8 Grade Intensive Math - iMater Charter Middle/High School · 2014-08-25 · Lesson 1 Curriculum...

8th Grade

Intensive Math

Ready Florida MAFS

Student Edition

August-September 2014

L1: Properties of Integer Exponents2©Curriculum Associates, LLC Copying is not permitted.

Properties of Integer ExponentsLesson 1 Part 1: Introduction

In the past, you have written and evaluated expressions with exponents such as 53 and x2 1 1. Now, take a look at this problem.

Multiply: 1 33 2 1 34 2

Explore It

Use the math you know to answer the questions.

What do the expressions 1 33 2 and 1 34 2 have in common?

Write a multiplication expression without exponents that is equivalent to 33.

How many factors of 3 did you write?

Write a multiplication expression without exponents that is equivalent to 34.


Write a multiplication expression without exponents that is equivalent to 1 33 2 1 34 2 .


Write an expression with exponents to complete this equation: 1 33 2 1 34 2 5

What is the relationship between the exponents of the factors and the exponent of the product in your equation?

Use words to explain how to multiply 1 33 2 1 34 2 .

Develop Skills and Strategies

MAFS8.EE.1.1

L1: Properties of Integer Exponents 3©Curriculum Associates, LLC Copying is not permitted.

Lesson 1Part 1: Introduction

Find Out More

You have seen one example of how to multiply powers with the same base. Here are two more:

1 58 2 1 55 2 5 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 5 5815 5 513

1 x6 2 1 x2 2 5 x • x • x • x • x • x • x • x 5 x 612 5 x 8

In general, for the product of powers with the same base 1 na 2 1 nb 2 5 na1b, where n ≠ 0.

You can also use the meaning of exponents to divide powers with the same base.

Divide 412 ··· 45 .

412 ··· 45 5 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 ························ 4 · 4 · 4 · 4 · 4 412 is twelve 4s multiplied together.

5 4 ·· 4 • 4 ·· 4 • 4 ·· 4 • 4 ·· 4 • 4 ·· 4 • 4 • 4 • 4 • 4 • 4 · 4 • 4 45 is five 4s multiplied together.

5 1 • 1 • 1 • 1 • 1 • 4 • 4 • 4 • 4 • 4 • 4 • 4 Any non-zero number divided by itself is 1.

5 4 • 4 • 4 • 4 • 4 • 4 • 4 Seven 4s multiplied together is 47.

5 47

So, 412 ··· 45 5 47. What is the relationship between the exponents of the dividend, divisor, and

quotient? The exponent of the quotient is the exponent of the dividend minus the exponent

of the divisor. 12 2 5 5 7.

In general, for the quotient of two powers with the same base, na ·· nb 5 na2b, where n ≠ 0.

Reflect

1 Explain why 510 ··· 52 equals 58.

Lesson 1

©Curriculum Associates, LLC Copying is not permitted.

L1: Properties of Integer Exponents4

Part 2: Modeled Instruction

Read the problem below. Then explore how to find the product of powers with the same base and the same exponent.

Simplify: (32)4

Model It

You can write it another way.

1 32 2 4 5 means 3 squared, multiplied as a factor 4 times.

1 32 2 4 5 32 • 32 • 32 • 32

1 32 2 4 is the product of 4 powers, each with the same base (3) and the same exponent (2).

Solve It

You can apply the associative property of multiplication.

1 32 2 4 5 32 • 32 • 32 • 32 1 32 2 4 is the product of four 32s multiplied together.

5 1 32 • 32 2 1 32 • 32 2 Apply the associative property of multiplication.

5 1 34 2 1 34 2 This is the product of powers with the same bases.

5 3414 Add the exponents.

5 38

Lesson 1

©Curriculum Associates, LLC Copying is not permitted.5L1: Properties of Integer Exponents

Part 2: Guided Instruction

Connect It

Now you will explore the concept from the previous page further.

2 Simplify: 1 32 2 4 5

3 Describe the relationship between the exponents of 1 32 2 4 and the exponent of 38.

4 Complete these examples of products of powers that have the same base and the same exponent.

1 58 2 6 5 58 • 58 • 58 • 58 • 58 • 58 5 581818181818 5 58 • 6 5

1 9537 2 3 5 9537 • 9537 • 9537 5 95371717 5 9537 • 3 5

5 In general, for a product of powers that have the same base and the same exponent, 1 na 2 b = , where n ≠ 0.

Now look at how to simplify a product of powers when the bases are different and the exponents are the same.

Simplify: 1 23 2 1 43 2

6 Write an expression without exponents that is equivalent to 1 23 2 1 43 2 .

7 Apply the associative and commutative properties of multiplication to write your expression as the product of groups of 2 • 4.

8 How many groups of 2 • 4 do you multiply together to get 1 23 2 1 43 2 ?

9 Complete this equation: 1 23 2 1 43 2 5 (2 3 4) 5 3

10 In general, for a product of powers that have different bases and the same exponent, 1 an 2 1 bn 2 5 , where a ≠ 0 and b ≠ 0.

Try It

Use what you just learned to solve these problems. Write your answers using exponents.

11 Simplify: 1 218 2 8 5

12 Simplify: 1 49 2 1 259 2 5 , or



Lesson 1

Read the problem below. Then explore simplifying expressions with exponents equal to zero.

Simplify: 50

Model It

You can write it another way.

It doesn’t make sense to ask yourself, “What is zero 5s multiplied together?” We will need to approach this problem another way.

So far, you have worked with powers where the exponents are counting numbers (1, 2, 3, . . .). The rules for working with powers are the same when the exponent is 0.

You have seen that when you multiply powers with bases that are the same you add the exponents.

1 50 2 1 54 2 5 5014 5 54

Solve It

You can apply the identity property of multiplication.

You know that 1 times any expression is equivalent to that expression by the identity property of multiplication.

1 1 2 1 54 2 5 54

Because 1 1 2 1 54 2 5 54

and 1 50 2 1 54 2 5 54,

50 must therefore be equal to 1.



Lesson 1Part 3: Guided Instruction

Connect It

Now you will explore the concept from the previous page further.

13 Simplify: 50 5

14 Complete these examples:

120 5

5 1

1 27 2 0 5

15 In general, for a power where the exponent is equal to 0, n0 5 , where n ≠ 0.

The rules for products of powers also apply when the exponent is a negative integer.

16 Complete this equation: 1 65 2 1 625 2 5 6 5

17 You already know that a number times its reciprocal equals 1. For example, 3 • 1 ·· 3 5 3 ·· 3 5 1.

Now complete this equation: 65 • 1 ··

65

5 5

18 Since 65 • 625 5 and 65 • 1 ·· 65 5 , then 625 5 .

19 Complete these examples:

1026 5

(234)27 5

5 1 ····· 14213

20 In general, for a power where the exponent is a negative integer, n2a 5 ,

where n ≠ 0.

Try It

Use what you just learned to solve these problems. Write your answers using exponents where appropriate.

21 Simplify: 4550 5

22 Simplify: 1924 5

Lesson 1



Part 4: Guided Practice

Student Model

Study the student model below. Then solve problems 23–25.

Simplify: 24 • 227

Look at how you could show your work.

24 • 227 product of powers with equivalent bases

5 241(27) add exponents

5 223 power with a negative integer exponent

5 1 ·· 23 reciprocal with positive exponent

Solution:

23 Simplify: 1 32 • 42 2 5

Show your work.

Solution:

If x and a are counting numbers, is x2a less than or greater than 1? Explain.

Pair/Share

Does 59 • 67 5 (30)16? Justify your answer.

Pair/Share

In this problem, you have to apply more than one rule of working with exponents.

Remember the order of operations. Simplify the expression within the parentheses first.

24 • 227 5 1 ·· 23

Lesson 1



24 Simplify: 928 • 1 ··

93 . Write your answer with a positive exponent.

Show your work.

Solution:

25 Which expression is equivalent to 4523 ···· 453 ?

A 4521

B 450

C 1 ··· 456

D 456

Isaac chose A as the correct answer. How did he get that answer?

Talk about the problem and then write your answer together.

Pair/Share

Describe the value of the expression 025.

Pair/Share

The expression is a quotient of powers.

Remember what you know about adding negative numbers.

Lesson 1



Solve the problems.

1 Which expression is equivalent to (2425)0?

A 1

B (24)5

C 1 ····· 1 24 2 5

D 15 ··· 24

2 Which expression is equivalent to (72)5 ···· 726 ?

A 7

B 74

C 713

D 716

3 Which expression is equivalent to 1 }} 49

? Select all that apply.

A 721 3 721

B 78 3 726

C 725 3 73

D 77 3 729

E 722 3 74

4 Write 168 as a power with a base of 4.

Part 5: MAFS Practice

Lesson 1


5 Look at the equations below. Choose True or False for each equation.

A 24 3 34 5 46 True False

B 52 4 53 5 1 } 5 True False

C (63)4 5 (64)3 True False

D 32 }}

37 5 32 3 327 True False

E 80 }}

824 5 824 True False

F 410 4 45 5 42 True False

6 Write each of these numbers as the product of a whole number and a power of 10. Then describe the relationship between place value and exponents.

3,000 5 ______________

300 5 ______________

30 5 ______________

3 5 ______________

0.3 5 ______________

0.03 5 ______________

0.003 5 ______________

Go back and see what you can check off on the Self Check on page 1.Self Check


Course 3 • Chapter 1 Real Numbers 3

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.Lesson 2 ReteachPowers and Exponents

Example 1Write each expression using exponents. a. 7 � 7 � 7 � 7

7 � 7 � 7 � 7 = 74 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent.

b. y � y � x � y � x

y � y � x � y � x = y � y � y � x � x Commutative Property

= (y � y � y) · (x · x) Associative Property

= y3 � x2 Defi nition of exponents

To evaluate a power, perform the repeated multiplication to fi nd the product.

Example 2

Evaluate (-6)4.

(-6)4 = (-6) � (-6) � (-6) � (-6) Write the power as a product.

= 1,296 Multiply.

The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction.

Example 3Evaluate m2 + (n - m)3 if m = -3 and n = 2.

m2 + (n - m)3 = (-3)2 + (2 - (-3))3 Replace m with -3 and n with 2.

= (-3)2 + (5)3 Perform operations inside parentheses.

= (-3 � -3) + (5 � 5 � 5) Write the powers as products.

= 9 + 125 or 134 Add.

ExercisesWrite each expression using exponents.

1. 8 � 8 � 8 � 8 � 8 2. a � a � a � a � a � a 3. 5 � 5 � 9 � 9 � 5 � 9 � 5 � 5

Evaluate each expression.

4. 24 5. (-3)5 6. (

3 −

4 )

3

ALGEBRA Evaluate each expression if a = 5 and b = -4.

7. a2 + b2 8. (a + b)2 9. a + b2

The product of repeated factors can be expressed as a power. A power consists of a base and an exponent. The exponent tells how many times the base is used as a factor.

001_022_CC_A_RSPC3_C01_662332.indd Page 3 26/05/11 11:18 PM s-60user001_022_CC_A_RSPC3_C01_662332.indd Page 3 26/05/11 11:18 PM s-60user /Volumes/110/GO00864/NATIONAL/ANCILLARY/RETEACH_AND_SKILLS_PRACTICE_COURSE1-3/É/Volumes/110/GO00864/NATIONAL/ANCILLARY/RETEACH_AND_SKILLS_PRACTICE_COURSE1-3/

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Course 3 • Chapter 1 Real Numbers

NAME __________________________________________ DATE ____________ PERIOD _______

Lesson 2 Extra Practice

Powers and Exponents

Write each expression using exponents.

1. 4 • 4 • 4 • 4 44 2. 34 •

34 •

34 •

34 •

34

3. 4 • 4 • 4 • 4 • 4 • 5 • 5 • 5 • 5 • 5 • 5 • 5 • 5 45 • 58 4. 3 • 2 • 56 •

56 •

56 • 2 • 2 • 2 • 3 •

56

5. b • b • b • b • c • c • c • c • c • c b4 • c6 6. 5 • 7 • 2 • 2 • 7 • 5 • 2 • 2 • 7 • 7 24 • 52 • 74 Evaluate each expression.

7. 43 64 8. 62 36 9. 3

2

5

8

125

10. 52 • 62 900 11. 3 • 24 48 12. 104 • 32 90,000 13. 53 • 19 125 14. 22 • 24 64 15. 2 • 32 • 42 288

16. 73 343 17. 3

1

2

• 45 128 18. 35 • 42 3,888

19. 72 • 34 3,969 20. (2)4 16 21. (5)3 125


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.Lesson 3 ReteachMultiply and Divide Monomials

The Product of Powers rule states that to multiply powers with the same base, add their exponents.

Example 1Simplify. Express using exponents.

a. 23 � 22

23 � 22 = 23 + 2 The common base is 2.

= 25 Add the exponents.

b. 2s6(7s7)

2s6(7s7) = (2 � 7)(s6 � s7) Commutative and Associative Properties

= 14(s6 + 7) The common base is s.

= 14s13 Add the exponents.

The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.

Example 2Simplify k

8 −−

k . Express using exponents.

k8 −−

k1 = k8 -1 The common base is k.

= k7 Subtract the exponents.

Example 3

Simplify (-2)10 � 56 � 63

−−−−−−−−

(-2)6 � 53 � 62 .

(-2)10 � 56 � 63

−−−−−−−−−

(-2)6 � 53 � 62 =

(

(-2)10

−−−−

(-2)6 )

� (

56 −−

53 )

� (

63 −−

62 )

Group by common base.

= (–2)4 � 53 � 61 Subtract the exponents.

= 16 � 125 � 6 or 12,000 Simplify.

ExercisesSimplify. Express using exponents.

1. 52 � 55 2. e2 � e7 3. 2a5 � 6a 4. 4x2(–5x6)

5. 79 −−

73 6. v

14 −−

v6 7. 15w7

−−−−

5w2 8. 10m8

−−−−

2m

9. 25 � 37 � 43

−−−−−−

21 � 35 � 4 10. 4

15 � (-5)6

−−−−−−

412 � (-5)4 11. 6

7 � 76 � 85 −−−−−−

65 � 75 � 84 12. (-3)6 � 105

−−−−−−−

(-3)4 � 103


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NAME __________________________________________ DATE ____________ PERIOD _______


Multiply and Divide Monomials

Simplify using the Laws of Exponents.

1. 23 • 24 27 or 128 2. 56 • 5 57 or 78,125 3. t4 • t2 t6

4. y5 • y3 y8 5. (3x3)(2x2) 6x5 6. b12 • b b13 7. 35 • 38 313 or 1,594,323 8. (2y3)(5y7) 10y10 9. (6a5)(3a6) 18a11 10. (x)(6x3) 6x4 11. (3x2)(2x5) 6x7 12. (6y2)(2y5) 12y7

13. x11

x2 x9 14. a6

a3 a3 15. 79

76 73 or 343

16. 26

22 23 or 8 17. 16x3

4x2 4x 18. 25y5

5y2 5y3

19. 48y3

8y 6y2 20.

12y5

3y2 4y3 21. 39x7y5

3x3y 13x4y4


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.Lesson 4 ReteachPowers of Monomials

Example 1 Simplify (53)6.

(53)6 = 53 · 6 Power of a power

= 518 Simplify.

Example 2

Simplify (-3m2n4)3.

(-3m2n4)3 = (-3)3 · m2 ·

3 · n4 ·

3 Power of a product

= -27m6n12 Simplify.

ExercisesSimplify.

1. (43)5 2. (42)7 3. (92)4

4. (k4)2 5. [(63)2]2 6. [(32)2]3

7. (5q4r2)5 8. (3y2z2)6 9. (7a4b3c7)2

10. (-4d3e5)2 11. (-5g4h9)7 12. (0.2k8)2

Power of a Power: To fi nd the power of a power, multiply the exponents.

Power of a Product: To fi nd the power of a product, fi nd the power of each factor and multiply.


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NAME __________________________________________ DATE ____________ PERIOD _______


Powers of Monomials

Simplify using the Laws of Exponents.

1. (23)2 26 or 64 2. (43)3 49 or 262,144 3. (62)4 68 or 1,679,616 4. (a4)3 a12 5. (m7)8 m56 6. (k5)7 k35 7. [(32)2]3 312 or 531,441 8. [(42)2]2 48 or 65,536 9. [(23)2]3 218 or 262,144 10. (6z4)5 7,776z20 11. (8c8)3 512c24 12. (3a5b12)5 243a25b60


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.Lesson 5 ReteachNegative Exponents

Example 1Write each expression using a positive exponent.

a. 7−3

7−3 = 1 −−

73 Defi nition of negative exponent

b. a−4

a−4 = 1 −−

a4 Defi nition of negative exponent

Example 2Evaluate each expression.

a. 5−4

5−4 = 1 −−

54 Defi nition of negative exponent

= 1 −−−

625 54 = 5 · 5 · 5 · 5

b. (−3)−5

(−3)−5 = 1 −−−−

(−3)5 Defi nition of negative exponent

= 1 −−−−

−243 (−3)5 = (−3) · (−3) · (−3) ·

(−3) · (−3)

Example 3Write 1 −−

65 as an expression using a negative exponent.

1 −−

65 = 6−5 Defi nition of negative exponent

Example 4Simplify. Express using positive exponents.

a. x−3 · x5

x−3· x5 = x(−3) + 5 Product of Powers

= x2 Add the exponents.

b. w−5 −−−

w−7

w−5 −−−

w−7 = w−5 − (−7) Quotient of Powers

= w2 Subtract the exponents.

ExercisesWrite each expression using a positive exponent.

1. a−8 2. 6−3 3. n−4

Evaluate each expression.

4. 7−2 5. 9−3 6. (−2)−5

Write each fraction as an expression using a negative exponent.

7. 1 −−

57 8. 1 −−

36 9. 1 −−

x8

Simplify. Express using positive exponents.

10. 4−2 · 4−4 11. r−3 · r5 12. h−2 −−−

h4

Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multiplicative inverse of the number to the nth power.


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NAME __________________________________________ DATE ____________ PERIOD _______


Negative Exponents

Write each expression using a positive exponent.

1. 53 153 2. 610

1610 3. (2)5

1(2)5

4. (3)2

1 (3)2 5. m6

1m6 6. g2

1g2

7. n9 1n9 8. r8

1r8 9. h7

1h7

Write each fraction as an expression using a negative exponent other than 1.

10. 145 45 11.

134 34 12.

1 (3)3 (3)3

13. 1

(6)5 (6)5 14. 164 26 or 43 or 82 15.

149 72

16. 1

243 35 17. 1

625 54 or 252 18. 1

216 63

Simplify.

19. 32 • 37 243 20. 53 • 55 25 21. x5 • x3 1x8

22. a4 • a7 1a3 23. a2b3 • a5b

b4

a7 24. x2y2 • x5y3 yx2

25. 72

76 2,401 26. x4

x5 1x9 27.

24a3

6a2 4a

28. 18y4

3y10 6y6 29. 42x5

45x2 x7

64 27. 62a4

63a2 6a2

©Curriculum Associates, LLC Copying is not permitted.L2: Square Roots and Cube Roots12

Lesson 2

Square Roots and Cube RootsPart 1: Introduction

In Lesson 1 you learned the properties of integer exponents. Now, take a look at this problem.

The length of each side of a square measures s inches long. The area of the square is 49 in.2 What is the length of one side of the square?

s

s s

s

Explore It

Use the math you know to answer the question.

Describe in words how to find the area of the square given that each side is s inches long.

Write a multiplication expression using the variable s to represent the area of the square.

Write an expression using the variable s and an exponent to represent the area of the square.

Write an equation setting your expression equal to the area of the square given in the problem.

Consider the factors of 49. Explain what the two sides of the equation have in common when you write each as the product of two factors.


MAFS8.EE.1.2

©Curriculum Associates, LLC Copying is not permitted.L2: Square Roots and Cube Roots 13


Find Out More

The number 49 is one of a set of numbers called perfect squares. A perfect square is a number that results from multiplying an integer by itself. The first 15 square numbers are shown.

12 5 1 42 5 16 72 5 49 102 5 100 132 5 169

22 5 4 52 5 25 82 5 64 112 5 121 142 5 196

32 5 9 62 5 36 92 5 81 122 5 144 152 5 225

Look at the equation you wrote on the previous page, s2 = 49. How do you solve an equation where a variable squared is equivalent to a perfect square? You have solved equations before by using inverse operations. You solved addition equations by subtracting. You solved division equations by multiplying. What is the inverse operation of squaring a number?

The inverse operation of squaring is finding the square root. A square root of a number is any number that you can multiply by itself to get your original number. For example, 3 is a square root of 9, because 3 • 3 = 9. Another square root of 9 is 23, because (23) • (23) 5 9.

The symbol Ï·· means positive square root. So, Ï·· 9 5 3.

Reflect

1 What is the difference between dividing 16 by 2 and finding the square roots of 16?

s2 5 49

Ï·· s2 5 Ï··· 49

Ï·· s2 5 Ï·· 72

s 5 7

The inverse of squaring is finding a square root.

Find the square root of both sides.

49 is a perfect square.

The length of one side of the square is 7 inches.

Lesson 2


L2: Square Roots and Cube Roots14


Read the problem below. Then explore how to solve equations with cubes and cube roots.

Each edge of a cube measures a feet long. The volume of the cube is 125 ft3. What is the measure of each edge of the cube?

Picture It

Draw and label the cube.

a

aa

Volume 5125 ft3

The length, width, and height of the cube each measure a feet.

Solve It

You can apply the formula for the volume of a cube.

The volume of the cube is the product of its length, width, and height.

a • a • a 5 V length 5 a, width 5 a, and height 5 a

a3 5 V Substitute the given volume of the cube for V.

a3 5 125

You can use this equation to find the value of a.

Lesson 2

©Curriculum Associates, LLC Copying is not permitted.15L2: Square Roots and Cube Roots


Connect It

Now you will solve the problem from the previous page.

2 Complete the prime factorization of 125.

125

25

3 Write 125 as the product of three factors.

4 Write 125 as a power of base 5.

5 What does 125 have in common with a3 when 125 is written as a power?

The product of an integer multiplied together three times is a perfect cube. Finding the cube root is the inverse of cubing a number. The cube root of a number is the number that is multiplied together three times to produce the original number. The symbol

3 Ï·· means

find the cube root.

6 Look at Solve It on the previous page. The equation shows a variable cubed equal to a perfect cube. Use the cube root to complete the solution.

Solution: Each edge of the cube is feet long.

Try It

Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

7 Solve: y3 5 8

8 Solve: x3 5 27

a3 5 125

3 Ï·· a3 5 3 Ï····· 1250

3 Ï·· a3 5 3 Ï···· 533

5

Lesson 2



Read the problem below. Then explore how to use square roots and cube roots to solve word problems.

City Park is a square piece of land with an area of 10,000 square yards. What is the length of the fence that encloses the park?

Picture It

You can draw a diagram to help solve the problem.

The park is a square. The fence runs along the outside edge of the park.

FenceArea 510,000 yd2 City park

The length of the fence is the perimeter of the square.

Solve It

To find the perimeter of the square park, you need to know the length of one side of the square.

Let f be the length of one side of the square.

A 5 10,000 Area of the park is 10,000 yd2

f 2 5 10,000 Area equals the length of one side squared.


Lesson 2


Connect It

Now you will solve the problem from the previous page.

9 What number squared equals 10,000?

10 Look at Solve It on the previous page. Solve the equation for f.

f 2 5 10,000

11 What is the length of each side of the park?

12 Write and solve an equation to find the perimeter of the park.

13 What is the length of the fence that encloses the park?

14 The park’s rectangular garden area is 450 square yards. Its length is twice its width. Find the dimensions of the garden. Begin with the equation (2w)(w) 5 450.

Rewrite the equation using exponents.

Divide both sides by 2.

Solve and write the garden’s dimensions.

Try It

Use what you just learned about square roots and cube roots to solve these problems.

15 The volume of a cube is 1,000 cm3. What is the length of an edge?

16 A gift box in the shape of a cube has a volume of 216 cm3. What is the area of the base of the box?

17 A scientist finds the temperature of a sample at the beginning of an experiment is t°C. After 1 hour, the temperature is t2 °C. If the temperature after 1 hour is 81°C, what are two possible original temperatures? What is the difference between the possible original temperatures?


Lesson 2



Student Model



The distance in feet that a freely falling dropped object falls in

t seconds is given by the equation d ··

16

5 t2.

How long does it take a dropped object to fall 64 feet?

Look at how you could solve this problem.

The given equation is: d ··· 16 5 t2

Substitute 64 for d: 64 ··· 16 5 t2

Simplify: 4 5 t2

Take the square root of both sides: Ï·· 4 5 Ï·· t2

t 5 2

Solution:

18 The area of the top face of a cube is 9 square meters. What is the volume of the cube?

Show your work.

Solution:

How far does an object fall in 1 second?

Pair/Share

The cube has 6 faces. What does the expression 6 • 9 describe?

Pair/Share

In this problem, you will divide before you find the square root.

What information do you need to calculate the volume of a cube?

The object takes 2 seconds to fall 64 feet.

Lesson 2



19 The length of each side of a cube is x centimeters. If x is an integer, why can’t the volume of the cube equal 15 cm3?

Show your work.

Solution:

20 Yesterday, there were b milligrams of bacteria in a lab experiment. Today, there are b2 milligrams of bacteria. If there are 400 milligrams today, how many milligrams of bacteria were there yesterday?

A 20 milligrams

B 200 milligrams

C 1,600 milligrams

D 160,000 milligrams

Eva chose B as the correct answer. How did she get that answer?


Pair/Share

Are all perfect cubes also multiples of 3? Are all multiples of 3 also perfect cubes? Discuss.

Pair/Share

Do you square a number or find the square root to solve the problem?

Write an equation showing a variable expression for volume is equal to 15.

Lesson 2



Solve the problems.

1 Solve a3 5 64.

A a 5 4

B a 5 8

C a 5 21

D a 5 32

2 Which number is a perfect square?

A 8

B 18

C 200

D 225

3 The fractions below are the values of x in the given equations. Write the correct fraction inside the box for each equation.

9 }} 8 1 }

2 3 }

4 2 }

3

A x2 5 4 } 9

B x3 5 27 }} 64

C x2 5 81 }} 64

D x3 5 1 } 8


Lesson 2


4 Use the numbers shown to make the two equations true. Each number can be used only once. Write the number in the appropriate box for each equation.

3 6 100 36 1,000 1,000,000

Ï·············· 5

3

Ï·············· 5

5 If x is a positive integer, is Ï·· 1 ·· x2 greater than, less than, or equivalent to

3 Ï·· 1 ·· x3 ?

Show your work.

Answer

6 Describe how you could use inverse operations to solve the equation Ï·· x 5 4.




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.Lesson 8 ReteachRoots

A square root of a number is one of its two equal factors. A radical sign, √

� is used to indicate a positive square root. Every positive number has both a negative and positive square root.

ExamplesFind each square root.

1. √

�

1 Find the positive square root of 1; 12 = 1, so √

�

1 = 1.

2. -

√

��

16 Find the negative square root of 16; (-4)2 = 16, so -

√

��

16 = -4.

3. ±

√

��

0.25 Find both square roots of 0.25; 0.52 = 0.25, so ±

√

��

0.25 = ±0.5.

4.

√

��

-49 There is no real square root because no number times itself is equal to -49.

Example 5Solve a2 = 4 −

9 . Check your solution(s).

a2 = 4 −

9 Write the equation.

a = ±

√

�

4 −

9 Defi nition of square root

a = 2 −

3 or -

2 −

3 Check 2 −

3 · 2 −

3 = 4 −

9 and

(

- 2 −

3 )

(

- 2 −

3 )

= 4 −

9 .

The equation has two solutions, 2 −

3 and -

2 −

3 .

ExercisesFind each square root.

1. √

�

4 2. √

�

9

3. -

√

��

49 4. -

√

��

25

5. ±

√

��

0.01 6. -

√

��

0.64

7. √

��

9 −−

16 8.

√

��

-1 −−

25

ALGEBRA Solve each equation. Check your solution(s).

9. x2 = 121 10. a2 = 3,600

11. p2 = 81 −−−

100 12. t2 = 121 −−−

196


PDF Pass


NAME __________________________________________ DATE ____________ PERIOD _______


Roots

Find each square root.

1. 9 3 2. 81 9 3. 625 25 4. 36 6 5. 169 13 6. 144 12 7. 961 31 8. 324 18 9. 225 15

10. 4 2 11. 529 23 12. 484 22 13. 0.04 0.2 14. 2.25 1.5 15. 0.01 0.1

16. 0.09 0.3 17. 0.49 0.7 18. 1.69 1.3

19. 49

23 20. 81

64 98 21.

2581

59


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.Lesson 9 ReteachEstimate Roots

Most numbers are not perfect squares or cubes. You can estimate roots for these numbers.

Example 1Estimate √

�� 204 to the nearest integer.

• The largest perfect square less than 204 is 196.

• The smallest perfect square greater than 204 is 225.

196 < 204 < 225 Write an inequality.

142 < 204 < 152 196 = 142 and 225 = 152.

√

��

142 < √

��

204 < √

��

152 Find the square root of each number.

14 < √

��

204 < 15 Simplify.

So, √

��

204 is between 14 and 15. Since 204 is closer to 196 than 225, the best whole number estimate for √

��

204 is 14.

Example 2Estimate 3 √

��

79.3 to the nearest integer.

• The largest perfect cube less than 79.3 is 64.

• The smallest perfect cube greater than 79.3 is 125.

64 < 79.3 < 125 Write an inequality.

43 < 79.3 < 53 64 = 43 and 125 = 53.

3 √

��

64 < 3 √

��

79.3 < 3 √

��

125 Find the cube root of each number.

4 < 3 √

��

79.3 < 5 Simplify.

So, 3 √

��

79.3 is between 4 and 5. Since 79.3 is closer to 64 than 125, the best whole number estimate for 3 √

��

79.3 is 4.

ExercisesEstimate to the nearest integer.

1. √

�

8 2. √

��

37 3. √

��

14

4. 3 √

��

30 5. 3 √

��

750 6. 3 √

��

200

7. √

��

103 8. √

��

141 9. √

��

14.3

10. √

��

51.2 11. 3 √

��

340.8 12. 3 √

��

7.5


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NAME __________________________________________ DATE ____________ PERIOD _______


Estimate Roots

Estimate to the nearest integer.

1. 229 15 2. 63 8 3. 290 17 4. 27 5 5. 333 18 6. 23 5 7. 96 10 8. 200 14 9. 117 11

10. 47 7 11. 1.3 1 12. 8.4 3 13. 18.35 4 14. 25.7 5 15. 14.1 4

16. 15.3 4 17. 32.7 6 18. 55.2 7

©Curriculum Associates, LLC Copying is not permitted.L3: Understand Rational and Irrational Numbers22

Understand Rational and Irrational NumbersLesson 3 Part 1: Introduction

Focus on Math Concepts

What are rational numbers?

Rational numbers are numbers that can be written as the quotient of two integers. Since the bar in a fraction represents division, every fraction whose numerator and denominator is an integer is a rational number.

Any number that could be written as a fraction whose numerator and denominator is an integer is also a rational number.

Every integer, whole number, and natural number is a rational number.Think

You can write every integer, whole number, and natural number as a fraction. So they are all rational numbers. The square root of a perfect square is also a rational number.

1

1 4 9 16

2 3 4

3 5 3 ·· 1

25 5 25 ·· 1

0 5 0 ·· 1

Ï··· 25 5 5 or 5 ·· 1

Every terminating decimal is a rational number.Think

You can write every terminating decimal as a fraction. So terminating decimals are all rational numbers.

You can use what you know about place value to find the fraction that is equivalent to any terminating decimal.

0.4 four tenths 4 ·· 10 5 2 ·· 5

0.75 seventy-five hundredths 75 ··· 100 5 3 ·· 4

0.386 three hundred eighty-six thousandths 386 ····· 1,000 5 193 ··· 500

Ï···· 0.16 5 0.4 four tenths 4 ·· 10 5 2 ·· 5

MAFS8.NS.1.1

8.NS.1.2

23©Curriculum Associates, LLC Copying is not permitted.L3: Understand Rational and Irrational Numbers


Every repeating decimal is a rational number.Think

You can write every repeating decimal as a fraction. So repeating decimals are all rational numbers.

As an example, look at the repeating decimal 0. ·· 3 .

Let x 5 0. ·· 3

10 • x 5 10 • 0. ·· 3 The repeating pattern goes to the tenths place. Multiply both 10x 5 3. ·· 3 sides by 10.

10x 2 x 5 3. ·· 3 2 0. ·· 3 Subtract x from the left side and 0. ·· 3 from the right side.

9x 5 3 The equation is still balanced because x and 0. ·· 3 are equivalent.

9x ·· 9 5 3 ·· 9

x 5 3 ·· 9 or 1 ·· 3

0. ·· 3 5 1 ·· 3

Here’s another example of how you can write a repeating decimal as a fraction.

x 5 0. ··· 512

1,000x 5 512. ··· 512 The repeating pattern goes to the thousandths place. Multiply by 1,000.

1,000x 2 x 5 512. ··· 512 2 ····· 0.512 Subtract x from the left side and the repeating decimal from the right side.

999x 5 512

x 5 512 ··· 999

Reflect

1 What fraction is equivalent to 5.1? Is 5.1 a rational number? Explain.

You can write and solve an equation to find a fraction equivalent to a repeating decimal.

Part 2: Guided Instruction Lesson 3


L3: Understand Rational and Irrational Numbers24

Explore It

What numbers are not rational? Let’s look at a number like Ï·· 2 , the square root of a number that is not a perfect square.

2 Look at the number line below. The number Ï·· 2 is between Ï·· 1 and Ï·· 4 . Since Ï·· 1 5 1 and Ï·· 4 5 2, that means that Ï·· 2 must be between what two integers?

1

1 4 9 16

2 3 4

3 Draw a point on the number line where you would locate Ï·· 2 . Where did you draw the point?

4 Calculate: 1.32 5 1.42 5 1.52 5

5 Based on your calculations, draw a point on the number line below where you would locate Ï·· 2 now. Where did you draw the point?

1 1.5

1

2

4

6 Calculate: 1.412 5 1.422 5

7 Based on these calculations, Ï·· 2 is between which two decimals?

8 You can continue to estimate, getting closer and closer to the value of Ï·· 2 . For example, 1.4142 5 1.999396 and 1.4152 5 2.002225, but you will never find an exact number that multiplied by itself equals 2. The decimal will also never have a repeating pattern.

Ï·· 2 cannot be expressed as a terminating or repeating decimal, so it cannot be written as a fraction. Numbers like Ï·· 2 and Ï·· 5 are not rational. You can only estimate their values. They are called irrational numbers. Here, irrational means “cannot be set as a ratio.” The set of rational and irrational numbers together make up the set of real numbers.

Now try this problem.

9 The value p is a decimal that does not repeat and does not terminate. Is it a rational or irrational number? Explain.


©Curriculum Associates, LLC Copying is not permitted.25L3: Understand Rational and Irrational Numbers

Talk About It

You can estimate the value of an irrational number like Ï·· 5 and locate that value on a number line.

10 Ï·· 5 is between which two integers? Explain your reasoning.

11 Mark a point at an approximate location for Ï·· 5 on the number line below. Ï·· 5 is between which two decimals to the tenths place?

2 2.22.1 2.3 2.5 2.7 2.92.4 2.6 2.8

4

3

9

12 Calculate: 2.222 5 2.232 5 2.243 5

Based on your results, Ï·· 5 is between which two decimals to the hundredths place?

13 Draw a number line from 2.2 to 2.3. Label tick marks at tenths to show 2.21, 2.22, 2.23, and so on. Mark a point at the approximate location of Ï·· 5 to the hundredths place.

Try It Another Way

Explore using a calculator to estimate irrational numbers.

14 Enter Ï·· 5 on a calculator and press Enter. What is the result on your screen?

15 If this number is equal to Ï·· 5 , then the number squared should equal .

16 Clear your calculator. Then enter your result from problem 14. Square the number. What is the result on your screen?

17 Explain this result.

Part 3: Guided Practice Lesson 3


L3: Understand Rational and Irrational Numbers26

Connect It

Talk through these problems as a class, then write your answers below.

18 Illustrate: Show that 0. ·· 74 is equivalent to a fraction. Is 0. ·· 74 a rational or irrational number? Explain.

19 Analyze: A circle has a circumference of 3p inches. Is it possible to state the exact length of the circumference as a decimal? Explain.

20 Create: Draw a Venn diagram showing the relationships among the following sets of numbers: integers, irrational numbers, natural numbers, rational numbers, real numbers, and whole numbers.

Lesson 3

©Curriculum Associates, LLC Copying is not permitted.27L3: Understand Rational and Irrational Numbers

Put It Together

Use what you have learned to complete this task.

21 Consider these numbers:

Ï··· 50 3.4 ··· 56 0 Ï·· 4 ·· 9 0.38 Ï··· 81 2p Ï···· 1.69 Ï·· 2 ·· 9

A Write each of the numbers in the list above in the correct box.

Rational Numbers Irrational Numbers

B Circle one of the numbers you said was rational. Explain how you decided that the number was rational.

C Now circle one of the numbers you said was irrational. Explain how you decided that the number was irrational.

D Draw a number line and locate the two numbers you circled on the line. Write a comparison statement using <, 5, or > to compare the numbers.

Part 4: Performance Task


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.Lesson 1 ReteachRational Numbers

To express a fraction as a decimal, divide the numerator by the denominator.

Example 1Write 3 −

4 as a decimal.

3 −

4 means 3 ÷ 4.

The fraction 3 −

4 can be written as 0.75, since 3 ÷ 4 = 0.75.

Example 2Write -0.16 as a fraction in simplest form.

-0.16 = -

16 −−−

100 0.16 is 16 hundredths.

= -

4 −−

25 Simplify.

The decimal -0.16 can be written as -

4 −−

25 .

Example 3Write 8.

−

2 as a mixed number in simplest form.

Assign a variable to the value 8. −

2 . Let N = 8.222… . Then perform the operations on N to determine its value.

N = 8. −

2 or 8.222….

10(N) = 10(8.222) Multiply each side by 10 because 1 digit repeats.

10N = 82.222… Multiplying by 10 moves the decimal point 1 place to the right.

-N = 8.222… Subtract N = 8.222… to eliminate the repeating part.

9N = 74 10N - 1N = 9N

9N −−

9 =

74 −−

9 Divide each side by 9.

N = 8 2 −

9 Simplify.

The decimal 8. −

2 can be written as 8 2 −

9 .

ExercisesWrite each fraction or mixed number as a decimal.

1. 2 −

5 2. 3 −−

10 3. 7 −

8 4. 2 16 −−

25

5. -

2 −

3 6. -1 2 −

9 7. 6 2 −

3 8. -4 3 −−

11

Write each decimal as a fraction or mixed number in simplest form.

9. 0.8 10. -0.15 11. 0. −

1 12. 1. −

7


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NAME __________________________________________ DATE ____________ PERIOD _______


Rational Numbers

Write each fraction or mixed number as a decimal.

1. 25 0.4 2. 2

311 3. 3

4 0.75

4. 57 5.

34 0.75 6. 2

3

7. 711 8.

12 0.5 9.

56

10. 135 1.6 11. 2

14 2.25 12.

89

Write each decimal as a fraction or mixed number in simplest form.

13. 0.5 12 14. 0.8

89 15. 0.32

825

16. 0.75 34 17. 2.2 2

29 18. 0.38

3899

19. 0.486 243500 20. 20.08 20

225 21. 9.36 9

925

22. 10.18 101790 23. 1.24 1

625 24. 5.7 5

79


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.Lesson 10 ReteachCompare Real Numbers

Numbers can be classifi ed by identifying to which of the following sets they belong.

Whole Numbers 0, 1, 2, 3, 4, … Integers …, -2, -1, 0, 1, 2, …

Rational Numbers numbers that can be expressed in the form a −−

b , where a and b are

integers and b ≠ 0

Irrational Numbers numbers that cannot be expressed in the form a −−

b , where a and b

are integers and b ≠ 0

ExamplesName all sets of numbers to which each real number belongs.

1. 5 whole number, integer, rational number

2. 0.666… Decimals that terminate or repeat are rational numbers, since they can be expressed as fractions. 0.666… 2 −

3

3. -

√

��

25 Since -

√

��

25 = -5, it is an integer and a rational number.

4.

√

��

11 √

��

11 ≈ 3.31662479… Since the decimal does not terminate or repeat, it is an irrational number.

To compare real numbers, write each number as a decimal and then compare the decimal values.

Example 5Replace with <, >, or = to make 2 1 −

4 √

� 5 a true statement.

Write each number as a decimal.

2 1 −

4 = 2.25

√

�

5 ≈ 2.236067…

Since 2.25 is greater than 2.236067…, 2 1 −

4 > √

�

5 .

ExercisesName all sets of numbers to which each real number belongs.

1. 30 2. -11

3. 5 4 −

7 4. √

��

21

5. 0 6. -

√

�

9

7. 6 −

3 8. -

√

��

101

Replace each with <, >, or = to make a true statement.

9. 2.7 √

�

7 10. √

��

11 3 1 −

2 11. 4 1 −

6 √

��

17 12. 3. −

8 √

��

15


PDF Pass


NAME __________________________________________ DATE ____________ PERIOD _______


Compare Real Numbers

Name all sets of numbers to which each real number belongs.

1. 6.5 rational 2. 25 3. 3 irrational 4. 7.2 rational natural, whole, integer, rational

5. 0.61 rational 6. 12 rational 7.

164 8. 27 irrational

natural, whole, integer, rational Fill in each with <, >, or = to make a true statement.

9. 7 2.8 < 10. 213 2.3 = 11. 121 11 =

12. 30 5.6 < 13. 9.45 2.4 > 14. 5 2.23 >

15. 6.25 212 = 16. 5

13 30 < 17. 2.9 8 >

Order each set of numbers from least to greatest. Verify your answer by graphing on a number line.

18. 12, 3, 3

12, 3.5

3, 12, 312,

19. 2, 140%, 1.45, 1.4 140%, 2, , 1.45

20. 10, 11, 3.5, 3

13

3.5, 313, 11, 10

©Curriculum Associates, LLC Copying is not permitted.L4: Scientific Notation28

Scientific NotationLesson 4 Part 1: Introduction

You’ve learned about place value and investigated multiplying and dividing by powers of 10. Now, take a look at this problem.

The planet Venus is more than 60,000,000 miles from the Sun. Write this number as the product of two factors:

(a number greater than or equal to 1 but less than 10) 3 (a power of 10)

Explore It

Use the math you already know to solve the problem.

Write 60,000,000 in words.

Fill in the missing factor: 60,000,000 5 6 •

Write the second factor in the equation as a product of 10s.

Write the second factor as a power of 10. Explain your reasoning.

Explain how you could write 60,000,000 as the product of a number greater than or

equal to 1 that is multiplied by a power of 10.


MAFS8.EE.1.3

©Curriculum Associates, LLC Copying is not permitted.29L4: Scientific Notation


Find Out More

Scientists often work with very large numbers, such as the distance from Venus to the Sun or the number of cells in a human body. Writing and calculating with very large numbers can be tedious and inconvenient.

When you wrote 60,000,000 as 6 3 107, you used scientific notation. Scientific notation uses exponents to make it easier to work with very large or very small numbers. To write a number using scientific notation, write it as a product of two factors:

To write the number 1,850,000 in scientific notation,

The power doesn’t tell you the number of zeros in your answer. Rather, it tells you how many place values the digits increase.

To write the number 3.54 3 105 in standard form, we move each digit in 3.54 up 5 place values because we are multiplying by 105.

To translate between scientific notation and standard notation, change the place values of the digits according to the power of 10.

Reflect

1 Write 6.85 3 108 in standard form. Show your work.

6 3 107a number that is greater than or equal to 1 but less than 10

a power of 10

1,850,000 5 1 8 5 0 0 0 0

5 1.85 3 106

Move the decimal point to get a number between 1 and 10.

The power of 10 is equal to the number of place values that the digits increase.

3.54 3 105 5 3.5 4 0 0 0 = 354,000

Lesson 4


L4: Scientific Notation30


Read the problem below. Then explore how to write very small numbers using scientific notation.

Seven nanoseconds is equivalent to 7 one-billionths of a second, or 0.000000007 second. Write 0.000000007 in scientific notation.

Picture It

Look at the patterns in the chart below.

100 10 • 10 102

10 10 101

1 1 100

0.1 1 ·· 10 5 1 ··· 101 1021

0.01 1 ······ 10 ? 10 5 1 ··· 102 1022

0.001 1 ········· 10 ? 10 ? 10 5 1 ··· 103 1023

0.0001 1 ············ 10 ? 10 ? 10 ? 10 5 1 ··· 104 1024

Model It

You can write the decimal as a fraction.

0.000000007 5 7 ··········· 1,000,000,000

5 7 • 1 ··········· 1,000,000,000

Solve It

You can write the number as the product of a number that is greater than or equal to 1 but less than 10. When you multiply by a number by a power of 10, the decimal point moves to the right, or the place value of each digit moves up.

0.0 0 0 0 0 0 0 0 7 Move the decimal point 9 places to the right.

Lesson 4



Connect It

Now look at different ways to solve the problem.

2 Write 1,000,000,000 as a product of 10s.

3 Write 1,000,000,000 as a power of 10.

4 Look at the table on the previous page. Write 1 ··········· 1,000,000,000 as 10 to a power to complete

the equation: 7 • 1 ··········· 1,000,000,000 5 3

5 Look at Solve It. When you have to move the decimal point to the right to express a number in scientific notation, will the power of 10 be positive or negative?

6 When a number is written in scientific notation, a exponent means the number is greater than 1 and a exponent means the number is between 0 and 1.

7 Is 2.14 3 1025 greater than 1 or between 0 and 1? Explain.

8 Write 2.14 • 1025 in standard form.

2.14 3 1025 5 214 ··· 100 3 1 ··· 105

5

5

9 Explain how to write 2.14 • 1025 in standard form by moving the decimal point.

Try It

Use what you just learned to solve these problems. Show your work.

10 Write 63,120,000 in scientific notation.

11 Write 9.054 3 1026 in standard form.

Lesson 4



Read the problem below. Then explore how to compare numbers written in scientific notation.

Earth is about 1.5 3 108 kilometers from the Sun, while the planet Neptune is almost 4.5 3 109 kilometers from the Sun. The distance from Neptune to Earth is about how many times the distance from the Sun to Earth?

Model It

You can write the distances of the planets from the Sun in standard form and compare them.

1.5 3 108 kilometers 5 150,000,000 kilometers

4.5 3 109 kilometers 5 4,500,000,000 kilometers

To find how many times as great 4,500,000,000 is than 150,000,000, divide.

4,500,000,000 ÷ 150,000,000

Model It

You can compare the distances of the planets from the Sun using scientific notation.

To compare 1.5 3 108 and 4.5 3 109:

First, compare 1.5 and 4.5.

4.5 is how many times as great as 1.5?

Then, compare 108 and 109.

109 is how many times as great as 108?


Lesson 4


Connect It

Now you will solve the problem using standard form and scientific notation.

12 Look at the first Model It on the previous page. 4,500,000,000 is how many times the value of 150,000,000?

13 Look at the second Model It. 4.5 is how many times the value of 1.5? Explain your reasoning.

14 109 is how many times the value of 108? Explain your reasoning.

15 Look at your answers to problems 13 and 14. 4.5 3 109 is how many times the value of 1.5 3 108? Give your answer in both scientific notation and standard form.

16 Which method of comparing the numbers would you use? Explain.

Try It


17 6 3 10–5 is how many times the value of 3 3 10–8?

18 Star A is about 3.4 3 1018 miles from Earth. Star B is 6.8 3 1016 miles from Earth. Star A is how many times as far from Earth as Star B?


Lesson 4



Student Model



Write 0.0000408306 in scientific notation.


In scientific notation, the solution will look like n • 10a. n must be greater than or equal to 1 and less than 10. a must be an integer.

To write 0.0000408306 in scientific notation, first move the decimal point 5 places to the right. Then multiply that number by a power of 10. The exponent in that power of 10 will be negative 25, which is found by counting the number of places the decimal is moved to the right.

Solution:

19 The mass of Earth is about 5,974,000,000,000,000,000,000,000 kilograms. Write this number in scientific notation.

Show your work.

Solution:

What is another method you could use to write the number in scientific notation?

Pair/Share

Explain why the procedure used to write a number in scientific notation works.

Pair/Share

The student moved the decimal point the number of places necessary to get a number greater than or equal to 1 and less than 10.

Do you move the exponent to the right or to the left to write the number in scientific notation.

0.0000408306 5 4.08306 3 1025

Lesson 4



20 Use the information in the table to solve the problem.

Orbiting Body Approximate Distance from the Sun (in miles)

Mercury 36,300,000

Mars 142,000,000

Neptune 2,800,000,000

Pluto 3,670,000,000

Show your work.

Write each distance in scientific notation.

Mercury

Mars

Neptune

Pluto

Neptune is about how many times as far from the Sun as Mars is from the Sun?

Solution:

21 Which is equivalent to 8.03 3 1028?

A 2803,000,000

B 20.0000000803

C 0.0000000803

D 803,000,000

Eva chose D as the correct answer. How did she get that answer?


Pair/Share

How does writing numbers in scientific notation make numbers easier to work with?

Pair/Share

Will the solution be a negative number or positive number?

Will the exponent be positive or negative?

Lesson 4



Solve the problems.

1 Which of the following expressions is equivalent to 5,710,900?

A 5.7109 3 1026

B 57109 3 102

C 5.7109 3 103

D 5.7109 3 106

2 The average distance from Pluto to the Sun is about 6 3 109 kilometers. The average distance from Mars to the Sun is 2 3 108 kilometers. The average distance from Pluto to the Sun is about how many times as great as the average distance from Mars to the Sun?

times

3 Last year a business earned 4.1 3 106 dollars in income. This year the business earned 2.05 3 108 dollars in income. Which best describes how this year’s earnings compare to last year’s earnings?

A This year the business earned about 0.5 times as much as it did last year.

B This year the business earned about 2 times as much as it did last year.

C This year the business earned about 50 times as much as it did last year.

D This year the business earned about 100 times as much as it did last year.


Lesson 4


4 Write the following numbers in order from least to greatest.

5 3 1026 29 3 1023

20.0000002 0.00007

Least Greatest

5 Cara was using her calculator to solve a problem. The answer that displayed was 1.6E+12. She knows that she entered all of the numbers correctly. Why did the calculator give the answer it did? What is the answer to Cara’s problem?

6 The length of a city block running north to south in New York City is about 5 3 1022 miles. The distance from New York City to Mumbai, India, is about 7.5 3 103 miles. The distance from New York City to Mumbai is about how many times the length of a New York City north-south block?

Show your work.

Answer




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.Lesson 6 ReteachScientifi c Notation

Example 1Write 8.65 × 107 in standard form.

8.65 × 107 = 8.65 × 10,000,000 107 = 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 or 10,000,000

= 86,500,000 The decimal point moves 7 places to the right.

Example 2Write 9.2 × 10–3 in standard form.

9.2 × 10-3 = 9.2 × 0.001 The decimal point moves 3 places to the left.

= 0.0092

Example 3Write 76,250 in scientific notation.

76,250 = 7.625 × 10,000 The decimal point moves 4 places.

= 7.625 × 104 Since 76,250 is >1, the exponent is positive.

Example 4Write 0.00157 in scientific notation.

0.00157 = 1.57 × 0.001 The decimal point moves 3 places.

= 1.57 × 10–3 Since 0.00157 is <1, the exponent is negative.

ExercisesWrite each number in standard form.

1. 5.3 × 101 2. 9.4 × 103

3. 7.07 × 105 4. 2.6 × 10-3

5. 8.651 × 10-2 6. 6.7 × 10-6

Write each number in scientific notation.

7. 561 8. 14

9. 56,400,000 10. 0.752

11. 0.0064 12. 0.000581

A number in scientifi c notation is written as the product of a factor that is at least one but less than ten and a power of ten.

��

��

��

��


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NAME __________________________________________ DATE ____________ PERIOD _______


Scientific Notation

Write each number in standard form.

1. 4.5 103 4,500 2. 2 104 20,000 3. 1.725896 106 1,725,896 4. 9.61 102 961 5. 1 107 10,000,000 6. 8.256 108 825,600,000 7. 5.26 104 52,600 8. 3.25 102 325 9. 6.79 105 679,000 10. 3.1 104 0.00031 11. 2.51 102 0.0251 12. 6 101 0.6 Write each number in scientific notation.

13. 720 7.2 102 14. 7,560 7.56 103 15. 892 8.92 102 16. 1,400 1.4 103 17. 91,256 9.1256 104 18. 51,000 5.1 104 19. 0.012 1.2 102 20. 0.0002 2 104 21. 0.054 5.4 102 22. 0.231 2.31 101 23. 0.0000056 5.6 106 24. 0.000123 1.23 104

©Curriculum Associates, LLC Copying is not permitted.L5: Operations and Scientific Notation38

Part 1: IntroductionLesson 5

Operations and Scientific Notation

In Lesson 4 you learned to express and compare numbers using scientific notation. Now, take a look at this problem.

Evaluate the following expression.

950,000 1 (4.6 3 107)

Explore It

Use the math you know to answer the question.

What form is 950,000 written in?

What form is 4.6 3 107 written in?

Write 4.6 3 107 in the same form as 950,000.

4.6 3 107 5 4.6 3

5

Write the original addition expression with all numbers in standard form.

Explain how you would simplify your expression. What is the sum?


MAFS8.EE.1.4

©Curriculum Associates, LLC Copying is not permitted.39L5: Operations and Scientific Notation


Find Out More

When you add very large (or very small) numbers expressed in standard form, it can be difficult to keep track of all the zeros and make sure the numbers are aligned by place value. One way to deal with these problems is to express each number in scientific notation.

Convert to scientific notation: 950,000 5 9.5 3 100,000 5 9.5 3 105

Remember that there is a link between place value and powers of 10. Before you can add numbers in standard form, you must align them by place value. Likewise, before you can add numbers in scientific notation, each power of 10 must have the same exponent.

4.6 3 107 5 4.6 3 (102 3 105) Apply the product of powers property. 5 (4.6 3 102) 3 105 Apply the associative property of multiplication. 5 460 3 105 Multiply.

Now that both numbers are expressed with the same exponent, you can find the sum.

(460 3 105) 1 (9.5 3 105) 5 (460 1 9.5) 3 105 Apply the distributive property.

5 469.5 3 105 Add.

5 (4.695 3 102) 3 105 Express in scientific notation.

5 4.695 3 (102 3 105) Apply the associative property of multiplication.

5 4.695 3 107 Apply the product of powers property.

Reflect

1 Paul says that 3.14 3 105 1 2.53 3 104 5 5.67 3 105. Is Paul correct? Explain.

Lesson 5


L5: Operations and Scientific Notation40


Read the problem below. Then explore how to subtract numbers expressed in scientific notation.

Find the difference: 5.1 3 1012 2 6,300,000,000

Solve It

Start by converting 6,300,000,000 to scientific notation.

6,300,000,000 5 6.3 3 1,000,000,000

5 6.3 3 109

Picture It

Make a table to help you compare powers of 10.

You cannot subtract numbers expressed in scientific notation unless the powers of 10 have the same exponent. Create a table to help you express the numbers in the problem in scientific notation and compare the exponents.

5.1 3 1012 6.3 3 109

5 5.1 3 1012 5 0.0063 3 1012

5 51 3 1011 5 0.063 3 1011

5 510 3 1010 5 0.63 3 1010

5 5,100 3 109 5 6.3 3 109

Any pair of numbers from the table with powers of 10 that have the same exponents can be used to solve the problem.

Lesson 5



Connect It

Now solve the problem from the previous page.

2 Look at Solve It on the previous page. Write the problem with both numbers expressed in scientific notation.

3 Look at the Picture It on the previous page. Use the table to rewrite the expression you wrote for problem 2. Rewrite that expression so that both terms are written with the same exponent.

4 Use the distributive property to simplify the expression you wrote for problem 3.

5 Write your expression as the product of a decimal times a power of 10.

6 Write your solution in scientific notation.

Try It

Use what you just learned to solve these problems. Show your work.

7 Evaluate: (7.4 3 1015) 2 (9.9 3 1013)

8 Evaluate: (8.9 3 105) 1 (6.5 3 106)

Lesson 5



Read the problem below. Then explore how to multiply numbers expressed in scientific notation.

Multiply: (5.78 3 105) 3 0.0804

Estimate It

You can round the factors to estimate the product.

Round 5.78 3 105 to 6 3 105. Then round 0.0804 to 0.08. The estimated product is: (6 3 105) 3 (0.08) 5 0.48 3 105 5 4.8 3 104

You can compare your calculated answer to this estimate to check your solution.

Solve It

You can convert both terms to scientific notation.

Write 0.0804 in scientific notation.

0.0804 5 8.04 ···· 100

5 8.04 3 1 ··· 100

5 8.04 3 1 ··· 102

5 8.04 3 1022

Write the problem with both factors in scientific notation. (5.78 3 105) 3 (8.04 3 1022)


Lesson 5


Connect It

Now solve the problem from the previous page.

9 Complete the equation by applying the associative property to group the decimals and to group the powers of 10.

(5.78 3 105) 3 (8.04 3 1022) 5

10 Multiply the decimals and multiply the powers of 10.

11 Apply the properties of exponents to write your solution in scientific notation.

12 Look at Estimate It on the previous page. Is your solution reasonable? Explain.

13 Why is it unnecessary to make the exponents the same before multiplying numbers expressed in scientific notation?

Try It


14 The world’s thinnest computer chip is 7.5 3 1023 millimeters thick. What would be the height of a stack of 3 3 109 chips?

15 The speed of a garden snail is about 8.3 3 1026 miles per second. If a garden snail moves at this speed in a straight line for 3.6 3 103 seconds, how far would the snail travel?


Lesson 5




A hardware factory produces 3.6 3 105 bolts in 2,400 minutes. What is the factory’s unit rate of production in bolts per minute?


Solution:

16 A company spends a total of $64,500,000 on salaries for its workers. If the company has 1.5 3 103 workers, what is the average salary per worker?

Show your work.

Solution:

Student Model


Would you rather solve this problem with both numbers expressed in standard form or in scientific notation? Explain.

Pair/Share

Do you need to write each number with the same exponent before you can divide? Explain.

Pair/Share

In this problem you will need to divide numbers expressed in scientific notation.

Which operation will you need to use to solve this problem?

Express 2,400 in scientific notation.

The quotient of products equals the product of quotients.

Subtract the exponents to find the quotient of powers.

2,400 5 2.4 3 103

total bolts ··········· total minutes 5 unit rate in bolts per minute

3.6 3 105 ········ 2.4 3 103 5 3.6 ··· 2.4 3 105

··· 103

5 1.5 3 105 2 3

5 1.5 3 102

The factory produces 1.5 3 102 bolts per minute.

Lesson 5



17 Stalactites are cone-shaped formations that hang from the ceilings of underground caverns. Stalactites can grow at the rate of about 0.005 inch per year. At this rate, what is the length of a stalactite that grows for 7.5 3 104 years?

Show your work.

Solution:


Pair/Share

Compare the stalactite’s rate of growth with a child’s rate of growth.

Pair/Share

How would you express the volume of Venus in scientific notation?

Would it be easier to solve this problem with numbers in scientific notation, fractions, or as they are written?

18 The volume of the planet Venus is about 928,000,000,000 km3. The volume of the planet Mercury is about 6.08 3 1010 km3. What is the combined volume of Mercury and Venus?

A 9.888 3 1010 km3

B 1.536 3 1011 km3

C 9.888 3 1011 km3

D 1.536 3 1012 km3

Maya chose D as the correct answer. How did she get that answer?

Lesson 5



Solve the problems.

1 A rancher uses a water bowl for her dog that holds 8,500 milliliters and a water trough for her horse that holds 2.7 3 105 milliliters. How many milliliters of water will the rancher use to completely fill both the bowl and the trough?

A 1.12 3 105 ml

B 2.785 3 105 ml

C 5.8 3 105 ml

D 1.12 3 109 ml

2 The Moon takes about 28 days to orbit the Earth, going a distance of about 2.413 3 106 kilometers. About how many kilometers does the Moon travel during one day of its orbit around the Earth?

A 8.6 3 104 km

B 2.8 3 106 km

C 1.16 3 107 km

D 6.8 3 107 km

3 Jackie incorrectly simplified the following expression.

(4 3 1026)(2 3 103) 1 1,000

Select each step that shows an error based solely on the previous step.

A Step 1. (4 3 1026)(2 3 103) 1 103

B Step 2. (4 3 1026)(3 3 103)

C Step 3. (4 3 3)(1026 3 103)

D Step 4. 12 3 1023

E Step 5. 1.2 3 1024


Lesson 5


4 In October 2009, there were approximately 5 3 107 members of a website. In January 2013, there were approximately 2 3 108 members. How many more members were there in January 2013 than in October 2009? Write your answer in scientific notation. Select from the given digits to complete the sentence.

1 2 3 4 5 6 7 8 9

There were . 3 10 more members in January 2013 than in October 2009.

5 Toshi and Owen need to add 4.9 3 109 and 4.1 3 107. Toshi says they must use the equation (490 3 107) 1 (4.1 3 107), but Owen says they must use the equation (4.9 3 109) 1 (0.041 3 109). Are neither, one, or both students correct? Explain.

6 Evaluate (7.3 3 106) 1 (2.4 3 107)

}}}}}}}}}}}} (4 3 104)

.

Show your work.

Answer




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.Lesson 7 ReteachCompute with Scientifi c Notation

Example 1Evaluate (3.4 × 105)(2.3 × 103). Express the result in scientific notation.

(3.4 × 105)(2.3 × 103) = (3.4 × 2.3)(105 × 103) Commutative and Associative Properties

= (7.82)(105 × 103) Multiply 3.4 by 2.3.

= 7.82 × 105 + 3 Product of Powers

= 7.82 × 108 Add the exponents.

Example 2

Evaluate 2.325 × 104 −−−−−−−−

3.1 × 102 . Express the result in scientific notation.

2.325 × 104

−−−−−−−

3.1 × 102

=

(

2.325 −−−−

3.1

)

(

104 −−−

102 )

Associative Property

= (0.75) (

104 −−−

102 )

Divide 2.325 by 3.1.

= 0.75 × 104 – 2 Quotient of Powers

= 0.75 × 102 Subtract the exponents.

= 0.75 × 102 Write 0.75 × 102 in scientifi c notation.

= 7.5 × 10 Since the decimal point moved 1 place to the right, subtract 1 from the exponent.

Example 3Evaluate (5.24 × 105) + (8.65 × 106). Express the result in scientific notation.

(5.24 × 105) + (8.65 × 106) = (5.24 × 105) + (86.5 × 105) Write 8.65 × 106 as 86.5 × 105.

= (5.24 + 86.5) × 105 Distributive Property

= 91.74 × 105 Add 5.24 and 86.5.

= 9.174 × 106 Write 91.74 × 105 in scientifi c notation.

ExercisesEvaluate each expression. Express the result in scientific notation.

1. (6.7 × 104)(2.9 × 105) 2. (4.3 × 104) + (5.21 × 105)

3. 5.46 × 105 −−−−−−−

8.4 × 103 4. (9.6 × 105) – (3.7 × 103)

You can use the Product of Powers and Quotient of Powers properties to multiply and divide numbers written in scientifi c notation.

�


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NAME __________________________________________ DATE ____________ PERIOD _______


Compute with Scientific Notation

Evaluate each expression. Express the result in scientific notation.

1. (3.2 104)(1.4 102) 2. (6.1 105)(8.2 104) 3. (5.2 103)(7.4 105) 4.48 106 5.002 1010 3.848 103 4. (4.8 106)(3.9 108) 5. (9.3 105)(2.7 102) 6. (4.3 102)(5.6 104) 1.872 101 2.511 106 2.408 105

7. 4.55 107

1.3 104 3.5 103 8. 8.84 105

3.4 102 2.6 103 9. 8.05 104

2.3 102 3.5 106

10. 7.56 106

4.2 107 1.8 101 11. 2.016 107

8.4 103 2.4 103 12. 1.175 103

1.25 106 9.4 102

13. (2.7 103) + (3.4 102) 14. (7.2 106) + (1.25 105) 15. (8.4 105) (7.9 103)

3.04 103 7.325 106 8.321 105 16. (9.2 103) (9.6 102) 17. (6.5 1012) + (3.1 1011) 18. (2.6 109) (7.4 107) 8.24 103 6.81 1012 2.526 109

Course 3 • Chapter 2 Equations in One Variable 23

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.

ExampleSolve 4 −

7 x = 16. Check your solution.

4 −

7 x = 16 Write the equation.

(

7 −

4 )

� 4 −

7 x =

(

7 −

4 )

� 16 Multiply each side by the multiplicative inverse of 4 −

7 , 7 −

4 .

7 −

4 � 4 −

7 x = 7 −

4 � 16 −−

1 Write 16 as 16 −−

1 . Divide out common factors.

x = 28 Simplify.

Check 4 −

7 x = 16 Write the original equation.

4 −

7 (28) � 16 Replace x with 28.

4

−

7

(

28 −−

1 )

� 16 Write 28 as 28 −−

1 . Divide out common factors.

16 = 16 � This sentence is true.

Solve each equation. Check your solution.

1. 1 −

6 x = 4 2. 5 −

6 n = 15 3. 2 −

3 d = 14 −−

15

4. 3 −

4 w =

21 −−

30 5. 3 −

5 t = 12 6. 1 −

8 a = 1 −

3

7. -

1 −

6 x = -5 8. 9 −

4 r = -

27 −−

32 9. -

2 −

5 m = 4

To solve an equation when the coefficient is a rational number, multiply each side by the multiplicative inverse of the fraction.

Lesson 1 ReteachSolve Equations with Rational Coefficients

1

4

1 1 1

1 1 4

023_034_CC_A_RSPC3_C02_662332.indd Page 23 01/06/11 11:55 AM s-077023_034_CC_A_RSPC3_C02_662332.indd Page 23 01/06/11 11:55 AM s-077 /Volumes/110/GO00864/NATIONAL/ANCILLARYÉÉ/Volumes/110/GO00864/NATIONAL/ANCILLARYÉÉ

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Course 3 • Chapter 2 Equations in One Variable

NAME __________________________________________ DATE ____________ PERIOD _______


Solve Equations with Rational Coefficients


1. 0.5m = 3.5 7 2. 1.8 = 0.6x 3 3. 0.4y = 2 5 4. 1.86 = 6.2z 0.3 5. 1.67t = 10.02 6 6. 0.9x = 4.5 5

7. 113a = 2

32 or 1

12 8. 8

9x = 24 27 9. 38r = 36 96

10. 34t =

12

23

12 11. 16 =

14h 64 12.

18m = 12 96

13. 58n = 45 72 14. 10 =

110b 100 15. 1

7x = 7 49

16. 5 = 15y 25 17.

43m = 28 21 18.

23z = 20 30

19. 19c = 81 729 20.

49f = 16 36 21.

158 x = 225 120


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.Lesson 2 ReteachSolve Two-Step Equations

A two-step equation contains two operations. To solve a two-step equation, undo each operation in reverse order.

Example 1Solve 2a + 6 = 14. Check your solution.

2a + 6 = 14 Write the equation.

-6 = -6 Subtraction Property of Equality

2a = 8 Simplify.

2a −−

2 = 8 −

2 Division Property of Equality

a = 4 Simplify.

Check 2a + 6 = 14 Write the equation.

2(4) + 6 � 14 Replace a with 4 to see if the sentence is true.

14 = 14 � The sentence is true.

The solution is 4.

Sometimes it is necessary to combine like terms before solving an equation.

Example 2Solve 5 = 8x - 2x - 7. Check your solution.

5 = 6x - 7 Write the equation.

5 + 7 = 6x - 7 + 7 Addition Property of Equality

12 = 6x Simplify.

12 −−

6 = 6x

−−


2 = x Simplify.

The solution is 2. Check this solution.

ExercisesSolve each equation. Check your solution.

1. 2d + 7 = 9 2. 11 = 3z + 5 3. 2s - 4 = 6

4. -12 = 5r + 8 5. -6p - 3 = 9 6. -14 = 4x - 2

7. 2c + 2 = 10 8. 3 + 9n = 21 9. 21 = 5 - r

10. 8 - 5b = -7 11. -10 = 6 - 4m 12. -3t + 4 = 19

13. 2 + a −

6 = 5 14. -

1 −

3 q - 7 = -3 15. 4 - v −

5 = 0


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NAME __________________________________________ DATE ____________ PERIOD _______


Two-Step Equations


1. 2x + 4 = 14 5 2. 5p 10 = 0 2 3. 5 + 6a = 41 6

4. x3 7 = 2 27 5. 18 = 6q 24 7 6. 18 = 4m 6 6

7. 3r 3 = 9 4 8. 2x + 3 = 5 1 9. 0 = 4x 28 7 10. 2x + 6 = 10 8 11. 3z + 5 = 14 3 12. 3x 15 = 12 9 13. 9a 8 = 73 9 14. 2x 3 = 7 5 15. a + 1 = 15 14 16. 2y + 10 = 22 6 17. 15 = 2y 5 10 18. 3c 4 = 2 2 19. 6 + 2p = 16 5 20. 8 = 2 + 3x 2 21. 4b + 24 = 24 0 22. 5x 6 = 19 5 23. 2x 6 = 14 10 24. 3x 9 = 18 3


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.Lesson 3 ReteachWrite Two-Step Equations

Some verbal sentences translate into two-step equations.

Example 1Translate each sentence into an equation.

Sentence Equation

Four more than three times a number is 19. 3n + 4 = 19

Five is seven less than twice a number. 5 = 2n - 7

Seven more than the quotient of a number and 3 is 10. 7 + n −−

3 = 10

After a sentence has been translated into a two-step equation, you can solve the equation.

Example 2Translate the sentence into an equation. Then find the number.Thirteen more than five times a number is 28.

Words Thirteen more than five times a number is 28.

Variable Let n = the number.

Equation 5n + 13 = 28 Write the equation.

-13 = -13 Subtraction Property of Equality

5n = 15 Simplify.

5n −−

5 = 15 −−


n = 3

Therefore, the number is 3.

Exercises Define a variable. Then translate each sentence into an equation. Then find each number.

1. Five more than twice a number is 7.

2. Fourteen more than three times a number is 2.

3. Seven less than twice a number is 5.

4. Two more than four times a number is -10.

5. Eight less than three times a number is -14.

6. Three more than the quotient of a number and 2 is 7.


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NAME __________________________________________ DATE ____________ PERIOD _______


Write Two-Step Equations

Translate each sentence into an equation. Then find each number.

1. Seven more than three times a number is 16. 3x + 7 = 16; 3

2. Seven more than the quotient of a number and 2 is 6. x2 + 7 = 6; 2

3. Six more than twice a number is 20. 2x + 6 = 20; 7 4. Two less than five times a number is equal to 8. 5x 2 = 8; 2 5. Twice a number plus 5 is 3. 2x + 5 = 3; 4 6. The product of a number and 3 plus 1 is 19. 3x + 1 = 19; 6 7. The product of a number and 4 plus 2 is 14. 4x + 2 = 14; 3

8. Eight less than the quotient of a number and 3 is 5. x3 8 = 5; 39

9. The difference of twice a number and 3 is 11. 2x 3 = 11; 7 10. The sum of 3 times a number and 7 is 25. 3x + 7 = 25; 6


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.Lesson 4 ReteachSolve Equations with Variables on Each Side

Some equations, like 3x – 9 = 6x, have variables on each side of the equals sign. Use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation.

Example 1

Solve 3x - 9 = 6x. Check your solution.

3x - 9 = 6x Write the equation.

3x - 3x - 9 = 6x - 3x Subtraction Property of Equality

-9 = 3x Simplify by combining like terms.

-3 = x Mentally divide each side by 3.

To check your solution, replace x with -3 in the original equation.

Check 3x - 9 = 6x Write the equation.

3(-3) - 9 � 6(-3) Replace x with -3.

-18 = -18 � The sentence is true.

The solution is -3.

Example 2

Solve 4a - 7 = 5 - 2a.

4a - 7 = 5 - 2a Write the equation.

4a + 2a - 7 = 5 - 2a + 2a Addition Property of Equality

6a - 7 = 5 Simplify by combining like terms.

6a - 7 + 7 = 5 + 7 Addition Property of Equality

6a = 12 Simplify.

a = 2 Mentally divide each side by 6.

The solution is 2. Check this solution.

ExercisesSolve each equation. Check your solution.

1. 6s - 10 = s 2. 8r = 4r - 16 3. 25 - 3u = 2u

4. 14t - 8 = 6t 5. k + 20 = 9k - 4 6. 11m + 13 = m + 23

7. -4b - 5 = 3b + 9 8. 6y - 1 = 27 - y 9. 1.6h - 72 = 4h - 30

10. 8.5 - 3z = -8z 11. 10x + 8 = 5x - 3 12. 16 - 7d = -3d + 2


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Equations with Variables on Each Side


1. 6x + 10 = 1x 2 2. 2a 5 = 3a 1 3. 7a 5 = 2a 1

4. 4a + 7 = 10 + a 1 5. 8x + 3 = 2x 12 6. 6x 3 = 18 + x 3

7. 3a 1 = 2a 1 8. 8a 2 = 12 + a 2 9. 3x + 6 = x 3 10. 2x + 7 = 11 2x 1 11. 8x + 10 = 3x 2 12. 7a + 4 = 3a 1 13. 7x + 8 = 11x 2 14. 21x + 11 = 10x 1 15. 5x + 5 = 14 + 2x 3 16. 7b 4 = 2b + 16 4 17. 2y 3 = 5 2y 2 18. 3m = 2m + 7 7 19. 9t + 1 = 4t 9 2 20. 2a + 3 = a 12 5 21. 3x = 9x 12 2 22. 2c + 3 = 3c 4 7 23. s 3 = 5 s 4 24. 3w 5 = 5w 7 1

©Curriculum Associates, LLC Copying is not permitted.L13: Solve Linear Equations with Rational Coefficients116

Solve Linear Equations with Rational Coefficients

Part 1: IntroductionLesson 13

You’ve learned how to solve linear equations with variables on one side of the equation. In this lesson, you’ll learn how to solve linear equations with variables on both sides of the equation. Take a look at this problem.

The square and equilateral triangle shown have

x 1 3 3x 2 1

the same perimeter. What is the value of x?

Explore It

Use the math you already know to solve this problem.

The perimeter of the square is 4(x 1 3), and the perimeter of the triangle is 3(3x 2 1). Because the perimeters are the same, 4(x 1 3) 5 3(3x 2 1). Try substituting 3 for x in the equation. Do you get a true statement? Explain.

Use the distributive property to transform 4(x 1 3) 5 3(3x 2 1) into a simpler form without parentheses.

Substitute 3 for x in your new equation. Do you get a true statement? Explain.

You can transform 4x 1 12 5 9x 2 3 again into a simpler form by subtracting 4x from both sides to get 12 5 5x 2 3. If you substitute 3 for x in 12 5 5x 2 3, do you still get a true statement?

You can continue to transform 12 5 5x 2 3 into a simpler form by adding 3 to both sides to get 15 5 5x. When x 5 3, do you get a true statement?

Finally, divide both sides of 15 5 5x by 5. What is the result?


MAFS8.EE.3.7b

©Curriculum Associates, LLC Copying is not permitted.117L13: Solve Linear Equations with Rational Coefficients


Find Out More

In Explore It, you transformed the original equation into simpler and simpler forms. Each time, substituting 3 for x resulted in a true statement. The transformations don’t change the solution to the equation.

You transform equations to get to the form x 5 a because that gives you the solution to the equation you started with. You can add and subtract variables from both sides of the equation just as you do with constants.

Look at a step-by-step solution of 4(x 1 3) 5 3(3x 2 1).

4(x 1 3) 5 3(3x 2 1) 4x 1 12 5 9x 2 3

12 5 5x 2 3

15 5 5x

3 5 x

There is often more than one way to solve an equation.

4(x 1 3) 5 3(3x 2 1) 4x 1 12 5 9x 2 3

25x 1 12 5 23

25x 5 215

x 5 3

Reflect

1 How do you solve multi-step equations that have variables on both sides?

Apply the distributive property.Subtract 4x from both sides so that the variable occurs on just one side.

Add 3 to both sides.


Apply the distributive property.Subtract 9x from both sides so that the variable occurs on just one side.

Subtract 12 from both sides.


2 4x 2 4x

1 3 1 3

5 5

2 9x 2 9x

2 12 2 12

25 25

Lesson 13


L13: Solve Linear Equations with Rational Coefficients118


Read the problem below. Then explore different ways to solve an equation.

On a math quiz, Elise and Kaitlyn solved the equation 1 ··

2 (5x 2 6) 5 3x in different

ways, but each student arrived at the correct answer. Describe the steps that each

student took to solve the equation. An explanation of the first step for each method

has been provided.

Solve It

Elise solved the problem in this way.

1 ·· 2 (5x 2 6) 5 3x

5 ·· 2 x 2 3 5 3x

2 5 ·· 2 x 2 5 ·· 2 x

23 5 1 ·· 2 x

1 ·· 2 1 ·· 2

26 5 x

Solve It

Kaitlyn solved the problem in this way.

1 ·· 2 (5x 2 6) 5 3x

2• 1 ·· 2 (5x 2 6) 5 2• 3x

5x 2 6 5 6 x

26 5 x

Step 1 Apply the distributive property.

Step 2 Subtract 5 ·· 2 x from each side.

Step 3 Divide both sides by 1 ·· 2 .

Step 1 Multiply both sides by 2.

Step 2 Subtract 5x from both sides. 25x 25x

Lesson 13



Connect It

Now you will analyze how each student solved the equation.

2 Look at Elise’s solution method. She took three steps to solve the equation. Describe

Step 2.

Why do you think Elise took that step?

3 Describe Step 3 in Elise’s solution.

Could Elise have used a different step? Explain.

4 Look at Kaitlyn’s solution method. Describe Step 2 in her solution.

Why do you think she took that step?

5 Which method do you prefer? Explain your thinking.

6 Explain how to check the solution to an equation. Then show how to check the solution to the equation on the previous page.

Try It

Use what you learned about different ways to solve linear equations. Show your work.

7 Solve the equation and check your solution: 10 5 1 ·· 3 (x 2 15).

Student Model

Lesson 13




Solve the following equation for r.

16.5 1 1.5r 5 12 1 2r

Look at how you can show your work.

Solution:

8 One fifth of a number plus three times the number is equal to twice the number plus 42. What is the number?

Show your work.

Solution:

How can you check your answer?

Pair/Share

What equation can you write to solve the problem?

Can you solve this equation in another way?

Pair/Share

The student multiplied each side of the equation by 10 to transform the equation into one without decimals.


9 5 r

16.5 1 1.5r 5 12 1 2r

10(16.5 1 1.5r) 5 10(12 1 2r) 165 1 15r 5 120 1 20r 2 15r 2 15r

165 5 120 1 5r 2120 2120

45 5 5r 5 5

Lesson 13


How would you help Haley understand her error?

Pair/Share

Justify the steps you took to solve the equation.

Pair/Share

What properties and operations can you use to simplify both sides of an equation?

Can you multiply both sides of the equation by a number to get a simpler equation without fractions?

9 Show two different ways to solve 1 ·· 4 x 2 5 5 3 ·· 4 x 2 12.

Show your work.

Solution:

10 Which equation has the same solution as 1 ·· 2 (6 2 x) 1 3x 5 1 ·· 2 x 2 8? Circle the letter of the correct answer.

A 3 1 2x 5 1 ·· 2 x 2 8

B 6 2 x 1 3x 5 x 2 16

C 3 1 5 ·· 2 x 5 1 ·· 2 x 2 8

D 6 2 x 1 3x 5 x 2 8

Haley chose A as the correct answer. How did she get that answer?


Lesson 13



Solve the problems.

1 Find the solution of 5 1 3(y 2 4) 5 5(y 1 2) 2 y.

A 10 1 ·· 2 C 21

B 7 1 ·· 3 D 217

2 Which equation does NOT have the same solution as 1 ·· 3 (9 2 2x) 5 x 1 1?

A 2 5 1 2 ·· 3 x C 9 2 2x 5 3x 1 3

B 3 2 2x 5 x 1 1 D 25x 5 26

3 Three students solved the equation 2(3x 2 8) 5 32 in different ways, but each student arrived at the correct answer. Select all of the solutions that show a correct method for solving the equation.

A 1 } 2 • 2(3x 2 8) 5 32 • 1 }

2

3x 2 8 5 16

3x 5 24

x 5 8

B 2(3x 2 8) 5 32

5x 2 8 5 32

5x 5 40

x 5 8

C 2(3x 2 8) 5 32

6x }} 6 2 16 }}

6 5 32 }}

6

x 5 48 }} 6

x 5 8

4 Consider the equation 2(4x 2 5) 5 ax 1 b. From the list of digits provided, determine the values of a and b that make the solution to this equation x 5 3.

0 1 2 3 4

a 5 b 5


Lesson 13


5 Solve the equation x 1 0.7 5 1 2 0.2x in two different ways. Then check your answer.

Show your work.

6 A square and an equilateral triangle have the same perimeter. Each side of the triangle is 4 inches longer than each side of the square. What is the perimeter of the square?

Show your work.

Answer The perimeter of the square is _______ inches.



©Curriculum Associates, LLC Copying is not permitted.L14: Solutions of Linear Equations124

Solutions of Linear EquationsLesson 14 Part 1: Introduction

You’ve learned how to solve linear equations and how to check your solution. In this lesson, you’ll learn that not every linear equation has just one solution. Take a look at this problem.

Jason and his friend Amy are arguing. Jason says that a linear equation always has just one solution. Amy says that some linear equations have more than one solution. Who’s right? Amy asked Jason to explore solutions to the following equation.

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

Explore It

Use the math you already know to solve this problem.

Remember that a solution to an equation is a number that makes the equation true. To check to see if a number is a solution to this equation, replace x with its value.

Is 6 a solution to the equation? Show your work.



Can an equation have more than one solution? Explain. Who is right—Jason or Amy?


MAFS8.EE.3.7a

©Curriculum Associates, LLC Copying is not permitted.125L14: Solutions of Linear Equations


Find Out More

Look at how you could solve Amy’s equation.

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

First, simplify each side:

Use the distributive property. 2x 1 1 1 x 5 3x 2 6 1 7

Combine like terms. 3x 1 1 5 3x 1 1

Once both sides of an equation are in simplest form, you can say a lot about the solution without actually solving the equation. You can just look at the structure.

Think about how you might solve this equation with pictures. Look at the pan balance below. The left pan represents 3x 1 1. So does the right pan.

x x x x x x

If you take away 3x from both sides, you end up with 1 5 1, a true statement. If you take away 1 from both sides, you end up with 3x 5 3x, a true statement. You can replace x with any number and you will always get a true statement. The pan will remain balanced. This equation has infinitely many solutions.

You’ve seen that a linear equation can have one solution or, in a case like this, infinitely many solutions. You will also see that a linear equation can have no solution.

Reflect

1 Once both sides of an equation are in simplest form, how can you tell if it has infinitely many solutions?

Part 2: Modeled Instruction Lesson 14


L14: Solutions of Linear Equations126

Read the problem below. Then explore how to identify when an equation has one solution, infinitely many solutions, or no solution.

Yari and her friends, Alyssa and David, were each given an equation to solve.

Yari: 2(x 1 10) 2 17 5 5 1 2x 2 2

Alyssa: 5x 1 3 2 3x 5 2(x 1 3) 2 5

David: 2(x 2 3) 1 9 5 5 1 x 2 1

Whose equation has one solution? Infinitely many solutions? No solution?

Model It

You can use properties of operations to simplify each side of Yari’s equation.

2(x 1 10) 2 17 5 5 1 2x 2 2

2x 1 20 2 17 5 3 1 2x

2x 1 3 5 3 1 2x

Model It

You can use properties of operations to simplify each side of Alyssa’s equation.

5x 1 3 2 3x 5 2(x 1 3) 2 5

2x 1 3 5 2x 1 6 2 5

2x 1 3 5 2x 1 1

Model It

You can use properties of operations to simplify each side of David’s equation.

2(x 2 3) 1 9 5 5 1 x 2 1

2x 2 6 1 9 5 4 1 x

2x 1 3 5 4 1 x

x x x x

x x x x

x x x

The variable terms are the same on both sides of the equation but the constants are different. There is no value for x that will make the equation true.

The variable terms are different. There is only one value for x that will make the equation true.

The variable terms and the constants are the same on both sides of the equation. No matter what value you choose for x, the equation will always be true.



Connect It

Now you will use the models to solve this problem.

2 Look at Model It for Yari’s equation. What do you notice about both sides of the equation? What equation do you get if you subtract 2x from both sides of the equation?

3 Look at Model It for Alyssa’s equation. How is it different than Yari’s equation? How is it similar?

4 Look at the pan balance for Alyssa’s equation. Is there any way to balance the pan? Explain. What equation do you get if you subtract 2x from both sides of the equation?

5 Look at the Model It for David’s equation. Are the variable terms on each side of the equation the same or different? Solve David’s equation.

6 Explain how you know when an equation has one solution, no solution, or infinitely many solutions.

Try It

Use what you just learned about equations with one solution, no solution, or infinitely many solutions. Show your work on a separate sheet of paper.

Replace c and d in the equation cx 1 d 5 8x 1 12 with the given values. Explain why the equation has one solution, no solution, or infinitely many solutions.

7 c 5 6 and d 5 34 ___________________ c 5 8 and d 5 6 ____________________8




Student Model


Michelle looks at the equation 26x 2 30 5 6x 2 30 and says there is no solution. Is she correct? Explain.

Look at how you can show your work.

Solution:

9 Draw lines to match each linear equation to its correct number of solutions.

Show your work.

5(4 2 x) 5 25x 1 20 no solution

25(4 2 x) 5 25x 1 20 infinitely many solutions

5(5 2 x) 5 25x 1 20 one solution

How could you convince Michelle that there is a solution to this equation?

Pair/Share

When the variable terms on both sides of an equation are the same, what does that tell you about the solution(s) to the equation?

Pair/Share

The student solved the equation to find that 0 is a solution.

How can you tell when a linear equation has no solution?

26x 2 30 5 6x 2 30

26x 2 30 2 6x 5 6x 2 30 2 6x

212x 2 30 5 230

212x 2 30 1 30 5 230 1 30

212x 5 0

212x ····· 212 5 0 ···· 212

x 5 0

No; There is one solution to this equation, x 5 0.



10 Write a number in the box so that the equation will have the type of solution(s) shown.

no solution

1 ·· 3 x 1 5 5 1 ·· 3 x 1

infinitely many solutions

1 ·· 3 x 1 5 5 1 ·· 3 x 1

one solution

1 ·· 3 x 1 5 5 x 1 5

11 What is the solution to the equation 3(x 2 4) 5 2(x 2 6)?

A x 5 0

B x 5 1

C There are infinitely many solutions.

D There is no solution.

Brian chose D as the correct answer. How did he get that answer?

How could you explain the correct response to Brian?

Pair/Share

What do you notice about the solution to equations that have the same variable term on each side of the equation?

Pair/Share

How can you simplify this equation to justify the correct answer?

What is the difference between an equation with no solution and an equation with infinitely many solutions?

Lesson 14



Solve the problems.

1 How many solutions does the equation 2(2x 2 10) 2 8 5 22(14 2 3x) have?

A one solution

B two solutions

C no solution

D infinitely many solutions

2 How many solutions does the equation 10 2 3x 1 10x 2 7 5 5x 2 5 1 2x 1 8 have?

A one solution

B two solutions

C no solution

D infinitely many solutions

3 For each linear equation in the table, shade in the appropriate box to indicate whether the equation has no solution, one solution, or infinitely many solutions.

Equation No Solution One SolutionInfinitely Many

Solutions

8x 1 16 5 8x 2 16

23x 2 17 5 2(17 1 3x)

9x 1 27 5 27

2x 2 6 5 6 1 2x

4 Which equation has an infinite number of solutions? Select all that apply.

A 3x 2 2(x 1 10) 5 x 2 20 D 5 } 2 x 2 2 5 9 }

2 x 2 2(x 1 1)

B 5x 1 2(x 2 3) 5 5x 1 2(3 2 x) E 7 } 2 x 1 x 5 x 1 7 }

4

C x } 2 1 1 5 3x }}

10 1 3


Lesson 14


5 Consider the equation 2(5x 2 4) 5 ax 1 b.

Part A

Find a value for a and a value for b so that the equation has one solution. Explain your reasoning.

Show your work.

a 5

b 5

Part B

Find a value for a and a value for b so that the equation has no solution. Explain your reasoning.

Show your work.

a 5

b 5

Part C

Find a value for a and a value for b so that the equation has infinitely many solutions. Explain your reasoning.

Show your work.

a 5

b 5




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.Lesson 5 ReteachSolve Multi-Step Equations

Example 1

Solve 2(4a - 5) = 30.

2(4a - 5) = 30 Write the equation.

8a - 10 = 30 Distributive Property

8a - 10 + 10 = 30 + 10 Addition Property of Equality

8a = 40 Simplify.

8a −−

8 = 40 −−


a = 5 Simplify.

Example 2

BOOKS Roland has

p + 11 p + 11 p + 11 p + 11

hardcover hardcover hardcoverhardcover

p p p

paperback paperbackpaperback 3 paperback books and 4 hardcover books. Each hardcover book is worth $11 more than each paperback book. If the value of all of his books is $79, what is the cost of one paperback book?

Write an equation to represent the bar model.

3p + 4(p + 11) = 79 Write the equation.

3p + 4p + 44 = 79 Distributive Property

7p + 44 = 79 Simplify.

7p + 44 + (-44) = 79 + (-44) Addition Property of Equality

7p = 35 Simplify.

7p

−−

7 = 35 −−


p = 5 Simplify.

So, the cost of one paperback book is $5.

Exercises


1. 2(3b - 1) = 40 2. 49 = -7(t + 1) 3. 5(1 - n) = 75

4. 4(x - 2) = 3(x - 3) 5. -5(p + 2) = 2(2p - 15) + p 6. 4z - 6 = 6(z + 2) + 8


PDF Pass


NAME __________________________________________ DATE ____________ PERIOD _______


Multi-Step Equations


1. 6(m 2) = 12 4 2. 4(x 3) = 4 4 3. 5(2d + 4) = 35 1.5 4. w + 6 = 2(w 6) 18 5. 3(b + 1) = 4b 1 4 6. 7w 6 = 3(w + 6) 6

7. 4(k 6) = 6(k + 2) 18 8. 3(x 0.8) = 4x + 4 6.4 9. 59(g + 18) =

16g + 3 18

10. 4(c + 12) = 2c + 18 15 11. 7(d 2) = 5(d + 2) 12 12. 5p 17 = 2(2p 7) 3

13. 4(3z 2) = 9z 7 13 14. 7s + 2 = 4(s + 1)

23 15. 6(k + 1) = 2k + 7

14

16. 6(n 1) = 2(n + 1) 24 17. 14y 3 = 5 2y 24 18.

23(3q + 6) = 8 2

8 Grade Intensive Math - iMater Charter Middle/High School · 2014-08-25 · Lesson 1 Curriculum...

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Transcript of 8 Grade Intensive Math - iMater Charter Middle/High School · 2014-08-25 · Lesson 1 Curriculum...