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Chapter 1:

Foundations for Algebra

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Unit 1: Vocabulary

1) Natural Numbers

2) Whole Numbers

3) Integers

4) Rational Numbers

5) Irrational Numbers

6) Real Numbers

7) Terminating Decimal

8) Repeating Decimal

9) Commutative Property

10) Associative Property

11) Identity

12) Inverse Property

13) Distributive Property

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Day 1: Classification of Real Numbers

N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the

product of a non-zero rational number and an irrational number irrational.

Warm-Up

Rewrite each fraction as a decimal.

2

1

4

1

3

1

9

1

How are the decimals of the first two fractions different from the decimals of the second two fractions?

All the numbers that we know – positives, negatives, radicals, decimals, and fractions – are called real

numbers and they can be placed in different groups.

Natural numbers (ℕ), or counting numbers, are the numbers we count with: 1, 2, 3, …

Whole numbers (𝑊) are the natural numbers plus zero: 0, 1, 2, 3, …

Integers (ℤ) are whole numbers and their opposites: …-2, -1, 0, 1, 2….

Rational numbers can be expressed in the form of a fraction (or ratio).

Rational numbers include terminating and repeating decimals.

Terminating decimals end, such as: 0.57, 2.3, or 0.05

Repeating decimals repeat the same digit or set of digits, like 0.333…. or 0.123123123….

Irrational numbers cannot be expressed in the form of a fraction.

In decimal form, these decimals are nonterminating and nonrepeating.

Note: Since all integers can be written as fractions, integers are also rational. Also, since the square roots of perfect squares

are integers, they are also rational.

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The following Venn diagram can help you classify any real number:

Exercise

1) Classify the following numbers. Remember that a number may belong to more than one category.

a) 4

b) 0 c) 5

d) - 5

e) 4

f) - √16

5

g)

3

4

h) 0.23 i) √3

j) π k)

4444.0

l)

2

5

Quick Check for Understanding

Classify the following numbers. Remember that a number may belong to more than one category.

a) 5

2

b) 121 c) 0.85

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Putting it All Together

Let’s recall some of the number sets we did today.

Closing Activity

1) Explain the difference between a rational number and an irrational number.

2) Give an example of a number that is an integer but not a whole number. Explain your choice.

3) Tell all of the sets to which the number ½ belongs.

Homework Chapter 1, Day 1

Vocabulary – Using the definitions in your packet, write and define these terms on page 2.

1) Natural numbers 5) Irrational Numbers

2) Whole numbers 6) Real Numbers

3) Integers 7) Terminating decimal

4) Rational Numbers 8) Repeating decimal

7

Homework continued

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Homework Continued

17)

18)

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Day 2: Properties of Rationals and Irrationals

N-RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational.

Warm-Up

Give an example of each type of number.

a) A rational number that is a fraction __________________

b) An irrational number that is a fraction ________________

c) A rational number that contains a radical sign ______________

d) An irrational number that contains a radical sign _____________

e) A rational number that is written in decimal form ______________

f) An irrational number that is written in decimal form _____________

Exploration 1: Rationals + Rationals

Expression Sum Rational or Irrational?

1

2+

5

6

√25 + √9

0.8888̅ + 0.252525 …

Conclusion 1: The sum of two rational numbers must be _____________________________.

Exploration 2: Rationals × Rationals

Expression Product Rational or Irrational?

3

4∙

1

5

3√49 ∙ √81

0. 5̅ × 0. 1̅

Conclusion 2: The product of two rational numbers must be___________________________________

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Quick Check for Understanding

The sum of 2.236 and 7.555…. must be

(A) rational

(B) irrational

(C) an integer

(D) a whole number

Exploration 3: Rationals + Irrationals

Expression Sum Rational or Irrational?

√2 + 9

√3

2+ √25

7 + 𝜋

Conclusion 3: The sum of a rational and an irrational number must be ____________________.

Why do you think that is? ___________________________________________________________

Exploration 4: Rationals × Irrationals

Expression Product Rational or Irrational?

6√5

10 ∙√3

2

0.1234767 … × 0. 4̅

Conclusion 4: The product of a rational and an irrational number must be _______________________

Why do you think that is? ___________________________________________________________

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Quick Check for Understanding

1)

2)

Summary

Summary

The sum of two rational numbers is a rational number.

The product of two rational numbers is a rational number.

The sum of a rational and an irrational number is irrational.

The product of a rational number and an irrational number is irrational.

Homework Chapter 1, Day 2

p. 35 #7-10

p. 36 #31-42

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Day 3: Evaluation of Expressions

6. EE. 2. Write, read, and evaluate expressions in which letters stand for numbers.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic

operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

Warm-Up

To evaluate an expression means to substitute (plug in) numbers for its variables and determine a numerical

answer.

Rule #1: The number you substitute must always go in parentheses.

Evaluate each expression when x = 5, y = - 4, z = -2.

1) 𝑥 + 𝑦

2) 𝑧 − 𝑦 3) −𝑧 − 𝑦𝑥

4) 𝑦2

5) −𝑥2 6) 𝑧2 − 5𝑧

7) 𝑥3

8) 4 + 𝑦3 9) −𝑧3 − 2𝑦

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Quick Check for Understanding

Evaluate 3𝑎2 − 𝑎3 when 𝑎 = −2.

Rule #2: Remember to close the parentheses at the appropriate time.

Evaluate each expression when 𝑎 = −3, 𝑏 = −4, 𝑐 = 2. Round all values to the nearest hundredth, if

needed.

1) |𝑎 + 3|

2) |𝑎| + 3 3) −|𝑏|

4) √𝑐

5) √𝑐 − 𝑎 6) √𝑐 − 1

7) 3

𝑐+5

8) 𝑏 −1

2 9)

𝑎

𝑏−𝑐

Quick Check for Understanding

Evaluate √𝑐 − 𝑏 when 𝑐 = 4 and 𝑏 = −12.

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Evaluation Skills and Problem Solving

Evaluate each expression for the value of the variable.

Expression Value(s) Calculations/Answer

1) 15(𝑥 − 40) + 400 42

2) −5𝑥2 − 12𝑥 + 4 -3

3) 𝑥2 + 46 - 5

4) (𝑥 −5

2)

2

2

5) 25𝑡+1

4

6) −1

225𝑥2 +

2

3𝑥

75

7) √3𝑉

2𝜋

10

8) 2𝑥 + 1

3

9) 1

3𝑥 + 9

36

10) 7

3(𝑥 +

9

28)

19.25

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11) Is −3𝑥 + 7 − 5𝑥 < 15 true when x = -2? Explain.

12) Evaluate each expression for the value of the variable. Tell which expression has the greater value.

Write (1) or (2).

(1) (2)

𝑥 = 3𝑥 2𝑥 + 5 Which is greater?

-1

2

-3

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HW Chapter 1 Day 3

Page 43 # 33-41 #76, 77

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Day 4: Identify and use Properties of Real Numbers

6.EE. 3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce

the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Warm up

1)

2)

Directions: Classify each number as: real, rational, irrational, whole, natural, and integer.

3) 4

3 _____________________________________________

4) –56 _____________________________________________

5) 5 _____________________________________________

6) 0.935935935 _____________________________________________

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Simplifying Expressions Using Properties

1) Examples of Properties of Real Numbers

Commutative Property

Commutative means that the order does not make

any difference.

Associative Property

Associative means that the grouping does not make

any difference.

Distributive Property

The process of distributing the number on the

outside of the parentheses to each term on the inside.

Additive Identity

Multiplicative Identity

Inverse Properties

Additive Inverse (Opposite)

Multiplicative Inverse (Reciprocal)

2) How might the commutative property help you with the following problem?

6 + 3 + 18 + 1 + 8 + 4 + 2 + 7 = ___________

3) How might the associative property help you with the following problem?

(6 + 3) + 7 = ______________

4) Why do you think 0 is called the identity element for addition?

5) Why do you think 1 is called the identity element for multiplication?

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6) Model Problems Identify the property illustrated by each statement.

a) (x + y) + z = x + (y + z)

b) (x + y) + z = z + (x + y)

c) 2(x + y) = 2x + 2y

d) x + 0 = x e) xx 1

f) x + (-x) = 0 g) 11

xx

Quick Check for Understanding

7) Identify the property illustrated by each statement.

a) (48)3 = 3(48)

b) (5 + 9) + 13 = 5 + (9 + 13)

c) 5(x + 9) = 5x + 45

d) 5

41

5

4

e) 011 f) 3 + 0 = 3

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8) Rewrite each expression using the property given.

2 + 3 = ___________________________________ (commutative)

4 + (x + y) = _______________________________ (associative)

a(x + 2) = _________________________________ (distributive)

Summary

Exit Ticket

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Homework Chapter 1 Day 4

Word Bank: Commutative, Distributive, Associative, Additive /Multiplicative Inverse,

Additive/Multiplicative Identity

1) Rewrite using the commutative property:

3 + 7 = _____________ a + (b + c) = _______________

2) Rewrite using the associative property:

7*(4*6) = _______________ (5 + 1) + 4 = ___________________

3) Rewrite using the distributive property:

2(3 + 5) = _______________________ x(d + m) = ________________________

4) A method for solving 5(x – 2) - 2(x- 5) = 9 is shown below.

Identify the property used to obtain each of the two indicated steps.

Identify which property is illustrated for each example.

5) x (yz) = x (zy) ____________________ 6) x(yz) = (xy)z

____________________

7) 2(x + y) = 2x + 2y ____________________ 8) x + (–x) = (–x) + x ____________________

9) 1(x) = x

____________________ 10) (x + y) + z =

x + (y + z)

____________________

11) x + 0 = x ____________________ 12) 1(x) = (x)1 ____________________

13) (x + y) + z = (y + x) + z

____________________ 14) x(y – z) = xy – xz ____________________

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Homework Continued

15)

16)

17)

18) [Use your calculator to answer this question]

19) [Use your calculator to answer this question]

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Day 5 Unit 1: Review

Classifying Numbers

1)

Place a check mark for each set that the number is part of.

Whole Number Integer Rational

Number

Irrational

Number

Real Number

- 7

¾

0

5

0.398

3.0

5.1823159……..

5

2

5

35

55

27

3

2) True or False? If false, explain why.

a) All integers are rational.

b) If a number is rational, then it must be a whole number.

c) Some irrational numbers are integers.

d) All irrational numbers are real numbers.

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3) Is the set of rational numbers closed under

addition?

Write an example to support your answer.

4) Is the set of rational numbers closed under

multiplication?

Write an example to support your answer.

5) Is the set of irrational numbers closed under

addition?

Write an example to support your answer.

6) Is the set of irrational numbers closed under

multiplication?

Write an example to support your answer.

Expressions with Exponents and Absolute Value

Simplify each expression.

7) -82

8) (-7)2 9) –(-11)

2 10) −|– (6)2|

11) -5(-4)2

12) 2(-3)3 13) –(-4)

3 14) 8 − |– (−3)2|

15) |2 − 4| + |6 − 3|

16) |4 + 5| − |1| 17) |−2| + |3| − |−4| 18) |10 − 6| + |6 − 10|

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Evaluating Polynomial Expressions

19) Evaluate x2 – 6x + 3 when x = -3.

20) Find the value of –x2 – 3x when x = 2.

21) Evaluate w – 3x2 when w = -6 and x = 3.

22) What is the value of the expression 3a2 + 4b

2

when a = -3 and b = 4?

23) Name the property illustrated by each statement.

a. 5 · 1 = 5 b. (3 + 5) + 4 = 3 + (5 + 4) c. abc = 1abc

g. 7 + (-7) = 0 h. 9 + 0 = 9 i. x + 9 = 9 + x

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24) The statement 2 + 0 = 2 is an example of the use of

which property of real numbers?

(1) associative (3) additive inverse

(2) additive identity (4) distributive

25) Which property is illustrated by the equation

ax + ay = a(x + y)?

(1) associative (3) distributive

(2) commutative (4) identity

26) Which property is illustrated by the equation

6 + (4 + x) = 6 + (x + 4)

(1) associative property of addition

(2) associative property of multiplication

(3) distributive property

(4) commutative property of addition

27) Which property of real numbers is illustrated by the

equation 52 + (27 + 36) = (52 + 27) + 36?

(1) commutative property

(2) distributive property

(3) associative property

(4) identity property of addition

28) What is the additive inverse of the expression

a - b?

(1) a + b (3) -a + b

(2) a - b (4) -a - b

29) Which equation illustrates the associative property?

(1) a a( )1 (3) a b c ab ac( ) ( ) ( )

(2) a b b a (4) ( ) ( )a b c a b c

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