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Chapter 1 Real Numbers
Before You Read
Before you read the chapter, think about what you know about real numbers. List three things
you already know about them in the first column. Then list three things you would like to learn
about them in the second column.
K
What I know…
W
What I want to find out…
Foldable
Construct the Foldable on page FL3 in your workbook, tape it down in the space provided.
1
Essential Question
Why is it helpful to write numbers in different ways?
Brainstorm:
Key points
Scan the pages in the chapter and write at least one specific fact concerning each lesson.
Lesson
1-1 Rational Numbers
Fact
1-2 Powers and Exponents
1-3 Multiply and Divide Monomials
1-4 Powers of Monomials
1-5 Negative Exponents
1-6 Scientific Notation
1-7 Compute with Scientific Notation
1-8 Roots
1-9 Roots of Non-Perfect Squares
1-10
Compare Real Numbers
2
1-1 Rational Numbers
Lesson Objective
_____________________________________________________________________________________
_____________________________________________________________________________________
Vocabulary
Match each term with its definition.
Rational Number
The decimal form of a rational number.
Repeating Decimal
zero.
A repeating decimal where the repeating digit is
Terminating Decimal
Numbers that can be written as the ratio of two
integers in which the denominator is not zero.
Key Concept
A rational number is a number that can be written as the _____________
of two _______________ in which the ___________________________ is not zero.
Use a dictionary to define the underlined words.
1.)
2.)
3.)
3
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Write each fraction or mixed number as a decimal.
1.
9
16
Work:
=
29
5
2. −1 40=
3. 4 6=
Work:
Work:
Write each decimal as a fraction or mixed number in simplest form.
4. 0.32=
5. −0. 7̅=
Work:
Work:
4
6. Give an example of a repeating decimal where two digits repeat. Explain why your number is
a rational number.
______________________________________________________________________________
______________________________________________________________________________
7. Explain why any rational number is either a terminating or repeating decimal.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
̅̅̅̅, and 0.157 and 0. ̅̅̅̅̅
8. Compare 0.1 and 0. 1̅ , 0.13 and 0. 13
157when written as fractions. Make
a conjecture about expressing repeating decimals like these as fractions.
Define the underlined word using a dictionary
Conjecture:
______________________________________________________________________________
______________________________________________________________________________
Write each decimal as a fraction and list them from least to greatest in the box below.
Make a conjecture about expressing repeating decimals like these as fractions.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
5
1-2 Powers and Exponents
Lesson Objective
_____________________________________________________________________________________
_____________________________________________________________________________________
Vocabulary
Match each term with its definition.
Power
In a power, the number that is the common factor.
Base
A product of repeated factors using an exponent
and a base.
Exponent
In a power, the number of times the base is used as
a factor
Key Concept
A _________________ of repeated ________________can be expressed as a power, that is,
using an exponent and a base.
Use a dictionary to define the underlined words.
1.)
2.)
Read and Write Powers
Power
Words
Factors
31
32
33
34
3𝑛
6
Practice
Write each expression using exponents.
1. (−11)(−11)(−11)
2. 2 ∗ 2 ∗ 2 ∗ 3 ∗ 3 ∗ 3
3. 𝑟 ∗ 𝑠 ∗ 𝑟 ∗ 𝑟 ∗ 𝑠 ∗ 𝑠 ∗ 𝑟 ∗ 𝑟
When working out an example, write what you are doing to solve the problem next to each
step.
Evaluate each expression.
1 3
4. 26
5. (−4)4
6. (7)
Work:
Work:
Work:
7. Write an expression with an exponent that has a value between 0 and 1.
______________________________________________________________________________
8. Describe the following pattern
34 = 81, 33 = 27, 32 = 9, 31 = 3. Then use a similar pattern to predict the value of 2−1 .
7
1-3 Multiply and Divide Monomials
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Vocabulary
Define the vocabulary term.
Monomial______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Key Concept
Product of Powers
To multiply powers with the same base, add their exponents.
32 ∗ 34 =
Quotient of Powers
To divide powers with the same base, subtract their exponents.
57
54
=
Use a dictionary to define all of the underlined words.
1.)
2.)
8
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Simplify using the Laws of Exponents.
1. 45 ∗ 43 =
Work:
4.
24𝑘 9
=
6𝑘 6
Work:
𝑦8
2. -2a(3𝑎4 )=
3. 45 =
Work:
Work:
5.
22 ∗33 ∗45
2∗3∗44
=
Work:
6.
(−3)4 ∗(−4)3 ∗52
(−3)2 ∗(−4)∗5
=
Work:
7. Write a multiplication expression with a product of 513 .
______________________________________________________________________________
8. Is
3100
399
greater than, less than, or equal to 3? Explain your reasoning to a classmate.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
9. What is twice 230 ? Write using exponents. Explain your reasoning.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
9
1-4 Powers of Monomials
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Key Concept
Power of a Power
To find the power of a power, multiply the exponents.
(64 )5 =
Power of a Product
To find the power of a product, find the power of each factor and multiply.
(3𝑎2 )5 =
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Simplify using the Laws of Exponents.
1. (32 )5 =
2. (ℎ6 )4 =
3. [(23 )2 ]3 =
Work:
Work:
Work:
4. (7𝑤 7 )3 =
5. (5𝑔8 𝑘12 )4 =
6. (−6𝑟 5 𝑠 9 )2 =
Work:
Work:
Work:
10
7. The table gives the area and volume of a square and cube, respectively, with side lengths
shown.
a.) Complete the table.
b.) Describe how the area and volume are each affected if the side length is doubled. Then
describe how they are each affected if the side length is tripled.
Side Length
(units)
Area of Square
(𝑢𝑛𝑖𝑡𝑠 2 )
Volume of a cube
(𝑢𝑛𝑖𝑡𝑠 3 )
x
2x
3x
𝑥2
(𝑥 3 )
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
8. (7𝑥 )3 = 715
x=________
9. (−2𝑚3 𝑛4 )𝑥 = −8𝑚9 𝑛12
x=________
11
1-5 Negative Exponents
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Key Concept
Zero and Negative Exponents
Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is
the multiplicative inverse of its nth power.
Multiply and Dived with Negative Exponents
The Product of Powers and the Quotient of Powers rules can be used to multiply and divide
powers with negative exponents.
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Write each expression using a positive exponent.
1. 2−4
2. 4−3
3. 𝑎−4
4. 𝑔−7
Work:
Work:
Work:
Work:
Write each fraction as an expression using a negative exponent other than -1.
1
5. 34
Work:
1
1
6. 𝑚5
Work:
1
7. 16
8. 49
Work:
Work:
Simplify.
𝑝−2
9. 3−3 ∗ 3−2=
10. 𝑟 −7 ∗ 𝑟 3=
11. 𝑝−12 =
Work:
Work:
Work:
12
12. Without evaluating, order 11−3 , 112 , 𝑎𝑛𝑑 110 from least to greatest. Explain your
reasoning.
Least ___________________________________ Greatest
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
1
13. Write and expression with a negative exponent that has a value between 0 and 2.
______________________________________
14. Select several fractions between 0 and 1. Find the value of each fraction after it is raised to
the -1 power. Explain the relationship between the -1 power and the original fraction.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
13
1-6 Scientific Notation
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Vocabulary
Define the vocabulary term.
Scientific Notation______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Key Concept
Scientific Notation is when a number is written as the product of a _____________________
and an ________________ power of 10. The factor must be greater than or equal to 1 and less
than 10.
Use a dictionary to define the underlined words.
1.)
2.)
𝑎 ∗ 10𝑛 , 𝑤ℎ𝑒𝑟𝑒 1 ≤ 𝑎 < 10 𝑎𝑛𝑑 𝑛 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Write each number in standard form.
1. 9.931 * 105 =
2. 6.02 * 10−4=
Work:
Work:
14
Write each number in scientific notation.
3. 8,785,000,000 =
4. 0.524 =
Work:
Work:
5. Determine whether 1.2 * 105 or 1.2 * 106 is closer to one million. Explain.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
6. Compute and express each value in scientific notation.
(130,000)(0.0057)
0.0004
(90,000)(0.0016)
(200,000)(30,000)(0.00012)
15
1-7 Compute with Scientific Notation
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Key Concept
Multiplication and Division with Scientific Notation
You can use the Product of Powers and Quotient of Powers properties to multiply and divide
numbers written in scientific notation.
Addition and Subtraction with Scientific Notation
When adding or subtracting decimals in standard form, it is necessary to line up the place
values. In scientific notation, the place value is represented by the exponent. Before adding or
subtracting, both numbers must be expressed in the same form.
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
8.37∗108
1. (2.6 ∗ 105 )(1.9 ∗ 102 ) =
2.
Work:
Work:
2.7∗103
=
3. (8.9 ∗ 109 ) + (4.2 ∗ 106 ) =
4. (9.64 ∗ 108 ) − (5.29 ∗ 106 ) =
Work:
Work:
16
5. Enrique is finding
6.63∗10−6
5.1∗10−2
=. Circle his mistake and correct it.
Enrique’s Work
6.63 ∗ 10−6 6.63 10−6
=
∗
5.1 ∗ 10−2
5.1 10−2
= 1.3 ∗ 10−6−2
= 1.3 ∗ 10−8
Your Work
17
1-8 Roots
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Vocabulary
Match each term with its definition.
Square Root
One of three equal factors of a number.
Perfect Square
A rational number whose cube root is a
whole number.
Radical Sign
A symbol used to indicate a positive
square root.
Cube Root
One of two equal factors of a number.
Perfect Cube
A rational number whose square root is
a whole number.
Key Concept
Square Root
A square root of a number is one of its two equal factors.
Cube Root
A cube root of a number is one of its three equal factors.
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Find each square root.
49
1. -√1.69=
2. ±√144=
3. √-1.44
Work:
Work:
Work:
18
Solve each equation. Check your solution(s).
1
4. 𝑝2 = 36
5. 𝑡 2 = 9
6. 6.25=𝑟 2
Work:
Work:
Work:
8.
9.
Find each cube root.
7.
216
Work:
-125
Work:
-8
Work:
Find each value.
25
10. (√36)²
11. (√81)²
12. (√x)²
Work:
Work:
Work:
13.Explain why the square root of 64 has a positive and a negative value.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
19
1-9 Estimate Roots
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Key Concept
You know that the √8 is not a whole number because 8 is not a perfect square.
The number line below shows that √8 is between 2 and 3. Since 8 is closer to 9 than 4, the best
whole number estimate for √8 is 3.
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Estimate to the nearest integer.
1. √28≈
2. √135≈
3. √38.7≈
Work:
Work:
Work:
4.
5.
6.
51≈
Work:
200≈
Work:
95≈
Work:
20
7. Find two numbers that have square roots between 7 and 8. One number should have a
square root closer to 7 and the other number should have a square root closer to 8. Justify your
answer.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
8. Jasmine is estimating √200. Find her mistake and correct it.
Jasmine’s Work
√200≈100
9. If 𝑥 4 = y, then x is the fourth root of y. Explain how to estimate the fourth root of 30. Find the
fourth root of 30 to the nearest whole number.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
21
1-10 Compare Real Numbers
Lesson Objective
______________________________________________________________________________
______________________________________________________________________________
Vocabulary
Define the vocabulary terms.
Irrational Number______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Real Number______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Key Concept
Real numbers
Rational Number:
A rational Number is a number that can be
𝑎
expressed as the ration 𝑏, where a and b are
integers and b ≠ 0.
Compare and order Real Numbers
Irrational Number:
An irrational number is a number that cannot
𝑎
be expressed as the ratio 𝑏, where a and b
are integers and b ≠ 0.
You can compare and order real numbers by writing them in the same notation. Write the
numbers in decimal notation before comparing or ordering them.
Practice
When working out an example, write what you are doing to solve the problem next to each
step.
Name all sets of numbers to which each real number belongs.
1. 0.050505…
2. -√64
3. √17
22
Fill in the blank with <,>,or = to make a true statement.
4. √15 ___ 3.5
5. √2.25 ___ 150%
6. √6.2 ___ 2. 4̅
Work:
Work:
Work:
7. Order the set {√5, 220%, 2.25, 2. 2̅} from least to greatest. Verify your answer by graphing on
a number line.
8. Give a counterexample for the statement All square roots are irrational numbers. Explain
your reasoning.
counterexample
______________________________________________________________________________
______________________________________________________________________________
23
9. Tell whether the following statements are always, sometimes, or never true. If a statement is
not always true, explain.
a.) Integers are rational numbers.
______________________________________________________________________________
b.) Rational numbers are integers.
______________________________________________________________________________
c.) The product of a non-zero rational number and an irrational number is irrational.
______________________________________________________________________________
24
Chapter 1 Real Numbers
Before the Test
Review the ideas you listed in the table at the beginning of the chapter. Cross out any incorrect
information in the first column. Then complete the table by filling in the third column.
K
What I know…
W
What I want to find out…
L
What I learned…
Essential Question
Why is it helpful to write numbers in different ways?
25
Chapter 1 Extended Response Questions
Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to
include all relevant drawings and justify your answers. You may show your solutions in more
than one way or investigate beyond the requirements of the problem. If necessary, record your
answer on another piece of paper.
26
Chapter 1 Power Up
27
Chapter 1 Performance Task
28
29
Are You Ready for the Chapter 1 Test?
Use this checklist to help you study.
o
o
o
o
o
o
o
o
I used my Foldable to review
I completed the Chapter 1 Review
I completed the Chapter 1 Extended Response Questions
I completed the Chapter 1 Power Up
I completed the Chapter 1 Performance Task
I used the online resources for additional review options
I reviewed my homework assignments and made corrections to incorrect problems
I reviewed all vocabulary from the chapter and their definitions
30