Download Computing with Signed Numbers and Combining Like Terms

Section 1.1
Computing with Signed Numbers
and Combining Like Terms
Pre-Activity
Preparation
One of the distinctions between arithmetic and algebra is
that arithmetic problems use concrete knowledge; you know
each of the components and can calculate an answer based
on complete information. For example, an arithmetic problem
might be stated as:
“What is the sum of 7 and 3?”
In algebra, however, problems often use information given
in general or abstract terms. The previous problem might be
changed to,
“What is the sum when any number is added to 3?”
Learning Objectives
• Begin to learn the fundamental language of algebra
• Learn the foundational properties of algebra
• Add, subtract, multiply, and divide signed numbers
• Apply the Distributive Property to combine like terms
Terminology
New Terms
to
Learn
absolute value
number line
coefficient
opposite
constant
product
dividend
quotient
divisor
reciprocal
expression
signed number
factor
simplify
integer
term
like terms
variable
negative
Chapter 1 — Evaluating Expressions
Building Mathematical Language
Addition and Subtraction Expressions
Addition and subtraction expressions have the following elements: addition signs, subtraction signs,
constants, variables and symbols of enclosure. (Check the table on the following page for definitions
of these terms.) Below are simple examples.
4 + (–7)
“four plus negative seven”
–11+ 7
“negative eleven plus seven” or “negative eleven plus positive seven”
a – (–6)
“a minus negative six”
–17 – (–12)
“negative seventeen minus negative twelve”
x+7
“x plus seven”
The result of adding two numbers is called a sum. Subtracting one number from another is called a
difference.
Multiplication and Division Expressions
Multiplication and division expressions have the following elements: multiplication signs, division signs,
constants, variables and symbols of enclosure. Below are simple examples.
–15 × 2
2 • (–3)
2 (–3)
–30 • (–8)
–30 (–8)
(–30)(–8)
90 ÷ (–9)
12x
xy
“negative fifteen times two” or “negative fifteen times positive two”
4
“two times negative three” or “positive two times negative three”
4
“negative thirty times negative eight”
“ninety divided by negative nine” or “the quotient of 90 and negative nine”
“twelve x” or “twelve times x”
“x times y” or “xy”
The result of a multiplication is called the
product; the numbers being multiplied
are most often called factors.
The result of a division is called the
quotient. The dividend is divided by
the divisor.
(factor) × (factor) = product
dividend ÷ divisor = quotient
quotient
or divisor dividend
)
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Common Language of Algebra
Language
Definition
Constant
A symbol (usually a number) that
does not change its value
Variable
A symbol (usually a letter) that
represents an unspecified or
unknown number (value). Each
variable represents a single
value, even if the variable
appears more than once in the
problem.
Term
Part of an expression made up
of constant and variable factors.
Terms are separated by + or
­– signs; that is, terms are added
or subtracted.
Example
7x + 2
π is an example of a
constant that is not a
numeric symbol. It is
a Greek letter used to
represent the constant
ratio of the circumference
to the diameter of a circle.
x, y, z, a, b, c, etc.
Two key ideas:
1. An unknown value that
makes an equation true
or
2. A placeholder in an
expression that can
assume any chosen
value
The sign preceding a term
The expression
is used as the sign of the
2x + 5y – 7c + 2c
coefficient.
has four terms: 2x, 5y, –7c, and 2c.
Connect the idea of
The terms –7c and 2c
terms with addition and
subtraction.
have the same variable
components; we call them
like terms.
7 in 7x, –3 in –3a3, π in πd
Coefficient
Any constant or variable factor
of another variable
Integer
All whole numbers, zero, and all
the opposites (negatives) of the
whole numbers
Factor
Constants, variables, or
expressions that are multiplied
Key Observations
For the expression 4xy2:
4 is the coefficient of xy2,
4y2 is the coefficient of x; and
4x is the coefficient of y2
…–3, –2, –1, 0, 1, 2, 3,…
4x (3)(6)
7xyz
(x + 2)5 z(2 + z)
???
Why can we do this?
Most often the word
coefficient refers to
the constant factor, the
numerical coefficient.
Integers are zero and the
signed numbers, both
positive and negative.
Connect the idea of
factors with multiplication
???
Terms are separated by + or – signs. Since subtraction means addition of the
opposite, an expression can always be assumed to be an addition of its terms.
Consequently, each subtraction sign preceding a term becomes the negative sign of the coefficient of that
term. That is, always that the + or - sign before a term as the sign of its numerical coefficient.
Why can we do this?
For example, 4x + 7y – 2c is the same as 4x + 7y + (–2c) and has the numerical coefficients 4, 7,
and –2 in its terms 4x, 7y, and –2c.
Caution: The word term can also refer to either the numerator or denominator of a fraction. Reduce to
lowest terms refers to dividing out common factors from the terms of the fraction.
Chapter 1 — Evaluating Expressions
Signed numbers
Positive numbers are greater than zero and negative numbers are less than zero. All signed numbers,
positive and negative, can be represented on a number line, such as the one below on which the integers
from –7 through +7 are marked:
-7
-5
-6
-4
-3
-2
-1
0
1
2
3
4
5
6
7
As you move from left to right on the number line, the numbers grow larger: –5 > –7 or 3 > 2
Or you can say that as you move left, the numbers get smaller: –2 < 0 or 1 < 4
Opposites
Aside from zero, which is neither positive nor negative, every signed number has an opposite number
that is equally distant from zero, but in the opposite direction. For example, 7 is the opposite of –7 and
–3.5 is the opposite of 3.5, etc. Find the opposite of a number by taking the negative of the number.
Operations
Addition, subtraction, multiplication, and division are operations used to combine numbers, variables
or expressions. Subtraction and division can be written in terms of addition and multiplication.
• Subtraction is addition of the negative (additive inverse) of a number
• Division is multiplication by the reciprocal (multiplicative inverse) of a number
1
For example: 2a – 3b = 2a + (–3b) and 5 ' x = 5 $
x
Use of the Dash
Mathematicians use the dash (–) to represent both the negation of a number and the subtraction of one
number from another. We know what the meaning of the dash is from the context in which it appears. For
example, in 0 – 7 the dash indicates subtraction while in (–7) the dash indicates negation. The two short
expressions have the same value, but the dash has different meanings in the two expressions.
Absolute Value
Special enclosure symbols are used to indicate the distance a signed number is from zero: a = − a = a.
Since distance is a non-negative concept, the absolute value returns a non-negative value. (Nonnegative means the same as “positive OR zero.”)
A number and its opposite have the same absolute value: -2.31 = 2.31 = 2.31 .
The numbers –2.31 and 2.31 are the same distance from zero but in opposite directions on the number line.
Caution: The opposite of the opposite of a number n is –(–n) or n, itself. Therefore, 7 and –(–7) are
different representations of the same number.
7 to its opposite, –7
-7
0
–7 to its opposite, –(–7), or 7
7
-7
0
7
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Properties and Principles
Algebra follows the basic properties and principles of mathematics. The following are applied as needed
to solve problems and develop new concepts.
Property
Identity for
Addition
Symbolic
Representation
Numeric Example
Key Observations
a+0=0+a=a
–8 + 0 = –8
Use this fact when a particular term
is “missing” from an expression;
add a zero term as a placeholder.
a + (–a) = (–a) + a = 0
3.5 + (–3.5) = 0
The additive inverse can also
be called the “opposite of”
or the “negative of.”
Identity for
Multiplication
a $ 1 =1$ a = a
3 3 2 6
= $ =
4 4 2 8
You can multiply by 1 in any
form anytime it is needed.
Inverse of
Multiplication
a$
Inverse of
Addition
Zero Factor
Zero Divisor
1 a
= =1
a a
a$ 0 =0$a =0
a ÷ 0 = undefined
(a + b) + c = a + (b + c)
Associative
(a $ b) $ c = a $ (b $ c)
Commutative
Distributive
Trichotomy
4$
1 4
= =1
4 4
99 × 0 = 0
12 ÷ 0 = undefined
However, 0 ÷ 12 = 0
(7 + 2) + 3 = 7 + (2 + 3)
9 +3 = 7 +5
12 = 12
(2 × 3) × 4 = 2 × (3 × 4)
6 × 4 = 2 ×12
24 = 24
a+b=b+a
7 + (–2) = (–2) + 7
a $b =b$a
(–5) × 2 = 2 × (–5)
a(b + c) = ab + ac
3(4 + 5) = 3 × 4 + 3 × 5
3(9) = 12 +15
27 = 27
Given any two numbers
a and b, one and only
one of the following
statements is true:
a) a is equal to b; a=b
b) a is less than b; a<b
c) a is greater than b; a>b
Given: –3.2 and 2.3
a) –3.2 = 2.3 False
b) –3.2 < 2.3 True
c) –3.2 > 2.3 False
Every non-zero number has a
multiplicative inverse, also called
its reciprocal.
No exceptions
Know the difference between
division BY 0 and division INTO 0.
Always validate: Is there a number
multiplied by 0 that is 12? No.
Is there a number multiplied by 12
that is 0? Yes (0).
Grouping does not matter for
addition or multiplication.
Order does not matter for addition
or multiplication.
Used extensively; left to right
to multiply through to remove
parentheses and right to left to
break out common factors.
If you can rule out two of the
relationships, the third one is true.
This property is most often used in
formal proofs.
Chapter 1 — Evaluating Expressions
Methodology
Adding and Subtracting Signed Numbers
►
►
Example 1: Subtract: –19–(–7)
Example 2: Subtract: –21 –32
Try It!
Steps in the Methodology
Step 1
Identify the intended
operation as addition
or subtraction.
Step 2
•If the operation
identified in Step 1
is addition, skip to
Step 3.
•If the operation
identified in Step 1 is
subtraction, change
it to addition AND
change the sign of
the second term
(change TWO signs).
Step 3
Determine the signs
of the two terms to be
combined.
Step 4
•If the signs are not
the same, skip to
Step 5.
•If the signs are
the same, add the
absolute values of
the two terms and
attach the common
sign. Then go to
Step 6.
The process hinges
on whether the
operation is addition
or subtraction.
Subtraction is the
same as addition of
the opposite. Once
a problem has been
changed to addition by
also changing the sign
of the second term,
follow the process for
addition.
This step splits the
process into two
possibilities: either
the signs are the
same—both positive or
both negative, or the
signs are different—
one positive and one
negative.
• If the signs are not
the same, skip to
Step 5.
• If the signs are
the same, add the
absolute values of
the two terms and
attach the common
sign, then go to
Step 6.
Example 1
Subtraction is the
operation.
–19 – (­­–7)
1
change to
addition
2
change sign of
second term
–19 + [ +7)]
= –19 + 7
???
Why can we do this?
19 is negative and
7 is positive
Signs are not the same.
Go to Step 5.
Example 2
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Steps in the Methodology
Step 5
If the signs are opposite,
subtract the smaller
absolute value from
the larger absolute
value. Attach the sign
corresponding to the
larger absolute value.
Step 6
Validate by performing
the opposite operation.
Example 1
Is one positive and one
negative? If so, take
the absolute values
of both numbers
and then subtract.
Reattach the sign of
the number with the
larger absolute value.
If the original
operation was addition,
subtract the original
second addend from
the answer. If it was
subtraction, add the
answer to the original
second addend. The
result in either case
should match the
original first number.
Example 2
-19 > 7
-19 - 7
= 19 - 7 = 12
Attach the sign of the –19:
Answer: –12
The original was
subtract, so add:
–12 + (–7)
(signs are the same,
so add absolute values)
12 + 7 (attach the
common sign)
–19 (and compare)
–19 = –19 
???
Built into the reasoning of Step 2 is an application
Why can we do this?
of both the Identity for Addition and the Inverse
of Addition properties. The simple problem at right illustrates the use of these
properties to change 3 – 7 to 3 + (–7); that is, add zero to the original expression
in the form of 7 + (–7).
inverses
sum = 0
3 − 7 + 7 + ( −7 )
3 − 7 + 7 + ( −7 )
inverses
sum = 0
3 + (− 7)
Models
Model 1
Add or subtract as needed: –42 + (–9)
-42 + -9 = 51
Step 1
Addition problem
Step 4
Add absolute values:
Step 2
Skip to Step 3
Attach common sign: Answer: –51
Step 3
Both numbers are negative.
Step 6
Validate by subtracting: –51 – (–9)
Change two signs:
Opposite signs so
subtract absolute values:
Attach the sign of the
number with the larger
absolute value: = –51 + ( + 9)
-51 - 9 = 42
–42 
Chapter 1 — Evaluating Expressions
Models
Model 2
Add or subtract as needed: –13 – (–18)
Step 1
Subtraction problem
Step 2
Change two signs:
Step 3
Opposite signs
Step 4
Subtract absolute values:
Step 5
Number with the larger absolute value is positive; Answer: 5
Step 6
Validate by adding:
–13 + (+18)
18 - -13 = 5
5 + (–18)
Opposite signs so
subtract absolute values:
-18 - 5 = 18 - 5 = 13
Attach the sign of the
number with the larger
absolute value: –13 
Model 3
Add or subtract as needed: –12 + 3 + (–7) –5 – (–6)
Step 1
Mixed addition and subtraction problem
Step 2
Change two signs for all subtractions:
–12 + 3 + (–7) + (–5) + (+ 6)
Steps 3–5
Add left to right:
= -9 + (-7 )+ (-5 )+ 6
= -16 + (-5 )+ 6
= -21 + 6
Answer: − 15
Step 6
Validate: -15 + (-6) + 5 - (-7) - 3 = -12
= -15 + (-6) + 5 + (+7) + (-3)
= -21 + 5 + 7 + (-3)
= -16 + 7 + (-3)
= -9 + (-3)
= -12 
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Methodology
Since we cannot carry out the multiplication of two variables whose values are not known (such as
y
x and y), we indicate their product simply as xy. Similarly, dividing x into y is shown as y/x, y ÷ x, or
.
x
Multiplying and Dividing Two Quantitites Containing Constants and Variables
►
►
Example 1: Divide: (–27x) ÷ (–3)
Example 2: Divide: (– 48a ) ÷ (7)
Steps in the Methodology
Step 1
Determine the sign of
the answer:
Positive (+) if both
quantities have the
same sign
Negative (–) if the
two quantities have
opposite signs
Step 2
Identify the
intended operation
as multiplication or
division.
The sign of the answer
to multiplication or
division is independent
of what the quantities
are; it hinges only
on whether the signs
match (a positive
answer) or do not
match (a negative
answer).
If the process is
division, there is an
extra step, so make
this determination
from the outset.
If multiplication, skip
to Step 4.
Step 3
If the operation
identified in Step 2
is division, change it
to multiplication AND
invert the second
quantity (multiply by
the reciprocal).
The Inverse Property
of Multiplication is
applied here. Two
changes occur: the
operation changes to
multiplication AND the
divisor changes to its
multiplicative inverse.
Try It!
Example 1
Example 2
(–27x) ÷ (–3)
Both signs are negative
so the answer is positive.
(–27x) ÷ (–3)
The operation
is division.
(–27x) ÷ (–3) =
 1
= (−27 x) :  − 
 3
That is, multiplying
by the multiplicative
inverse is the same as
dividing.
continued on next page
Chapter 1 — Evaluating Expressions
10
Steps in the Methodology
Step 4
Calculate the product
of the absolute values
of the quantitites.
Attach the sign
determined in Step 1.
Step 5
Validate by following
the methodology
for the opposite
operation.
Example 1
Example 2
1
3
1
= (27 x) :
3
27 x
=
3
Answer: +9x
Multiply, disregarding
the sign; you have
already determined
the sign of the answer
in Step 1.
= -27 x : -
= 9 x : -3
1 Negative answer
2 Multiplication; skip to
Step 4
4 Multiply absolute
values and make the
result negative:
9 x : -3
For division, multiply
the answer by the
original divisor; if it
was multiplication,
divide the product by
either factor to get the
other factor.
= 9 x : 3 = 27 x
-27 x 
Models
Model 1
Take a closer look
Multiply: –2.3c • 5
Step 1
The signs are opposite, so the answer is negative.
Step 2
The operation is multiplication: Skip to Step 4.
Step 4
Multiply absolute values:
2­ .3c • 5
can change to
2.3 • 5 • c
–2.3c • 5
by applying the
commutative property.
2.3c • 5 = 11.5c
Answer: –11.5c
The original problem used multiplication so
Validate
use the division methodology to validate:
Step 1 Choose either factor: –11.5c ÷ 5 (signs are opposite; answer
will be negative)
Step 2 Operation is division
Step 5
1
Step 3 = -11.5c :
5
1
= 2.3c
5
Answer: –2.3c 
Step 4 Multiply absolute values:-11.5c :
11
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Model 2
Multiply: (–3)(5x)(–2y)(–7z)
???
Step 1
The sign of the answer is negative. How do you know this?
Step 2
All operations are multiplication; skip to Step 4
Step 4
Multiply absolute values from left to right: 3 : 5 x : 2 y : 7 z
= 15 x : 2 y : 7 z
= 30 xy : 7 z
= 210 xyz
Answer: –210xyz
The original problem used multiplication so
Validate:
use the division methodology to validate:
-210 xyz ÷ (-7 z )= 30 xy
Step 5
Take a closer look
(-210 xyz ) ' (-7 z ) =
30
210 xyz 210 xy z
=
=
= 30 xy
1
7z
7 z
30 xy ' (-2 y ) =
15
30 x y
1
-2 y
30 xy ÷ (-2 y )= -15 x
(-15 x )÷ 5 x = -3
= -15 x
( -15 x) ' 5 x =

3
=
-15 x - 15 x
=
= -3
5x
5x
Model 3
Divide:
2 -5
÷
-3 -7
Step 1
The sign of the answer is negative.
Step 2
The operation is division.
Step 3
Two changes:
Step 4
Multiply absolute values: Step 5
Validate:
???
How do you know this?
2
−7
:
-3 −5
2 7 14
: =
3 5 15
14
Answer: −
15
The original problem used division so
use the multiplication methodology to validate:
-
14 -5
:
15 -7
Step 1
Negative answer
Step 2
Multiplication; skip to Step 4.
Step 4
14
5 2
Multiply absolute values:3 : =
7 3
15
2
Answer: −
2

3
Chapter 1 — Evaluating Expressions
12
???
How do you know this?
Determining the sign of the answer for multiplication of signed numbers (or
division, after changing to multiplication of the reciprocal) is independent of the
values of the numbers. You can pair up the signs of negative factors, recalling
that (−a )(−b) = ab . Every time an even number of negative factors is present,
the answer is positive because all of the signs can pair up. If the number of
negative factors is odd, the answer will always be negative, because one negative
factor will not be able to pair up.
Methodology
Expressions containing variable factors can often be simplified by applying the Associative, Commutative
and Distributive properties to combine like terms. Like terms contain the same variable factors. Below is a
methodology for combining like terms.
Combining Like Terms
►
►
Example 1: Combine like terms: 12x – 2y + 7x – y
Example 2: Combine like terms: 8x – 3y + x – 5y
Steps in the Methodology
Step 1
Use the
Commutative
Property to sort
the expression
so that like terms
are together.
Step 2
Use the
Associative
Property to
group like terms
together within
parentheses.
Scan through the
problem to make
sure like terms are
identified. You can
highlight each set of
like terms with color
or underscoring.
Always convert
subtraction to
addition before
grouping within
parentheses to
eliminate sign errors.
Try It!
Example 1
12x – 2y + 7x – y
= 12x + 7x – 2y – y
12x + 7x –2y – y
= 12x + 7x + (–2y) + (– y)
= (12x + 7x) + ((–2y) + (– y))
Example 2
13
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Steps in the Methodology
Step 3
For each group
of like terms, use
the Distributive
Property to
factor out
the common
variable(s).
Step 4
Add or
subtract within
parentheses.
Step 5
Rewrite addition
of a negative as
subtraction.
Step 6
Validate
Example 1
Factoring is
“un”-multiplying.
The Distributive
Property distributes
multiplication over
addition, but the
reverse action also
applies.
(12 + 7)x + (–2 + (–1))y
Follow the
methodologies
presented previously
for combining terms
within parentheses.
(12 + 7)x + (–2 + (–1))y
Why do this? This
is a step that
technically doesn’t
need to happen,
but it adds to the
sophistication of the
answer, making it
simpler to read.
19x – 3y
Validation is best
done after some time
has elapsed. Come
back to the problem
and rework it from
the beginning.
12x – 2y + 7x – y
19x + (–3)y
= 12x + 7x + (–2y) + (–y)
= (12 + 7)x + (–2 + (–1))y
= 19x – 3y
Example 2
Chapter 1 — Evaluating Expressions
14
Models
Model 1
Combine like terms: 5s + 3t + c –2s –7t + 2c
Step 1
Sort by like terms:
Step 2
Group within parentheses after converting subtraction to addition:
5s + 3t + c –2s –7t + 2c
5s – 2s + 3t –7t + c + 2c
(5s + (– 2)s) + (3t + (–7)t) + (c + 2c)
Step 3
Factor: (5 + (–2))s + (3 + (–7))t + (1 + 2)c
Step 4
Add or subtract in parentheses: 3s+ (–4)t + 3c
Step 5
Rewrite as subtraction: Answer: 3s – 4t + 3c
Step 6
Validate: (Reworked) 
???
Why do this?
Validate:
5 + (–2) = 3
3 – (–2) = 5 
3 + (–7) = –4
–4 – (–7) = 3 
1+2=3
3–2=1 
???
Why do this?
Any variable without a written numerical coefficient has an implied
coefficient of 1. In the above model, c is understood to mean 1c. The 1
is not customarily written; however, you may elect to write the implied
1 to remember to include the term.
Model 2
Combine like terms: 3x + 4y –7 –x + 3y
Step 1
Sort:
3x + 4y –7 –x + 3y
3x – x + 4y + 3y –7
Step 2
Group:
(3x + (–1)x) + (4y + 3y) + (–7)
Step 3
Factor: (3 + (–1))x + (4 + 3)y + (–7)
Step 4
Combine: 2x + 7y + (–7)
Step 5
Rewrite: Answer: 2x + 7y –7
Step 6
Validate: (Reworked) 
Validate:
3 + (–1) = 2
2 – (–1) = 3 
4+3=7
7–3=4 
15
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
Addressing Common Errors
Issue
Incorrect Process
Misinterpreting
absolute
value
symbols
Resolution
Correct Process
Validation
12 is 12 units from
zero on a number
line.
12 = -12
Absolute value
returns a nonnegative value,
not the opposite
value.
Misunderstanding
division of
zero by a
number
0
= undefined
-4
0 ' x = undefined
Division BY zero
is undefined.
Division INTO
zero is 0.
0
=0
-4
0 ' x =0
Changing
only one
sign when
subtracting
4 + 6 – (–8)
Always change
two: apply
the Inverse of
Addition Property
to make the
changes.
4 + 6 – (–8)
18 + (–8) – 6
= 4 + 6 + (+8)
= 10 – 6
=4+6+8
=4 
= 10 + (–8)
=2
12 = 12
and
-12 = 12
Does 0 • 4 = 0? Yes
Does x • 0 = 0? Yes.
= 10 + 8
= 18
Using the
Associative
Property
with
subtraction
2–7–8–3
= (2 – 7) – (8 – 3)
= –5 – 5
= –10
Change
subtraction
to addition of
the opposite
before using
the Associative
Property.
Once the
problem has
been changed to
addition, group
as desired.
Forgetting
a variable
factor when
multiplying
3z • 5 • 2y • (–2x)
= (3 • 5 • 2 • (–2))yx
= –60xy
Write the
variable factors
in alphabetical
order. Each factor
must be included
in the product.
Mark off each
factor as it is
used.
2–7–8–3
–16 + 3 + 8 + 7
= 2 + (–7) + (–8) + (–3)
= –13 + 8 + 7
= [2 + ( − 7) ]+ [( − 8) + ( − 3) ]
=
=
−5
+
− 16
− 11
↓
↓
↓
3z • 5 • 2y • (–2x)
= 3 • 5 • 2 • (–2)xyz
= 3 • 5 • 2 • (–2)xyz
= 15 • 2 • (–2)xyz
= 30(–2)xyz
= – 60xyz
↓↓↓
= –5 + 7
=2 
Is each variable
included in the
product? Yes 
Is each numeric
factor included?
Check numeric
factors by
dividing:
–60 ÷ (–2) = 30
30 ÷ 2 = 15
15 ÷ 5 = 3 
Chapter 1 — Evaluating Expressions
16
Issue
Leaving
out an
unmatched
term when
combining
like terms
Incorrect Process
7x –2y + 3z – 3y + x
= 7x + x – 2y – 3y
= 8x – 5y
Resolution
Highlight or
underline like
terms. In order
for equality to
prevail, every
term must be
included in the
simplified answer.
Circle any terms
that do not
combine with
other terms so
that you don’t
forget to include
them.
Forgetting
to combine
a term
with no
coefficient
Correct Process
7x – 2y + 3z–3y + x
= 7x + x–2y–3y + 3z
= (7x + x) + ((–2)y + (–3)y) + 3z
= (7 + 1)x + (–2 + (–3))y + 3z
= 8x + (–5)y + 3z
= 8x – 5y + 3z
8–1=7 
–5 – (–3) = –2 
2a + 3b + 5 + a + 2b – 1
= 2a + a + 3b + 2b + 5
= 2a + 3b + 5 + 1a + 2b – 1
= (2a + a) + (3b + 2b) + 5
= (2 + )a + (3 + 2)b + 5
= 2a + 5b + 5
Come back and
rework this
problem after the
next problem.
If you rework
it immediately,
your eyes may be
tricked into seeing
combinations that
are not true.
Validate addition:
2a + 3b + 5 + a + 2b – 1 Write in the
implied 1
coefficient.
For a more
sophisticated
answer, you can
rewrite it without
the 1 coefficient.
Validation
= 2a + 1a + 3b + 2b + 5
Validate by
reworking after
working the next
problem.
= (2a + a) + (3b + 2b) + 5 – 1
Validate addition:
= (2 + 1)a + (3 + 2)b + 5 – 1
3–1=2 
= 3a + 5b + 4
5–2=3 
4+1=5 
Preparation Inventory
Before proceeding, you should be able to do each of the following:
Use and understand the basic vocabulary of algebra.
Apply basic properties and principles of algebra.
Perform and validate calculations with signed numbers.
Combine like terms by applying the Distributive Property.
Section 1.1
Computing with Signed Numbers
and Combining Like Terms
Activity
Performance Criteria
• Demonstrating the use of appropriate algebraic
language
– correct use of terms in oral and written
communication
– correct spelling of terms in written
communication
• Computing with signed numbers
– accuracy
– correct sign for the result
• Simplifying expressions by combining like terms
– correct use of Distributive Property
– accuracy in signed number operations
Activity
A Model for Multiplication―Multiplication on a Grid
Supplies: square tiles, grid paper, problem set, instructions
If you have tiles provided by your instructor, use them to make models of the following integer multiplication
problems. If you do not have tiles provided, you can make some by using squares of paper or other material.
(Alternatively, you can draw more grids and color in the tiles for each problem.)
negative times a positive
- × +
positive times a positive
I
II
+ × +
8
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
1 2 3 4 5 6 7 8
III
IV
- × -
negative times a negative
Locate the two crossed number lines. They intersect at
zero, dividing the grid into four parts, called quadrants.
The white portion indicates positive regions: quadrants I
and III; light gray indicates negative regions: quadrants
II and IV. To model a problem, the first number indicates
how many tiles in a row, either in the positive (right)
direction or the negative (left) direction. The second
number indicates the number of rows, in either the
positive (up) or negative (down) direction. In the
example, five tiles in the positive direction for each of
three rows in the negative direction. The tiles are in the
light gray area and indicate a negative product.
Example: Show the product of +5 × –3
+ × -
positive times a negative
First number: +5
“five tiles in a row towards the right”
Second number: –3
“three rows of 5 tiles each, downward”
+5 × –3 = –15 as shown by
15 tiles in the negative region.
17
Chapter 1 — Evaluating Expressions
18
- × +
8
7
6
5
4
3
2
1
Use tiles to show the following
products:
–2 × 4
3×2
–4 × –2
–1 × 5
3 × –2
–2 × –3
–1 × 6
Note: Additional blank grids may be
found at the end of this activity.
+ × +
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
1 2 3 4 5 6 7 8
- × Critical Thinking Questions
1. Can 0 be the coefficient of a term? Illustrate your answer with an example.
2. What is a variable factor?
3. What information does the coefficient of a variable impart?
+ × -
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
19
4. How do you find the absolute value for a variable?
5. What is the difference between the “opposite of” a number and the “negative of” a number?
6. Is there an associative property for subtraction? (Illustrate your answer with an example.)
7. When would you insert a “+” (positive sign) and when would you insert “( )” (parentheses) without
changing the value of an expression?
8. Why is reworking the problem the preferred method to validate combining like terms?
9. How do you validate that every term is accounted for in combining like terms?
Tips
for
Success
• Mathematicians agree that two signs together, such as 4 + –7, may be ambiguous. Therefore, unless it is
the first term in the expression or the denominator of a division problem, a negative number is written
within parentheses: 4 + (–7).
• When combining like terms, count the number of terms to be sure no term is overlooked. Underline or
highlight every term in the original problem and make sure each term is accounted for as either combining
with another term or left unchanged in the final answer.
Chapter 1 — Evaluating Expressions
20
Demonstrate Your Understanding
1. Perform the indicated computation and identify the property used or illustrated.
Problem
a)
0 + (–3)
=
b)
19 + (–19)
=
c)
36 ÷ 0
=
d)
(–39) • 0
=
e)
(–209) • 1
=
f)
12 :
1
12
=
g)
0 ÷ 16
=
Validation
Property
21
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
2. Add or Subtract the following:
Problem
Problem
Validation
g) –24 + 5
a) 2 – 5
=
=
h) –23 – 12
b) 8 – (–3)
=
=
c) 0 – (–7)
i)
=
–9 – 14
=
d) 18 + (–17)
j)
33 + (–23)
=
=
e) 16 + (–10)
k) 2 – (–5)
=
f)
Validation
=
–26 – 14
l)
=
–8 – (–32)
=
3. Perform the indicated multiplication or division:
Problem
a) 25 ÷ (–5)
Validation
Problem
b) (–12)(–1)
=
=
c) –2 • –7
d) –7 • 4
=
=
e) 63 ÷ (–1)
=
g) –42 ÷ (–7)
=
f)
54 ÷ (–9)
=
h) –2 • 4 • (–5)
=
Validation
Chapter 1 — Evaluating Expressions
22
4. Apply the distributive property. Example: 5(a + b) = 5a + 5b
a) –3(m + n)
b) –2(3x – 1)
c) 7(x + 2)
answer:
answer:
answer:
e) 3(2x –3y)
f) – ( 2x + 3)
g) –(–2x – 2)
answer:
answer:
answer:
5. Use the distributive property to combine like terms:
Problem
a) 7x + 3 + 4x – 9
b) ab + c + 2ab – 3c
c) x – 3y + 5 – 5 – 3x – y
d) –3x + 5x – 2x – x – 4x + x
e) 7x + 3x – x + 4x – x + 9
Like Terms
Worked Solution
Validation
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
23
7. Multiplication and division are opposite operations. That means that every multiplication problem can be
written as a division problem and every division problem can be written as a multiplication problem. In
fact, we validate multiplication with division and validate division with multiplication.
So 3 × 4 = 12 and 12 ÷ 3 = 4 and 12 ÷ 4 = 3 are all related.
a. Use this idea to explain why a positive number, 100, divided by a negative number, –25, yields a
negative number, – 4.
b. Show why a negative number (–15) divided by a negative number (–3) is a positive number (5).
c. Show why a negative number (–22) divided by zero (0) is undefined and not equal to zero.
Chapter 1 — Evaluating Expressions
24
Identify
and
Correct
the
Errors
Identify and correct the errors in the following problems.
Incorrect Translation
1) If -27 = 27 , then 27 = -27
2) 101 ÷ 0 = undefined,
then 0 ÷ 101 = 0.
3) 6 – (– 5)
= 6 + (–5)
=1
Validate: 1 – (–5) = 6 
4) 3 + 5 – 2 + 7
= (3+5) – (2+7)
=8–9
= ­–1
Validate: –1 + 9 = 8 
5) 3x2yz(–3)
= 3 • 2(–3)xy
= –18 xy
Validate: –18 ÷ (–3) = 6
and 6/3 = 3 
6) 3x + 5y + x + 2y
= 3x + x + 5y + 2y
= (3)x + (5+2)y
= 3x + 7y
List the Errors
Correct Process
Validation
25
Section 1.1 — Computing with Signed Numbers and Combining Like Terms
- × +
+ × +
- × +
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
+ × +
1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
- × -
+ × -
- × -
+ × -
- × +
+ × +
- × +
+ × +
8
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
- × -
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
+ × -
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
- × -
1 2 3 4 5 6 7 8
+ × -
Chapter 1 — Evaluating Expressions
26
- × +
+ × +
- × +
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
+ × +
1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
- × -
+ × -
- × -
+ × -
- × +
+ × +
- × +
+ × +
8
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
- × -
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8
+ × -
-8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
- × -
1 2 3 4 5 6 7 8
+ × -