate and Simplify:

18) The difference of 8 and -9 19) The quotient of -36 and 12 20) The sum of -5 and -8

21) An elevator began on the fourth floor. It went up 6 floors, dropped 3 floors, and then went up another two

floors. What floor did the elevator stop on?

22) On Monday afternoon the temperature was 6°. That night it dropped 8°. What was the temperature on

Tuesday morning?

23) Explain the difference between the way you represent “7 decreased by x” and “7 less than x”. Is there any

value of x for which the two representations are the same?

16

Unit 1 - Lesson 5

Aim: I can evaluate algebraic expressions

Warm Up: Simplify

1) -43 + 6 + 13 2)-5 + 1 – 8 + 7 3) 7 · 5 + 62 4) 60 – 3 · 42

Guided Practice:

Algebraic Expression vs. Algebraic Equation

Parts of an Expression

Name the coefficient: _______ Coefficient - ________________________________________

Name the variable: _______ Variable – __________________________________________

Name the exponent:_______ Exponent – _________________________________________

Name the constant: ________ Constant - ________________________________________

Name the base: __________ Base - ________________________________________________

Evaluating Algebraic Expressions

1 - __________________________________________________________________________

2 - __________________________________________________________________________

Evaluate the following expressions:

1) 3x - 2 for x = 4 2) 5(x + 3) when x = 2 3) -13 - 5x if x = -2

4) 2a5 if a = 3 5) (2a)5 when a = 3 6) 3𝑎 + 6

2𝑏 −3𝑐 if a = 8, b = 6, c = 2

3𝑥2 + 5

17

Conversion Formulas

Fahrenheit Celsius Celsius Fahrenheit

)32(9

5 FC 32

5

9 CF

Convert the following temperatures:

7) 50 ° C = _____ F 8) 113 ° F = _____ C

Problem Set:

9) 5x – 2y if x = 3; y = 6 10) 2

5 x + 3 for x = 20 11) 3x + 5y - 8 for x = 3.1; y = -.8

12) 19 - 2m for m = -6 13) ab4 if a = 3; b = -2 14) 2x2 if x = -3

15) (2x)2 when x = -3 16) (4x + 3y)2 + 9 for x = -1

2 and y = -

2

3 17)

3𝑎−8

2𝑏+4 for a = -2 and b = -3

Convert the following temperatures:

18) 77 ° F = _____ C 19) 35 ° C = _____ F

18


1) −𝑥2 for x = -5

2) 6x – 4y for x = −2

3, y = -3

3) 3x+2y

2x+y for x = -4, y = 3

4) Given the formula h = 60t – 5t2, to answer the following question. If an object is shot upward from the

ground, what is its height (h) above the ground after 5 seconds (t)? (t = 5)

5) The formula for area of a trapezoid is 1

2ℎ(𝑏1 + 𝑏2). Find the area of a trapezoid when h = 8,

𝑏1 = 5, 𝑎𝑛𝑑 𝑏2 = 7

6) Given: A = 1

4 B = -4

Which of the following results as an integer?

A) A + B B) A – B C) AB D) A ÷ B

7) What is 194 ° F in Celsius? 8) What is 120 ° C in Fahrenheit?

19

Unit 1 – Lesson 6

Aim: I can express the perimeter of a polygon algebraically

Warm Up:

1) How much fencing does Jeremy need to make a dog run in his yard?

10ft 8ft

Simplifying algebraic expressions: (Combine like terms)

2) 6x + 3x 3) 5x + x 4) 2x + 7 5) -9x + 4x

6) 9y – 3 + 6y – 8 7) 9x + 4y 8) -7x + 7x + 3 9) -4x – -3x

Guided Practice:

Perimeter of polygons ______________________________________________________________________

__________________________________________________________________________________________

Express the perimeter in simplest terms of x.

1) 2) 3)

x

3x - 1

2x + 3

4x + 3

6x + 2

5

9x

20

Problem Set:

4) 5)

2x – 1

3x + 2 x

6) Express the perimeter of a triangle whose sides are x + 8, x + 5, and 2x – 6 in simplest terms of x?

7) Find the missing side of the triangle if the perimeter is 6x + 8

represented as 15x + 12 2x - 1

21


Express the perimeter in simplest terms of x:

1) 2)

3) 4)

5) Express the perimeter of a quadrilateral whose sides are 7x - 10, 4x + 8, 2x + 12, and 5x – 10 in simplest

terms of x.

6) Evaluate 5x – 2y – 7 for x = -2, y = 4 7) When is the sum of a positive integer and a negative

integer negative?

8) Which expressions are equivalent?

A) 6x + 7y – 10x B) 7y +7x + 3x C) 8x + 2x + 7y D) -4x + y + 6y

Rational or Irrational

9) 2

9 10) -98 11) √16 12) √20 13) 5𝜋

2x + 3

5x – 7y

4x + 3y

x

2x + 3

2x + 1

2x + 1

22

Unit 1 - Lesson 7 Aim: I can simplify algebraic expressions using the Distributive Property

Warm Up:

1) Express the perimeter in terms of x: 2) Simplify and Evaluate: when x = 2 and y = 3

6 + 7x + 9y + 3x – 2 – 4y

Guided Practice

Distributive Property: Express in simplest terms of x.

1) 2(x + 3) 2) 2(4x – 5y) 3) 3(4x + 5y – 6z + 8)

4) - (-2x + 4) 5) -6(7k + .5) 6) 1

3 (15x + 27)

7) 3

4 (7n + 1) 8) -4(1 + 11x) + 20x 9) -3(5x – 1) -8x

23

Problem Set:

10) 9(3 – 10n) – 3(10n + 1) 11) 3 – 2(3x + 5) 12) 8 – 3(2x – 7) + 5x

13) -4(3x – 3) + 9(x + 1) 14) 10.8(x - 3.6) 15) – 5

4 ( 4n – 3) 16) -2( -8x – 10)

Area of rectangles : Express the area in simplest terms of x

17) 18) 18) 5 7 5

2x + 3 x + 6

24


Express in simplest terms of x.

1) 5(3x – 4) 2) 3(2x + 7y) 3) -6(1 + 11b) 4) -10(a – 5)

5) - (-6x + 9) 6) -2(3x + .6) 7) 4(7 – 5n) – 2(3n + 4) 8) 7(2x2 – 3x ) + 10x

9) -9(3 – 10n) – 3 10) 1

3 (7n + 1) 11) 2(1 – 4.3k) – 2

12) Express the area of the rectangle. 13) Express the perimeter of the triangle.

14) It the width of a rectangle is 6 and length is (x + 7), express the area in simplest terms of x.

15) If Lisa’s yard has a length of 9 and a width of x – 2.

A) Express the amount of fence she would need to enclose her yard in

terms of x.

B) Express the amount fertilizer she will need in terms of x.

9x + 7

4 5x - 6 2x - 8

3x

25

Unit 1 Review

Write the number that best describes the situation:

1) A gain of 20 yards 2) A withdrawal of 100 dollars

Compare: Use < or >

3) -10 _____ -9 4) -1 _____ -7

Simplify (round to the nearest tenth if necessary)

5) -15 + 8 6) 5 – (-6) 7) 10 + ( -6) 8) - 22 – 13 + - 6

9) - 92

3 – (-3) 10) (-4)(-5

1

2) 11) (-1)(5)(-3) 12)

0

8

13) 26 ÷ - 13 14) (-0.27)(-0.6) 15) – 25

8 ÷ -

4

12

16) The elevator begins on the fifth floor and goes up 5 floors, and goes down 7 floors. What floor is the

elevator on now?

Simplify (Show all work)

17) -25 + 3 – 7 18) 52 + (-5)(7) 19) 3 ∙ 22 + 24 ÷ 8 20) (15 ÷ 3)2 + 9 ÷ 3

21) Evaluate 3x + 8y for x = -1 and y = -4 22) Evaluate 3

5x - 6y for x = -5 and y = -4

23) Evaluate 6

95

x

x for x = 3 24) Evaluate: 3.3p + 2 for p = 9

26

Given the following formulas: C = 𝟓

𝟗 (F-32) F =

𝟗

𝟓𝐂 + 32

25) Convert 9 degrees Celsius to Fahrenheit. 26) Convert 59 degrees Fahrenheit to Celsius.

(Round to the nearest tenth)

Simplify each algebraic expressions

27) -5x – 7x 28) 6x – 9y + 4x – y 29) 10x – 6x – 4x

30) 3(2x + 5) – 5 31) – (5x – 8) 32) 25 – (-5x + 10) 33) 6(10x + 2) + 3x

34) 3

4(3𝑥 − 4) 35) 3 – 2(4x – 8) + 5x 36) -3(4x – 9) + 2x

37) Express the area of a rectangle 38) Express the perimeter in simplest terms of x:

in simplest terms of x,

whose width is 4 and length is 2x – 8

7x + 2

3x – 8

27

39) A fitness club charges $100 to join and $33 for each month. Write an algebraic expression for the total cost

of the fitness club for m months.

Write an algebraic expressions for each phrase.

40) 3 less than the product of 8 and a number 41) the quotient of a number and 5 subtracted from 3

Identify each umber using: integer, rational, irrational

42) √49 43) -4 44) −1

7 45) √24 46) 17

28

Math 8

Unit 2 - Solving Equations

2019-2020


1 1-Step and 2-Step Equations Lesson 1 – Page 31

2 Like Term Equations Lesson 2 – Page 35

3 Distribute Equations Lesson 3 – Page 38

4 Multi-Step Equations Lesson 4 – Page 41

Quiz

5 Variable on Both Sides Lesson 5 – Page 44

6 Variable on Both Sides Lesson 6 – Page 48

7 Word Problems - Translating Lesson 7 – Page 52

8 Equation Solutions Lesson 8 – Page 55

9 Decimal & Fractional Equations Lesson 9 – Page 58

Review Study!

Test Good Luck!

29

Unit 2 – Lesson 1

Aim: I can solve one and two step equations

Warm Up: Evaluate the following expressions for when 𝑥 = −1

(a) −2𝑥 + 5 (b) 𝑥2 − 1 (c) 3𝑥

3+ 6𝑥

Guided Practice: Solving one and two step equations-

When solving a two-step equation, our main goal is to _______________________ the variable.

Step 1- We will always __________________ or ___________________ the constant to the other side

Step 2- Isolate the variable by either _________________________ or __________________________

Exercise 1- Solve the following equations to determine the value of the variable:

1) 𝑥 + 3 = 4 𝑐ℎ𝑒𝑐𝑘 2) ℎ − 18 = 25 𝑐ℎ𝑒𝑐𝑘

3) 3𝑚 = 27 𝑐ℎ𝑒𝑐𝑘 4) 𝑥

2= 15 𝑐ℎ𝑒𝑐𝑘

5) 6 = x + 2 check 6) 12 + x = -10 check

30

Exercise 2- Solve the following equations to determine the value of the variable. Check your solutions!

1. –𝟔 +𝒙

𝟒= −𝟓

2. 𝟗𝒙 − 𝟕 = −𝟕

3. – 𝟏𝟓 = −𝟒𝒎+ 𝟓

4. 𝟐𝒏 + 𝟏𝟎 = −𝟐

Problem Set: Determine the value of each variable. Show all steps and check your answer!

1. 𝒗

𝟑+ 𝟗 = 𝟖 2. 𝟎 = 𝟒 +

𝒏

𝟓

3. 𝟎. 𝟐𝒏 + 𝟑 = 𝟖. 𝟔 4. 𝟏𝟔𝒃 − 𝟔. 𝟓 = 𝟗. 𝟓

31

Unit 2- Lesson 1 Homework

Solve & Check

1) 6𝑥 + 1 = 19 2) 2𝑥 + 12 = 18 3) −8 = 4 + 12𝑎

4) 3 − 4𝑥 = 23 5) 𝑥

3 – 21 = −56 6) 0.4𝑎 + 0.7 = 55

7) 0.7𝑦 = 4.2 8) 51

8+𝑥

5= 2

1

5 9)

𝑥

9 + 12 = 9

32

Unit 2 – Lesson 2

Aim: I can solve equations by combining like terms.

Warm Up: Simplify the following expressions by combining like terms-

(a) 7𝑥 + 𝑥 − 9 (c) 9𝑥 + 8 + 3𝑥 − 2

(b) −2𝑥 + 8𝑥 − 6 + 9 + 𝑥2 (d) 5𝑥 − 1 − 7𝑥 + 4

Guided Practice:

1) 3ℎ − 5ℎ + 11 = 17 Check

2) 5𝑥 − 27 + 4𝑥 = 81 Check

3) Find the value of x if the perimeter of the triangle is 30.

33

Problem Set: For each problem, solve for x, using the properties of equality and check that your solution is

correct.

1) 𝑥 + 3 + 𝑥 − 8 = 55 Check

2) −5𝑥 + 8 + 14𝑥 = −73 Check

3) 3𝑥 – 8 + 5𝑥 = −32 Check

4) 2𝑥 − 5𝑥 + 6.3 = −14.4 Check

34

5) The three sides of a triangle backyard measure 5𝑥 − 3, 3𝑥 − 1, and 4𝑥. If the perimeter of the backyard is

146 feet, determine the length of each side.

6) Tom was told to solve the following problem 𝟏𝟗𝒂 − 𝟏𝟐 − 𝟒𝒂 = −𝟓𝟕. Below is Tom’s work. After

reviewing his work, determine where Tom went wrong.

Lesson Summary:

When combining like terms, it is important to include the _________________ that’s attached with the number!

Once we simply the expression, then we can _______________ or _______________ the constant to the other

side and then isolate the variable by either ______________________ or _____________

__________________________________________________________________________________________________

__________________________________________________________________________________________________

__________________________________________________________________________________________________

__________________________________________________________________

35

Unit 2 – Lesson 2 Homework

Directions- Solve each equation for the unknown variable:

1. 3𝑦 + 4 + 𝑦 = −20

2. −8𝑥 − 7 + 2𝑥 − 2 = 3

3. 4𝑘 + 8 – 5𝑘 = 5

4. 4𝑥 − 6 + 15 = 45

5. 7𝑝 − 12𝑝 − 15 = −65

6. 6 = −𝑥 − 2 − 𝑥

36

Unit 2 - Lesson 3

Aim: I can solve equations containing parenthesis

Warm Up: Simplify the following expressions-

(a) 2(3𝑥 − 7) − 1 (c) 3(𝑥 + 8) + 6𝑥

(b) −5(2𝑥 − 6) (d) −(6𝑥 − 7)

Guided Practice: Solving an equation using the distributive property

1. −2(3 − 2𝑥) = 10 Check 2. 4(𝑥 − 2) = 40 Check

3. −5(𝑥 + 4) = 35 Check 4. 2(x + 5) = 20 Check

37

Problem Set:

Solve and Check

5. 2(5𝑥 + 15) = 50 6. 100(5𝑥 + 10) = 1500

7. 3(7𝑥 + 9) = 48 8. −2(−𝑥 – 5) = 18

9. 2(2𝑥 – 4) = 32 10. 6(3𝑥 − 1) = −42

11. 2(16𝑥 − 20) = 24 12. 5(3𝑥 + 2) = 55

38


Directions- solve the following equations for x: (Check your answers)

1) 4(𝑥 – 2) = 20 2) −5(𝑥 + 4) = 15

3) 2 (6 + 2𝑎) = 24 4) 7(3𝑥 – 4) = 14

5) −5(𝑏 + 4) = 15

39

Unit 2 – Lesson 4

Aim: I can solve multi-step equations

Warm Up: Simplify

1. -5(3x + 2) 2. 7(-6x – 4) 3. –(7x -6)

Guided Practice:

1) 2𝑐 − 8 − 7 = − 21 2) 3𝑑 − 10 + 4𝑑 = − 31

3) 16 = 8 + 2(𝑧 + 4) 4) − 4(8 + 3𝑛) = 16

5) − 2(9𝑎 − 3) = − 30 6) 60 = 3(9𝑠 + 2)

7) 9 − 3(1 − 7𝑥) = 34 8) − 3(4𝑏 + 7) = 27

40

Problem Set:

Solve the Multi-Step Equations and check

1) 15x - 24 - 4x = -79 Check: 2) 105 = 69 - 7x + 3x Check:

3) 3(2x - 5) - 4x = 33 Check: 4) 3(4x - 5) + 2(11 - 2x) = 39 Check:

5) 9(3x + 6) - 6(7x - 3) = 12 Check: 6) 7(4x - 5) - 4(6x + 5) = -91 Check:

41


Simplify:

1) 6x + x 2) 5x + 7 – 2 3) 6x + 2y – 4x + y 4) 3(x + 7) 5) -5(2x + 4)

6) Is 4 the solution to the equation -2n + 5 = -3? Explain how you know.

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

7) Yolanda has $38 in a bank account. She wants to make two equal deposits so that her account will have a

balance of $100. How much money does Yolanda need to deposit each time?

A. $19

B. $31

C. $38

D. $62

8) Sylvie’s age is 5 years less than half Katie’s age. If Sylvie is 11 years old, what is Katie’s age?

A. 8 years old

B. 12 years old

C. 27 years old

D. 32 years old

9) What value of t make this equation true? 6t – 8 = 4

A. t = -3

B. t = 1

C. t = 2

D. t = 5

Solve for x:

10) 3x + 15 = 40 – 10 11) 16310

x

12) x – 22 + 3x = 1 + 1 13) -4(2x - 4) = 80

42

Unit 2 – Lesson 5

Aim: I can solve equations with variables on both sides of the equal sign

Warm Up: Discuss with your partner what the next algebraic step would be to finish solving the equation

below-

Guided Practice: solving with equations with variables on both sides (Check your answers)

1) 2𝑦 + 8 = 6𝑦 + 20 2) 3𝑏 – 8 = 14 – 8𝑏

3) 3𝑥 + 4 = 2𝑥 + 14 4) 3𝑥 + 1 = 9 – 𝑥

5) 7𝑥 + 8 = 4𝑥 + 17 6) 2𝑥 = 8𝑥 + 21

7) 6𝑥 – 3 = 11𝑥 + 7

43

Problem Set:

1) 15𝑥 = 10𝑥 − 30 2) 12𝑥 − 40 = 10𝑥 − 30

3) 5𝑥 + 5 = 3𝑥 − 17 4) 9𝑥 + 6 = 6𝑥 + 24

5) 14𝑥 + 9 = −1 + 12𝑥 6) 𝑥 − 7 = 13 − 4𝑥

44

Unit 2 -Lesson 5 Homework

Directions: Solve the following equations for the unknown variable

1. 6𝑟 + 7 = 13 + 7𝑟

2. 𝑛 + 2 = −14 – 𝑛

3. 5𝑥 + 4 = 8𝑥 – 5

4. 4𝑛 – 40 = −14𝑛 + 14

5. 7𝑥 – 11 = 17 – 7𝑥

6. 8𝑘 – 36 = −4𝑘

45

Unit 2 – Lesson 6

Aim: I can solve equations with variables on both sides of the equal sign

Warm up:

Solve and check your work

9y – 7 = 5y + 21

Guided Practice: “Don’t Call Me After Midnight” approach to solving an equation!

Exercise 1- Solve the following problem, for x

𝑥 + 4 (2 + 3𝑥 ) = 21


−2𝑥 − 3(6𝑥 + 6) = 142


5𝑥 − 5(3𝑥 − 4) = 140

46

Problem Set:

1. 2x + 3(4 – 2x) = 2(x+3) Steps:

a) ________________________________________

b) _______________________________________

c) _________________________________________

d) _______________________________________

2. 14 – x = 2(x+4) 3. 6(x+2) = 4 – (3 – 2x) – 1

4. x – 2(4 – 7x) = 22 5. x + 30 = 2(6x – 7)

6. 9x = -2(x – 22) 7. 8x – 3(-5x + 2) = 40

8. 7(x – 7) = 8(x + 5)

47

If you finish, try these:

Morgan solved the linear equation 2𝑥 − 3 − 9𝑥 = 14 + 𝑥 − 1. Her work is shown below. When she checked

her answer, the left side of the equation did not equal the right side. Find and explain Morgan’s error, and then

solve the equation correctly. 2𝑥 − 3 − 9𝑥 = 14 + 𝑥 − 1

−6𝑥 – 3 = 2𝑥 + 13

+𝟑 + 𝟑

−6𝑥 = 2𝑥 + 16

+ 2𝑥 + 2𝑥

−4𝑥 = 16

𝑥 = −4

Challenge: Marco is trying to find the width of a rectangle. The length of the rectangle is 6 inches and the width

is only explained as four less than twice some number ‘p’. If the area of the rectangle is 48 inches, help Marco

determine the width. Explain how you found your answer. *Draw a picture!

48


Solve:

1) x + 9 = 7x + 9 + 6x 2) 0.7x + 0.06 = 0.5x – 0.29

3) – (–3w – 2) = 11 4) 2(x + 2) = 4 – (3 – 3x) + 5

5) a – 22 + 3a = 1 + 1 6) 3

4(12𝑥 + 4)+ 8 =

1

5𝑥

7) 8

3x + 5

If the area of the rectangle is 112, what is the value of x?

49

Unit 2 – Lesson 7

Aim: I can translate and solve algebraic equations.

Warm Up: Determine as many words associated with the given categories below:

Addition Subtraction Multiplication Division

Guide Practice: Translating Algebraic Expressions

An __________________________ is a mathematical phrase containing numbers, operations, and/or variables.

Expressions can be __________________ or ___________________, but they can’t be _____________.

An __________________________ is a mathematical sentence that states two expressions are

_________________. There’s an __________ sign in between each expression. Equations can be solved.

When translating verbal expressions/equations be careful of the following phrase:

- “more than”

- “less than” “fewer than”

- “subtracted from”

Remember, the following phrases indicate that it’s an equation (which means you need an equal sign)

Is, are, were, equivalent to, same as, yields, and gives

The phrases to the left, switch the order in the way the

expression/equation is created.

Example: “7 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑥” is written as “𝑥 − 7”

50

Exercise 1- Translate the following verbal expressions/equations into an algebraic expression. Use the variable

“x” for the unknown.

1. Five times a number increased by seven 1. _________________________

2. Three times a number x subtracted from 24 2.__________________________

3. The square of a number x decreased by six 3.__________________________

4. The product of five and a number x plus half the number x 4.__________________________

5. Twice Don’s age increased by 8 5.__________________________

6. 19 less than a number cubed 6.__________________________

Exercise 2- Translate each of the following into an equation, and then solve the equation.

7. The sum of number and 12 is 30. Determine the number.

8. The sum of a number and 2 is 12. Determine the number.

9. The difference of a number and 12 is 30. Determine the number.

10. If 2 is subtracted from a number, the result is 4. Determine the number.

11. If three times a number is increased by 4, the result is -8. Determine the number.

Exercise 3-

12. You buy some roses for 14.70 and 9 carnations. Your total cost is $40.08. How much does each carnation

cost?

13. Optimum charges $64.95 per month and $4.95 per On Demand movie. If Ms. Gregory’s Optimum bill for

one month was $84.75, write an equation that can be used to determine the number of On Demand movies, m,

she rented.

14. You have $12.50 in a savings account. You deposit $7.25 more each week. Your friend has $32.50 in a

savings account. She deposits $5.25 more each week. In how many weeks will they have the same amount?

51

Problem Set:

Directions- Translate the following expressions into an algebraic expression. Use the variable “x” for the

unknown.

1. $70 more than the regular price 1.__________________________

2. The quotient of the number of days and 30 2.__________________________

3. Twice the number of sports fans diminished by 5 3.__________________________

4. 7 less than a number x 4._________________________

5. An admission fee of $10 plus an additional $2 per game 5._________________________

6. The square of the sum of a number and eight 6._________________________

7. If 5 is added to the sum of twice a number and three times the number, the result is 25. Determine the number

8. Five less than 2 times a number is 7. Determine the number.

9. One half of a number is 24. Determine the number.

10. A number is one tenth less than 1.54. Determine the number.

11) Nine fewer than half a number is five more than four times the number. Find the number

12) Jamie attends a carnival. The admission fee is $6 and each ride cost $2. Write an equation that Jamie can

use to determine the number of rides, r, she can buy if she has $40.

13) Sarah downloads 3 movies on Friday and 11 on Saturday. Each download costs the same amount. If Sarah

spends $97.86, how much was each movie?

52


1) The table shows ticket prices for the local Duck’s baseball team for fan club members and non-members.

For how many tickets is the cost the same for club and non-club members?

Ticket Prices

If a Club Member pays a total of $75, write an equation to represent how many tickets, t, they purchased.

2) A phone company charges twenty dollars per month plus ten cents per minute for a long distance call. If a

phone bill for the month is $185, how many minutes were on long distance calls?

3) Bob is a mechanic. He charges $110 for supplies and $10 for each hour, h, he works. Write an equation that

represents to find the total number of hours Bob worked if he charged a customer $230.

4) Lauren buys three notebooks for school. Each notebook is the same price. Lauren uses a coupon that is worth

$2 off her total purchase. She pays a total of $7 with the coupon. Write an equation that can be used to find the

cost of one notebook, n?

*5) A wireless company offers two cell phone plans. Plan A charges $24.95 per month plus $0.10 per minute

for calls. Plan B charges $19.95 per month plus $0.20 per minute. At how many minutes will the two plans

cost the same?

Club

Members

Non-Club

Members

Membership Fee

(one-time)

$30 none

Ticket Price $3 $6

53

Unit 2 - Lesson 8

Aim: I can determine how many solutions an equation has

Warm up: Copy down the vocabulary words.

Guided Practice:

Solving Special Solutions

Vocabulary:

No Solution - ___________________________________________________________________________

One Solution - __________________________________________________________________________

Infinite Solutions - ______________________________________________________________________

Examples: (Solve for the variable and identify the type of solution)

1) 5x + 8 = 5(x + 3) 2) 9x = 8 + 5x 3) 6x + 12 = 6x + 12

Type of Solution Description Word Algebraic Form

No Solution None a = b

One Solution One x = a

Infinite Solutions Many a = a

2x – 4 = 2(x + 1) 2x – 4 = -x – 1 2x – 4 = 2(x – 2)

2x – 4 = 2x + 2 +x +x 2x – 4 = 2x – 4

-2x -2x 3x – 4 = -1 -2x -2x

-4 = 2 (no solution) +4 +4 -4 = -4 (infinite solutions)

3x = 3

3 3

x = 1 (one solution)

54

Problem Set:

Solve for the variable and identify the type of solution

1) 5x = 9 + 2x 2) x + 9 = 7x + 9 – 6x 3) 22y = 11(3 + y)

4) -3y + 1 = y + 9 – 4y 5) 16z – 24 = 8(2z – 3) 6) -5w = 7 – 4w + 8

55


Solve for the variable and identify the type of solution

1) 4r + 7 – r = 3(5 + r) 2) 4(x – 2) = -2x + 12 + 5x 3) 2x + 18 = (x + 9)2

4) 2(5 + x) = 22 5) 6x + 11 – 4x = 10 + 2x + 1 6) 9y – 24 = 3(3y – 8)

7) 8 – 5x + 22 = -5(x – 6) 8) -4(2m + 6) = 16 9) 14 – x + 12x = 32 + 11x

10) Which best describes the solution for 5b – 25 = 25? 11) Which best describes the solution for the

a. no solution equation: 6y – 3 = -6y + 3

b. 0 a. no solution

c. 10 b. infinitely many solutions

d. 80 c. 0.5

d. 2

56

Unit 2 – Lesson 9

Aim: I can solve equations with decimals and fractions

Warm Up:

Solve

5(2𝑥 + 1) – 7𝑥 = 20

Steps:

1. Distribute (If possible)

2. Combine Like Terms on Each Side

3. Isolate The Variable (Inverse Operations)

Guided Practice:

Decimals

1) . 5 + .2𝑥 = .9 2) 4.5 + 𝑥 = 12 3) . 9 − 10𝑥 = −9.1

4) . 3𝑥 = 9 5) −20 = .2(10𝑥 − 30)

Fractions

1) 5

1

5

3j 2)

8

1

8

3h 3)

9

4

9

1g 4) 8

6

1

6

5 xx

5) 50 = 3

2(3x + 6) 6) 54 =

3

2 7)

2

1(2x + 2) = 48 8)

3

1(9x 12) = 25

57

Problem Set:

Decimals

1) 𝑧 + 1.25 = −9.54 2) 𝑐 − 14.59 = −88.22 3) 14.9 − 𝑥 = 15.1

4) 7𝑎 = 1.4 5) 3𝑥 = −2.4 6) . 25(12𝑥 + 8) = 17

Fractions

1) 5

8+ 𝑥 =

3

4 2) ℎ +

15

25=13

50 3) 𝑥 −

30

40=

5

20 4) 2x +

1

4 =

1

8

5) 4

1(12x + 8) = 17 6) 20 =

5

1(10x 30) 7)

6

1(6x 18) = 4 8) 20 =

2

1(4x + 8)

58


1) 9 − 79.2 = 𝑥 2) −1.30 + 𝑣 = −9.3 3) 𝑏 + 4 = 25.65

4) 𝑛 − 14 = −7.7 5) 𝑞 + 11.25 = 5.3 6) −.4𝑥 = 16

7) 138.75 = 9.25(−6 + 𝑡) 8) 21 = .5(4𝑥 + 6) 9) −.2(10𝑥 – 15) = 9

10) 𝑓 +1

7= −

1

7 11) 𝑥 +

6

15=

5

15 12) 𝑚 −

3

4=1

2

13) 1

2= 𝑑 +

5

12 14)

1

4+ 𝑝 =

3

20 15)

1

4𝑦 +

1

3=

1

12

Review:

16) Sal did the following work: 17) Today it is 25○. Last month, it was −15○.

Explain his error. What was the difference in temperature?

9y – 2 + 4y

9y – 4y + 2

5y + 2

Write and solve an equation for each:

18) A tile man is laying an 84 inch border using 12 inch tiles. How many tiles would need to be placed?

19) Student Government sold 175 bags of popcorn at the dance. If they made $306.25, how much was the cost

of each bag of popcorn?

59

Unit 2 Review

Unit 1: Integers

1) 0

7 2)

7

0 3) (-1)(5)(-3) 4) (-3)3 5) 5 – (-6) 6) (-0.27)(-0.6)

7) Compare using > or < to make each sentence true.

a) 8 _____ - 3 b) -5_____-2

8) Find the new temperature if it was 28 degrees at 1 pm and it dropped 30 degrees by 5 pm.

Order of Operations

Simplify.

9) 13 − 14 + 7 − 3 + 2 10) 4(−3)2 − 18 11) −3.5(6.1−11.3)

5−7

Evaluating

Given the values the x = -2, y = 3, z = -1 and k = 5, evaluate the following problems

12) x(y + z) + k 13) 𝑘𝑦2− 𝑥3

𝑧 14)

4

3(𝑥𝑦𝑧) 15) 𝑦4 ÷ (𝑘 + 𝑥)

16) The formula C = 5

9(F-32) is used to find the Celsius temperature (C) for the given Fahrenheit temperature

(F). What Celsius temperature is equal to 104° Fahrenheit?

60

Combine Like Terms & Distributive Property

Simplify

17) x + x 18) -5x – 7x 19) 6x – 9y + 4x – y 20) 10x – 6x – 4x

21) 3(2x + 5) – 5 22) -(5x – 8) 23) 25 – (-5x + 10)

24) Express the area of the rectangle in simplest terms of x.

25) The pentagon building in Washington D.C. is a regular pentagon. If the length of one side is represented by

3n + 8, express the perimeter as a binomial.

26) Give an example of each:

Monomial _________ Binomial________________ Trinomial_______________________

𝟐𝐱−𝟔

𝟔

61

Unit 2: Equations

Solve the following equation for the missing variable, otherwise determine solution type.

27) 5x - 3 = -8 28) 3

8𝑥 = 6 29)

4

5𝑥 − 3 = 9

30) 0.8 - 2x = 10 31) 7x - 2 + 5x = 10 32) -15 + 4x = 3x + 5

33) 4x – 4 + 2x = 3x + 17 34) -13x + 8 = -13x + 70 35) -1.2 + 4x = 2x + 6.8

36) 4

3𝑥 −

10

3 =

1

2𝑥 37) -0.1x + 0.4x = 15.3 38) 5(3x + 6) = -10x + 30

39) 3

5+

1

4𝑥 =

1

2 40) 2(9x + 3) = 6(3x + 1) 41) -4(2x – 7) = 3x – (x + 12)

42) Jackie wants to buy a TV for her mother that cost $300. She plans to make a down payment of $135. She

will pay the rest of the cost in 6 equal payments. How much will each payment be? Only an algebraic solution

will be accepted.

62

Translating & Word problems

43) Write an algebraic equation for each sentence DO NOT SOLVE

a) 8 more than a number is 12.

b) Ten is four times a number plus two.

c) The sum of five times a number and 3 times the same number is 24

d) Seven less than twice a number is 18

44) Jonas is working with his Uncle Tim as an apprentice electrician wiring a new house. For each job that

Jonas works, he is paid a one-time service fee of $50.00 plus $25.00 an hour. When the house was completely

wired, Uncle Tim paid Jonas $2250.00.

Write an equation that can be used to determine the number of hours Jonas worked wiring the house.

45) Wonka candy bars cost $1.50 each. You have a total of $18 to spend on Wonka bars. Write an algebraic

equation that can be used to determine how many Wonka bars you can buy.

46) What value of the constant, c, in the equation below will result in and infinite number of solutions?

6x + 21 = c(2x + 7)

47) Solve: 3.7x + 8.4 = 15(x - 3

5)

63

Math 8

Unit 3 – Exponents & Scientific Notation

2019-2020

Date Lesson Topic

Homework

1 Introduction to Exponents Lesson 1 – Page 66

2 Zero and Negative Exponents Lesson 2 – Page 70

3 Multiplying Exponents Lesson 3 – Page 73

4 Power to a Power Lesson 4 –Page 76

5 Dividing Exponents Lesson 5 – Page 79

Quiz Review Study!

Quiz

6 Introduction to Scientific Notation Lesson 6–Page 86

7 Converting Scientific Notation Lesson 7- Page 90

8 Compare and Order Scientific Notation Lesson 8 – Page 93

9 Multiply and Divide Scientific Notation Lesson 9 – Pages 98-99

10 Applications and Word Problems Lesson 10 – Page 109

Review Study!

Test Good Luck!

64

Unit 3 – Lesson 1

Aim: I can identify and simplify exponential expressions.

Warm Up: Answer Questions in boxes.

4x² + 7 2 6

1) Name the variable _______ 4) Name the base ______ 1) What is the base _________

2) Name the coefficient _______ 5) Name the constant ______ 2) What is the exponent ________

3) Name the exponent ________

Guided Practice:

Base – When a number is raised to a power, the number that is used as a factor is the base.

Exponential Form – A number written with a base and an exponent.

Expanded form – A number written as the sum of the values of its digits.

Compute – Solve. Get an answer.

Examples: Write the following in exponential form:

1) 5 × 5 × 5 × 5 × 5 × 5 ____________ 2) 7

9

7

9

7

9 ____________

3) −9 ∙ −9 ∙ −9 ∙ −9 ∙ −9 ∙ −9 ∙ −9 ∙ −9 ∙ −9 ____________ 4) 𝑥 ∙ 𝑥 ∙ 𝑥 ____________

Exponent Notation: Check for Understanding

5) 4 ×⋯× 4⏟ 7 𝑡𝑖𝑚𝑒𝑠

= ______ 6) 3.6 ×⋯× 3.6⏟ _______ 𝑡𝑖𝑚𝑒𝑠

= 3.647

7) (−11.63) × ⋯× (−11.63)⏟ 34 𝑡𝑖𝑚𝑒𝑠

= _________ 8) 12 × ⋯× 12⏟ _______𝑡𝑖𝑚𝑒𝑠

= 1215

9) (−5) × ⋯× (−5)⏟ 10 𝑡𝑖𝑚𝑒𝑠

= ________ 10) 7

2×⋯×

7

2⏟ 21 𝑡𝑖𝑚𝑒𝑠

= ______

11) (−13) × ⋯× (−13)⏟ 6 𝑡𝑖𝑚𝑒𝑠

= ________ 12) (−1

14) × ⋯× (−

1

14)⏟

10 𝑡𝑖𝑚𝑒𝑠

= __________

13) 𝑥 ∙ 𝑥⋯𝑥⏟ 185 𝑡𝑖𝑚𝑒𝑠

= _______ 14) 𝑥 ∙ 𝑥⋯𝑥⏟ _______𝑡𝑖𝑚𝑒𝑠

= 𝑥𝑛

65

Is it necessary to do all of the calculations to determine the sign of the product? Why or why not?

15) (−5) × (−5) ×⋯× (−5)⏟ 95 𝑡𝑖𝑚𝑒𝑠

= (−5)95 16) (−1.8) × (−1.8) × ⋯× (−1.8)⏟ 122 𝑡𝑖𝑚𝑒𝑠

= (−1.8)122

Find the missing exponent

17) 3𝑛 = 81 18) 2𝑛 = 64 19) 5𝑛 = 125

Problem set:

Write in expanded form and compute the value:

1) 36 2) (−2)4 3) (−

4

11)5

4) 92 5) -42 6) −(

1

2)3

Write the following in exponential form:

7) 1

4×1

4 8)

7

9

7

9

7

9 9)

1

𝑥×1

𝑥×1

𝑥×1

𝑥×1

𝑥

Write in expanded form.

10) (-2)3 11) (x)2 12) (7)4

13) What do exponents represent? What’s a base? What’s a coefficient?

66


Write the following in exponential form:

1) 333 2) 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 3) 3

7 ∙ 3

7 ∙ 3

7 ∙ 3

7

Compute:

4) 13 + 43 5) 22 − 41 + 23 6) (1

3)4

Find the value of n:

7) 3𝑛 = 81 8) 4𝑛 = 64 9) 5𝑛 = 15, 625

10) A square has a length of 6.2 square feet. 11) Find the value of x: 0.4𝑥 − 2(0.5𝑥 + 9) = −3

If the area of a square formula is 𝐴 = 𝑠2,

what is its area?

Fill in the blanks about whether the number is positive or negative.

12) If 𝑛 is a positive even number, then (−55)𝑛 is __________________________.

13) If 𝑛 is a positive odd number, then (−72.4)𝑛 is __________________________.

14) Write an exponential expression with (−1) as its base that will produce a positive product.

15) Write an exponential expression with (−1) as its base that will produce a negative product.

67

Unit 3- Lesson 2

Aim: I can simplify negative and zero exponents.

Warm Up: Express in expanded form.

1) (-6)5 2) (x)7 3) (2

5)3

Express in exponential form.

4) 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 5) (8)(8)(8)

Guided Practice:

The Zero Exponent Rule

Rule: Any number or variable raised to the ________________ power will ALWAYS be ________.

Note this works when 𝑥 ≠ 0

Exercise 1- Evaluate the following

1) 50 ________ 2) x0 ________ 3) 5,9280 ________ 4) (-2)0 ________

5) -20 ________ 6) (3x)0 ________ 7) 3x0 ________ 8) 3(xyz)0 ________

9) (x4y)0 ________ 10) 8(x2y3)0 ________ 11) 4(x9y5)0________

68

The Negative Exponent Rule

Rule:

Exercise 2- Write each expression using a positive exponent

a) (4)−4 ________ b) (3

4)−1

________ c) 8-1 ________

d) (5

6)−2

= ________ e) (𝑥

𝑦)−5

= ________ f) x-9 ________

Exercise 3- Convert the exponent from a negative to a positive when both parts of the fraction have negative

exponents

a) 5−4

2−6 ________ b)

1−5

3−2 ________ c)

𝑥−3

𝑦−8 ________

d) 𝑥−5

9−5 = ________ e)

14−6

1−7 = ________ f)

12−1

𝑦−6 ________

Exercise 4- Special Situations

Rule: Move only the negative exponent to the opposite part of the fraction and then make the

exponent positive.

a) 54

8−6 ________ b)

6−3

32 ________ c)

3−9

52 ________ d)

𝑥−3

𝑦 8 _______

e) 𝑥5

92 ________ f)

4−4

3−7 ________

1) Write as a fraction if needed.

2) Write the reciprocal of the base.

3) Make the exponent positive.

69

Problem Set:

Simplify each expression and re-write with a positive exponent

1) -50 2) 4x0 3) (4x)0 4) 3(ab)0 5) (-3)0

6) -30 7) 3(y)0 8) (3y)0 9) 149x0 10) 2(a3b7c4)0

Write the following as a positive exponent:

11) (1

2)−3

12) 9−2 13) 5−2 14) (𝑥)−4

15) 6−2

7−3 16)

𝑥−4

𝑦6 17)

73

2−8 18)

(5)−2

(2)−5

Lesson Summary:

Anything raised to the zero power is always __________.

When you have negative exponents, in order to make them positive you:

70


Simplify and write as a positive exponent if needed

1) 0)8( x 2)

08x 3) 9(y3)0

4) 2(mn)0 5) 9

0 6)

0

9

7) 5−3 8) 1

8−2 9) (

2

3)−4

10) 4−3

7−2 11) 12−1 12) (

1

2)−3

Compute:

13) 93 14) 50 15) 32 + (1

9)0

16) 22 + 23 + 91 17) 52 − 33 18) 60 + 4

19) Which is 2(x0) in standard form? 20) Which shows 9-3 in standard form?

A. 0 B. 1 A. -729 B. -27

C. 2 D. 2x C. 1

27 D.

1

729

21) Which is -70 in standard form? 22) Which shows (-3)2 in standard form?

A. -7 B. -1 A. 9 B. 6

C. 0 D. 1

7 C.

1

9 D.

1

6

23) Which is 5(xy)0 in standard form? 24) Which shows (2

3)−4 in standard form?

A. 5 B. 5x A. (2

3)4 B. (

2

3)

C. 1 D. 0 C. (3

2)−4 D. (

3

2)4

71

Unit 3 – Lesson 3

Aim: I can multiply exponents with like bases.

Warm Up: What is another way you can abbreviate each expression?

(a) 3 + 3 + 3 + 3 + 3 (b) 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3

Guided Practice:

Exercise 1- For each of the following expressions, name the constant, coefficient, base, variable, and exponent:

Expression Constant Coefficient Base Variable Exponent

6𝑥2 − 5

42

10𝑥3 + 1

𝑦2

Multiplying Exponents Discovery

Exercise 2- For each of the following expressions, simplify by expanding then re-write in exponential form

Expression to be simplified The expression in expanded form End result in exponential form

𝟑𝟐 ∙ 𝟑𝟒 (3 ∙ 3) ∙ (3 ∙ 3 ∙ 3 ∙ 3)

36

𝒙𝟓 ∙ 𝒙𝟑 (x ∙ x ∙ x ∙ x ∙ x) ∙ (x ∙ x ∙ x)

𝟓𝟔 ∙ 𝟓𝟒

𝒚𝟕 ∙ 𝒚𝟓

Rule: When multiplying terms with like _____________, you keep the base and _____________ the exponents.

72

Simplify the following expressions

(1) 𝑥4 ∙ 𝑥3

(2) (1

7)6 ∙ (

1

7)2

(3) 42 ∙ 410 ∙ 4−3

(4) 𝑘5 ∙ 𝑘

(5) 27 ∙ 2 ∙ 2−3

Problem Set:

1) 1013 ∙ 10−8 = _______ 2) 2−2 ∙ 2 = _______ 3) 1−3 ∙ 19 ∙ 14 = ______ 4) 84 ∙ 8−4 = _____

5) 6−7 ∙ 62 ∙ 6−4 = ______ 6) (1

2)6

∙ (1

2)2

= ______ 7) 𝑐−3 ∙ 𝑐9 = ______ 8) 𝑥2 ∙ 𝑥4 ∙ 𝑥 =___

9) Which is (-7)2 in standard form? 10) Which shows (-3)2 in standard form?

A) -7 C) 7 A) 9 C) -6

B) -49 D) 49 B) -9 D) 6

11) What is the value of 74 33 ? 12) The result of 8−4 comes from which multiplication

A) 3−3 C) 33 A) 83 ∙ 85 C) 34 ∙ 34

B) 9−3 D) 311 B) 83 ∙ 8−7 D) 34 ∙ 3−8

73


Find the product and express with a positive exponent.

1) 22 ∙ 25 2) 118 33 3) 44 ∙ 4 4) 6−3 ∙ 62

5) (1

2)−2

∙ (1

2)1

6) 𝑥4 ∙ 𝑥0 7) 3 ∙ 3−3 ∙ 3 ∙ 30 8) (2

3)−5

∙ (2

3)9

9) (𝑥

𝑦)4

∙ (𝑥

𝑦)−1

10) 7−8 ∙ 78 ∙ 7 11) 𝑦4 ∙ 𝑦9

12) Amy wrote these expressions: 63 35 102

Part A: Write these expressions in order from least to greatest. ________ ________ __________

Part B: Explain how you know your answer is correct

__________________________________________________________________________________________

__________________________________________________________________________________________

13) Which is -70 in standard form? 14) Which shows (2-2 )( 26 ) in exponential form?

a. -7 b. -1 a. 24 b. 2-4

c. 0 d. 1

7 c. 2-8 d. 2-12

15) Which of the following equations has no solution?

A) 3x + 15 = 3(x + 5) B) 3x + 7 = 2x + 7 C) 3x + 5 = 3(x+5) D) 3x + 9 = 15

Write the following in exponential form.

16) xxx 333 ___________ 17) yyyxx 55 ____________ 18) 3

7 ∙ 3

7 ∙ 3

7 ∙ 3

7 _________

Express in expanded form

19) 74 _________________ 20) (3x)3 ____________________ 21) x3y5 ______________________

Fill in the blanks about whether the number is positive or negative.

22) If n is a positive even number, then (−55)n is __________________________.

23) If n is a positive odd number, then (−72.4)n is __________________________.

74

Unit 3 – Lesson 4

Aim: I can simplify an exponential expression raised to a power.

Warm Up: Simplify the following.

1) 𝑥2 ∙ 𝑥7 2) 118 33 3) 9 ∙ 9 4) 𝑦−3 ∙ 𝑦2

Discovering the Laws of Exponents: Power to a Power Rule

What happens when you raise a power to a power? Look for a pattern as you fill in the table below.

Example Write in Expanded Form Rewrite using Exponents

(23)2

(32)4

(54)3

(72)2

[ (1

2)2 ]5

Rule: When you raise a power to a power, keep the ______________ and _________________ the exponents.

75

Discovering the Laws of Exponents: Product to a Power Rule

What happens when you raise a product to a power? Look for a pattern as you fill in the table below.

Example Write in Expanded Form Rewrite using Exponents

(2 ∙ 3)3

(2 ∙ 3) ∙ (2 ∙ 3) ∙ (2 ∙ 3) 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 3

2333

(4 ∙ 6)5

(𝑎)4

(7 ∙ 4 ∙ 11)2

Rule: When finding a product raised to a power, you find the power of each factor and then ______________

Problem Set:

Simplify the following expressions, do not evaluate. Rewrite as positive exponents if necessary.

(1) (52)3 (2) (𝑥5)4 (3) (𝑦4)3 (4) (62)2

(5) (73)4 (6) (2−1)0 (7) (−27)2 (8) (-35)2

(9) (2 ∙ 3 ∙ 4)3 (10) (6−2)3 (11) (𝑥4 ∙ 𝑥2)2 (12) (𝑎3𝑏−2)3

(13) The formula for the volume of a rectangular prism is 𝑉 = 𝐿𝑊𝐻. If the length is 84, the width is 8−2, and

the height is 80. Express the volume, in exponential form.

76


Simplify using the laws of exponents. Express as a positive exponent (Do not compute)

1) (84)3 = ______ 2) (21)0 = ______ 3) (𝑥3)4_____ 4) (72)2 = ______ 5) (810)2 = _____

6) (37)3 ∙ (32)4 = ______ 7) (90)6 = ______ 8)[(1

2)3

]2

= ______ 9) (124)4 = ____10) 62 99

_______

11) (𝑦15)2 = ______ 12) (𝑥5)4 = ______ 13) (𝑦3 ∙ 𝑦)3 = ____ 14) (𝑥7)2 _____

15) Which is (-7)2 in standard form? 16) What is the value of (3-2) ?

a. -49 b. −1

49 a. -9 b. 9

c. 49 d. 1

49 c. −

1

9 d.

1

9

17) In exponential form, what is the area of a square that has length of 43 ?

18) Determine the missing (?) value in each:

a) (5?)3 = 512 b) (−2𝑚3𝑛4)? = −8𝑚9𝑛12

19) The formula for the volume of a rectangular prism is V = LWH. If the L = 44 and W = 4−2

and the H = 40. What is the volume in exponential form?

20) The formula for the volume of a cube is V = s 3 where s is the length of the side of the square. Express the

volume of a cube in simplest terms of x whose side is x 8 .

77

Unit 3 – Lesson 5

Aim: I can divide exponents with like bases.

Warm up:

1. What does 53 represent?

2. What does ( 1

3)2 represent?

3. What does (-6)4 represent?

Guided Practice:

Dividing Exponents Discovery

Exercise 1- For each of the following expressions, simplify by expanding then re-write in exponential form

Expression to be simplified The expression in expanded form End result in exponential form

𝟓𝟔

𝟓𝟐

5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5

5 ∙ 5

54

𝒙𝟓

𝒙𝟐

x ∙ x ∙ x ∙ x ∙ x

x ∙ x

𝟒𝟐

𝟒𝟑

𝒙𝟕𝒚𝟏𝟎

𝒙𝟒𝒚𝟔

Rule: When dividing terms with like _____________, you keep the base and _____________ the exponents.

78

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

Problem Set: Simplify the following expressions

(1) 𝑥5𝑦4

𝑥2𝑦 (2)

510

52 (3)

68

6

(4) 𝑥6

𝑥4 (5)

𝑎6𝑏

𝑎4𝑏 (6)

59

56

(7) 𝑎9𝑏5

𝑎3𝑏4 (8)

816∙85

812 (9)

94

9

(10) 𝑚5𝑛4

𝑚2𝑛4

11. Bryan and Brendan simplify the following expression 123

127 below are their responses.

Bryan: 124 Brendan: 12−4

Determine which student, if any, got the correct answer and explain the mistake made, if any.

79


Simplify using the laws of exponents. Express all answers as a positive exponent.

1) 911

99 2)

710

79 3)

104

10−9 4)

46

43

5) 𝑥3

𝑥 6)

79

73 7)

𝑥4𝑦7

𝑥2𝑦3 8)

𝑦12

𝑦3

9) 99

95 10)

(2

5)7

(2

5)2

11) 𝑥15

𝑥3 12)

𝑥5

𝑥−2

13) 𝑥15

𝑥5 14)

25

2−9 15)

𝑥4

𝑥−5 16)

(1

3)9

(1

3)2 17)

2−4

2−7

Determine the missing (?) value in each:

18) 𝑥

𝑥?

6 = 𝑥4 19)

28

2? = 29 20)

𝑥?

𝑥5= 𝑥7

80

Unit 3 – Quiz Review

Let us summarize our main conclusions about exponents. For any numbers 𝑥, 𝑦 and any positive integers 𝑚, 𝑛,

the following holds:

1) 𝑥𝑚 ∙ 𝑥𝑛 = 𝑥𝑚+𝑛 Rule: ___________________________________________________

2) (𝑥𝑚)𝑛 = 𝑥𝑚𝑛 Rule: ___________________________________________________

3) (𝑥𝑦)𝑛 = 𝑥𝑛𝑦𝑛 Rule: ___________________________________________________

4) 𝑥𝑚

𝑥𝑛 = 𝑥𝑚−𝑛, 𝑚 > 𝑛 Rule: ___________________________________________________

5) 𝑥0 = 1 Rule: ___________________________________________________

6) 𝑥−2

𝑦−3 =

𝑦3

𝑥2 Rule: ___________________________________________________

Using Exponent Rules:

Remember: When you multiply powers with the same base, you add the exponents.

When you divide powers with the same base, you subtract the exponents.

Multiply

Write each answer with a positive exponent

1) x5 · x9 2) 102 · 103

3) x6 · x -4 4) x -2 · x -3

5) 73 1010

6) xx 4

81

Divide

7) 28 xx 8)

64 xx

9) 28 xx 10)

52 1010

Write each answer with a positive exponent

11) 52 xx 12)

46 66 13) 66 xx 14)

97 xx

15) 69 xx 16)

4xx 17) 55 xx 18)

54 xx

19) 22 33 20)

500500 xx 21) 0)3( x 22)

03x

82

Unit 3 – Exponents Study Guide

Zero Exponents

Anything raised to the zero power = 1.

x0 = 1 50 = 1

3(x0) = 3(1) = 3 (3x0) = 1

Negative Exponents

x –n = 1

𝑥𝑛

If the exponent is negative, switch its position to make it a positive exponent

If it is in the numerator, move it to the denominator

If it is in the denominator, move it to the numerator

9−3 ∙ 42 = 42

93

𝑥−3

𝑦−5= 𝑦5

𝑥3

Multiplying Exponents MADS (Multiply – Add exponents)

Without a Coefficient

1. Keep like bases

2. Add the exponents of like bases

**Remember: 3 𝑥4

x is the base. The base is what is being raised to a power.

3 is the coefficient. The coefficient is the number in front of

the variable.

4 is the exponent. The exponent tells you how many times you

multiply the base by itself.

(65)(62) = 67

83

Raising a Power to a Power MADS (Multiply – Add exponents)

1 - Rewrite in expanded form

2 - Keep the base

3 - Add the exponents

Dividing Exponents MADS (Division – Subtract Exponents)

Without a Coefficient

3. Keep the base

4. Subtract the exponents

95

93= 92

(65)3= (65)(65)(65) = 615

84

Unit 3- Lesson 6

Aim: I can understand scientific notation.

Warm up: 1, 2 3 4 . 5 6 7 8

Rounding : 4 and lower stay the same, 5 and higher go up.

1) Round the following to the nearest whole number: 2) Round the following to the nearest tenth:

754. 8132 = 8.9437 =

3) Round the following to the nearest hundredths: 4) Round the following to nearest tenth:

3987.42915 = 9534.0934 =

Guided Practice:

Scientific Notation- when you are dealing with very large or very small numbers, it is helpful to be able to

write them in a shorter form.

Scientific Notation Standard Form

2.59 × 1011 = 259,000,000,000

Rule: A number is in Scientific Notation if:

1) The first factor (coefficient) is a single digit followed by a decimal point

2) Multiplied by the second factor which is a power of 10.

Exercise 1- Determine if the following numbers are written in scientific notation:

(1) 3.2 × 104 (2) 78.96 × 104 (3) 456.1 × 10−8 (4) 9. × 10−5

Scientific Notation: Positive Exponents and Negative Exponents

A number in scientific notation with ______________________ exponents represents a number

________________ than one (whole number).

A number in scientific notation with ________________________ exponents represents a number between 0

and 1 (decimal).

Remember:

Positive Exponent ____________________________________

Negative Exponent ____________________________________

Exercise 2- Determine if the following numbers below will be whole numbers or decimals.

(1) 1.2 × 105 (2) 5.8 × 10−5 (3) 6.8 × 10−9 (4) 3 × 109

85

Scientific Notation: Real Life Situations

When is it appropriate to use scientific notation in real life?

Exercise 3- determine if the number in scientific notation would be written as a positive or negative exponent:

(1) The weight of 10 Mack Trucks (2) The width of a grain of sand

Problem Set:

1. Determine if the numbers below are written in scientific notation. Explain your answer.

(a) 4.1 × 1015 (b) 24.01 × 105 (c) 0.1 × 10−6

2. Determine if the number below are in whole numbers or decimals. Explain your reasoning.

(a) 2.1 × 1015 (b) 2.1 × 10−15 (c) 1.0 × 10−5

3. Determine if the number in scientific notation would be written with a positive or negative exponent.

Explain your reasoning.

(a) The size of a cheek cell (b) The mass of Earth

Examples of Large Numbers Examples of Small Numbers

86


Determine if the numbers below are written in scientific notation. Explain why you chose your answer.

1) 1.5 x 104 2) 1.50 x 105 3) 0.42 x 102 4) 4. 56 + 106 5) 134, 987

6) 9.5 x 10-3 7) 17 x 10-16 8) 75.9 x 106 9) 1.3 x 10-23 10) 65 x 102

Determine if the following number in scientific notation would be written as a positive or negative exponent.

11) How many seconds in a year 12) The width of a piece of thread (in feet)

13) The weight of a skyscraper (in pounds) 14) The weight of an electron (in pounds)

Review Work:

15) 7 3 x 7-6 16) (1

4) -3

17) 4x + x – 8 = 5x + 12 18) 18

-3

87

Unit 3 – Lesson 7

Aim: I can express numbers in scientific notation and standard form.

Warm Up: Tia was told to put her answer in scientific notation, so she wrote the following answer:

12.3 × 10−7. Is her response in the correct form? Explain.

Guided Practice:

Standard Form to Scientific Notation

1. Write the number placing the decimal point after the first non-zero digit

2. Write “ 𝑥 10”

3. Count the number of digits you moved the decimal point and write it as the exponent.

Remember:

If it is a whole number ______________________________________________________________

If it is a decimal ______________________________________________________________

Exercise 1- Convert from standard form to scientific notation

(1) 245,000,000 (3) 500,000

(2) .00084 (4) .000007643

88

Scientific Notation to Standard Form

1. Move the decimal point to the number the number of places indicated by the exponent

2. If it’s a positive exponent, move the decimal point to the right (Whole number- make the number larger)

If it’s a negative exponent, move the decimal point to the left (Decimal-make the number smaller)

Exercise 2- Convert from scientific notation to standard form

(1) 5.9 × 103 (3) 8.32 × 10−4

(2) 4.765 × 108 (4) 1.9 × 10−7

Making Sure a Number is written in Scientific Notation

Rule:

If a decimal point needs to move to the left, the exponent increases 48.6 × 103

If the decimal point needs to move to the right, the exponent decreases . 48 × 103

** Be careful when the exponent is negative!

Exercise 3- Write each in Scientific Notation, if necessary:

(1) 68.7 × 109 ______________________________ (2) 6 × 105 _____________________________

(3) . 725 × 108 ______________________________ (4) . 292 × 10−4 ____________________________

(5) 326 × 10−8 ______________________________ (6) 7.5 × 10−9 ____________________________

LARS

89

Problem Set:

1. The speed of light in a vacuum is 299,792,458 meters per second. Which number, written in scientific

notation, is the best approximation of the speed of light?

(1) 0.3 𝑥 107 meters per second




2. Expressed in scientific notation, 0.0000056 is equivalent to:

(1) 5.6 × 10−6

(2) 0.56 × 10−5

(3) 5.6 × 106

(4) 56 × 106

3. In 2013, JFK Airport had approximately 9.4 × 107 passengers pass through the airport. What is that number

written in standard form?

4. A virus is viewed under a microscope. It’s diameter is 0.00000002 meter. How would this length be

expressed in scientific notation?

5. Mrs. Volpe gave her class the following problem: Convert the following number from standard form to

scientific notation 1,742,103,000.

Anthony’s Answer: 17.42103 × 108

Is Anthony’s answer correct? Explain your reasoning and correct Anthony’s answer if needed.

90


Convert into scientific notation. Write the numbers below in standard form.

1) 3,400

5) 1.901 x 10-7

2) 0.000023 6) 8.65 x 10-1

3) 101,000 7) 2.30 x 104

4) 0.010

8) 9.11 x 103

91

Unit 3 – Lesson 8

Aim: I can compare and order numbers in scientific notation.

Warm Up: Are the following numbers written in scientific notation? If not, state the reason.

(a) 4.0701 + 107 (b) . 325 × 10−2

Investigation 1: Express each scientific notation in standard form-

(a) 4.3 × 102 (b) 2.5 × 103

Which number is smaller? (answers should be in scientific notation)

Which number is larger? (answers should be in scientific notation)

Do you notice anything?

Investigation 2: Express each scientific notation in standard form-

(a) 2.1 × 104 (b) 1.5 × 104

Which number is smaller? (answers should be in scientific notation)

Which number is larger? (answers should be in scientific notation)

Is there a relationship between the value of the coefficient and which number is larger or smaller?

92

Guided Practice: Steps for Comparing Numbers in Scientific Notation Form

1. To compare two numbers given in scientific notation, first compare the _____________________. The one

with the greater exponent will be _________________________.

2. If the exponents are _______________, compare the decimals.

Exercise 1: 1.06 × 1016 ________ 2.4 × 1015 Exercise 2: 2.78 × 107 ________ 278 × 107

Exercise 3- Order the countries according to the amount of money

visitors spent in the United States from least to greatest.

Problem Set:

For the following problems, use >,<, 𝑜𝑟 = to make the statement true.

(1) 9.74 × 1021 _______ 2.1 × 1022 (2) 5.28 × 1012 ______ 95.4 × 1012

(3) 2.33 × 1010 _______ 7.6 × 1010 (4) 4.4 × 107 ______ 44,000,000

(5) 548,000,000 _______ 5.48 × 107 (6) 1.2 × 10−3 ______ 4.7 × 10−3

(7) 6.23 × 1014 ______ 8.912 × 1012 (8) 5.15 × 10−4 ______ 6.35 × 10−5

(9) 3.28 × 1017 ______ 4.25 × 1017 (10) −1.2 × 105 ______ −1.7 × 105

(11) Compare the following problem. Be sure to explain your reasoning.

12.8 × 103 ______ 1.4 × 103

93


Directions- Compare the following numbers:

(1) 2.56 × 105 _______ 4.2 × 10−7

(2) 4.3 × 104 _______ 1.6 × 106

(3) 7.1 × 10−2 _______ 2.9 × 10−6

(4) 5.27 × 105 _______ 2.139 × 105

Directions-Order the numbers from least to greatest:

2.81 × 10−7; 2.01 × 103; 2.72 × 10−7; 9.45 × 10−4

Directions- The table lists the populations of five countries. List the countries from least to greatest.

94

Unit 3 – Lesson 9

Aim: I can find the product and quotient of numbers written in scientific notation.

Warm Up: Answer the following questions

(a) 2.7 × 3.4 (b) 8.04 ÷ 6.7

(c) 103 × 105 (d) 1012

104

Guided Practice:

To find the product of numbers that are in scientific notation:

1. Multiply the first factors (the numbers before the multiplication sign)

2. Keep the base of ten

3. Add the exponents

Exercise 1- Evaluate the following (7.2 × 103) (1.6 × 104)

Exercise 2- Evaluate the following (8.4 × 102) (2.5 × 106)

95

To find the quotient of numbers that are in scientific notation:

1. Divide the first factors (the numbers before the multiplication sign)

2. Keep the base of ten

3. Subtract the exponents

Exercise 3- Evaluate the following 9.45×1010

1.5×106

Exercise 4- Evaluate the following 1.14 × 106

4.8 × 103

Rules for Multiplying and Dividing Numbers in Scientific Notation

1 - Put your calculator in Sci. Not. Mode

2 - Type the problem into the calculator EXACTLY how it is written.

How to multiply and divide numbers in scientific notation:

You MUST use parentheses ( ) when inputting each number in

scientific notation!

To input an exponent, enter the base then hit ( ^ ) before entering

the exponent.

Hit (−) first if you need to make a number negative.

Simple numbers like (1.2 x 104) can be inputted like this:

( 1 . 2 x 10 ^ 4 =

96

Examples:

1) (3.4 x 103)(1.2 x 104)

Enter the following:

( 3 . 4 x 1 0 ^ 3 ) x ( 1 . 2 x 1 0 ^ 4 )

Answer: (3.4 x 103)(1.2 x 104) = 4.08 x 107

4.08 x1007

Problem Set:

1) (2.63 × 104) (1.2 × 10−3)

2) 7×106

2×103

3) 9 × 10−11

2.4× 108

4) (4 × 10)(2 × 103)

97

5) 9.8 × 103

2 × 102

6) (8 × 103)(3 × 104)

7) (51.3 × 108) ÷ (2.7 × 106) 8) (7 × 105)(6 × 107)

9) (4 × 102)(9 × 104) 10) (2 × 103)(5 × 104)

11. Neurons are cells in the nervous system that process and transit information. An average neuron is about

5 × 10−6meter in diameter. A standard table tennis ball is 0.04 meter in diameter. About how many times as

great is the diameter of a ball than a neuron?

98


Lesson 6

Determine if the number in scientific notation would be written with a positive or negative exponent.

1) Total weight of 18-wheel truck ______________ 2) Population in China ______________

3) Size of a computer pixel _____________ 4) Weight of an atom_____________

Write each number in correct scientific notation

5) 29 x 102 = ______________________ 6) .17 x 10-7 = ________________________

7) .052 x 10-4 = ______________________ 8) 386.4 x 10-6 = ______________________

Lesson 7

Write each of the following in scientific notation:

9) 25,000 __________________________________ 10) .000302 __________________________________

11) -4,700__________________________________ 12) 2 million__________________________________

Write each of the following in standard form:

13) 7104.2 x _______________________________ 14) 3108 x _________________________________

15) 4101.8 x _______________________________ 16) 51003.4 x ______________________________

What is the value of the missing exponent (n):

17) What is the value of n in the problem: 50,200,000 = nx 1002.5 n = ______

18) What is the value of n in the problem: 0.00032 = nx 102.3 n = ______

19) What is the value of n in the problem: 31,000 = nx 101.3 n = ______

20) What is the value of n in the problem: 0.0000082 = nx 102.8 n = ______

99

Lesson 8

Compare: Use < , >, or =

21) 6109.2 x 2,900,000 22) 3104.2 x 31041.2 x

23) 5101.2 x 7101.1 x 24) 5106.7 x 3108.4 x

Lesson 9

Multiply, or Divide

25) (2.8 x 107) (4.1 x 107) = ____________________ 26) (9.1 x 108) ÷ (3.8 x 108) =

_______________

27) (4.0 x 10-4) x (2.1 x 109) = ____________________ 28) (9.9 x 1010) ÷ (3.3 x 109) = ______________

Mixed Review

33) Which expression has the greatest value?

A) 1.045 x 10 2 B) 1.45 x 10 2 C) 8.4 x 10 2 D) -8.4 x 10 2

34) In the year 2000, approximately 169,000,000 personal computers were used in the United States. What is

this number expressed in scientific notation?

A) 1.69 × 10-8 B) 16.9 × 10-7 C) 16.9 × 107 D) 1.69 × 108

35) A butterfly weighs only about 5.0 x 10 -5 of a kilogram. What is the number written in standard form?

A) 0.00005 B) 0.000005 C) 50,000 D) 500,000

36) The average distance from Pluto to the Sun is 3.65 × 10 9 miles. What is this number written in

standard form?

A) 365,000,000 B) 3,650,000,000 C) 36,500,000,000 D) 365,000,000,000

100

Unit 3- Lesson 10

Aim: I can apply scientific notation

Warm up:

1. Which one doesn’t belong? Explain your reasoning.

14.28 x 10 9 (3.4 x 106)(4.2 x 103) 1.4 x 109 (3.4)(4.2) x 10 (6 + 3)

Problem Set:

1. What is the product of , , and expressed in scientific notation?

1) 2) 3) 4)

2. What is the quotient of and ?

1) 2) 3) 4)

3. What is the value of in scientific notation?

1) 2) 3) 4)

4. If the mass of a proton is gram, what is the mass of 1,000 protons?

1) g

2) g

3) g

4) g

101

5. The areas of the world’s oceans are listed in the table. Order the oceans according to their area from least to

greatest.

Ocean Area (ml2)

Atlantic 2.96 x 107

Arctic 5.43 x 106

Indian 2.65 x 107

Pacific 6 x 107

Southern 7.85 x 106

6. Mr. Murphy’s yard is 2.4 x 102 feet by 1.15 x 102 feet. Calculate the area of Mr. Murphy’s yard.

7. Every day, nearly 1.30 x 109 spam E-mails are sent worldwide! Express in scientific notation how many

spam e-mails are sent each year.

8. In 2005, 8.1 x 1010 text messages were sent in the United States. In 2010, the number of annual text

messages had risen to 1,810,000,000,000. About how many times as great was the number of text

messages in 2010 than 2005?

9. A micron is a unit used to measure specimens viewed with a microscope. One micron is equivalent to

0.00003937 inch. How is this number expressed in scientific notation?

1) 3.937 𝑥 105 3) 3937 𝑥 108

2) 3937 𝑥 10−8 4) 3.937 𝑥 10−5

10. The distance from Earth to the Sun is approximately 93 million miles. A scientist would write that

number as

1) 93 𝑥 107 3) 9.3 𝑥 106

2) 93 𝑥 1010 4) 9.3 𝑥 107

102


1. By the year 2050, the world population is expected to reach 10 billion people. When 10 billion is written

in scientific notation, what is the exponent of the power of ten?

2. The table shows the mass in grams of one atom of each of several elements. List the elements in order

from the least mass to greatest mass per atom.

Element Mass per Atom

Carbon 1.995 x 10-23

Gold 3.272 x 10-22

Hydrogen 1.674 x 10-24

Oxygen 2.658 x 10-23

Silver 1.792 x 10-22

3. A music download Web site announced that over 4 x 109 songs were downloaded by 5 x 107 registered users.

What is the average number of downloads per user?

4. Sara’s bedroom is 2.4 x 103 inches by 4.35 x 102 inches. How many carpeting would it take to cover her

floor? Express your answer in scientific notation.

103

Unit 3 Review

Exponents

1) Write in expanded form 2) Write the following in exponential form.

a. 𝑥5 b. (3

4)4

a. 2

3 ∙

2

3 ∙

2

3 ∙

2

3 b. 𝑎 ∙ 𝑎 ∙ 𝑎

________________ _______________ _____________ ______________

3) Write the following as POSITIVE exponents. 4) Compute the value (evaluate)

a. 5−3 b. 𝑥−1

9−5 c. (

2

3)−2

a. (−2)4 b. (4

5)−2

c. 52 − 34 + 70

______ _ _______ _______ _______ _______ _______

Will this product be positive or negative? 7) Find the value of n:

How do you know?

a. 2n = 64 b. 3n = 27 c. 5n = 125

5) (−1)14 6) (−1)283

_______ _______ _______

8) Rewrite 8 in exponential notation using 2 as the base. 9) Rewrite 81 in exponential notation using 3 as the base.

10) Multiply. Exponents must be put into positive exponential form.

a. 8−3 ∙ 87 = ______ b. 44 ∙ 4−13 ∙ 4 = ______ c. 𝑥6 ∙ 𝑥19 = ______ d. 3−5 ∙ 3−9 = ______

e. (6x)0 = ______ f. 6x0 = ______ g. (x4y)0 = ______ h. 6(x3y7)0 = ______

11) Which is equal to 82? 12) Write (-7) 2 in standard form. 13) The result of 5−4 comes from which

multiplication?

a) 28 b) 26 a) 53 ∙ 52 b) 34 ∙ 34

c) 92 d) 82 c) 53 ∙ 5−7 d) 32 ∙ 3−6

Raising a Power to a Power: Multiply and put answers into positive exponential form.

14) (54)3 = ______ 15) (𝑦)4 = ______ 16) (𝑥−2 𝑦)3 = ______

17) (𝑥8𝑦6)−2 = ______ 18) (𝑥9)2 = ______ 19) (7−8)2 ∙ 74 = ______

104

Dividing exponents with the same base. Put answers in positive exponential form.

20) 614

65 = ______ 21)

𝑥4

𝑥7 = ______ 22) 35 ÷ 33 = _______

23) 87 ÷ 813 = _______ 24) (−2)5

(−2)5 = _______ 25)

(2

5)7

(2

5)2 = _______

26) 26∙ (2)7

25∙ (2)4 = _______ 27)

𝑥10

𝑥−6 = _______

28) A square has a length of 𝒙𝟐 .

(a) Find the area of the square in exponential form. (b) If x =3, what is the area?

Scientific Notation Write the following in Standard Form:

29) 6.3 × 107 30) 5.23 × 10−4 31) 8.08 × 100 32) 4.2 × 10−1 33) 9.24 × 1010

Write the following using Scientific Notation:

34) 120 35) 65,002,000 36) 0.0000233 37) .345 x 104 38) 523 x 109

105

Find the value of the following. Write your answer in Scientific Notation.

39) (4.3 × 107)( 2.2 × 103) 40) (5 × 1012)(4.77 × 10−5) 41) (3.6 × 10−5)3

42) 6.2×109

2×102 43) (3.45 × 106) ÷ (8.01 × 10−5) 44)

1.6332×1011

1.6332×1011

Compare using < > =

45) 5.3 x 103 4.5 x 103 46) 2300 2.3 x 103

Word Problems

47) How many times larger is 9.8 x 106 than 6.32 x 105?

Unit 1: 48) Perform the indicated operations and evaluate 49) Given a = -2 ; b = 3 and c = -1, evaluate the following.

a. 72 − (−2)4 b. 1

2 ÷

3

4+

2

3 𝑐(𝑎 + 𝑏)

50) Distribute and/or Combine like terms

a. 3

5 (5𝑥 −

2

3) b. 5x + 6 – 3x + 8 c. 5(2y – 11) – 6y d. 4 – 2(4x + 3)

106

51) The following figure contains a rectangle.

a. Determine the area of the rectangle. b. If you put a fence around the rectangle

how much fence would you need?

52) Using the equations to the right, determine the following: 𝐶 = 5

9(𝐹 − 32) 𝐹 =

9

5𝐶 + 32

a. If C = 10, find F. b. If F = 68, find C.

Unit 2: Solve the following equation for the missing variable, otherwise determine no solution or infinitely many solutions.

53) 0.8 - 2x = 7.6 54 ) 4

5𝑥 − 3 = 9 55) 3x + 12 = 3(4 + x)

56) -3(2x – 1) = 3x + x + 53

57) 6(2𝑥 − 8) = 12(𝑥 + 3) 58) 3

5+

1

4𝑥 =

1

2

𝟑𝐱 − 𝟐

𝟓

107

Math 8

Unit 4 – Graphing Linear Equations

2019-2020


1 Writing Equations of a Line Lesson 1 – Page 109

2

Creating a Table Lesson 2 – Pages 112-

113

3 Graphing – Table Method Lesson 3 – Page 116

4 Finding slope and y-intercept from an equation Lesson 4 – Page 119

5 Graphing – Slope/Intercept Method Lesson 5 – Page 123

Quiz

6 Graphing Systems Lesson 6 – Page 129

7 Graphing Systems (Types of Solutions) Lesson 7 – Page 133

8

Graphing – Table and Slope/Intercept Method

Project

Project!

Review Study!

Test Good Luck!

108

Unit 4 -Lesson 1

Aim: I can write an equation of a line in slope – intercept form.

Warm up: Solve

1) 2x + 7 = 15 2) 2x -2x + 7 – 3 = 29

Guided Practice:

Vocabulary:

Slope – Intercept Form _____________________________________________________________________

Examples:

Rewrite the equation in slope-intercept form (y = mx + b)

1) 3x + y = 8 2) -x + y = 12 3) 4x – 2y = 8 4) 2x - y = 8

5) 4x = y + 8 6) y – 3x = 0 7) 3x + 3y = 15 8) 3y - 2x = 27

Problem Set:


1) 5x - y = 10 2) y - 7x = 0 3) 12x + 2y = 20 4) -15x + 3y = -3

5) -x + 2y = 8 6) 4x + y = 6 7) -2x + 4y = -28 8) -x + y = 15

9) 5x - y = 8 10) y - 12 = 16 11) 3y - 2x = 9 12) 2(y - 3) = 10

13) 25 = 10x + 5y 14) x = y - 4 15) 3y + 21 = 3x 16) 3x + 2 = y

109



1) -x + y = 6 2) x + y = -2 3) -x + y = -2 4) -2x + y = -4

5) 3x - y = 1 6) -2x + y = 0 7) 4x + 2y = 8 8) -9x + 3y = -6

9) -2x + 3y = 3 10) 2x + y = 6 11) x + 4y = -20 12) -x + y = 7

13) 3x - y = -6 14) y - 2 = 8 15) 3y - x = 12 16) 2(y - 4) =8

17) 10x + 5y = 25 18) y - 8 = x 19) 3y + 12 = 9x 20) 6x + 4 = y

Review Work:

21) Hawaii’s total shoreline is about 210 miles long. New Hampshire’s shoreline is about 27 miles long.

About how many times longer is Hawaii’s shoreline than New Hampshire’s?

22) Evaluate each expression for n = 3

a. 2n + 5 – n b. 3n+18

3n c.

24

4 − n · n

110

Unit 4 -Lesson 2

Aim: I can create a table from an equation.

Warm up: Substitute: x = 2, y = 3

1) (x + y)2 2) 3x + 2y 3) 2x2 + y

Guided Practice:

Vocabulary:

Equation of a Line ____________________________________________________________

Rules:

1) Solve for y (y = mx + b) if needed

2) Pick x - values for the table (watch for special cases)

3) Solve to get the y - values

4) List the points of the line ((x,y), ordered pair)

Examples: Complete each table

111

Problem Set: Complete each table

112


Review Work:

8) Find the area of the rectangle.

9) Simplify: (3x – 5) – (x + 3) + (-2x + 7) 10) Simplify: 42

3 – (-3

3

4)

11) Simplify: 64 - 42 ÷ 8 12) Solve for x: 5x + 20 = 5(x + 4)

Write as a positive exponent:

13) 6 -4 14) 25x 4 ÷ 5x 8

3x

2x - 8

113

“Things you should know”

1) x axis

2) y axis

3) Origin (0,0)

4) Quadrants (I, II, III, IV)

5) A point is always written (x, y)

6) How to plot a point

7) How to name a point

Examples:

Give the quadrant or the axis of each point.

1) (3, 1) ____ 7) (-8, -4) ____

2) (-8, 2) ____ 8) (0, -2) ____

3) (7, -4) ____ 9) (-5, 5) ____

4) (-9, -2) ____ 10) (6, 0) ____

5) (2, -9) ____ 11) (2, 2) ____

6) (0, 0) ____ 12) (1, -8) ____

Plot each of the following points and label.

J (2, 3) N (8, 0)

K (-5, 1) P (0,0)

L (-4, -6) Q (-6, 0)

M (5, -3) R (0, 5)

114

Unit 4 – Lesson 3

Aim: I can graph a line using the table method.

Warm up: Complete the table

y = 3x + 4

Guided Practice: 1) Solve for y (y = mx + b) if needed

2) Pick values for the table

3) Solve for all y values

4) List the points of the line

5) Graph and label the points

6) Connect the points with a ruler and put arrows on your line

7) Label the line

Examples: Complete the table and Graph the Line

1) y = 2x – 5 2) y = 3

1x + 2

x y (x,y)

x y (x,y)

x y (x,y)

115

Problem Set:

1) y = 2x + 2 2) 52

1 xy

3) 3x + y = 9 4) 2x – y = 4

x y (x,y)

x y (x,y)

x y (x,y)

x y (x,y)

116

Unit 4 -Lesson 3 Homework Complete the table and Graph the Line

1) y = 2x + 3 2) y = -x + 1

Review Work:

3) Write x-6 as a positive exponent.

4) Solve for x: 4(2x – 6) = 8(x – 3)

Simplify:

5) 8

0 6)

0

9 7) x-2 · x 8 8) (22 )3 9) 5x0

10) Write 3x + 5y = 15 in y form

x y (x,y)

x y (x,y)

117

Unit 4 - Lesson 4

Aim: I can find the slope and y-intercept from an equation.

Warm up: Rewrite the equation in slope-intercept form (y = mx + b)

1) 3x + y = 8 2) -x + y = 12

Guided Practice:

Vocabulary:

Slope - ______________________________________________________________________________

y-intercept - __________________________________________________________________________

Rule:

1 – Write the equation in slope - intercept form

2 – Find m and b (write m = and b = )

Examples: Rewrite the equation in slope - intercept form (y = mx + b) and then find m and b

1) y = 3x + 8 2) y = 1

2 x - 2 3) y = -2x 4) y = -4x - 7

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

5) 2x + y = 11 6) -x + y = 6 7) 6x - 3y = 15 8) 5x - y = 1

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

9) 4x = y + 8 10) y - x = 0 11) 6x + 6y = 36 12) 2y - x = 8

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

118

Problem Set: Rewrite the equation in slope - intercept form (y = mx + b) and then find m and b

1) y = -x + 6 2) y = x - 2 3) -x + y = -2 4) -2x + y = -4

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

5) 3x - y = 1 6) -2x + y = 0 7) 4x + 2y = 8 8) -9x + 3y = -6

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

9) -2x + 3y = 3 10) 2x + y = 6 11) x + 4y = -20 12) -x + y = 7

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

13) 3x - y = -6 14) y - 2 = 8 15) 3y - x = 12 16) 2(y - 4) =8

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

17) 10x + 5y = 25 18) y - 8 = x 19) 3y + 12 = 9x 20) 6x + 4 = y

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = ____

119


Rewrite the equation in slope - intercept form (y = mx + b) and then find m and b

1) 5x + y = 16 2) 4x + y = -12 3) -x + y = 6 4) -2x + y = 15

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

5) 5x - y = 10 6) y - 7x = 0 7) 12x + 2y = 20 8) -15x + 3y = -3

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

9) -x + 2y = 8 10) 4x + y = 6 11) -2x + 4y = -28 12) -x + y = 15

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

13) 5x - y = 8 14) y - 12 = 16 15) 3y - 2x = 9 16) 2(y - 3) = 10

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

Review Work:

17) Solve for x: -1.2 + 4x = 2x + 6.8

Simplify:

18) 13 − 15 + 8 − 1 + 4 19) 3(−4)2 − 12

120

Unit 4 - Lesson 5

Aim: I can graph a line using the slope/y-intercept method.

Warm Up: 1) 5x + y = 16 2) 4x + y = -12 3) -x + y = 6 4) -2x + y = 15

m = _____ m = _____ m = _____ m = _____

b = _____ b = _____ b = _____ b = _____

Guided Practice:

Rules:

1) Solve for y (y = mx + b) if needed

2) m = b =

3) Graph b on the y line

4) Make more points using m (extend as far as possible)

5) Connect the points with a ruler and put arrows on your line

6) Label the line

Examples:

Graph the following using slope/y-intercept

1) y = 2x - 3 2) y = -2x + 4

m = m =

b = b =

Remember:

Always write the slope (m)

as a fraction!!!

121

3) y = 7 4) x = 9

Problem Set:

1) y = 1

2 x - 5 2) y =

−2

3 x + 8

m = m =

b = b =

122

*3) y – 5 = -2x 4) y = 3x

m = m =

b = b =

123

Unit 4- Lesson 5 Homework 1) y = 2x - 10 2) y = -3x + 8

3) y = x - 5 4) y = -x + 6

5) 72

1

xy 6) 4

3

1 xy

124

Unit 4 - Lesson 6

Aim: I can graph Systems of Equations.

Warm up: Solve

1) 2x + 2 = 2x – 6 2) 4x – 2 = -x + 8

Guided Practice:

Rules:

1) Graph each equation on the same set of axes.

2) Find and Label the coordinates of the point of intersection of the lines.

3) Check both equations.

Examples:

1) Graph and Check:

y = -x + 2

y = 3x - 2

Check: What is the solution? _________

125

2) Graph and Check:

3x - 2y = 4

x + y = 3


126

3) Graph and Check:

y = 2x + 2

y = 2x – 6


127

Problem Set: 1) Graph and Check:

y = 4x - 2

y = -x + 8


128

2) Graph and Check:

y = -3x + 7

y = 2x – 3


129


1) Graph and Check: 2) Graph and Check:

y = -4x + 5 y = x + 6

y = 3x - 9 y = -2x

3) Graph and Check: 4) Graph and Check:

y = 4x - 3 y = 3

2 x - 3

y = -2x + 9 y = x - 2

130

Unit 4 - Lesson 7

Aim: I can graph systems of equations and identify types of solutions (One Solution, No Solution, Infinitely

Many Solutions)

Warm up:

1) Graph and Solve

y = -x + 2

y = 3x - 2

Solve for x:

2) 2x – 4 = -x – 1 3) 2x + 4 = 2(x + 1) 4) 2x + 4 = 2(x + 2)

Guided Practice:

When you graph two lines, there are three things that can happen.

Intersect Once Never Intersect Same Line

One Solution No Solution Infinite Solutions

131

Examples:

Graph each and determine the type of solution. (one solution, no solution, or infinite solutions).

1) y = 2x + 4 2) y = x + 6 3) y = 2x + 1

y = 2x - 6 y = -2x 2y = 4x + 2

Type of Solution: _____________ Type of Solution: ______________ Type of Solution: ___________

How does it look graphically? How does it look graphically? How does it look graphically?

__________________________ __________________________ _____________________

What do you notice about the What do you notice about the What do you notice about the

slopes and the y- intercepts? slopes and the y- intercepts? slopes and the y- intercepts?

__________________________ __________________________ ________________________

Finding types of slopes algebraically (By Inspection)

Use your knowledge of slopes and y-intercepts to determine the type of solution.

(one solution, no solution, or infinite solutions).

4) y = 2x + 8 5) y = 3x + 8 6) y = 2x + 3

y = 2x – 7 y = -2x – 4 3y = 6x + 9

__________________________ __________________________ ________________________

132

Problem Set:

Solve graphically and state the type of solution:

1) y = 1

2x – 3 2) 3y – 6x = 12

2y = x + 4 y = 2x + 4

State the type of slope algebraically and explain how you got the answer.

3) y = x + 4 4) y = 2x + 3 5) y = 5x + 7

3y = 3x + 12 y = -2x + 9 y = 5x + 4

__________________________ __________________________ ________________________

__________________________ __________________________ ________________________

__________________________ __________________________ ________________________

__________________________ __________________________ ________________________

6) y = 3x + 2 7) 2y = 4x + 16 8) 5x – 2y = 8

y = 3x + 6 y – 2x = 8 5x – 2y = -8

__________________________ __________________________ ________________________

__________________________ __________________________ ________________________

__________________________ __________________________ ________________________

__________________________ __________________________ ________________________

133

Unit 4 Lesson 7 - Homework

State the type of solution and then prove it graphically:

1) y = 4

3 x + 3 2) y = 2x + 4

y = −2

3 x – 3 2y – 8 = 4x

Type of Solution ____________________ Type of Solution ____________________

State the type of solution and then explain how you got it.

3) 2y = 6x – 10 4) y = 2x - 8

3y – 9x = 12 y = 2x

Type of Solution ____________________ Type of Solution ____________________

Explanation:________________________ Explanation:________________________

___________________________________ ___________________________________

134

Unit 4 -Lesson 8 (project)

Aim: I can graph a line using both table and slope/ y intercept method.

Warm up: Complete each table. Then graph each solution on your project.

1) 𝑦 = −1

2𝑥 − 4 2) 𝑦 =

3

2𝑥 + 12

5) 𝑦 =1

2𝑥 + 4 6) 𝑦 =

1

2𝑥 − 4

x y (x,y)

-4

-2

0

2

4

x y (x,y)

-4

-2

0

2

4

x y (x,y)

-4

-2

0

2

4

x y (x,y)

-4

-2

0

2

4

135

Unit 4 Review

Rewrite the equation in slope- intercept form (y = mx + b)

1) 5x + y = 4 2) 3x - y = 9 3) -x + y = 17 4) 5y - 2x = 15 5) 2(y - 4) = 8

y = 5x – 10 y = -2x + 5

6) What is the slope?_____ 8) b = _______

7) What is the y-intercept?_____ 9) m = ______

Graph the following lines using the table method:

10) y = -3x + 5 11) x + y = 2

x

y (x,y)

x

y (x,y)

136

Graph the following lines using the slope-intercept method:

12) y = 3x – 5 13) y = 1

2x – 3 14) y = -x + 4

15) 2x + y = 5 16) Graph any method: 17) Graph any method:

y = 2 x = -3

18) Draw 2 lines representing the following number of solutions:

a) No Solutions b) One Solution c) Infinite Solutions

19) How many solutions does each system of equations represent?

(one solution, no solution, or infinite solutions)

a) y = -5x + 9 b) y = x + 5 c) y = 4x + 2

y = -5x – 6 3y = 3x + 15 y = 3x – 8

137

20) Graph the system of equations:

y = x + 4

y = -2x + 7

21) What is the solution?__________

22) Check the solution:

Unit 3: Simplify. Exponents must be placed in positive exponential form.

23) 3−5 ∙ 3−7 24) 37 ∙ 21 25) 5−2 26) x0

27) -20 28) (x4)(x−2)

x 29) (a-4bc2)3

Unit 2: Solving Equations Solve the following equation for the missing variable, otherwise determine solution type

30) 5x – 6 = -41 31) 4(-3x + 2) = 44 32) 12 – 4x = 18

33) 3x + 5 = -15 + 4x 34) 0.7x – 0.2 + 0.5x = 1 35) 3

5+

1

4𝑥 =

1

2

Unit 1: Simplify

36)

a) Find area of rectangle b) Find the perimeter of the rectangle

3x - 4

2x

sagamore middle school math 8 quarter 1 · quarter 1 name:_____ #____ 1 math 8 unit 1- integers...

Documents

sagamore middle school math 8 quarter 1 · quarter 1 name:_ # 1 math 8 unit 1- integers...