Download Pre-Algebra, Unit 1: Variables, Expression, and Integers

Pre-Algebra 8 Notes – Unit One: Variables, Expressions, and Integers
Evaluating Algebraic Expressions
Syllabus Objective: (1.1) The student will evaluate variable and numerical expressions using
the order of operations.
A variable is defined as a letter or symbol that represents a number.
Examples: a, b, c, _, , ,...
A variable (algebraic) expression is an expression that consists of numbers, variables, and
operations.
Examples: a  b, 2 x  7 y, 5  3(m  2)  1
General Strategy
To evaluate algebraic expressions, substitute the assigned number (or value) to each variable and
simplify the resulting arithmetic expression. When expressions have more than one operation,
mathematicians have agreed on a set of rules called the Order of Operations.
Order of Operations (PEMDAS or Please excuse my dear Aunt Sally’s loud radio)*
1.
2.
3.
4.
Do all work inside the Parentheses and/or grouping symbols.
Evaluate Exponents.
Multiply/Divide from left to right.*
Add/Subtract from left to right.*
*Emphasize that it is NOT always multiply-then divide, but rather which ever operation
occurs first (going from left to right). Likewise, it is NOT always add-then subtract, but
which of the two operations occurs first when looking from left to right.
Example 1: Evaluate a  b when a  5 and b  4.
a  b 5  4
9
Example 2: Evaluate 2 x  7 y when x  3 and y  6.
2 x  7 y  2  3  7  6 
 6  42
 48
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 1 of 13
Example 3: Evaluate 5  3( m  2)  1 when m  6.
5  3( m  2)  1  5  3(6  2)  1
 5  3(4)  1
 5  (12)  1
 17  1
 16
Exponential Notation
An exponent is the superscript which tells how many times the base is used as a factor.
exponent
base
=
222
{
2
3
The base 2 is used
as a factor 3 times.
factors
An exponent tells you how many times to write the base as a factor.
23 is read “2 to the third power” or “2 cubed”. 2 is the base and 3 is the exponent. Students
must be able to convert numbers into exponential form and vice versa.
Examples:
(a) Write 81 in exponential form.
(b) Evaluate 53.
81  9 9
53  5  5  5
3 3 3 3
 125
3
4
Let us look at a pattern that will allow you to determine the values of exponential expressions
with exponents of 1 or 0.
24  2 2 2 2  16
23  2 2 2  8
22  2 2  4
2
2
34  3 3 3 3  81
4 4  4 4 4 4  256
33  3 3 3  27
4 3  4 4 4  64
32  3 3  9
4 2  4 4  16
21  ? 21  2
31  ? 31  3
41  ? 41  4
20  ? 20  1
30  ? 30  1
40  ? 40  1
Any number to the power of 1 is equal to the number. That is, n1 = n.
Any number to the power of 0 is equal to one. That is, n0 = 1.
Also point out to students that if there is no exponent, the exponent is always 1.
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 2 of 13
Integers
Syllabus Objective: (1.2) The student will compare integers. (1.3) The student will order
integers.
Integers are defined as whole numbers and their opposites.
Integers = . . .  3,  2,  1, 0, 1, 2, 3, . . .
The negative numbers lie to the left of 0 on the number line
and the positive numbers lie to the right of 0. As you move
from left to right on the number line, numbers are always
increasing in value. Likewise, as you move from right to
left, the numbers are decreasing in value.
6 5 4 3 2 1
0
1
2
3
4
5
6
Examples: Compare the following. Use > or <.
(b) −13 __ 2
(a) 24 __ 5
24 is to the right of 5
(c) −8 __ −12
−8 is to the right of −12
2 is to the right of -13
−13 < 2
24 > 5
−8 > −12
(d) List the following integers in order from least to greatest.
7, −12, −5, 0, 3, 15, −20, 2, −2
Answer: −20, −12, −5, −2, 0, 2, 3, 7, 15
The absolute value of a number is the distance from zero on a number line without regard to the
sign. The symbol for absolute value is
and is read as “the absolute value of ”.
(a) 7  7
Examples:
(b) 5  5
(c) 0  0
Let’s take a look at the graph of x  1 . What numbers are one unit from zero without regard to
sign?
When we work with signed numbers, we are often working with two different signs that look
exactly alike. They are signs of value and signs of operations. A sign of value tells you if the
number you are working with is greater than zero (positive) or less than zero (negative). Signs of
operations tell you to add, subtract, multiply, or divide.
(3)

sign of value
(2)
sign of value
sign of
operation
McDougal Littell, Chapter 1
or
( 3)
sign of value
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010

sign of
operation
( 2)
sign of value
Page 3 of 13
Notice that the sign of value and the sign of operations are identical. Please note that numbers
written without a sign are assumed to be positive; for example, 3  3, +2  2 .
Operations with Integers
Adding Integers
Syllabus Objective: (1.4) The student will add integers.
Sometimes we refer to integers as “directed numbers”, because the sign of the number tells us to
move a certain number of units in a specific direction along the number line. Positive numbers
tell us to move to the right on the number line or in a positive direction; negative numbers tell us
to move to the left on the number line or in a negative direction. Our starting point will always
be zero or the origin.
Now, let’s look at the previous problem mentioned, (+3) + (+2). We start at 0 on the number
line and move in a positive direction 3 units. From the point +3, we move another 2 units in a
positive direction. Our ending point is the sum of (+3) + (+2) = +5 or 5.
We now have our first rule for adding integers.
Rule 1: A positive number plus a positive number is equal to a positive number. Try
a few sample problems with the students.
Let’s look at what happens when we add a negative number to a negative number. We will
begin with  3   2 . We start at 0 on the number line and move in a negative direction (left)
3 units. From the point 3 , we move another 2 units in the negative direction. Our ending point
is the sum of  3   2  5 .
Rule 2: A negative number plus a negative number is equal to a negative number.
Try a few sample problems with the students.
You should devote some time reviewing the commutative property of addition. In the previous
two examples, it should be clear to the students that (+3) + (+2) = (+2) + (+3) and that (−3) +
(−2) = (−2) + (−3).
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 4 of 13
It is now time to look at what happens when we add a positive number to a negative number or a
negative number to a positive number.
Let’s start with the problem (−3) + (+3)
We start at 0 on the number line and move in a negative direction 3 units. From the point −3, we
now move 3 units in a positive direction. As you can see from the above picture, you end at
zero, which was our starting point: (−3) + (+3) = 0.
Our next example is (−2) + (+5):
We start at 0 on the number line and move in the negative direction 2 units. From the point −2,
we now move 5 units in a positive direction. (−2) + (+5) = +3.
We have one more addition to look at, (+2) + (−5):
We start at 0 on the number line and move in a positive direction 2 units. From the point +2, we
now move 5 units in a negative direction, (+2) + (−5) = −3.
If we play with the above examples and enough similar type problems, we’ll come up a rule that
will allow us to add positive and negative numbers without the use of the number line.
Rule 3: When adding one negative number and one positive number, find the
difference between their absolute values and use the sign of the number with the
greatest absolute value.
Examples:
(−12) + (+12) = 0
(−12) + (+16) = +4
(+7) + (−5) = +2
(−7) + (+5) = −2
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 5 of 13
Subtracting Integers
Syllabus Objective: (1.5) The student will subtract integers.
Remember that subtraction is the opposite operation for addition. So if we subtract on the
number line, we want to do the opposite of what the number tells us to do. So let’s look at our
first example.
Example 1:
 2   8  ?
We will use zero as our starting point just like we did in addition and move to the left, 2 units to
−2. We want to subtract +8 from −2. But remember, we are subtracting. Moving 8 units in the
positive direction is what we would do if we were adding. Since we are subtracting and
subtraction is the opposite of addition, we will do the opposite and move 8 units in the negative
direction.
 2   8  10
Example 2:
 1   5  ?
We start at zero and move 1 unit in a negative direction to −1. The second number is −5, but
because we are subtracting, we move in a positive direction 5 units.
 1   5  4
Example 3:
6   13  ?
We start at zero and move 6 units in a positive direction to +6. The second number is 13, but
because we are subtracting, we move in a negative direction 13 units.
6   13  7
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 6 of 13
Let’s go back and look at the above examples on the number line.
Example 1:  2   8  10 looks like the addition problem  2   8  10
Example 2:
 1   5  4 looks like the addition problem  1   5  4
6   13  7 looks like the addition problem 6   13  7
Example 3:
A close examination of our examples and additional practice problems leads us to a rule for the
subtraction of integers.
Rule 4: When subtracting integers, change the sign of the problem from subtraction
to addition, change the sign of the subtrahend (second number), and use the rules
for addition of integers. Another way to state this is “add the opposite”.
Multiplying Integers
Syllabus Objective: (1.6) The student will multiply integers.
Multiplying integers is a mathematical operation that is an abbreviated process of adding an
integer to itself a specified number of times.
For example,
5  3  3  3  3  3  3 (five groups of 3)
or
5  3  5  5  5 (three groups of 5)
A couple of points to be made here:
1. Review of the commutative property of multiplication
5 3  3 5
2. The above is the product of positive numbers
Positive  Positive = Positive  P  P  P 
So, let’s take a look at the product of a negative number times a positive number. By our
definition of multiplication,
5  3   5   5   5  15
or
5   3   3   3   3   3   3  15
Notice that in each of the above examples, we are looking at a negative number added a positive
number of times. Since we cannot use our definition of multiplication to show a number added
to itself a negative number of times, we will use pattern development.
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 7 of 13
43 
42 
4 1 
40 
4  ( 1) 
4  ( 2) 
4  ( 3) 
12
8
4
0
−4
−8
−12
5 3 
5 2 
5 1 
5 0 
5  ( 1) 
5  ( 2) 
5  ( 3) 
15
10
5
0
−5
−10
−15
83 
82 
8 1 
80 
8  ( 1) 
8  ( 2) 
8  ( 3) 
24
16
8
0
−8
−16
−24
By now, as you work with your students, you will have done enough problems to establish the
following rules of multiplication:
Positive  Positive = Positive  P  P  P 
Positive  Negative = Negative  P  N  N 
Let’s continue the pattern development to show what the sign is when you multiply a negative
number times a negative number. We know the following to be true:
4  3 
4  2 
4  1 
4  0 
−12
−8
−4
0
5 3 
5 2 
5 1 
5 0 
15
10
5
0
83 
82 
8 1 
80 
24
16
8
0
Be prepared to discuss the identity property and multiplication by zero.
You are now ready to discuss that a negative number times a negative number equals a positive
number. Continuing the pattern development:
4  ( 1) 
4  ( 2) 
4  ( 3) 
4
8
12
5  ( 1) 
5  ( 2) 
5  ( 3) 
5
10
15
8  ( 1) 
8  ( 2) 
8  ( 3) 
8
16
24
Therefore, the rules for the multiplication of integers are:
Positive  Positive = Positive  P  P  P 
Positive  Negative = Negative  P  N  N 
Negative  Positive = Negative  N  P  N 
Negative  Negative = Positive  N  N  P 
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 8 of 13
Division of Integers
Syllabus Objective: (1.7) The student will divide integers.
Division is the opposite operation to multiplication, but we can still use pattern development to
explain the rules of division for integers. The pattern development is a little trickier, but we can
do it. Notice that the examples below are tied to the problems used in the pattern development
for multiplication. If necessary, each problem can be rewritten as a multiplication problem
which has been used with the students.
12  4 
84 
44
04 
4  4 
8  4 
12  4 
15  5 
10  5 
55 
05
5  5 
10  5 
15  5 
3
2
1
0
−1
−2
−3
3
2
1
0
−1
−2
−3
24  8 
16  8 
88 
08 
8  8 
16  8 
24  8 
3
2
1
0
−1
−2
−3
The patterns developed in the above examples give the students two rules for division:
 P  P  P
Negative  Positive  Negative  N  P  N 
Positive  Positive  Positive
Making a few slight modifications, we can develop patterns to show the last two rules of
division:
12  ( 4) 
8  ( 4) 
4  ( 4) 
0  ( 4) 
4  ( 4) 
8  ( 4) 
16  ( 4) 
15  ( 5) 
15  ( 5) 
55 
05
5  5 
10  5 
15  5 
−3
−2
−1
0
1
2
3
−3
−2
−1
0
1
2
3
24  8 
16  8 
88 
08 
8  8 
16  8 
24  8 
−3
−2
−1
0
1
2
−3
Positive  Negative  Negative  P  N  N 
Negative  Negative  Positive  N  N  P 
Now is the time to compare the rules for multiplication and division:
McDougal Littell, Chapter 1
PP  P
P PP
P N  N
PN N
N P N
N PN
N N P
N N P
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 9 of 13
Further investigations and discussion will lead us to two rules which can be used with both
multiplication and division:
Rule 5: When multiplying or dividing, like signs equal a positive.
Rule 6: When multiplying or dividing, unlike signs equal a negative.
To review our rules:
Addition
Rule 1: A positive number plus a positive number is equal to a positive number.
Rule 2: A negative number plus a negative number is equal to a negative number.
Rule 3: When adding one negative number and one positive number, find the difference
between absolute values and use the sign of the number with the greatest absolute
value.
Subtraction
Rule 4: When subtracting integers, change the sign of the problem from subtraction to
addition, change the sign of the subtrahend (second number), and use rules 1, 2, or
3 for addition of integers. Another way to state this is “add the opposite”.
Multiplication and Division
Rule 5: When multiplying or dividing, like signs equal a positive.
Rule 6: When multiplying or dividing, unlike signs equal a negative.
When working with these rules, we must understand the rules work for only two numbers at a
time. In other words, if I asked you to simplify  3 4 5 , the answer would be –60.
The reason is  3 4  12, and then  12 5  60 .
Have students investigate what happens to the sign if you have more than 3 numbers by
experimenting with several possibilities. Students should discover that if you have an even
number of “−” signs, the product is positive. If you have an odd number of “−” signs, the
product is negative.
In math, when we have two parentheses coming together without a sign of operation, it is
understood to be a multiplication problem. We leave out the “  ” sign because in algebra it
might be confused with the variable x.
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 10 of 13
Stay with me on this! Often times, for the sake of convenience, we also leave out the “+” sign
when adding integers.
Example:
 8   5
can be written without the sign of operation
8  5, still
equals 13 or 8  5  13 .
Example:  8   5 can be written without the sign of operation
equals 13 or  8  5  13 .
Example:
 8   5
can be written without the sign of operation
8  5, still
8  5 , still
equals 3 or  8  5   3 .
For ease, we have eliminated the “  ” sign for multiplication and the “+” sign for addition. That
can be confusing. Now the question is: “How do I know what operation to use if we eliminate
the signs of operation?”
The answer: If you have two parentheses coming together as we do here,  5 3 , you need to
recognize that as a multiplication problem.
A subtraction problem will always have an additional sign, the sign of operation. For
example, 12   5 , you need to recognize the negative sign inside the parentheses is a sign of
value, the extra sign outside the parentheses is a sign of operation. It tells you to subtract.
Now, if a problem does not have two parentheses coming together and it does not have an extra
sign of operation, then it’s an addition problem. For example, 8  4,  12  5, and 9  12 are all
samples of addition problems. Naturally, you would have to use the rule that applies.
The Coordinate Plane
Syllabus Objectives: (1.8) The student will identify points in a coordinate plane.
(1.9) The student will plot points in a coordinate plane.
A coordinate plane is formed by the intersection of a horizontal number line called the x-axis and
a vertical number line called the y-axis.
The x-axis and y-axis meet or intersect at a point called the origin.
The coordinate plane is divided into six parts: the x-axis, the y-axis, Quadrant I, Quadrant II,
Quadrant III, and Quadrant IV. Refer to the diagram on the next page. Hint: one way to
remember the order of the quadrants is to think of writing a “C” (for coordinate plane) around
the origin. To create the “C” you start in quadrant I and move counterclockwise (and so does the
numbering of the quadrants).
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 11 of 13
y-axis
Quadrant II
Quadrant I
y
origin (0, 0)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
(
x-axis
1 2 3 4 5 6 x
Quadrant III
Quadrant IV
The coordinate plane consists of infinitely many points called ordered pairs. Each ordered pair
is written in the form of (x, y). The first coordinate of the ordered pair corresponds to a value on
the x-axis and the second number of the ordered pair corresponds to a value on the y-axis. Our
movements in the coordinate plane are similar to movements on the number line. As you move
from left to right on the x-axis, the numbers are increasing in value. The numbers are increasing
in value on the y-axis as you go up.
To find the coordinates of point A in Quadrant I, start from the origin and move 2 units to the
right, and up 3 units. Point A in Quadrant I has coordinates (2, 3).
y
B
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
C
-3
-4
-5
-6
A
1 2 3 4 5 6x
D
To find the coordinates of point B in Quadrant II, start from the origin and move 3 units to the
left, and up 4 units. Point B in Quadrant II has coordinates (−3, 4).
To find the coordinates of the point C in Quadrant III, start from the origin and move 4 units to
the left, and down 2 units. Point C in Quadrant III has coordinates (−4, −2).
To find the coordinates of the point D in Quadrant IV, start from the origin and move 2 units to
the right, and down 5 units. Point D in Quadrant IV has coordinates (2,−5).
McDougal Littell, Chapter 1
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
Page 12 of 13
Here are a few phrases that teachers have used with students to help them remember how to
determine the coordinates of a point. “Taxi before you take off” implies moving right or left
before you move up or down. Same with “Run before you jump”.
Notice that there are two other points on the above graph, one point on the x-axis and the other
on the y-axis. For the point on the x-axis, you move 3 units to the right and do not move up or
down. This point has coordinates of (3, 0). For the point on the y-axis, you do not move left or
right, but you do move up 2 units on the y-axis. This point has coordinates of (0, 2). Points on
the x-axis will have coordinates of (x, 0) and points on the y-axis will have coordinates of (0, y).
The first number in an ordered pair tells you to move left or right along the x-axis. The second
number in the ordered pair tells you to move up or down along the y-axis.
Plot the following points on the coordinate plane given below.
1. A (2, 5) 2. B (−3, 8)
McDougal Littell, Chapter 1
3. C (4, −5)
4. D (−3, −2)
5. E (0, −4) 6. F (3, 0)
Pre-Algebra 8, Unit 01: Variables, Expressions, Integers
Revised 2010
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