Download Chapter 1 - White Plains Public Schools

Chapter 1:
Foundations for Algebra
1
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
2
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.
1
2
1
4
1
3
1
9
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 (𝑄) 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….


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.
Irrational numbers cannot be expressed in the form of a fraction.
In decimal form, these decimals are nonterminating and nonrepeating.
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The following Venn diagram can help you classify any real number:
Model Problems
1) Classify the following numbers. Remember that a number may belong to more than one category.
a) 4
d)
3
4
b) 0
c)
e) 0.23
f) - √8 (try on calculator)
-5
4
Independent Practice (7 minutes)
Classify the following numbers. Remember that a number may belong to more than one category.

a)
2
5
d) π
b)
121
e)  0.4444
c) 
5
2
f) 4
Discussion (7 minutes)
Use your keyword signs to participate in the discussion:

I agree.

I disagree.

I need further explanation.

I am confused.

I want to extend that.
5
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.
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Homework Chapter 1, Day 1
Vocabulary – Using the definitions in your packet, write and define these terms on page 2.
1)
2)
3)
4)
Natural numbers
Whole numbers
Integers
Rational Numbers
5) Irrational Numbers
6) Real Numbers
7) Terminating decimal
8) Repeating decimal
7
17)
18)
8
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 5
+
2 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 1
∙
4 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
+ √25
2
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? ___________________________________________________________
10
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) 𝑦 2
5) −𝑥 2
6) 𝑧 2 − 5𝑧
7) 𝑥 3
8) 4 + 𝑦 3
9) −𝑧 3 − 2𝑦
12
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
1
8) 𝑏 − 2
9)
𝑎
𝑏−𝑐
Quick Check for Understanding
Evaluate √𝑐 − 𝑏 when 𝑐 = 4 and 𝑏 = −12.
13
Evaluation Skills and Problem Solving
Evaluate each expression for the value of the variable.
Expression
Value(s)
1) 15(𝑥 − 40) + 400
42
2) −5𝑥 2 − 12𝑥 + 4
-3
3) 𝑥 2 + 46
-5
5 2
4) (𝑥 − 2)
2
5) 25𝑡+1
4
6) −
7) √
1
225
2
𝑥2 + 𝑥
1
3
7
10
3𝑉
2𝜋
3
36
𝑥+9
10) (𝑥 +
3
75
3
8) 2𝑥 + 1
9)
Calculations/Answer
9
)
28
19.25
14
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).
𝑥=
(1)
(2)
3𝑥
2𝑥 + 5
Which is greater?
-1
2
-3
4
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)
3
4
_____________________________________________
4)
–56
_____________________________________________
5)
5
_____________________________________________
6) 0.935935935
_____________________________________________
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Simplifying Expressions Using Properties
1) Examples of Properties of Real Numbers
Commutative Property
Associative Property
Commutative means that the order does not make
any difference.
Associative means that the grouping does not make
any difference.
Distributive Property
Additive Identity
The process of distributing the number on the
outside of the parentheses to each term on the inside.
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?
17
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) x 1  x
f) x + (-x) = 0
g) x   1
1
x
Quick Check for Understanding
7) Identify the property illustrated by each statement.
a) (48)3 = 3(48)
d)
4
1  4
5
5
b) (5 + 9) + 13 = 5 + (9 + 13)
c) 5(x + 9) = 5x + 45
e) 1   1  0
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
19
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
____________________
20
Homework Continued
15)
16)
17)
18) [Use your calculator to answer this question]
19) [Use your calculator to answer this question]
21
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
0.3
5.1823159……..
5
5
2
5 3
5 5
3
27
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?
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.
Write an example to support your answer.
Expressions with Exponents and Absolute Value
Simplify each expression.
7) -82
8)
(-7)2
9)
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|
–(-11)2
10) −|– (6)2 |
23
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 + 4b2
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
24
24) The statement 2 + 0 = 2 is an example of the use of 25) Which property is illustrated by the equation
which property of real numbers?
ax + ay = a(x + y)?
(1) associative
(2) additive identity
(3) additive inverse
(4) distributive
26) Which property is illustrated by the equation
6 + (4 + x) = 6 + (x + 4)
(1)
(2)
(3)
(4)
associative property of addition
associative property of multiplication
distributive property
commutative property of addition
28) What is the additive inverse of the expression
a - b?
(1) a + b
(2) a - b
(3) -a + b
(4) -a - b
(1) associative
(3) distributive
(2) commutative (4) identity
27) Which property of real numbers is illustrated by the
equation 52 + (27 + 36) = (52 + 27) + 36?
(1)
(2)
(3)
(4)
commutative property
distributive property
associative property
identity property of addition
29) Which equation illustrates the associative property?
(1) a(1)  a
(3) a(b  c)  (ab)  (ac)
(2) a  b  b  a (4) (a  b)  c  a  (b  c)
25
Day 6: Review
1) Use the list of numbers to answer the questions:
7
1
, 16 ,
,  42 , 3.666........ , 4
4
8
a) Identify any integer(s).
30 ,
b) Identify any rational numbers.
c) Identify any irrational number(s).
2) Are whole numbers, integers, or rational numbers the most reasonable for each situation?
a) Your shoe size
b) The number of siblings you have
c) The number of quarts of paint you need to buy to paint a room
d) The number of quarts of paint you use when you paint a room
e) The score in a football game.
f) The amount of snowfall on a given day.
3)
26
Evaluating Expressions
Evaluate each expression for the value of the variable.
Expression
Value(s)
4) −5𝑥 2 − 12𝑥 + 4
-2
5) 𝑥 2 + 46
-5
5 2
6) (𝑥 − 2)
2
7) 25𝑡+1
5
8) −
9) √
1
225
2
𝑥2 + 𝑥
Calculations/Answer
75
3
10
3𝑉
2𝜋
10) 2𝑥 + 1
3
7
9
3
28
11) (𝑥 +
)
15.25
27
12) Answer the following questions using the numbers form the following list below.
5,
1
, 0,
2
2,  2,
1
,
2
Based on the above information, which of the statements is ALWAYS true, which is SOMETIMES true, and
which is NEVER true?
a) The sum of a rational number and a rational number is rational.
b) The sum of a rational number and an irrational number is irrational.
c) The sum of an irrational number and an irrational number is irrational.
d) The product of a rational number and a rational number is rational.
e) The product of a rational number and an irrational number is irrational.
f) The product of an irrational number and an irrational number is irrational.
Summary:
28
Properties
29
30