Math 8 Quarter 1mrsleonardo1.weebly.com/uploads/5/6/8/7/56870649/math_8_quarte… · Quarter 1 Unit 1 – Integers Unit 2 – Solving Equations Unit 3 – Exponents Unit 4 – Graphing

Math 8

Quarter 1

Unit 1 – Integers

Unit 2 – Solving Equations

Unit 3 – Exponents

Unit 4 – Graphing Linear Equations

Name: ___________________

Period: _______

2

3

Math 8

Unit 1- Integers

Math 8 Remind Code:

Google Classroom Code:

Extra Help Schedule: Tuesdays at 7:40am

Date Lesson Topic Homework

9/4 Class Procedures /Growth Mindset

9/5 1 Review of Integers Lesson 1 - Page 6

9/6 2 Operations with Integers Lesson 2 - Page 11

9/9 3 Applying Order of Operations Lesson 3 - Page 14

9/10 4 Translating Algebraic Expressions Lesson 4 - Page 17

9/11 5 QUIZ - Integer Rules

Evaluating Algebraic Expressions

Lesson 5 - Page 20

9/12 6 Combining Like Terms Applications Lesson 6 - Page 23

9/13 7 Distributive Property Applications Lesson 7 - Page 25

9/16 Review Please study 😊

9/17 TEST

4

Lesson 1 – Introduction to Integers

Aim: I can identify and 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?

Define below and give an example.

Natural/Counting Numbers: ________________________________________________________

______________________________________________________________________________________

Examples –

Whole Numbers: ____________________________________________________________________

______________________________________________________________________________________

Examples –

Integers: __________________________________________________________________________

______________________________________________________________________________________

Examples –

Rational Numbers: __________________________________________________________________

______________________________________________________________________________________

Examples –

Irrational Numbers: _________________________________________________________________

______________________________________________________________________________________

Examples –

Real Numbers: ______________________________________________________________________

______________________________________________________________________________________

Examples –

5

GUIDED PRACTICE

INDEPENDENT PRACTICE

State the Integer using a positive “+” or negative “-” sign along with the number.

1) 5 degrees above 0 ________ 4) a deposit of $25 ________

2) a loss of 2 yards ________ 5) 3 inches of rainfall ________

3) a withdrawal of $25 ________

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

6

Unit 1 Lesson 1 –Homework

Write the number that describes each situation using a positive “+” or negative “-” sign.

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

7

Lesson 2 – Add and Subtract Integers

Aim: I can add and subtract 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

1) -2 + -3 2) -2 + 8

3) -6 + 1 4) -4 + 4

5) 6 + (-3) 6) -8 + 2

Rules for Adding Integers

Signs Same Signs Different

1) _____________________________________

2) _____________________________________

1) _____________________________________

2) _____________________________________

Subtracting Integers

1) 6 – 10 2) -3 – 8

3) 4 - -6 4) -2 – (-8)

5) -2 - -2 6) 7 + -3

Rules for Subtracting Integers

1) ______________________________________________________________________________

2) ______________________________________________________________________________

8

Multiplying and Dividing Integers

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

Rules for Multiplying and Dividing Integers

Odd number of negative signs Answer – Negative

Even number of negative signs Answer – Positive

Steps for Multiplying and Dividing Integers

1) ______________________________________________________________________________

2) ______________________________________________________________________________

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 Under the answer is U__________________________________

9

Using a Calculator for Fractions

Steps 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 = _________

10


Adding and Subtracting Integers

1) 8 + -7 2) 10 + -5 + 3 3) -1 + -7 4) -3 + (-3)

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

9) 5 - 12 10) 8 – (-9) 11) -7 – (-10) 12) 8 – 9 + 3 – 7

13) -8 + 12 – 9 + 4 14) 5 – 6 + 2 – 11

Multiplying and Dividing Integers

15) (-4)2 16) -42 17) -28

2 18) (-8)(-5)

19) -3

0 20) (-6)(7) 21)

0

5 22) -2 ÷ 0

23) 0

8 24)

8

0 25)

-32

4 26) (4)(-9)

Operations with Fractions

27) 3

10+

7

20 28) −

5

4−

3

4 29) −

2

3×5

4 30)

1

9÷ −1

1

3

11

Unit 1 Lesson 2 - Homework

Simplify:

1) -4 + (-4) 2) 8 + (-8) 3) -3 - (-8) 4) 5 - 9

5) (-24)(−5

8) 6) (−

2

3)(-18) 7) (3)(-1)(6)(-2) 8) (-4)(-23)

9) -55

5 10) -18 ÷ -9 11)

0

7 12)

3

0

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

21) −8 ∙23

2 22) −14 ∙ −

6

7 23) 7 ∙ −

1

7 24) 18 ∙

2

3

25) 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?

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

Tuesday morning?

27) 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?

28) Circle all the numbers that are NOT integers and explain why you chose them.

12

Lesson 3 – Order of Operations

Aim: I can simplify numerical expressions

Warm Up:

1) The recipe for mint chocolate chip ice cream requires 2 1

4 cups of cream for 5 people. You need

ice cream for 8 people. How much cream will you need?

2) 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?

Order of Operations

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 −+−

13


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

14

Unit 1 Lesson 3 – Homework

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

15

Lesson 4 – Translating Algebraic Expressions

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

Algebraic Expression vs. Algebraic Equation

__________________________________ __________________________________

Example___________________________ Example___________________________

Common Expressions

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….

GUIDED PRACTICE

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

___________________

16

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

Match each phrase with the correct algebraic expression

___ 5) n increased by 11 A) n – 19

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

___ 7) the sum of n and 19 C) n + 19

___ 8) 11 more than n D) n – 11

___ 9) n increased by 19 E) 19 – n

F) 11 - n

INDEPENDENT PRACTICE Write each as an algebraic expression

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.

17

Unit 1 Lesson 4 - Homework

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

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?

18

Lesson 5 – Evaluating Algebraic Expressions

Aim: I can translate verbal expressions to algebraic expressions

Warm Up:

1) Which answer is greater? (3x)² or 3x² when x = 2

GUIDED PRACTICE

Parts of an Expression

Name the coefficient: _______ Coefficient – ________________________________________

Name the variable: _______ Variable – __________________________________________

Name the exponent:_______ Exponent – _________________________________________

Name the constant: ________ Constant – __________________________________________

Name the base: __________ Base – _____________________________________________

Steps for 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

19

Conversion Formulas

Fahrenheit → Celsius Celsius → Fahrenheit

)32(9

5−= FC 32

5

9+= CF

Practice Conversions:


7) 5x – 2y if x = 3; y = 6 8) 2

5 x + 3 for x = 20 9) 3x + 5y - 8 for x = 3.1; y = -.8

10) 19 - 2m for m = -6 11) ab4 if a = 3; b = -2 12) 2x2 if x = -3

13) (2x)2 when x = -3 14) (4x + 3y)2 + 9 for x = -1

2 and y = -

2

3 15)

3𝑎−8

2𝑏+4 for a = -2 and b = -3

96 ° F

35 ° C

20


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)?

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

Convert the following temperatures using the given formulas

𝐶 =5

9(𝐹 − 32) 𝐹 =

9

5𝐶 + 32

7. 50 ° F 8. 56 ° C

21

Lesson 6 – Combine Like Terms Applications

Aim: I can express the perimeter of a polygon algebraically

Warm Up:

1) What is the coefficient of “p” in a - 7p + 21? ______ What is the constant? ________________

2) What are the like terms of 7r + 5 + 3r? ________________ What is the constant? ______________

3) What is the coefficient of “a” in the expression 4c + 5 + a? __________

Simplify each algebraic expression (#4 – 7)

4) -9x + 4x 5) 9y – 3 + 6y – 8 6) -2a + 2a 7) -7x + 7x + 3

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

10ft 8ft

Review:

Classify each polynomial as either a monomial, binomial, or trinomial.

1) 2x² + 3x - 1 2) 6xy 3) -7m5 4) 5y² - 2

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

22


Simplify by combining like terms

1) x + 9 + x – 8 + 3c 2) 8xy + 4a – 9xy – 6a – 7a 3) 6g + 14 + 3g + g – 5

Find the perimeter

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 - 10

23


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

24

Lesson 7 – Applications using the Distributive Property

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


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

17) 18) 19)

5 7 5

2x + 3

x + 6

25


Express in simplest terms.

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

5) - (-6x + 9) 6) -2(3x + .6) 10) 4(7 – 5n) – 2(3n + 4) 11) 7(2x2 – 3x ) + 10x

9) -9(3 – 10n) – 3 7) 1

3 (7n + 1) 8) 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

26

Unit 1 Integers – 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

27

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 expression

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) (a) Express the area of a rectangle in simplest terms of x, whose width is 4 and length is 2x - 8.

28

(b) Express the perimeter in simplest terms of x:

7x + 2

3x – 8

38) You can estimate the stopping distance, needed for a car going at a speed of x mph by adding the

reaction distance, x, and the braking distance ( 𝑥2

30). The expression 𝑥 + (

𝑥2

30) is used to represent the

stopping distance. How much longer is the stopping distance for a race car going 120 mph than one

going 90 mph?

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 expression 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

42) Check all that apply

Natural Number Whole Number Integer

6

0

-4

29

Math 8

Unit 2 - Solving Equations

Extra Help Schedule: _______________________

_______________________


1 1-Step and 2-Step Equations Lesson 1 – Page 33

2 Like Term Equations Lesson 2 – Page 37

3 Distribute Equations Lesson 3 – Page 40

4 Multi-Step Equations Lesson 4 – Page 43

QUIZ

5 Variable on Both Sides Lesson 5 – Page 46 & 47

6 Variable on Both Sides (Multi-step) Lesson 6 – Page 50

7 Word Problems - Translating Lesson 7 – Page 54

8 Equation Solutions Lesson 8 – Page 57

9 Decimal & Fractional Equations Lesson 9 – Page 60

Review Please study 😊

TEST

30

Unit 2 – Lesson 1 − Solving 1-Step and 2-Step Equations

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

31

Exercise 2- Solve the following equations to determine the value of the variable:

1. – 𝟔 +𝒙

𝟒= −𝟓

3. 𝟗𝒙 − 𝟕 = −𝟕

2. – 𝟏𝟓 = −𝟒𝒎+ 𝟓

4. 𝟐𝒏 + 𝟏𝟎 = −𝟐

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

1. 𝒗

𝟑+ 𝟗 = 𝟖 2. 𝟎 = 𝟒 +

𝒏

𝟓

32

3) -x = -10 4) 4x = 2 5) -25 = -5x

6) 𝑥

3 = 7 7)

𝑥

4 = -8 8) 2x + 8 = 20

9) 𝑥

4 – 3 = 27 10) 13 = 3x – 8 11)

𝑥

4 + 12 = -4

12) 16 – 4y = -8 13) 4x – 3 = -15 14) 6 + 𝑥

−3 = 8

Write an equation and solve

15) The $54 selling price of a sweater is the cost 16) If you multiply John’s age by 4 and then

increased by $18. Find the cost. subtract 2, you get 10. What is John’s age?

33

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) 10 + 2a = 22

7) – 19 – 6y = 21 8) -x + 3 = -15 9) 𝑥

9 + 12 = 9

10) 𝑥

−2 = -7 11) 5x = 5 12) 3x = 0

13) 4 posters cost $7.40. If each poster costs the 14) If you double your weight and add 6, your

same, how much does each poster cost? weight is 200 lbs. How much do you weight?

34

Unit 2 – Lesson 2 – Combining Like Terms Applications

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

Steps to Solving Equations with Like Terms

1) _________________________________________________________________________________

2) _________________________________________________________________________________

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.

35

Independent Practice: 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

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.

36

Write an equation and solve

6) We rent 3 video games on Friday and 11 on Saturday. Each video game costs the same amount. If we spend

$8.40, how much was each game?

7) 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 simplify the expression, then we can _______________ or _______________ the constant to

the other side and then isolate the variable by either ______________________ or _____________

___________________________________________________________________________________________________

___________________________________________________________________________________________________

___________________________________________________________________________________________________

37

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

Write an equation and solve:

7. If you sell 3 bags of candy, then 4 bags and finally 1 bag, how much is each bag if you collected $6.40?

38

Unit 2 – Lesson 3 – Applications using the Distributive Property

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. 4𝑥 + 2(3 + 2𝑥) = 10 2. 4(𝑥 − 2) = 40

3. −5(𝑥 + 4) = 35 4. 2(x + 5) = 3(x – 8)

39

Independent Practice:

1) 2(5𝑥 + 15) = 50 Check: 2) 100(5𝑥 + 10) = 1500 Check:

3) 3(7𝑥 + 9) = 48 Check: 4) −2(−𝑥 – 5) = 18 Check:

5) 7(3x – 4) = 14 6) 5(3𝑥 − 2) + 8 = 43

40

Unit 2 - Lesson 3 Homework

Solve the following equations for x:

1) 4(x – 1) = 20 Check: 2) 2(5 + x) = 22 Check: 3) -5(x + 4) = 15 Check:

4) 5(a + 4) = 10 Check: 5) -4(2m + 6) = 16 Check: 6) 3(2b + 3) = 27 Check:

7) −5(𝑥 + 4) = 15 Check: 8) 2(3 – y ) = 14 Check: 9) 5(x – 3) = 45 Check:

13) Which best describes the solution for this equation? 0.5(4x + 12) = 26

A. x = 5

B. x = 4

C. x = 7

D. x = 1

Convert the following temperatures:

14) 80 ° C = _____ F 15) 145 ° F = _____ C

41

Unit 2 - Lesson 4 – Multi-Step Equations

Aim: I can solve multi-step equations

Warm Up: Simplify

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

Steps for Solving Multi-step Equations

1) _________________________________________________________________________________

2) _________________________________________________________________________________

3) _________________________________________________________________________________

4) _________________________________________________________________________________

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

42


Solve 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:

43


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

44

Unit 2 - Lesson 5 – Solving Equations with Variables on Both Sides

Aim: I can solve equations with variables on both sides

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

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

3) Solve for x and show all algebraic steps.

2𝑥 + 3 = 3(𝑥 + 7) Property used here is: ___________________________

4) 4(2𝑝 − 8) − 6𝑝 = 20 + 4(𝑝 + 6) What property would you use for the first

algebraic step of this equation?

__________________________

45


1) 15𝑥 = 10𝑥 − 30 2) 4(3𝑥 − 10) = 10(𝑥 − 3)

3) 7𝑥 + 5 − 2𝑥 = 3𝑥 − 17 4) -4y – 13 + 9y = -18 + 6y

5) 7x + 8 = 4x + 17 6) 8 + 12x = 10x + 14

7) 4y + 11 = -y + 7 Check your solution for #7

46

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

47

5. 9𝑦 – 5 – 3𝑦 = 4(𝑦 + 1)

6. 3𝑥 – 4(2𝑥 + 5) = 3(𝑥 – 2) + 10

7. Marcy made an error when solving the equation below.

8m – 20 = 36

8m – 20 + 20 = 36

8m = 36

8𝑚

8=36

8

𝑚 = 44

8

A) Identify Marcy’s error to explain why it resulted in an incorrect solution.

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

B) Correctly solve 8m – 20 = 36 for m. Show your work.

48

Unit 2 - Lesson 6 – Solving Equations with Variables on Both Sides

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

Warm up:

Solve and check your work

1) 1

2(6𝑥 + 8) + 3𝑥 = 5𝑥 + 25

Guided Practice:

Use the acronym to solve equations

1) 3(x – 5) = 2(2x + 1) Check:

2) 2(3x + 4) + 2x = 3x + x + 20 Check:

Multiply to clear ALL

parentheses.

Combine like terms on the

SAME side of the equal sign.

Use inverse operations to move

variables on opposite sides of

equal sign.

to combine constants

to get variable alone

49

3) 14 – x = 2(x + 4) 4) x – 2(4 – 7x) = 22

5) 6(x + 2) = 4 – (3 – 2x) – 1 6) 3x + 2(x + 9) – 8 = 3(x – 5) + 7x – 10

Challenge Questions

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

Marco is trying to find the width of a rectangle. The length of the rectangle is 6 inches. 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!

50


Solve and check

1) x + 9 = 7x + 9 + 6x 2) x – 22 + 4x = 1 + 1 + x

3) – (–6w – 2) = 11 + 3w 4) 2(x + 2) = 4 – (3 – 3x) + 5

5) a – 22 + 3a = 1 + 1 6) 2x + 3(x + 6) – 2 = 2(x – 5) + 4x – 1

7) If the area of the rectangle is 112, 8) Find the value of x so that each pair of polygons

what is the value of x? has the same area.

8

3x + 5 x + 2

3

x + 4

5

51

Unit 2 - Lesson 7 – Solving Word Problems

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 these 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”

52

Translate the following verbal expressions 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.__________________________

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.

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?

53

INDEPENDENT PRACTICE:

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?

54


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

55

Unit 2 - Lesson 8 – Solving Special Solutions

Aim: I can determine how many solutions an equation has

Warm up: Copy down the vocabulary words.

Solving Special Solutions

Vocabulary:

No Solution -

___________________________________________________________________________

One Solution -

__________________________________________________________________________

Infinite Solutions -

__________________________________________________________________________

Algebraic examples

Guided Practice:

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)

56


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

57


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 11) Which best describes the solution for the

for 5b – 25 = 25? equation: 6y – 3 = -6y + 3

a. no solution a. no solution

b. 0 b. infinitely many solutions

c. 10 c. 0.5

d. 80 d. 2

58

Unit 2 – Lesson 9 – Solving Equations with Decimals and Fractions

Aim: I can solve equations with decimals and fractions

Warm Up:

Solve

5(2𝑥 + 1) – 7𝑥 = 20

Steps to remember when solving equations:

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

3=−j 2)

8

1

8

3=−h 3)

9

4

9

1=+g 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

59


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)

60


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?

61

Math 8 - Unit 2 Review

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)

62

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?

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

63

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

64

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.

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

65

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)

66

Math 8

Unit 3 – Exponents

Extra Help Schedule: _______________________

_______________________

Date Lesson Topic

Homework

1 Introduction to Exponents Lesson 1 – Page 69

2 Zero and Negative Exponents Lesson 2 – Page 74

3 Multiplying Exponents Lesson 3 – Page 77

4 Raising and Power to a Power Lesson 4 –Page 80

5 Dividing Exponents Lesson 5 – Page 83

Quiz Review Pages 87 & 88 as needed

Solutions posted

Quiz

6 Introduction to Scientific Notation Lesson 6 –Page 91

7 Converting Scientific Notation Lesson 7–Page 95

8 Compare and Order Scientific Notation Lesson 8–Page 98

9 Multiply and Divide Numbers in Scientific Notation Lesson 9 – Page 102

10 Applications of Scientific Notation Lesson – 10 Page 108

Review Please Study 😊

Test

67

Unit 3 – Lesson 1 – Introduction to Exponents

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. (Also considered Evaluate or Calculate)

Examples: Write the following in exponential form:

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

9

7

9

7

9 ____________

3) 2𝑥 ∙ 2𝑥 ∙ 2𝑥 ∙ 9 ∙ 9 ____________ 4) 4 ∙ 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

68

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

Independent Practice: Write in expanded form and compute the value:

17) 36 18) (−2)4 19) (−

4

11)5

20) 92 21) -42 22) −(

1

2)3

Write the following in exponential form:

23) 1

4×1

4 24)

7

9

7

9

7

9 25)

1

3×1

3×1

𝑥×1

𝑥×1

𝑥

Write in expanded form.

26) (2x)3 27) (3x)2 28) (7x)4

29) What do exponents represent?

30) What is a base? What is a coefficient? Give examples of each.

69


Write the following in exponential form:

1) 333 2) yyyxx 55 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.

70

Unit 3 – Lesson 2 – Zero and Negative Exponents

Aim: I can simplify zero and negative exponents.

Warm Up: Express in expanded form.

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

5)3

Express in exponential form.

4) 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ______________ 5) (8)(8)(8) ______________

Discovering the Zero Exponent Rule

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

evaluate each expression using what you know about dividing a number by itself.

Expression The Expression in

Expanded Form

Rewrite Using

Exponents

Evaluate

𝟓𝟔

𝟓𝟔

𝒙𝟓

𝒙𝟓

(−𝟒)𝟑

(−𝟒)𝟑

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

Note: this works when 𝒙 ≠ 𝟎

71

GUIDED PRACTICE: 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) 4x(x9y5)0________

Discovering the Negative Exponent Rule

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

Expression Expanded Form Rewrite as a Fraction Rewrite Using

Exponents

22

25

2 ∙ 2

2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

(−9)3

(−9)4

𝑎2𝑏5

𝑎4𝑏7

Definition: For any positive number 𝑥 and for any positive integer 𝑛, we define 𝒙−𝒏 = 𝟏

𝒙𝒏.

Note that this definition of negative exponents says 𝒙−𝟏 is just the reciprocal 𝟏

𝒙 of 𝒙.

As a consequence of the definition, for a positive 𝑥 and all integers 𝑏, we get 𝒙−𝒃 =𝟏

𝒙𝒃

Rules to convert from a Negative Exponent to a Positive Exponent:

Step 1) Write as a fraction if needed.

Step 2) Write the reciprocal of the base.

Step 3) Make the exponent positive.

Definition

72

Write each expression using a positive exponent

(a) 8−5 (b) 3−9 (c) 𝑧−2 (d) 𝑝−4

Rewrite as a positive exponent and evaluate each expression

(a) (−6)−2 (b) 3−3 (c) 2−4 (d) 5−3

Write each fraction as an expression using a negative exponent

(a) 1

29 (b)

1

64 (c)

1

𝑒5 (d)

1

74

1) 54

8−6 ________ 2)

6−3

32 ________ 3)

3−9

52 ________ 4)

𝑥−3

𝑦 8 _______

5) 𝑥5

92 ________ 6)

4−4

3−7 ________ *7)

2𝑥−4

𝑦−3 _______ *8)

𝑥4

6𝑦−3 _______


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) 3x(y)0 8) (3xy)0 9) 149x0 10) x(3a3b7c4)0

Special Situations

Move only the negative exponent to the opposite part of the fraction and then make the exponent positive.

73

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)

(5𝑥)−2

(2𝑦)−5

Lesson Summary:

• Anything raised to the zero power is always __________.

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

74


Simplify and write as a positive exponent if needed

1) 0)8( x 2)

08x 3) 9x(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)

𝑥8𝑦−3

𝑎𝑐−5

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 5x(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

75

Unit 3 – Lesson 3 – Multiplying Exponents

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.

76

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


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

77


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: Justify your answer. ________________________________________________________________

_________________________________________________________________________________________

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 __________________________.

78

Unit 3 – Lesson 4 – Raising a Power to a Power

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

Warm Up: Simplify the following.

1) -2x4(3x5 – 2x3) 2) (5x-3)(3x-4)

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

[ (1

2)2 ]5

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

79

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

(6𝑎)4

(7 ∙ 4 ∙ 11)2

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


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 ∙ 6−5

(5) (73)4 (6) (2−1)0 (7) (−27)2 ∙ (−2)−1 (8) (-3𝑦5)2

(9) (2𝑥3𝑦−2𝑧4)3 (10) (6−2)3 (11) (𝑥4 ∙ 𝑥2)2 (12) (2𝑎3𝑏−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.

80


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

1) (84)3 = ______ 2) (21)0 = ______ 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) (3𝑥5)4 = ______ 13) (3𝑦3 ∙ −2𝑦)3 = ______ 14) (5𝑥7)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) 28

2? = 29 c) (−2𝑚3𝑛4)? = −8𝑚9𝑛12

19) The formula for the volume of a rectangular prism is V = LWH. If the L = 84 and W = 8−2

and the H = 80. What is the volume in exponential form?

20) The formula for the volume of a cube is V = s3where s is the length of the side of the square. Express the

volume of a cube in simplest terms of x whose side is 6x8.

81

Unit 3 – Lesson 5 – Dividing Exponents

Aim: I can divide like bases with exponents.

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.

Independent Practice: Simplify the following expressions

(1) 𝑥5𝑦4

𝑥2𝑦 (2)

510

52 (3)

68

6

82

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

(4) 𝑥6𝑦8

𝑥4 (5)

𝑎6𝑏

𝑎4𝑏 (6)

59

26

(7) 8𝑎9𝑏5

12𝑎3𝑏4 (8)

816∙85

812 (9)

𝑎4𝑏𝑐6

𝑎4𝑏𝑐5

(10) 6𝑚5𝑛4

2𝑚2𝑛4

11. Bryan and Brendan simplify the following expression 𝑚3

𝑚7, below are their responses.

Bryan: 𝑚4 Brendan: 𝑚−4

Determine which student got the correct answer and explain the mistake made by the other student.

83


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

53 7)

𝑥4𝑦7

𝑥2𝑦3 8)

𝑤4𝑦12

𝑥7𝑦3

9) 99

95 10)

(2

5)7

(2

5)2 11)

𝑥15

𝑥3 12)

𝑥5

𝑥−2

13) 𝑥15𝑦3

𝑥5 14)

25

2−9 15)

𝑥4

𝑥−5 16)

(1

3)9

(1

3)2 17)

2−4

2−7

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

a) 𝑥

𝑥?

6 = 𝑥4 b)

28

2? = 29 c)

𝒙?

𝒙𝟓= 𝒙𝟕

84

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

Divide

7) 28 xx 8)

64 xx

9) 28 − xx 10)

52 1010 −−

85

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

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

**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.

86

M A D S

Multiplying Exponents MADS (Multiply – Add exponents)

1) Keep like bases

2) Add the exponents of like bases

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)

1) Keep the base

2) Subtract the exponents 95

93= 92

(65)(62) = 67

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

87

Quiz Review Homework

Write in expanded form. Write in exponential form:

1) 53𝑥2 2) (1

4𝑦)

4

3) 1

3 ∙ 1

3 ∙ 1

3 ∙ 1

3 4) 2 ∙ 2 ∙ 5 ∙ 5 ∙ 5 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎

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

5). 7−3 6) 𝑥−1

𝑦−5 7) (

2

3)−2

8) (−8)4 9) (2

5)−2

10) 32 − 64 + 40

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

How do you know?

11) (−1)14 12) (−1)283 13) 2n = 64 14) 3n = 27 15) 5n = 125

Multiply or Divide. Put answers in exponential form.

16) 3−3 ∙ 3 = ______ 17) 54 ∙ 5−13 ∙ 5 = ______ 18) 𝑥6 ∙ −𝑥19 = ______

19) 3−5 ∙ 3−9 = ______ 20) (6x)0 = ______ 21) 6x0 = ______

22) (x4y)0 = ______ 23) 6(x3y7)0 = ______ 24) (74)3 = ______

25) (𝑥−7𝑦)4 = ______ 26) (𝑥−5 𝑦)3 = _____ 27) (𝑥4𝑦6)−2 = _____

88

28) (−𝑥6)2 = ______ 29) (5−8)2 ∙ 5 = ______ 30) 614

65 = ______

31) 𝑥4

𝑥7 = ______ 32)

−𝑦11

𝑦9 = ______ 33)

𝑥4𝑦8𝑧

𝑥2𝑦8𝑧4 = _____

34) 35 ÷ 33 = ______ 35) 87 ÷ 813 = _____ 36) 𝑥7 ÷−𝑥3 = _______

37) (−2)5

(−2)5 = _______ 38)

(2

5)7

(2

5)2 = _______ 39)

26∙ 32∙ (2)7

25∙ (2)4 = _______

40) 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?

89

Unit 3 – Lesson 6 - Introduction to Scientific Notation

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 =

Scientific Notation

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 x 1011 = 259,000,000,000

Coefficient Power of 10

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.

For example: 1 x 10 8 1.54 x 10 -11 9.99 x 10 5 3.675 x 10 -5

GUIDED PRACTICE

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 a positive exponent represents a number larger than 1 (whole number).

A number in scientific notation with a negative exponent represents a number between 0 and 1 (decimal).

90

Remember

Positive Exponent ____________________________________

Negative Exponent ____________________________________

Scientific Notation: Real Life Situations

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

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


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

91

Unit 3- Lesson 6 Homework

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:

Simplify as a positive exponent

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

4) -3

Solve Simplify

17) 4x + x – 8 = 5x + 12 18) 18

-3

92

Unit 3 – Lesson 7 Converting Scientific Notation

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

Converting Standard Form → Scientific Notation

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

Step 2: Write “ x 10 ”

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

Remember:

If it is a whole number the exponent is __________________.

If it is a decimal the exponent is __________________.

Convert from standard form to scientific notation.

1) 245,000,000 = ______________________ 2) 0.00084 = ______________________

3) 500,000 = ______________________ 4) 0.000007643 = ______________________

Converting Scientific Notation → Standard Form

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

Step 2: If - Positive exponent: Move decimal point Right

If - Negative exponent: Move decimal point Left

Convert from scientific notation to standard form.

5) 5.93 x 103 = ______________________ 6) 1.9 x 10-7 = ______________________

7) 4.765 x 108 = ______________________ 8) 8.32 x 10-4 = ______________________

93

A positive, finite decimal 𝑠 is said to be written in scientific notation if it is expressed as a product 𝑑 × 10𝑛,

where 𝑑 is a finite decimal so that 1 ≤ 𝑑 < 10, and 𝑛 is an integer.

The integer 𝑛 is called the order of magnitude of the decimal 𝑑 × 10𝑛

Making Sure a Number is Written in Scientific Notation

L A R S ______________________________________________

If decimal point needs to move to the LEFT, the exponent increases ( 48.6 x 103)

If decimal point needs to move to the RIGHT, the exponent decreases (.48 x 103)

* Be careful when exponent is negative *

Write each in Scientific Notation if necessary:

1) 68.7 x 109 = ______________________ 2) 6 x 105 = ______________________

3) 0.725 x 108 = ______________________ 4) 0.292 x 10-4 = ______________________

5) 326 x 10-8 = ______________________ 6) 7.5 x 10-9 = ______________________


Write each of the following in scientific notation:

1) 650,000 _________________________________ 2) 23,500,000_______________________________

3) 0.00034 ________________________________ 4) 0.00758 _________________________________

Write each of the following in standard form:

5) 4.6 x 104 _______________________________ 6) 1.98 x 106 _______________________________

7) 6.23 x 10-7_______________________________ 8) 5.55 x 10-3 ______________________________

9) What is the value of n in the problem: 91,000 = 9.1 x 10n n = ______

10) What is the value of n in the problem: 0.0000027 = 2.7 x 10n n = ______

94

11) What is the value of n in the problem: 624,000 = 6.24 x 10n n = __________

12) If n = 7, find the value of 5.2 x 10n in standard form. ____________________

13) Which number is written in the correct scientific notation form?

a) 5,000 b) 0.5 x 102 c) 5.0 x 10-4 d) 50 x 105

14) 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?

(a) 0.3 𝑥 107 meters per second

(b) 0.3 𝑥 108 meters per second

(c) 3.0 𝑥 107 meters per second

(d) 3.0 𝑥 108 meters per second

15) Expressed in scientific notation, 0.0000056 is equivalent to:

(a) 5.6 × 10−6

(b) 0.56 × 10−5

(c) 5.6 × 106

(d) 56 × 106

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

written in standard form?

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

expressed in scientific notation?

95



1) 5,000,000 _____________________________ 3) 6,267 _______________________________

2) 0.046 ________________________________ 4) 0.000004____________________________


5) 2.0 x 103 _______________________________ 6) 5.14 x 106 ____________________________

7) 9.8 x 10-2_______________________________ 8) 3.75 x 10-9 ____________________________

Review: Write each in Scientific Notation if necessary

9) .98 x 103 = ______________________ 10) 79.02 x 108 = ______________________

11) 25 x 10-4 = ______________________ 12) 0.18 x 10-6 = ______________________

13) 7 x 10-4 = ______________________ 14) 925.4 x 1026 = ______________________

15) What is the value of n in the problem: 624,000 = 6.24 x 10n n = ______

16) If n = 4, find the value of 2.3 x 10n in standard form. ____________________

96

Lesson 8 – Compare and Order Scientific Notation

Aim: How do we compare numbers written 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?

Which number is larger?

Investigation 2: Express each scientific notation in standard form-

(a) 2.1 × 104 (b) 1.5 × 104

Which number is smaller?

Which number is larger?

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

97

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.


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

98

Lesson 8 Homework

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

(5) Order the numbers from least to greatest:

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

(6) The table lists the populations of five countries. List the countries from least to greatest.

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

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

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

Write each in Scientific Notation if necessary:

9) .345 x 107 = ______________________ 10) 22.2 x 104 = ______________________

11) 98 x 10-6 = ______________________ 12) 0.35 x 10-9 = ______________________

99

Unit 3 – Lesson 9 – Multiplying and Dividing Numbers in Scientific Notation

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

Rules for Multiplying and Dividing Numbers in Scientific Notation without a Calculator

1 - Multiply or Divide Coefficients – Using rules of multiplying or dividing decimals.

2 - Multiply or Divide powers of 10 by adding or subtracting the exponents.

3 – Make sure the answer is in correct scientific notation (LARS)

• If you have to move the decimal to the Left, INCREASE the exponent.

• If you have to move the decimal to the Right, DECREASE the exponent.

GUIDED PRACTICE

1) (3.5 x 103)(2 x 105) 2) (8.0 x 106) ÷ (2.5 x 103)

3) (7.2 x 105)(6.5 x 104) 4) (9.9 x 10-3) ÷ (3 x 102)

5) A paperclip factory produces 5 x 102 paperclips a day. In a period of 1.5 x 103 days, how many

can be produced?

100

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 =

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

Calculate

1) (2.63× 104) (1.2 × 10−3) 2) (4 × 10)(2 × 103) 3) 7×106

2×103 4)

9 × 10−11

2.4× 108

101


Calculate

1) 9.8 × 103

2 × 102

2) (8 × 103)(3 × 104)

3) (51.3 × 108) ÷ (2.7 × 106)

4) (7 × 105)(6 × 107)

5) (4 × 102) ÷ (9 × 104) 6) (2 × 103)(5 × 104)

7. Find the perimeter of the rectangle, if the area is 5.612 × 1014 cm2

102


1) (6.2 x 104)(3.2 x 103) 2) (19.5 x 105) 3) (1.1 x 10-5)(1.2 x 102)

(6.5 x 10-4)

4) 1.24 x 101 ÷ 4 x 105 5) (8 × 103)(3 × 104) 6) (51.3 × 108) ÷ (2.7 × 106)

7) (7 × 105)(6 × 107) 8) (4 × 102)(9 × 104) 9) 9.8 × 103

2 × 102

10) A newborn baby has about 26,000,000,000 cells. An adult has about 4.94 x 1013 cells. How many times

as many cells does an adult have then a newborn? Write your answer in scientific notation.

(Please show work)

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?

(Please show work)

103

Unit 3 – Lesson 9 Additional Review

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


9) 25,000 __________________________________ 10) .000302 __________________________________

11) -4,700__________________________________ 12) 2 million__________________________________


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 = ______


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


104

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 × 109 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

105

Unit 3 – Lesson 10 – Applications of Scientific Notation

Aim: I can apply scientific notation to solve problems

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)

Guided Practice:

Use the table below for questions 1-3. The table below shows the debt of the three most populous states and the

three least populous states.

State Debt (in dollars) Population

(2012)

California 407,000,000,000 3.8 x 107

New York 337,000,000,000 1.9 x 107

Texas 276,000,000,000 2.6 x 107

North Dakota 4,000,000,000 6.9 x 104

Vermont 4,000,000,000 6.26 x 104

Wyoming 2,000,000,000 5.76 x 104

1. What is the sum of the debts for the 3 most populous states? Express your answer in scientific notation.

2. What is the sum of the debts for the 3 least populous states? Express your answer in scientific notation.

3. How much larger is the combined debt of the three most populated states than that of the three least

populated states? Express your answer in scientific notation.

106

4. Here are the masses of the so-called inner planets of the Solar System.

Mercury: 3.3 x 1023 kg Earth: 5.9 x 1024 kg

Venus: 4.8 x 1024 kg Mars: 6.4 x 1023

What is the average mass of all four inner planets? Write your answer in scientific notation.

5. What is the difference of and written in scientific notation?

6. What is the sum of 12 and expressed in scientific notation?

1) 4.2 x 106

2) -4.2 x 106

3) 42 x 106

4) 42 x 107

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

1) 2) 3) 4)

8. What is the quotient of and ?

1) 2) 3) 4)

9. What is the value of in scientific notation?

1) 84 x 108

2) 8.4 x 109

3) 2 x 105

4) 8.4 x 108

1) 2) 3) 4)

107

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

11. If the number of molecules in 1 mole of a substance is , then the number of molecules in 100

moles is

12. If you could walk at a rate of 2 meters per second, it would take you 1.92 x 108 seconds to walk to the

moon. Is it more appropriate to report this time as 1.92 x 108 or 6.02 years?

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

greatest.

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

15. 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.

16. 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?

1) g

2) g

3) g

4) g

1) 2) 3) 4)

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

108


1. All planets revolve around the sun in elliptical orbits. Uranus’s furthest distance from the sun is

approximately 3.004 x 109 km, and its closest distance is approximately 2.749 x 109 km. Using this

information, what is the average distance of Uranus from the sun?

2. 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

3. 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

4. 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?

5. 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

6. 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?

7. 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.

109

Unit 3 Review

Unit 3 Exponents 1) Write in expanded form 2) Write the following in exponential form.

a. 53𝑥2 b. (3

4𝑦)

4

a. 2

3 ∙

2

3 ∙

2

3 ∙

2

3 b. 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 ∙ 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 = ______ e. 6x0 = ______ f. (x4y)0 = ______ g. 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) (𝑥−2𝑦)4 = ______ 16) (𝑥−2 𝑦)3 = ______

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

110

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

20) 614

65 = ______ 21)

𝑥4

𝑥7 = ______ 22)

−𝑦11

𝑦9 = ______ 23)

𝑥4𝑦8𝑧

𝑥2𝑦8𝑧4 = ______

24) 35 ÷ 33 = _______ 25) 87 ÷ 813 = _______ 26) 𝑥7 ÷−𝑥3 = _______ 27) (−2)5

(−2)5 = _______

28) (2

5)7

(2

5)2 = _______ 29)

26∙ 32∙ (2)7

25∙ (2)4 = _______ 30)

37∙ 711∙ 𝑦4

33∙ 7−7∙ 𝑦4 = _______ 31)

𝑥10

𝑥−6 = _______

32) 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:

30) 6.3 × 107 31) 5.23 × 10−4 32) 8.08 × 100 33) 4.2 × 10−1 34) 9.24 × 1010

Write the following using Scientific Notation:

35) 120 36) 65,002,000 37) 0.0000233 38) .345 x 104 39) 523 x 109

111

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

40) (4.3 × 107)( 2.2 × 103) 41) (5 × 1012)(4.77 × 10−5) 42) (3.6 × 10−5)3

43) 6.2×109

2×102 44) (3.45 × 106) ÷ (8.01 × 10−5) 45)

1.6332×1011

1.6332×1011

Compare using < > =

46) 5.3 x 103 4.5 x 103 47) 2300 2.3 x 103

Word Problems

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

Unit 1 Integers 49) Perform the indicated operations and evaluate 50) 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)

51) 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)

112

52) 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?

53) 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 Solving Equations

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

54) 0.8 - 2x = 7.6 55 ) 4

5𝑥 − 3 = 9 56) 3x + 12 = 3(4 + x)

57) -3(2x – 1) = 3x + x + 53

58) 6(2𝑥 − 8) = 12(𝑥 + 3) 59) 3

5+

1

4𝑥 =

1

2

𝟑𝐱 − 𝟐

𝟓

113

Math 8

Unit 4

Graphing Linear Equations

Extra Help Schedule: ______________________________

______________________________


1 Writing Equations of a Line Lesson 1 – Page 116

2 Creating a Table Lesson 2 – Page 119

3 Graphing – Table Method Lesson 3 – Page 123

4 Finding Slope and y-intercept from an Equation Lesson 4 – Page 126

QUIZ

5 Graphing – Slope/Intercept Method Lesson 5 – Page 130

6 Graphing Systems of Equations Lesson 6 – Page 136

7 Graphing Systems of Equations (Types of solutions) Lesson 7 – Page 141

8

Graphing – Table and Slope/Intercept Method

Project

Work on Project

Start Review

Review Please Study 😊

Project Due Tomorrow

UNIT TEST

114

Unit 4 – Lesson 1 – Writing Equations of a Line

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

Warm up: Solve

1) 2x + 7 = 15 2) -2x + 7 – 3 = 28

Vocabulary:

Equation of a Line ________________________________________

Equation of a Line is also called _____________________________

OR ____________________________

Equation of a Line

y = mx + b

Slope y-intercept

(How steep) (where line crosses y axis)

Guided Practice: Rewrite 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

y-intercept

115


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

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 + 3y = 15 17) 3y - 2x = 27

116


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

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

Review:

17) 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?

18) Evaluate each expression for n = 3.

a. 2n + 5 – n b. 3n+18

3n c.

24

4 − n · n

117

Unit 4 – Lesson 2 – Creating a Table

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

Vocabulary:

Equation of a Line _____________________________________

STEPS: 1) Solve for y (y = mx + b) if needed (5, -3)

2) Pick values for the table

3) Solve for all y values

4) List the points of the line

Guided Practice: Complete each table

Ordered pair

x-coordinate

5

y-coordinate

-3

118

Independent Practice: Complete each table

119


Review:

11) Find the area of the rectangle.

12) Simplify: (3x – 5) – (x + 3) + (-2x + 7) 13) Simplify: 42

3 – (-3

3

4)

14) Simplify: 64 - 42 ÷ 8 15) Solve for x: 5x + 20 = 5(x + 4)

Write as a positive exponent:

16) 6 -4 17) 25x 4 ÷ 5x 8

3x

2x - 8

120

REVIEW OF GRAPHING COORDINATES

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)

121

Unit 4 – Lesson 3 – Graphing – Table Method

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

Warm up: Complete the table

y = 3x + 4

Guided Practice:

Rules: 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)

122


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)

123


Complete the table and Graph the Line

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

Review:

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-intercept form (y = mx + b)

x y (x,y)

x y (x,y)

124

Unit 4 – Lesson 4 – Finding Slope and y-intercept from an Equation

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

Vocabulary:

Slope - ______________________________________________________________________________

y-intercept - __________________________________________________________________________

STEPS:

1 – Write the equation in function form (y= mx + b)

2 – Find m and b (write m = and b = )

Guided Practice: Rewrite the equation in function 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 = _____

125

Independent Practice: 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 = _____

126


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:

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

Simplify:

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

127

Unit 4 – Lesson 5 – Graphing – Slope/Intercept Method

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 = _____

STEPS:

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

2) m = b =

3) Graph b on the y line

4) Make three more points using m

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

6) Label the line

Guided Practice:

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!!!

128

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

m = m =

b = b =


5) y = 1

2 x - 5 6) y =

−2

3 x + 8

m = m =

b = b =

129

7) y = 7 8) x = 9

9) y – 5 = -2x 10) y = 3x

130


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

131

Unit 4 – Lesson 6 – Graphing Systems of Equations

Aim: I can graph Systems of Equations.

Warm up: Solve 1) 2x + 2 = 2x – 6 2) 4x – 2 = -x + 8

STEPS:

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.

Guided Practice:

1) Graph and Check:

y = -x + 2

y = 3x - 2

What is the solution? __________________ Check:

Type of Solution: _____________________

How does it look graphically? _________________________

What do you notice about the slopes? _________________________

132

2) Graph and Check:

y = 2x + 1

2y = 4x + 2


Type of Solution: _____________________



133

3) Graph and Check:

y = 2x + 2

y = 2x – 6


Type of Solution: _____________________



134


1) Graph and Check:

y = 4x - 2

y = -x + 8


Type of Solution: _____________________



135

2) Graph and Check:

y = -3x + 7

y = 2x – 3


Type of Solution: _____________________



LESSON SUMMARY:

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

One Solution No Solution Infinite Solutions

______________________ ______________________ ______________________

(slopes) (slopes) (slopes)

______________________ ______________________ ______________________

(y-intercepts) (y-intercepts) (y-intercepts)

136


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

137

Unit 4 – Lesson 7 – Graphing Systems of Equations (Types of solutions)

Aim: I can graph systems of equations (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)

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

The lines Intersect The lines Never Intersect The lines are the Same

=> One Solution => No Solution => Infinite Solutions

138

Guided Practice:

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?

_________________________________ ____________________________________ _________________________________

139

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

__________________________ __________________________ ________________________

How do you know? How do you know? How do you know?

__________________________ __________________________ __________________________


Solve graphically and state the type of solution:

1) y = 1

2x – 3 2) 3y – 6x = 12

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

Type of Solution __________________________ Type of Solution__________________________

140

State the type of solution algebraically (by inspection) 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

__________________________ __________________________ ________________________


__________________________ __________________________ __________________________

__________________________ __________________________ __________________________

Lesson Summary

What are the characteristics of the graph that tell us how many solutions a system has?

Slopes y-intercepts Lines

One solution: __________________________ __________________________ ________________________________

No solution: __________________________ __________________________ ________________________________

Infinite solutions: __________________________ __________________________ ________________________________

141


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 algebraically 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:_____________________________

___________________________________________ ___________________________________________

142

Unit 4 – Lesson 8 – Graphing – Table and Slope/Intercept Method (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

143

Unit 4 Review

Rewrite the equation in function 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)

144

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

145

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) (−3x4)(6x−2)

9x 29) (-3a-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

Documents

Math 8 Quarter 1mrsleonardo1.weebly.com/uploads/5/6/8/7/56870649/math_8_quarte… · Quarter 1 Unit 1 – Integers Unit 2 – Solving Equations Unit 3 – Exponents Unit 4 – Graphing