Download unit 2 - Algebra 1 -

NAME:
ALGEBRA I
UNIT 2:
REAL NUMBERS
(Resource Guide)
UNIT 2 – REVIEWING THE BASICS
ASSIGNMENT/ACTIVITY CHECKLIST
ASSESSMENT
Unit 2 – Pre-Test
SECTION 1 – Number Lines, Opposites, and Absolute Value
Topics: Number Lines, Opposites, and Absolute Value
Review the notes in the resource guide.
Complete the journal questions and KWL charts.
Review PowerPoint presentations.
Review video tutorials.
Review relevant websites and links.
Post initial response and responses to classmates for DQ #1.
Post initial response and responses to classmates for DQ #2.
Complete homework assignments (see wiki for details):
Section 1-8
Section 1-9
Square Root Worksheet
Number Family Worksheet
Word Problem Worksheet
Creating a Mathematical Timeline.
Complete the Unit 2 – Section 1 – Quiz.
SECTION 2 – Basic Properties and Distributing
Topics: Basic Properties and the Distributive Property
Review the notes in the resource guide.
Complete the journal questions and KWL charts.
Review PowerPoint presentations.
Review video tutorials.
Review relevant websites and links.
Post initial response and responses to classmates for DQ #1.
Post initial response and responses to classmates for DQ #2.
Complete homework assignments (see wiki for details):
Properties Worksheet #1
Properties Worksheet #2
Section 2-1
Section 2-5
Word Problem Worksheet
Unit Multipliers: Multiplying by 1
Complete the Unit 2 – Section 2 – Quiz.
SECTION 3 – Working the With Positive and Negative Numbers
Topics: Working with Positive and Negative Numbers
Review the notes in the resource guide.
Complete the journal questions and KWL charts.
Review PowerPoint presentations.
Review video tutorials.
Review relevant websites and links.
Post initial response and responses to classmates for DQ #1.
Post initial response and responses to classmates for DQ #2.
Complete homework assignments (see wiki for details):
Section 2-3
Section 2-4
Section 2-6
Section 2-9
Reviewing +/– Worksheet
Word Problem Worksheet #1
Word Problem Worksheet #2
Tracking Your Investments
Complete the Unit 2 – Section 3 – Quiz.
SECTION 4 – Review
Topics: Review for Unit 2
Review the notes for all sections in the resource guide. Be sure to review any of the PowerPoint
presentations, video tutorials, websites, old homework assignments, and so forth that would be helpful.
Post initial response and responses to classmates for the Wrap-up DQ.
Complete homework assignments (see wiki for details):
Review Worksheet
Problem-Solving Practice
Complete “Everyday Math: When will I ever use this stuff?” activity. (This can be done at any time
throughout the course of the unit.)
GROUP ACTIVITY
Unit 2 – Group Activity (Creating a Review Game)
ASSESSMENT
Unit 2 – Assessment – Written Test
Unit 2 – Assessment – Notes Quiz
Unit 2 – Assessment – Individual Assessment
Unit 2 – Assessment – Problem-Solving Activity
Unit 2 – Post-Test
UNIT 2 – SECTION 1
TEXTBOOK: Algebra: Structure and Method – Book 1 (McDougal Littell)
SECTIONS: 1-8; 1-9; (part of) 5-5; and (part of) 5-6
VOCABULARY
The following terms (among others) will appear in this section. You will
be expected to know what they mean:
 Number Line
 Inequality
 Opposite
 Absolute Value
 Square Root
 Number Family
K-W-L
K
W
KNOW: What do
you know about
the terms listed
above?
WOULD: What
would you like to
know after this
lesson?
GOALS AND EXPECTATIONS
In this section, you will be expected to:
 Identify the key characteristics of a number line.
 Utilize a number line to compare values.
 Use the inequality symbols to compare values.
 Define the opposite of number and determine the opposite of a given
value.
 Define the absolute value of a number and determine the absolute
value of a given value.
 Define the “square root” of a number. Calculate the square roots of
both perfect squares and non-perfect squares.
 Identify the key characteristics of numbers that are part of each
number family.
NOTES
NUMBER LINE – a visual representation of numbers that is organized along a line with smaller
numbers to the left and larger numbers to the right
 ORIGIN – the zero point on a number line
 POSITIVE SIDE – the part of the number line to the right of zero (these are numbers that are
larger than zero)
 A positive number is indicated with a plus sign (or no sign at all).
 NEGATIVE SIDE – the part of the number line to the left of zero (these are numbers that are
smaller than zero)
 A negative number is indicated with a minus sign.
 GRAPH – a visual representation of a number on a number line
 COORDINATE – a point (or the value) of a number on a number line
 The sketch below is an example of typical number line and the graph of a point:
coordinate
negative
–8

–7 –6 –5 –4 –3 –2 –1
origin
0
positive
1
2
3
4
5
6
7
8
INEQUALITY – a statement using inequality symbols that indicates how two unequal numbers
compare
 The inequality symbols include:
1) Not equal 
2) Less than 
3) Greater than 
4) Less than or equal to 
5) Greater than or equal to 
 Examples of inequalities include: 4  9; x  6; 2 + 8  8 – 2; etc.
 HINTS to remember which inequality is which:
1) The arrow always points to the smaller
number.
2) The “alligator” eats the bigger number.
OPPOSITE – the opposite of a number is the same
number with the other sign (positive or negative)
 Examples of opposites include:
1) –4 and 4
2) 19 and –19
3) 3.2 and –3.2
 The opposite of zero is…zero.
 Another word for opposite is the ADDITIVE INVERSE.
 The word opposite is synonymous with “negative” and “minus.”
 Simplify the following expression: –(–(–(–4)))
 This problem could be read as: the opposite of the opposite of the opposite of the opposite
of four…which is positive four.
 Another way to think about this problem is that two negatives will cancel out to a positive.
 An EVEN number of negative signs will result in a positive answer.
 An ODD number of negative signs will result in a negative answer.
ABSOLUTE VALUE – the distance that a number is from zero on a number line
 Absolute value is always a positive number.
 The absolute value of zero is zero.
 Absolute value is denoted by the symbol:
 The absolute value is a type of grouping symbol and should be done during the first step of
order of operations.
 Examples include:
1)  5  5
2) 7  7
3)   4  4
 Note in the last problem that the negative sign do NOT cancel out. Absolute value is done
before any operation outside of the lines.
SQUARE ROOT – the square root of a number (or
expression) is the number (or expression) that can be
multiplied by itself to get the given number (or
expression)
 Find the square root of the following numbers:
a) 64 = 8 …because 8  8 = 64
b) 144 = 12 …because 12  12 = 144
c) 10 = a weird decimal and is not a
PERFECT SQUARE
 PERFECT SQUARE – a number (or expression) that has a nice square root; in other words
the square root is not a weird decimal
 Do numbers that do not have perfect squares have square roots?
 The answer is yes. The catch is that square root is a weird decimal. In addition, that
decimal number is a non-terminating, non-repeating decimal. Therefore, you generally
round your answer off to an appropriate value. Here are some examples:
a) 10 = 3.16227766… …because 3.16227766…  3.16227766… = 10 The answer
would probably be rounded to about () 3.162
b) 35 = 5.916079783… …because 5.916079783…  5.916079783… = 35 The
answer would probably be rounded to about () 5.916
 Notes about square roots:
 You cannot take the square root of a negative number.
 Relatively few numbers are perfect squares.
 Numbers that have decimal square roots are okay, however the decimal is not the exact
square root.
NUMBER FAMILIES – groups of numbers that have similar characteristics
 COUNTING (natural) NUMBERS – nice, round numbers that start at one and go as high as you
can count
 There are no weird decimals or fractions in this group.
 This group includes: {1, 2, 3, 4, 5, 6…}
 WHOLE NUMBERS – nice, round numbers that start at zero and go as high as you can count
 There are no weird decimals or fractions in this group.
 This group includes: {0, 1, 2, 3, 4, 5, 6…}
 INTEGERS – nice, round numbers that include positive numbers, negative numbers, and zero
 There are no weird decimals or fractions in this group.
 This group includes: {…–4, –3, –2, –1, 0, 1, 2, 3, 4…}
 RATIONAL NUMBERS – numbers that can be written
as fractions
1
 Examples of numbers in this group include: ,
4
7
0.567, , 3, etc.
2
 IRRATIONAL NUMBERS – numbers that cannot be
written as fractions
 Common examples of this group are weird roots:
2 , 3 17 , , e, i, etc.
 REAL NUMBERS – any number that you can think of
that can be found on a number line
 The following diagram how the numbers families are related. If a family’s “box” is inside of
another box, then those numbers are a part of that group as well:
RATIONAL
INTEGERS
IRRATIONAL
 REALS (all numbers)
WHOLE
COUNTING
JOURNALING
Describe what the meaning of absolute value is.
Describe what it means for a decimal to be “non-terminating” and “non-repeating.” How do these
ideas fit with the definitions of rational and irrational numbers?
What do you think is the purpose of identifying different number families? Are there advantages to
having different groups of numbers? Is it possible for a number to be in more than one number
family? Explain.
If a “real” number is any number that can be graphed on a number line, then what is the purpose of
having this group in the first place?
K-W-L
L
LEARN: What did
you learn in this
lesson?
UNIT 2 – SECTION 2
TEXTBOOK: Algebra: Structure and Method – Book 1 (McDougal Littell)
SECTIONS: 2-1; 2-5; (part of) 2-6; and 2-8
VOCABULARY
The following terms (among others) will appear in this section. You will
be expected to know what they mean:
 Commutative and Associative Properties
 Equality Properties
 Identity
 Inverse
 Reciprocal
 Terms and Like Terms
 Distributive Property
K-W-L
K
W
KNOW: What do
you know about
the terms listed
above?
WOULD: What
would you like to
know after this
lesson?
GOALS AND EXPECTATIONS
In this section, you will be expected to:
 Define, identify examples of, and clearly verbal each of the basic
properties associated with real numbers and algebraic expressions.
 Review the concept of “opposite.”





Define the term “reciprocal” and determine the reciprocal of number.
Differentiate between the meanings of “terms” and “factors.”
Identify the key characteristics of “like terms.”
Simplify basic expressions involving like terms.
Utilize the distributive property to simplify expressions with
parentheses.
NOTES
COMMUTATIVE PROPERTY – this property says that it is okay to
switch the order of numbers when you are adding or multiplying
 Examples of the commutative property include:
1) 5 + 6 = 6 + 5
2) 8  3 = 3  8
3) x + 4 = 4 + x
4) 2(x) = (x)2
 The commutative property does NOT work for subtraction or
division.
ASSOCIATIVE PROPERTY – this property says that is
okay to regroup numbers (with parentheses) when you
are adding or multiplying
 Examples of the associative property include:
1) 5 + (6 + 2) = (5 + 6) + 2
2) 8(3  4) = (8  3)4
3) x + (y + z) = (x + y) + z
4) a(bc) = (ab)c
 The associative property does NOT work for
subtraction or division.
CLOSURE PROPERTY – this property says that when adding, subtracting, multiplying, or dividing,
you will get one unique answer
 Examples of the closure property include:
1) 3 + 8 ONLY equals 11
2) 2  5 ONLY equals 10
3) 20 – 11 ONLY equals 9
4) 80  16 ONLY equals 5
EQUALITY PROPERTIES – these properties illustrate the characteristics of an
equal sign
 The equality properties are:
 Reflexive Property
 Symmetric Property
 Transitive Property
REFLEXIVE – this equality property says that a number (or expression) is equal to itself
 Examples of this property include:
1) 5 = 5
2) 6x = 6x
3) x + 2 = x + 2
SYMMETRY – this equality property says that it is okay to switch
expressions around an equal sign
 Examples of this property include:
1) x = 5 and 5 = x
2) x + 2 = 6 and 6 = x + 2
3) 5x = x – 8 and x – 8 = 5x
TRANSITIVE – this equality property says that two things equal to the same things are equal to each
other
 Examples of this property include:
1) If a = b and b = c, then a = c
2) If 2x + 3 = y and y = 6, then 2x + 3 = 6
SUBSTITUTION – replacing an expression with another expression or number that is equivalent
(a.k.a. has the same value)
IDENTITY PROPERTIES – these properties identify certain
numbers and operations that can be performed without
changing the value of a number
 The two basic identity properties are:
 Additive Identity
 Multiplicative Identity
ADDITIVE IDENTITY – this is the number that can be added to (or subtracted from) any other
number and will not change the value of that number
0


ZERO
The additive identity is:
Examples of this property include:
1) 5 + 0 = 5
2) 0 + 6x = 6x
3) 2yz – 0 = 2yz
MULTIPLICATIVE IDENTITY – this is the number that can be multiplied to (or divided into) any other
number and will not change the value of that number
1


ONE
The multiplicative identity is:
Examples of this property include:
1) 5 × 1 = 5
2) 6x × 1 = 6x
3) 9 ÷ 1 = 9
INVERSE PROPERTIES – these properties identify certain numbers and operations that will cancel
out other numbers and operations
 The two basic inverse properties are:
 Additive Inverse
 Multiplicative Inverse
ADDITIVE INVERSE – this is the number that can be added to another number and will cause it to
cancel out to zero
 The additive inverse is same thing as the OPPOSITE of a number.
 Examples of this property include:
1) 5 + (–5) = 0
2) (–12) + 12 = 0
3) 4x + (–4x) = 0
MULTIPLICATIVE INVERSE – this is the number that can be multiplied to another number and will
cause it to cancel out to one
 The multiplicative identity is same thing as the RECIPROCAL of a number.
 RECIPROCAL – taking a number (written in fraction form) and flipping the numerator and
denominator
 Examples of reciprocals include:
1
a) The reciprocal of 2 is .
2
3 4
b) The reciprocal of is .
4 3
7
1
3
c) The reciprocal of 2 or
is .
3
7
3
 Examples of this property include:
3 4 12
 
1
1)
4 3 12
1 7
2) 7    1
7 7
3 4 7 4 28
1
3) 1    
4 7 4 7 28
MULTIPLICATIVE PROPERTY OF ZERO – this property states that any number that is multiplied by
zero will equal zero
 Examples of this property include:
1) 5 × 0 = 0
2) –4xyz × 0 = 0
3) 2789023456789011123453465794457810095456745787925 × 0 = 0
MULTIPLICATIVE PROPERTY OF –1 – this property states that any number that is multiplied by
negative one will equal the opposite of the number
 Examples of this property include:
1) 5 × –1 = –5
2) –7 × –1 = 7
3) x × –1 = –x
TERM – an individual piece of an addition or subtraction problem
 In the expression 2x + 8, 2x and 8 are the terms.
 In the expression x – 7y, x and –7y are the terms.
LIKE TERMS – these are parts in an expression (involving addition and subtraction) that can be
combined together when simplifying
 In order to be like terms, the pieces have to meet the following criteria:
1) the terms have to have the exact same variable(s)
2) the variables have to have the exact same exponents
 Simplify the following expressions by COMBINING LIKE TERMS:
a) 2x + 3 + 4x = 6x + 3
b) 2x + 7y + x – 5y = 3x + 2y
c) 4x + 8 – 2 – x = 3x + 6
 Notes about combining like terms:
 If an expression has “unlike terms,” they cannot be combined together.
 The sign in front of the term must stay attached to that term.
 Numbers (constants) are considered to be like terms.
FACTOR – the “pieces” (number or variables) that are being multiplied together in a multiplication
problem
 In the expression 2x, 2 and x are the factors.
 In the expression (4x)(2y), 4x and 2y are the factors.
 In the expression 5(2a), 5 and 2a are the factors.
 In order to simplify an expression involving multiplication:
1) Multiply the numbers together.
2) Combine the same variable together using exponents (will be discussed later).
3) Keep different variables separate.
 Examples of simplifying include:
a) 2(5x) = 10x
b) 4x(2y) = 8xy
c) (2x)(y)(3z) = 6xyz
DISTRIBUTIVE PROPERTY – an alternative way of simplifying a problem that involves multiplication
and addition
 Solve the following problem using ORDER OF OPERATIONS:
5 (4 + 2) 
5(4 + 2) = 5(6) = 30

Solve the following problem using the DISTRIBUTIVE PROPERTY:
5 (4 + 2) 
5(4 + 2) = 5(4) + 5(2) = 20 + 10 = 30

Notes about the distributive property:
1) The distributive property is especially useful when working with problems that have
variables.
2) In a problem with all numbers, the distributive property may not be the most efficient
method of simplifying.
3) Watch the signs on the numbers when distributing.
 Simplify the following problems:
a) 5(1 + 3) = 5(1) + 5(3) = 5 + 15 = 20
b) 4(2x + 3) = 8x + 12
c) –(x + 4) = –x – 4
d) –2(3 – 4x) = –6 + 8x
e) 2(2x + y – 5) = 4x + 2y – 10
JOURNALING
Although the properties discussed in this section seem to be somewhat obvious, why do you think
that it is important that these properties are identified?
Many times, people tend to confuse the “commutative property” and the “symmetric property of
equality”. What are some similarities between these properties? More importantly, what is different
about these properties?
Many times, people tend to confuse the “substitution property” and the “transitive property of
equality”. What are some similarities between these properties? More importantly, what is different
about these properties?
The distributive property is an alternative way for simplifying expression that involves a set a
parentheses being multiplied by another factor. If simplifying an expression using the distributive
property gets the same result as following the normal steps in order of operations, what is the
purpose of using this approach? Provide an example to illustrate your answer.
K-W-L
L
LEARN: What did
you learn in this
lesson?
UNIT 2 – SECTION 3
TEXTBOOK: Algebra: Structure and Method – Book 1 (McDougal Littell)
SECTIONS: 2-2; 2-3;2-4; 2-6; and 2-9
VOCABULARY
The following terms (among others) will appear in this section. You will
be expected to know what they mean:
 Rules for Adding Positive and Negative Numbers
 “Add the Opposite”
 Rules for Multiplying and Dividing Positive and Negative Numbers
 Range
 Function Notation and Arrow Notation
 Graphs (Broken-line graph, Bar graph, Scatterplot, etc.)
 Tables and Charts
 Consecutive Integers
K-W-L
K
W
KNOW: What do
you know about
the terms listed
above?
WOULD: What
would you like to
know after this
lesson?
GOALS AND EXPECTATIONS
In this section, you will be expected to:
 Add and subtract positive and negative numbers.
 Multiply and divide positive and negative numbers.
NOTES
ADDITIVE IDENTITY – the number that can be added to any other number without changing its value
 The additive identity is: ZERO.
ADDITIVE INVERSE – a number and its opposite
 The number and its opposite are called additive inverses because the sum of them is zero.
 In other words, a number and its opposite cancel each other out.
OPPOSITE OF A SUM – the opposite of a sum of real numbers is equal to the sum of the opposites
of the numbers
 Examples of this property include:
a) –(4 + 3) = –4 + (–3) = –7
b) –(a + b) = –a + (–b)
ADDING POSITIVE AND NEGATIVE NUMBERS
 OPTION #1 – Graphing on a Number Line
 On a number line, a positive sign means go to the right and a negative sign means go the
left.

Solve the following problems:
a) –5 + (–3)
–3
–5
–8
–7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
b) –5 + 9
+9
–5
–8 –7 –6 –5 –4 –3 –2 –1
c) 8 + (–9)
–9
+8
–8 –7 –6 –5 –4 –3 –2
–1
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
d) –5 + 1
+1
–5
–8 –7 –6 –5

–4 –3
–2 –1
OPTION #2 – Rules for Addition
 If two numbers have the SAME SIGN:
1) Ignore the signs and add the numbers together.
2) Keep the same sign as the original numbers.


If two numbers have DIFFERENT SIGNS:
1) Ignore the signs and subtract the numbers.
2) Use the sign of the “larger” number (a.k.a. the
number with the great absolute value).


Examples include:
a) –4 + (–5)  4 + 5 = 9  ANSWER = –9
b) 4 + 5  4 + 5 = 9  ANSWER = 9
Examples include:
a) –6 + 9  9 – 6 = 3  ANSWER = 3
b) –10 + 6  10 – 6 = 4  ANSWER = –4
c) 14 + (–19)  19 – 14 = 5  ANSWER = –5
These rules apply to pairs of numbers. If you have more
than two numbers:
a) you can work from left to right.
b) you can combine the positives together, the negative
together, and then solve.
SAME SIGN
ADD
and
KEEP SAME
SIGN
DIFFERENT
SIGN
SUBTRACT
and
USE
“BIGGER”
SIGN
SUBTRACTING POSITIVE AND NEGATIVE NUMBERS
 The rule for subtracting numbers is:
ADD THE OPPOSITE


After you add the opposite, follow the rules for addition.
Examples include:
1) 9 – 15 = 9 + (–15)  …  ANSWER = –6
2) –8 – (–12) = –8 + 12  …  ANSWER = 4

Remember that opposite, minus, negative, and so forth all mean the same thing. Therefore,
you can think of a subtraction problem (i.e. 9 – 15) as adding two numbers with different signs.
MULTIPLICATIVE IDENTITY – the number that can be multiplied to any other number without
changing its value
 The multiplicative identity is: ONE.
MULTIPLICATIVE PROPERTY OF ZERO – any number times zero is zero
MULTIPLICATIVE PROPERTY OF –1 – any number times –1 is the opposite of the number
MULTIPLYING POSITIVE AND NEGATIVE NUMBERS
 The rules for multiplying numbers are:
1) If two numbers have the SAME sign, the answer is POSITIVE.
2) If two numbers have DIFFERENT signs, the answer is NEGATIVE.



Examples include:
a) (–4)(–2) = 8
b) (–4)(2) = –8
c) (4)(–2) = –8
d) (4)(2) = 8
NOTE: These rules apply to pairs of numbers.
If there are more than two numbers in a problem, the rules are:
1) If there is an EVEN number of negative signs, the answer is POSITIVE.
2) If there is an ODD number of negative signs, the answer is NEGATIVE.

Examples include:
a) (–2)(–3)(–1)(–4) = 24
b) (–2)(–3)(–1) = –6
EVEN NUMBER OF NEGATIVES  POSITIVE
ODD NUMBER OF NEGATIVES  NEGATIVE
DIVIDING POSITIVE AND NEGATIVE NUMBERS
 The rules for dividing numbers are the same as multiplying:
1) If two numbers have the SAME sign, the answer is POSITIVE.
2) If two numbers have DIFFERENT signs, the answer is NEGATIVE.

Examples include:
a) (–4)  (–2) = 2
b) (–4)  2 = –2
c) 4  (–2) = –2
d) 4  2 = 2


NOTE: These rules apply to pairs of numbers.
If there are more than two numbers in a problem, the rules are:
1) If there is an EVEN number of negative signs, the answer is POSITIVE.
2) If there is an ODD number of negative signs, the answer is NEGATIVE.

Examples include:
12
a)
2
(3)  (2)
 20
b)
1
(4)  (5)
 You CANNOT divide by zero. Dividing by zero is UNDEFINED.
 However, you CAN divide zero by other numbers. The answer in this situation is always zero.
JOURNALING
Provide an explanation about why multiplying (or dividing) a number by –1 will give you the opposite
number.
Explain how the rule “add the opposite” works when you are subtracting numbers.
K-W-L
L
LEARN: What did
you learn in this
lesson?