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Ramp Up to Algebra Curriculum Updated 08/09
Katie Livermore
Christine Buckman
Ramp Up to Algebra
The objective of the Ramp Up to Algebra course is to prepare students
for success in Integrated Algebra and on the Integrated Algebra Regents
Examination. It is expected that students will take this course in the 9th
grade. This curriculum is designed for students that are not performing at
grade level in the area of mathematics.
This is a one year course, two periods per day. Completion will
provide students the prerequisite skills to ensure success with the further
study of mathematics. Successful completion will allow them to progress to
the New York State level of mathematics; Integrated Algebra.
Throughout this curriculum the text referred to is Ramp Up to
Algebra, provided by America's Choice. In order for many of the ~"" cent
standards to be addressed, additional lessons and materials were added.
While the writers realize that the order in which the topics are taught is open
to individual interpretation, it is recommended that teachers follow the
sequence as laid out in this document. This allows students to learn and
review the basic skills required for concepts later in the course.
Students should be awarded one math credit and one elective credit
upon:
• Completion of the course as outlined in this curriculum
• A cumulative final examination
• An average grade of at least 65%**
• Adherence to the Poughkeepsie High School attendance policy
**Students final grade for the course will be an average of the four quarter
grades and their grade on the final examination.
Ramp Up to Algebra Sequencing and Pacing Guide Unit
Number
Number of
Title
Days
,,,
,,
1
The Number System (RU Unit #2)
2
Negatives (RU Unit 115)
17
3
Factors and Fractions (RU Unit #4)
22 17 ..
4
Exponents & Polynomials (not in RU)
14
5
Using Equations to Solve Problems
(RU Unit #7 & 8)
17 6
Ratio and Proportionality (RU Unit 116)
21 7
GeomelIy and Measure (RU Unit #3)
15 8
Graphing (RU Unit #7)
14 9
Foundations of Algebm (RU Unit #1)
20 Total Days
157 ,,
In the unit outlines below, (*J indicates a lesson Ihal is being added 10 the
Ramp Up 10 Algebra curriculum, in this 08/09 updale.
Ramp Up 10 Algebra - Unit 1 - The Number System (RU Unit #2) D
ay
Goal
!
To
understan
d decimal
Ramp Up
Lesson
Intro !o Algebra
Lesson (Plus extra
worksheets)
#!
i Exponents Review
Place Value : Worksheet;
numbers
usmg
place
values
2
3
Place Values and
Rounding
Worksheet;
To use
#2
number
Number
lines to
Lines
represent"
order, and
estimate
whole
numbers,
fractions,
and
decimals
To
#3
identify
Classifying
and
Numbers
represent
Real Numbers
Worksheet
numbers,
whole
numbers.
integers,
and
rational
numbers
usmg
number
4
,,
114
Rational
Numbers
NYSAlgebra
Standard
Digits) Place A.N.!
Value, Power : Identify and
.pplytbe
• ofTen
. (Multiples of . properties of real
numbers (closure,
Ten),
Expanded
commutative,
Fonn,
associative.
: distributive,
: Decimals,
identity, inverse)
Tenths,
Hundredths,
etc. Fractions
Interval,
A.N.!
Inequality
(same as above)
«,
», Rounding
Objective 7.0
Worksheet
natural
lines and
diagrams
To
convert
fractions
Word Wall
Suggestions
Fractions aod
Decimals Skills
Review Worksheet
Naturnl
A.N.1
(same as above)
Numbers,
Wbole
Numbers,
Integers,
Rational
Numbers,
Numerator,
Denominator,
Classifying
(describing),
Venn
Diagrams,
Between)
Irrational
Equivalent,
Fraction Bar
I (means to
A.N.l
(same as abeve)
to
Always,
Sometimes, Never
Activity on
Number Systems
and Operations
tenninaHn
g or
repeating
deCimals,
to convert
decimals
to
fractions.
and to
identify
rational
numbers
5
To write H,
and Fractions
represent and
equivalent Decimals
fractions
Comparing
Decimals
Worksheet;
divide
numerator by
denominator),
Mixed
Number,
Improper
Fraction,
Terminating
Decimal,
Repeating
Decimal,
Irrational
Numbers,
Unit Fraction
A.N,1
Simplest
Form,
(same as above)
Compare
Comparing
on
: Fractions
numbers
• Worksheet',
lines and
to order fractions and decimals
6
7
To use
rational
numbers
to
measure
length and
weight
To add
and
subtract
whole
numbers
and
, decimals
#6
Measuring
Precise,
A.N.5
Approximatio Solve algebraic
ns, Density
problems arising
with
Rational
Numbers
from situations
that involve
#7
Riddle Worksheets
Adding and
Subtracting
On the
Number
Line
(2);
Objective 4,0
Worksheet;
Adding,
Operation,
Subtraction,
Commutative
Property,
Associative
SimnlitviM
• Property,
fractions,
decimals,
percents
(decreaselincreas
e and discount),
and
proportionality!di
reet variatlon
A.RP, I
Recognize that
mathematical
ideas can be
supported by a
variety of
strateRies
using
number
lines;
8
9
To use
mental
strategies
and stand
methods
to add and
subtract
To
multiply
and divide
whole
numbers
and
decimals
using
number
lines and
area
models;
#8
Adding and
Subtracting
Objective 20.0
Worksheet
multidigit
numbers
II
To use the
"guess
and
check"
Identity
Property,
Inverse
Operations,
Sum,
Difference
Mental
Strategies,
Intro. Unit 1,
Assignment 9 ­
and Dividing Evaluating
Expressions
#9
Multiplying
#10
Multiplying
Multidigit
Numbers
Intro. Unit 1,
Assignment lOA &
lOB - Simplifying
Expressions
MUltiple
Representation
Activity on
Division for
Problem Solving
To use
mental
strategies
and the
standard
method to
multiolv
10 To divide
Numerical
Expressions
Worksheet;
Regrouping,
Carrying
Repeated
Addition,
Repeated
Subtraction,
Dividend.
Divisor,
Quotient
A.CN.2
Understand the
corresponding
,.. .." edures for
,.
similar problems
or mathematical
concepts
Count on,
Doubling,
Halving,
Count Back,
Estimating,
Area
(Rectangle &
Parallelogram
IA=bh),
Patterns,
Conjectures
#11
Dividing
Multidigit
Numbers
#12
Estimating
Square
Roots
Objective 6.0
Worksheet
Multiple
Representation
Activity - Laws of
Exponents for
Multiplication and
Division
Base Ten Blocks
info.
Intro. Unit 10,
Lesson #5 ­ Square
Roots
Short
Division,
Long
Division,
Lining Up
Places
A.CN.2
Square,
Square Root,
Guess and
Check,
A.N.I
(same as above)
(same as above)
• strategy to
estimate
Accuracy,
Area \Square;
A=s)
square
roots
12 To
represent
percents
on.
number
line from
Ito
#13
Percents and
Number
Lines
Fractions,
Convert,
Decimals, and
Peroents
Worksheet;
Percent, %
Intro. Unit 4,
Lesson #1 ­
100%~
Percents to
with
decimals
and
fractions
Decimals;
Decimals to
Percents
from Oto
Multiple
Representation
Activity on
Percents
--.Deeimals-->Fract
ions
"of'means
Finding a Percent
"multiplIed
ofaNumber
Worksheets (2)
bt'
I
13 To
calculate
and use
AN.5
(same as above)
#14
Percents and
Decimals
A.N5
(same as above)
decimal
equivalent
sof
percents
I
l~ To solve
15 problems
16 that
, involve
Intro. Unit 4,
Lesson 112­
Percent ofa
Number
#15
Applying
Percent
Inlro. Unit 4,
Lesson #3 Mixed Review
• Intro. Unit 4,
Lesson #7
Percentages - Sales
• Tax
percent
Intro. Unit 4,
Lesson #9
Percentages - Sales
Tax
Intro. Unit 4
• Sales Tax,
Interest.
Percent-off,
: Discount,
A.N.5
(same as above)
Lesson #8
Percentages ~
Discounts
Intra. Unit 4,
Lesson #10
Percentages -
.
Simple Interest
Intra. Unit 4,
Lesson #11
Percentages , Commissions
•
•
•
·
, Percentagesi Mixed Problems
•
•
··
•
•
•
·
17 To review
the
number
system
concepts
#16
The Unit in
Review
Objective 17.0
Worksheet
Intro. Unit 4,
Lesscn#13
Percentages ~
Review
·
,.
Intro. Unit 4,
• Lesson #12
.
··•
,~--
All ofthe above
slllnilards
···
I
•
•
·•
Ramp Up to Algebra-Unit 2-Negatives (RU Unit #5) Day
Goal
1
To recognize
the need for
negative
numbers. and
to place
Ramp Up Lesson
6: Extending the
Number Line
negative
Intro to Algebra
Lesson (plus extra
worksheets)
Worksheet:
Developing skills
in Algebra
Book A
Graphing integers
sheet (page 21)
numbers on
the number
line
2
3
4-S
!
To compare
7: Putting
positive and
Numbers in Order
negative
numbers and
zero, using is
less than.(<)
is less than or
equalto,(:s) is
greater
than,(» and
is greater than
or equal to(?)
To multiply
13: Multiplying
and divide
and Dividing
positive and
negative
numbers
To use the
8: Adding with
number line to Negative
add positive
Numbers
and negative
numbers
Worksheet:
Developing skills
in Algebra
Book A
Directed Distance
sheet (page 19)
Intro Unit 1
Integers packet
Multiplicationassignment #6
Dividing integers~
Assignment # 11
Intro Unit 1
Integers packet
Addition­
Assignment #5
Worksheet:
Developing skills
in Algebra
Book A
Addition of
Integers
Word Wall
Suggestions
• Negative
numbers
• Positive nwnbers
• origin
NYS Algebra
Standard
A.N.l
Identify and
apply the
properties of
real numbers
(closure,
commutative,
associative,
distributive,
identity,
invers~)
A.CM.12
• positive direction
Understand and
• negative
use
appropriate
direction
language,
• ascending
representations,
• descending
and tenninology
• (the inequality
when describing
symbols)
objects,
relationships,
mathematical
solutions, and
rationale
A.PS.3
Observe and
explain patterns
to formulate
generalizations
and coni ectures
A.CM.S
• Addend
Communicate
• sum
logical
arguments
clearly,
showing why a
result makes
sense and why
the reasoning is
valid
6
7
To use the
: 9: Subtracting
number line to willi Negative
subtract
Numbers
positive and
negative
numbers
To understand
and use
equivalence
relationships
between
adding and
subtracting
I
Pages 27, 29, 31,
and 33
Riddle sheet #6­
• difl"erence
"Wh.t do you call a cow that won't
give milk.?'"
ACM.8
Reflect on
strntegies of
others in
relation to one~s
own strategy
Worksheet: Developing skills in Algebra Book A Subtracting Integers Page 35 10: Adding and
Subtracting
Riddle sheet #5­
ACN,1
Understand and
make
connections
among multiple
representations
of the same
mathematical
idea
"Where do
chickens go to
work?"
Worksheet
Developing skills
in Algebra
,,
;---------­
,,
,, 8
,,
,
,,
To model
addition and
subtraction of
positive and
negative
numbers
Book
Adding
and Subtracting
Integers
Page 37
A,R.l
Use physical
Objects,
diagrams.
charts, tables,
graphs,
symbols,
equations. or
object created
using
technology as
representations
of mathematical
11: Balloon
Model
,
concepts
9
12: Reviewing
To review
what you have Addition and
learned about Subtraction
Intro Unit I
Integers packet
Working with
I
A,CNA
Understand how
: concepts,
,,
,,
1 posjti~e and
,,
mathematical
results in one
area of
mathematics
can be used to
solve problems
in other areas of
mathematics
A.N.!
numbers
,,
,
,
,
,
,
10
14: Mixed operations To calculate
expressions
, involving
operstion.
using L'le
,,
distributive
property
11 , To apply the
: number
properties to
positive and
negative
: numbers
,,
,,
,,
15: Number Properties 13
To revIew
working with
16: Progress Check ,,
,,
,
do" Intra Unit 1 Integers Packet Quiz #2 positive and
negative
,
(See above)
Intro Unit 5 Distributive Property packet Lesson #11 Simplifying expressIons Joke Ul1 'Whata '
,
Matador tries to ,,
,
,,
,
12,
,
Intra Unit I Integers packet Worlcing with Integers-
Assignment #8 : mOre than one
l
procedures. and
Integers-
Assignment #7 : negative
I
,
A.CN.2
Understand the
corresponding
procedures for
similar
problems or
mathematical
concepts
,
,,
,
,,
, A.PS.!
, Use a variety of
,
problem solving
strategies to
understand new
numbers
: mathematical
14
17: Learning from
the Progress
To review and
learn from the
Progress
Check
Check
Intra Unit 5 Distributive Property packet Quiz 2 Unit 5 The Distributive Property ,
~"""~
......~ ....
,,
, content
A.PS,g
Determine
,,
information
,,
,,
required to
solve a
problem,
choose methods
for obtaining
the information~
and define
parameters for
I acceptable
,
,
15
To Graph with 18: It's Cold Up
positive and
There
negative
numbers
: solutions
A.CN.5
Understand how
quantitative
models connect
to various
16
To solve word 19: Word
problems
Problems
involving
positive and
physical models
and
.!epresentations
ACNA
(See above)
negative
numbers
*17 To review the
mathematical
concepts of
the writ
All standards
from previous
lessons in the
unit
Ramp Up to Algebra-Unit 3-Factors and Fractions (RU Unit #4) Day
I
Goal
To define
factor and
multiple, and
In understand
how factors
and multiples
appear in the
multiplication
tabl~ a..•.l ... n
: the number
line
Ramp Up Lesson
I: Multiples and
Factors
Intro to Algebra
Lesson (plus extra
worksheets)
•
•
•
•
Word Wall
Suggestions
NYS Algebra
Standard
multiple
factor
factorization
product
A.CM.2
Use
m.athematical
representations
to
communicate,
with appropriate
accuracy.
including
numerical
tables,
formulas,
functions,
equations,
charts, graphs,
Venn diagramsl
and other
2
To identify
prime
numbers and
composite
numbers
2: Prime and
Composite
Numbers
• pnme
• composite
• distinct
diagrams.
ACM.3
Present
organized
mathematical
ideas with the
use of
appropriate
standard
notations.
3
To writc any
3: Prime
natural
Factorization
number
greater than I
as a product of
including the
use ofsymbols
and other
representations
when sharing an
idea in verbal
: and written
: form
ACM
.pnme
factorization Communicate
logical
• Fundamental
arguments
Theorem of
clearly,
Arithmetic
its prime
factors
• Factor tree
4
To define and
understand
common
multiples
4: Common
Multiples
5
To understand
and use the
relationships
between the
greatest
common
factor (GCF)
of two
numbers and
their least
common
multiple
(LCM)
To review
factors,
multiples,
pnme
numbers,
pnme
factorization,
greatest
common
factor, and
least common
multiple
To solve realworld
problems
involving
factors and
multiples
To mUltiply a
fraction by a
whole
number, and
to write
5: GCF and LCM
i
6
7
8
• Common
multiple
• Least common
multiple
(LCM)
• Greatest
common
factor
(GCF)
6: Reviewing
Multiples and
Factors
7: Making Sizes
that Fit
10: Multiplying
bya Whole
Number
• Inverse
showing why a
result makes
sense and why
the reasoning is
valid.
ACM.5
(see above)
AN.I
Identify and
apply properties
of real numbers
ACN.4
Understand how
concepts,
procedures, and
mathematical
results in one
area of
mathematics
can be used to
solve problems
in other areas of
mathematics
A.CN.6
Recognize and
apply
mathematics to
situations in the
outside world
AN.I
(see above)
equivaJent
fractions using
division
9
To multiply
II: Multiplying
Fractions
• horizontal
• vertical
AN.I
(.eeabove)
To add and
subtract
8: Adding and
Subtracting
AN.!
($eeabove)
:fractions with
Fractions
• Equivalent
fractions
• fraction
• numel1ltor
• denominator
• mixed nwnber
• improper
fraction
• simplest form
fractions, and
to represent
products as
areas of
rectan~les
10
common
denominators
11
12,
13
To add
lhletions that
have different
denominators
To review the
concepts of
9: Adding with
Different
Denominators
12: Progress
Check
AN. I
(see above)
ACM.2
(seeahove)
fractions and
adding,
subtracting,
14
and
multiplying
fractions
To subtract
13: Finding
fractions with
Differences
different
denominators,
to see
subtraction as
the
inverse of
.
• ascending
order
• descending
order
• Inverse
equations
A.N.I
(see above)
addition, and
to use
subtraction to
order fractions
15
To review
14: Addition and
addition and
subtraction of
fractions and
the inverse
Subtraction as
Inverses
relationshin
AN.!
(see above)
16
17
.
18
19
20
21
22
between these
operations
To divide with
fractions
To calculate
with fraction
in order to
solve
problems
involving
measurement
and
calculations of
length and
distance
To apply the
number
properties to
operations
with fractions
TO,review
work based on
the skills and
concepts
discussed in
the unit
To reinforce
understanding
of the topics
covered in the
previous
lesson
To calculate
with fractions
in problems
involving unit
fractions and
enlargements
To review the
work of the
unit and
prepare for the
final
assessment
16: Dividing
Fractions
15: Shortest
Distance
• reciprocals
A.N.l
(see above)
A.R4
Select
appropriate
representations
to solve
problem
situations
17: Mixed
Operations
A.N.I
(see above)
18: Progress
Check
A.N.I
(see above)
19: Leanning
From the Progress
Check
All standards
from previous
lessons in the
unit
20: Shortest Time
A.RA
(see above)
21: The Unit in
Review
All standards
from previous
lessons in the
unit
Ramp Up to Algebra-Unit 4---Exponents & Polynomials (not in RU)
Day
Goal
Ramp Up
Lesson
*1
To apply properties
Not in RU
of exponents
involving products
Intro to
Algebra
Lesson (plus
extra
worksheets)
Arlington
ProjectExponents
and Their
Properties
Find a Match
Word Wan
Suggestions
Power,
exponent,
base, Product
of Powers
Property,
NYS Algebra
Standard
A.A.12.
Multiply and
divide
monomial
expressIons
Powerofa
with a common
Power
base, using the
Property,
Powerofa
exponent
properties of
Product
Property
*2
*3
To apply properties
of exponents
involving quotients
To use zero and
Not in RU
Not in RU
negative exponents
Arlington
ProjectExponents
Quotient of a
Powers
A.A.12.
Same as above
Property,
and Their
Powerofa
Properties
Quotient
Property
Can you Build
This?
Arlington
reciprocal
Project-Zero
A.A.12.
Same as above
and Negative
Exponents
How did the
Light Dress
up for the
Costume
Party?
Why did the
Farmer Open
a Bakery?
*4
To convert numbers
Not in RU
How did
Slugger
McFist Get a
Black Eye?
Intro. Lesson
Standard
A.NA.
• from standard form
#8 - Scientific Fonn,
Scientific
Notation;
Notation
to scientific notation
Scientific
Notation
Worksheet
. Multiple
Understand and
use scientific
notation to
compute
products and
quotients of
numbers
Representatio
n Activity on
Scientific
Notation
*5
To convert numbers
from scientific
notation to standard
form
. A.N.4.
Understand and
Not in RU
use scientific
notation to
compute
products and
quotients of
numbers
*6
To introduce and add
polynomials
NotinRU
Arlington
Monomial,
Project-
binomial~
Combining
Like Tenns
trinomial.
polynomial,
degree,
*7
To subtract
polynomials
Not in RU
WestSe.­
Addition of
Monomials
and
Polynomials
• Arlington
ProjectCombining
Uk. Terms
A.A.!3.
Add, subtract,
and multiply
monomials and
polynomials
leading
coefficient
AA13.
Add, subtract,
and multiply
monomials and
, polynomials
West SeaSubtraction of
Monomials
and
Polynomials
--.g
• To multiply
monomials by
monomials
Not in RU
Arlington
ProjectMUltiplying a
Polynomial
AA.I3. Add, subtract, and multiply monomials and
bya
Monomial
polynomials
West SeaMultipHcation
of Monomials
*9
To multiply a
monomial by a
polynomial
Not in RU
Arlington
ProjectMUltiplying a
Polynomial
bya
A.A. \3.
Add, subtract,
and multiply
monomials and
polynomials
Monomial
West SeaMultiplication
of
Polynomials
bya
Monomial
*\0
*11
*12
To mUltiply a
binomial by a
binomial
To multiply a
binomial by a
polynomial
To divide a
monomial by a
monomial
NotinRU
Arlington
ProjectMultiplying
Polynomials
Not in RU
West SeaMultiplication
of Binomials
Arlington
A.A.13.
Add, subtract,
and multiply
monomials and
polynomials
Project-
A.A.!3.
Add, subtract,
Multiplying
and mUltiply
Polynomials
monomials and
polynomials
A.A.14.
Divide a
polynomial by a
monomial or
Not in RU
binomiaL where
*13
To divide a
polynomial by a
monomial
Not in RU
Division of
Monomials
and
Polynomials
the quolienl has
no remainder
A.A. 14.
Divide a
polynomial by a
monomial or
binomial. where
the quoLient has
no remainder
Not in RU
To review concepts
ofexponents and
polynomials
,,• *14
West SeaAddition and
Subtraction of
Polynomials
,,
,,
West Sea~
Properties of ,
,,
Exponents
,,
,,
M
___
West Sea­
' Multiplication
: of
Polynomials
,,
All of the
standards above
,,
,
,
,
,
,
I
Name _____________________________ Date _____________________
A. Write as factors. Then, rewrite with exponents.
Factors _____
.~
_________
Exponents _____________
Factors ________________
Exponent. ____________
Factors ________________
Exponents ____________
B. Tell whether each equation is true or false.
the right side to nl.ake a true equation.
]f false,
correct
1. r'. r' = ,9 ___________________________________
2. x 5
•
3xy =
X 2 3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___
y
3. 4n'p' np2 = 4n'pZ ____________________________
C. Multiply or divide.
d'
2.
1. ~
);,0. a2
5. nz.. n- 1
3
7.
x
XS
8.
a- 2 b3
a- 3 b
D. Find each number named in scientific notation.
1. 1.44
56
x 10"
2. 6.23
Cha:pter 10· Exponents and Functions
x 10-'
3. 2.06 X 10- 2
Name _ _ _ _ _ _ _ _ _ _ _ _ __ Date _ _ _ _ _ _ _ _ __
~':Pivi:SIP"'ia;id'R\.Ile'<()f'EXpQnfJnts;>" ',' , ',··.:Eiif!r6is~;!)7'
LeSSOf/S
10,3 to to,5
A. Multiply or divide.
1. x S • x7
2.
0
6
Ii'
3. b4 cZ • bS
4­
alOif
a5 b& 5, 3x 3y2 • 2xy2 4 6
6. 20n4 c
4n c·
B. Find the missing term.
=
1. e' •
• 2a 3
3. 5. a 2 b 3
• _....
,16
=:
=
IOa7
a 4b 4
*x2.=xs
2. ~
4. 6y'
l8y'
• n 2x S = n 3x 5
6. C. Find the missing term.
a'
1. . . =
~
~
~
•
~
~
~
.~
0
~
~
0
E
~
•2
c
»
Q
"1>
Y
3 . ~=4"s
2n'
.. 4. 24x
10
•
=
4x9
6. ~=x3l
y
~
,
"
~
0
~
D
0
@
1"
2. ~ = y6
~
1
1• (13 ~
8
I
"
1
Chapter 10· Exponents and Funclions
57
Narn~ ~"'~''---,"
.. _'''':-''c:,-=::"-;c~c:.'-,,.'~C"'"-'.'~'"._ _--'----'­
,
".- .' ."
": ''',
Date _ _ _ _ _ _ _ _ _ _ _ __
.. - , ' .
'111' •..
8x+ 5y+ "17x= -9x+ 5y
1. 9x+ 4x=
14. 3.5y- 7.2y=
2.
15.
17x+ X=
A.7y- 2.3y=
3. m+ (4m) =
16. 30 + 5c - 90 =
4. 7x- 8x=
17. 2x- 9x+ 7 =
5.
18.
140-190=
7x-8-11x=
6. -0+ 90=
19. 3x- 3y- 9x+ 7y=
7. 6xy+5xy=
20.
17x+ 4 - 3x=
8. "9m- m =
21.
3x-7y- 12y=
9.
150+ (110) =
22.
110-130+ 150=
10.
'14x+ 13x=
23.
17x + 50 - 3x - 40 =
11.
5x'y+ 13x'y=
24.
6x+ 9y+ 2x- 8y+ 5 ~
12.
21xy+C9xy)=
25.
3xy + 4xy + 5x' Y + 6xy' =
13.
17x+ 1 ~
26. "25y - 17y + 6xy - 3xy ~
Published by lnst[lJCIIonoI Fair, Copyright protecled.
I
Page 12
Q-7<12<!-1788-J Mgetxo
J..
-
Name ________________________
Date ___~ ______________
I
.p
,
,:Jj
•
What's Not to Like? I Simplify ea~h expre~ion by combining like terrrls. Circle the expression in each ; problem that does not belong. Place the letter above the problem number below. ~
~
~~
~
~~
~~
~~
1. A. Sf + 3r + 91 - lOr
E. r+t-8r+ 13t
I. -r+4t+ 10f+8r
2, D. 12x- 3y+ x+ 2V
E. 3 (4x-3y) + x+ 3y
F. 4 (4x- 20 - 3x+ 7V
3. E, 4 (y - 7x) - V
O. 7 (4x + y) + 7y
I, -30x - (2x)
W. 4x+ y- 7x- lOy
4, U.6(x- v)-3(3x+)I)
V. 3 (3x- y) - 6y
5. Q. 3 (r-1) -4r+ 5 X. 2 (3 - 2r) - 4 (2 - "
Z. -r + 7 + 3r - 9 - 2r
M. 10 (x + )I) + x + y
N, "(x + y) - 2 (x + 0
B, 3 (0 + 2b) - (b + 2(1)
7, A. 3 (2b - a) - (20 - b)
8,
I. '0 (a- b) -2 (a-b) + 8 (a-b)
U. 6(a- b)- 4 (0- b) + (0- b) O. (0- b) -(0- b) + (0- b)
9. R. 3 (x - y) - 2 (y - x)
10. L
~4
C. 2 (a + 2b) - (0 - b)
S.2(x-y)-3(y-x)
(X+ 2 (5xy- x)
T. 3 (y - x) - 2
N. -2 (3x + 3 (lOxy+ 3:<)
M. -4 (x+ 5 (3xy+ x»
Two expressions in each problem are
2
5
8
1
4
7
Page 13
10
ex - y)
3
6
9
r
Dote _ _ _ _ _ _ _ _ _ _ __
Nome _ _ _ _ __
Exponents 1. Write in exponential form.
4· x· x· y. Y· y=4x2y3
The cube of c - 4
6.
2.
=(C_4)3
The quotient of 3 and 1I1e cube
of y+2
mn*mn-mn·mn
8, (-x) (-x) (-x)
4.
5(0+1)(0+1)(0+1)
5.
(a + b) squared
9. 3' ab' ab' ab' ab
10. The square of
=
= ='3
II. Evaluate each expression if x '1 , Y 2, Z
5X 2 j2 = 5 •
x· x· z· z = 5 • ~1
• ~1 • -3 • "3 = 45
6.
lOzS
3. 4y"z 5,
-(xyz)
10. llx2
Page 26
x2 y - 3
.ame ________________________
Date _________________
Adding and Subtracting Polynomials
(x" +2x'- 8x) - (2x 2 + 7x- 5) = x3 + 2x" -8x+2x"-7x+ 5 '" x" + 4x" -15x+ 5
1. (4x + 2) + (x - I) =
2. (5a-2b+4)+(2a+b+2)=
3. (30 + 2b) - (a - b) =
4.
(x" + y2 -
5.
(40 2
-
ab) - (x" _ V' + ab) =
5ab - 6ti) + (lOab - 6a 2
-
8b') =
6. (4x2 -2x-3)-(5x-4)= 7,
(4a 2 - 4ab - ti) + (0 2
- ti) + (2ab + a 2 + b 2 )
=
8, (4x' -6x2 + 3x- 1) - (8.0 + 4x2 - 2x+ 3) =
Q,
(a + 2b) + (3b- 4e) + (50- 7e) + 3b=
10. (x 2 _ 2XV + y2) _ (x2 _ 2XV + V') =
11. ex+ 3)1) + (3x- y) - (x- V) =
12. (2x' + Sy' -
22) -
ex' -
0 - 22) + (4x 2 -
Sy') =
13. (2x+3)+(2x"+x-5)=
14. (2y+ 3x-4) + (9- 8V-5x) + (3x+ 4y- 2) =
15.
c20 + 8) - (3y2- 4y- 6) =
16. (7y+ 4x+ 9) - (6x- 8y+ 11) =
Find the perimeter.
2x-6
17,
18.
,
x+3
x+3
2x- 6 x-2
3x-7
Page 27 Name _________________________
0016 ________________________
Monomial Quiz
Polly N. Omlal did not understand the rules of exponents when she
completed the Monomial Quiz. Find and correct the 10 errors Polly made.
Monomlol Quiz
1. (4C)2
i:
i
Name:
=8e'
Polly
1l. (_xy2)3 (2x' yi
=-4i'y"
2. 4(e)' =4c2
12. x 2 (2xt) (4z5) = -3'x3Z6
3. (2cPb) (40ti) =B0 3b3
13. (3pq2r) (g32 r) _- pq4r'
4.
14. (-x) (2):0 (3xyz) = -6x3 y2 z
(4pq) (_p2 q 3) = -4p 3 q '
5. 2x (_xy) (_y')
= 2x2 y3
15. 2x2 (xy2)' (XZ2)2 = 2X4 y' Z4
6.
(<looi = 16be'
16.
7.
-abc' (edl = -0005d 3
17. (-x) (_x 5 ) =
8. -2 (3x)' (xy) (2x) 2 = -12x3 Y
9.
0(20 2)3
=60 6
10. 35 (2s1)2 = -12s2f
lB.
(2u)' (u'v)' (>0 = 2u'v"w
:I'
(5x2) (7xy3) = '35x 3V3
19. (-x 3 y) (6xy)2 = -6x' V3
20. (<Iff') (2rf) (t2) = sr< f'
Explain to Polly how to calculate the power of a power, for example, (5x' l
Page 28
Name _ _ _ _ _ _ _ _ _ _ _ _ __
Date _ _ _ _ _ _ _ _ _ _ _ ___
Multiplying a Polynomial by a Monomial
-20 2 (9 - 0 - 402 ) = -2a' • 9 - (20'· a) ­ (20' • 4a') =-lSa' + 2a 3 + 80'
(x + 2) (2X2) = 2x' • x + 2X2 • 2 = 2i' + 4x'
l. 2 (x"
- xy+ Sy') = ';, =
2_ -2n (4 + 5n3 )
3. c'd (c 2 cfl + 2cd' + a) =
2xy' (2 - x- x2 y) = 4.
5_ (a' - 3ab - 2b 2 ) (2ab)
6_
an (Sn 2 -
7_
ew' z - 2wz + z) (-z') =
2n)
=
i: =
,,
.,
B_ -3ati' (a 3 b' - 2a'b) =
9. 4x'y(9x' - 6xy2 - 7) =
10_ -6k'rrf (2k - 3m + 4km - k'm') =
11. -n' (n + 4n') =
12_ (4x' - 7x) (-x)
=
2x'(i'-2x'+Bx-S) =
13.
14. (6x 3 ) (3x' - 1) =
15_ (6x- sx" + 8) (3x)
=
16_ -Sx' (2x 3 + 3x' - 7x + 9) =
Find the area.
17_
18.
x+4
2xL----1
A triangle has a base length (b)
of 2x + 4 and a height (h) of Sy.
1
(Area = - bh)
_I
2
Page 29
, .• '(,s' - s+ 3) - 2 (8 2 - 8 + 3)
i" S' 52 ­ S· 5 + 5' 3 - 2 • 5· - 2 (-5) - 2 • 3
~ 53 ­ 52 + 35 - 25 2 + 25 - 6
= 5' -35 2 + 55- 6
·
1. (z - 3)(z+ 3) =
'I!
"
"
i:
2,
(3t-2)(t-3)=
11.
(2x- 1) (x' + x+ 3)
=
3. (0+5) (0+ 5) =
4. (0+ b)(2x+ y) =
6, (4x - 5)(4x + 5) =
7,
(1.6n - 9)(O.2n - 5) =
8. (2e + d) (c' + 2c + 2d) =
17, (x 2 - 3) (2x' + 3x+ 5) =
Page 30
F
Name ______________________
·
"
Date _____________________
Multiplying Binomials Using FOIL 1
I.
4.
II I
1.
2.
3.
4.
first
Quter
Inner
last
(x+ 5)(x-3) =x' x+ x ('3) + 5' x+ 5 ('3) = X' - 3x+ 5x- 15 = X' + 2x- 15
I
"
I W I
il
.,,
2.
1. (X+ 2)(X+ 3) =
,
'
10. (2x+ 5)(4x- 3) =
,
'I
·• .
.'
2.
(y+ 7)(y+ 4) =
11. (n - 7) (3n - 2) =
3. (x- 8)(x+4) =
12. (5x+ 2) (3x-7) =
4. (x- 8)(x-4) =
13. (4x+5)(2x-3)=
5. (y- 4)(y+ 5) =
14.
(-x - 4) (4 + 3x) =
(x- 9)(x- 2) =
15.
ex + 2y)(2x+ 3y) =
7. (2x+4)(x+3)=
16.
(6x - y) (3x -
8, (3x+ 2)(2x+ 5) =
17. (4x + y)(3x - 4y) =
6.
9,
(4x- 9) (3x + 1) =
20 =
18. (50 + 3b)(40 - b) =
Page 31
Dole _ _ _ _ _ _ _ _ _ _ __
(x+2) (x~4) = x2-4x+ 2x- 8 = x'- 2x- 8
F
ColumnA
I
I l
Column B
1. (x-4)(x+ 3)
I. 25x 2 + 70x + 49
2. (x+ 2)(x- 6)
T. 6x 2 -7x- 3
3. (3x + 1)(2x - 3)
O. x 2 -x-12
4. (x+ 3) (6x-1)
A. 24x2 + x-lO
5. (5x+ 7) (5x+ 7)
l. O.6x' + 15.2x + 5
6. (4x-l)(4x+ 1)
R. x2-4x-12
7. (O.2x - 3) (O,3x - 2)
M. 6x 2 + 17x- 3
I
!,,
a
8. (3x + 1)(O.2x+ 5)
S. 24x2 + 58x+ 35
9. (4x + 5)(6x + 7)
I. O.06X2 - 1.3x + 6
10. (8x - 5) (3x + 2)
N,
16x' - 1
All but one of the answers Is
32561471089
What is special about the factored form of answer N - 16x' - 1? _ __
Page 32
Name _________________________
Dare ________________________
Special Products
(2x+ 5) (2x- 5)" 4x' - 10x+ lOx- 25 =4x'
(0 + b)(o - b)" 0' - b'
,
II
1. (x+ 3)(x- 3) = 10, (2X+ 9) (2x- 9) =
2, (y - lOHy + 10) = 11. (7x - 5)(7x+ 5) =
3, (0+4) (0 -4) = 12,
(x+ y) (x- y) =
13,
(5x- y) (5x+ y) =
4,
(x+ 7) (x- 7) = 5, (2X+ 1)(2x-l) " 6,
(5y - 6)(5y + 6)" 7,
(4x+3)(4x-3)=
8,
(3n+ 7)(3n-7)=
9,
(3e + 4)(3e - 4) =
14, (2x+ lly)(2x-l1y)"
15, (3x - 7y) (3x+ 7y) "
Thinking About It
19, The product of the sum and difference of two terms is equal to the _____
of the squares of the terms,
20. Why is the product of a sum ond difference of two terms a binomial and
not a trinomial? ________'____________________
Page 33
"]
Name _________________________
Date _ _ _ _ _ _ _ _ _ _ _ _ __
Squaring Binomials 1,
(x-sf= s,
(x-si=
I,
4, (Sn + 1i =
11, (4x- vi=
12, (6x- 5vi =
,
!i
, i:
5. (y-
lOi =
13.
(3V-5z)2=
14,
(70 + 2bi =
"
8.
(20+ 5)2 =
16. (50 + 3bi =
Page 34 Name: _ _ _ _ _ _ _ _ _ _ _ __
Dare;
----­
Exponents and Their Properties - Multiplying and Dividing Monomials Algebra 1 ­
Wben we want to express a product of the same number like3· 3·3·3 we can usc a shortcut notation
4
3
,-~
.....
The exponent teUs how many times the base is used as a faetur in the producl
Exercise #1: Write each of the following in the form ofan expanded product.
(al xl =
(bl 4' =
(c)
(2x)' =
(d) {x+5}' =
Exud:se #2; Express each of the following with an equivalent expression involving exponents.
(a) z·z-z·z·z= (b) 6-6-6 =
(d) x'x' y. y. y= (e) (x+ y}(x+ y)(x+ y)=
Exercise #3: Consider the product shown below:
(al Write both parts oftbi, product as extended products.
(b) Write the product x4
involving an exponent
x
3
as an expanded product and in tenns of an equivalent expression
Exercise #4: Express each of the following product<; as a single variable raised to a power.
(a) X' -,,'
(b)
x··x'
(d)
I-l-y"
Al'8e\># I,UIlitN6,,·~Algc:b&l-Ll
Tru:::Adiagtoo A.lg1:bto\ Projeo:t. LaGrangC\-ith;. NY 12540
\ <i !
We also must be able to divide monomial ~pressions that have the same base.
understanding how thi. process wow i.the following:
The key to
Any quantity divided by itself, except for zero, is equal to I.
,
Exercise #5: Consider the quotient x 3 •
X
(a) Fill in the following:
x' = x' . _ __
(c) Rewrite the quotient as the product ofwo
fiaCtiODS. one of them being equal to I.
(b) Rewrite the nwnerator of the quotient
using (a).
(d) Simplify the quotient using the Mulliplicotive
Identity Property of Real Numbers.
Exut:.ise #6: Fill in the blanks in the box below:
ExPoNENT PROPERTIES
For any real numbers a and b~
Exerch'e #7: Write each product or quotient in its simplest form.
(a).
0
4
·a4
;;::
(b)
i'·x'=
(e)
34 _3 2 ;;::
Cd)
y2'i=
(h)
(i)
2'·2·2'=
(1)
YS''''''y=
(g)
(x')' =
(k)
A!gebTa l.lfni!~-~ Ala<bla-L! The Arl~oll IJgd;ni P'rojr.::t, l..aOrangeville, NY t254D 6y' =
z,
=
"
0)
Ce) ..
lOy'
(1)
b'
=
b'
:;?y5
xSy2 =
(i'l' =
.'
Name: ,_ _ _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ _ __
Exponents and Their Properties - Multiplying and Dividing Monomials Algebra 1 Homework Skin
17.
Express the product with exponents.
I.
a·a·,a·b·b=
2.(2.)(2.)(2.) =
3.
(2x){2.)y.y=
18.
19.
Express the product in simplest form.
4. h' ·b=
Reasonlng
SimplifY.
5.
l·y') ;;; 6.
2 l
X 'X
7.
1'/,4
8.
y.y=
4
x'
20. ",=
21.
z(2z)'(2z)=
c>d
·x4 =
'n=
2
9.
0
10.
";.•'=
'a =
23.
7
,
x-·x
-x•
24.
II.
Z4 'Z4;;;;
Express the quotient in simplest fonn.
13.
x' =
25.
x'
Determine True or False for each.
State the reason for your answer.
15.
27.
16.
28.
Name:
____________
--------------------Exponenlll alld Their Properties - Multiplying alld Dividing Monomials
D~
Additional Practice Problems
Reasoning
Skill Find the product. I.
x'·x' ; 2.
y'Z'y=
3.
y'.y'.y';
4.
a·a·a·a=
af, _0 4
5.
6. 7.
o
4'
20. -;
2J.
-;
22.
-= 4
x' x
"
X' Simplify.
- ;
33.
--;
r
24.
-= 4x
13 _,2
,
34.
a.b3
-=
35.
(ab)'
---=
ab'
y' 23.
4x'
32.
b'
=:
8.
X 4 .x,5=
9.
l-y':::::
10.
/ . / =;
11.
w'~'W'W2=
12.
p~.p3.p~
B.
X4·~·X= h'·ll=
15.
X4
25.
26.
yz.y::.
17.
x-x-x"'"
18.
mlJ _m17 :::;:;
;c4y2
36.
x'y'
37.
4x4l
2xy' =
;
38.
xiJ 'x!;::::
I' 39.
--=
z'
-= z, z'
-= 5' 28.
-= 29.
p' --=
p'
30.
3l. b" ·b' = 40.
5
=
x'
x'
"
_Xl = 16.
t' "
27.
14.
19.
.
=
,'J .f7 ::::::;
rS ,,.3
Find each quotient
X'ly7
z=
xlyzJ
x4 / ,
x4y2
41.
---;
-= 42.
~--=
x" 43.
6' 6'
x lO -=
xy'
x'-y'
x',
1,' =
Alg.;bm l, Uoit.tl<i-Quadntle Algcbm -Lt
TheJufu>gtoo Algl:bra Proj~ ~11e, NY 1t540
\ 'I 'I
Name; _ _ _ _ _ _ _ _ _ _ _ __
Date;
----­
Zero and Negative Exponents Algebra 1 In our last lesson we leamed how to simplify products and quotients of monomials using Jaws of
exponents with positive integers. But, zero and negative exponents are also possible,
ExercIse #1: Recall that
~ = x.....
x
(a) Usit\g this exponent law, simplifY each ofthe following.
(b) What must eaeb of these quantities equal, assuming none of the variables equals zero?
Exercise #2: Simplify each of the following:
(al 1250 =
(b)
(c) 5xo =
(2y)" =
We can investigate negative exponents in a very similar fashion to the zero exponent The key is to
define a negative exponent in such a way that our fundamental rules for exponents don't need to
change.
2
Exercise #3: Consider the quotient -~y
x
(a) Write this quotient using the exponent Jaw
from Exercise # I.
<
(b) Write this quotient in its simplest form without a negative exponent Ex£rcise #4: Rewrite each ex.pression in simplest terms without the use ofnegative exp'onents.
(b)
x-' =
(c) 2-3 =
(d)
y-10
=
\~5 Exercise #5: Rewrite each ofthe foUowing monomials without the use of negative exponents.
(d)
y_,-, ~
x
NEGATIVE ANI> ZERO ExpONEN'IS
[r a is any integer- and
(l)
,r ¢
0 then
c~1..
(3)
x'
-
~
,·=1
5
Exercise #6: Which of the following is equivalent to x_s Y-l ?
x y
l
(I)
-=­
l.
(3)
,
x'y'
I
(4) ...,-;;
xy
(2)L
i'
Exercise #1: Rewrite the following expressions without negative or zero exponents.
(1)) 4' =
(e)
(g) 3x· ~
(i)
~, =
x
-r' =
(I) (-1)· =
, -2
(I) r t
=
Exercise #8: Evaluate each of the foUowing expressions using the -values a = -1 , b::::: 2 and c:z.: 3.
Use the Sl'ORE feature on your calculator to aid you.
(b)
(aber' =
Name:
--------------------­
Date: _ _ _ _ _ __
Zero and Negative Exponents
Algebra 1 Homework
SldIIs
For problems 1 tbrough 36,
rewrite without zero or 17. -3' = 33.
x'
2y-' =
negative exponents. 34.
-3;
y'" =
19. (-3f' =
xOy-3
35, ~=
36, 2x-'y-4
=
-I1'-' =
21. (
2,
6, 2-4
=
22.
/1)-' =
l"3
1
7. Z-, =
1
23. 1-6 =
Use the STORE feature on
your cakulator to belp
evaluate the following.
37, y-' for y=2
24. (-5" =
8. • =
4
9. (-3r'=
26. -2-' =
II. :5x-4=
x'
12. y-J
=
27. (-2f' =
,
40, (x+3r' forx=-4
28. (-2r' =
29, (-T'r' =
. 30.
15. T' =
41.
forx=-1,y=2
4
2
42. ( x, y ')' for x=-,y=-~
3
43,
16. (16x'y-'j' =
x~y
7
_
2
4
x""x:Y fo(x=-,y=-­
S
3
A1getna 1, U!litfl6~~ AlgdJI'lI-ll
Thc~AJgdlta l'tvjed..1..eGmn~r~
NY 11l4n
1'\ 1 Reasoning
Fill ill III. missing 0 for
oach of Ihe foRowing.
55. Evaluate each of the
following products:
(a) 2' ·2-' =
2
(b) 5
.,'=
63.
(-4)" =0
(0) 10'" ·10' =
46. _I ,,[J->
25
56. Which ofthe following
is correct?
Find the value of x that
makes each statement true.
48. 6-'=~
66.2'·2'=2"
49.
uP
1
10,000
Explain why the other
choice is incorrect.
True or False
Write the answer to each of
the following as a single
number.
57.
52. [-1+(5+2)"]'=
58.
GJ'
=2
(~r =-~
59. (-Zr' =.!.
4
A1gebn t, Unit 116 -~ A,lgebr./. - Ll
'1'b<! A~ll Algebrn ~ l.!!Gr.mgC'olUe, NY 11540
Name' _ _ _ _ _ _ _ _ _ _ _ __
Date;
Combining Like Terms Algebra 1 We have already seen the process of combining like terms when solving linear equations. In this
-lesson we will broaden our understanding of what constitutes like terms and how to combine them.
F"ust. we review the reasoning process behind combining like linear terms.
Exercise #1: Fill in tho blanks for each with the real number property that justifies the particular step.
(I)
(I) _ _ _ _ _ _ _ _ _ __
6x+2y+3x+4y~6x+3x+2y+4y
(2)
~(6+3)x+(2+4)y
(2)
=9x+6y
Exercise #2: Combine each of the following like tenns using the Distributive Property,
(a) 2x+7x=
(b) -5x'y+2?y=
Clearly like terms are those monomials in an expression that have the same variables raised to the
sarne power. We should be able to combine them mentally by fIrst identifying like teons and then
summing aU coefficients of those tenus.
Erercise #3: Combine an like terms in the following expressions.
(a) 8x+4-6x-7
(b) 4y'-20+3y'+S
(d) h'+3x-7+5x'-8x+3
(el
-3? +9x-6+4x' -2x-8
Exercise #4: Which of the following expressions cannot be simplified?
(1) 3x+6x
(3) 3x+6y
(2) 6y-3y
(4) 2x' + 7x'
Algebrn 1, u.ut #6 - Qu,adt.w.e AlgcbA - LJ
TM Arlington Algch!a: Projcd, La~1k, NY l2540
(e) 3w-2(3-5w)
We will oftentimes be asked to combine terms either in sums or differences. Differences can be
particularly tricky because subtntction is not commutative, mea.ning the order in which you do the
subtraction will change the result.
Eurcise #5: Which ofthe foUowingrepresents the sum of (3x1 -3x+8) and (_Sx 2 +4x+2) 1
(I) -8x' -x+1O
(3). 2x' -x+l0
(2) -2x' + x+ 10
(4) 8x'-7x+6
(I) 3x'+3x+3
(3) Hx' + 13x+ 3
(2) -3x' +3x+9
(4) 3x' +13x-9
"ExercJse#7: When (4~-8x-3) is subtrncted from (il-2x+l) the result is
(1) -3x' +6.>:+4
(3) 5x'+6x+4
(2) 3x' +6x-2
(4) -3x' -6x-2
- - - - - - - - - - Additional Classroom Practice - - - - - - - - - ­
Simplify by combining.
l. 5x+2y-I!x+7y= 2.
(2x-4)+(5x+9)=
).
(2x-Y)-(5x+y) =
4.
3x2 +4x+2+2x2-5x-1 = 5.
3x2-5x+1-4~ +2x+3=
6.
5x z +8x~(3x2- - 2x) =
7.
(x'-2x)-(3x'-7x) =
8. From 4x 2 + 2x- 3 subtract
9.
Add
to.
Subtract
4a 2 + 2ab+ 3b7 _a 2 _3ab+b 2
3/ +4y-5
5y'-Zy+!
Xl -
AJgcl!!a I, Unit fJ<i -QwdnI.I.kAJgwra-Ll
The Ailingtoo Algebra ho.l«t, UlGtangevilk!, NY IZS41J
3.
12.
How much less than 5x 2 - 3):+ 2 is
Y+5?
Name:~~
_ _ _ _ _ _ _ _ __ nate: _~~_ _ __
Combining Like Terms
Algebra 1 Homework
t5.combill~ 2x+4.-3;c+2.5+~
SkiD
Combine as indicated.
16.
I.
9%+7+2.1'-4=
2.
4x-3-(3x-2)~
3.
(x' -4x)+(3x-5) ~
4.
4t-3t' +5+{-2t' +3t-5);
5.
(12t-9) -(51-3)=
6.
eombiIic:
a 2 -Sa+25;-a2 +6a+9
19.
Subtract 5x+3y from 4x+ y
(0-60)-(40+80)=
20.
From 5.1'+3y subtract 4x+y
7.
(3XY' +3xy-4x'y)-2x'Y+ xy' =
Applications
8.
(3x-4x')+(-4x-.') =
21. Represent the perimeter of a sqWIJ:e whose
side length is given by the binomial 4x+6.
9.
1+3y'-5y'+6y'=
10.
Sx 2 _4xy+6y 2 _xi + 3xy+ 2y2 +xy;:;;
II.
a1. +3'a+ 5+ 2a'2 -4a-l-·5;: +20=
12.
_r2 s2 +2i2:VZ ~6r2$Z _5t'2v2 ::::
13.
a+8c+5b-c-Sa-5b ::::
22. Represent the perimeter of a rectangle
whose width is y and whose length is the
binomial 2y-7.
-
23. Write !he length of the arc ABC as a
binomial involvirig x, y, and z.
A
2xy-z
B
14.
[-4y+(1O-5y))+2- Y=
lxy+2z
Alzd,Im I, Unit /J6 - ~ Algd:IR - 1.)
Tb¢.Arllngtac Algdlra rroja:t, ~e«iU<:. NY 12540
c
24, Express the perimeter orthe triangle os.
29,
binomial,
HoW much greater than x 2 - xy is
Sx"+IO>:v?
.<-4y
Sx+8y
30.
What expression must be added to
3x2 ~ 5x + 4 to give the result
7xl -5x-<i'J
2:ct-3y
25. The perimeter of a triangle is given by the
expression 12x" -4.<+ 15, Find the third
side of the triangle if the other two sides
31. From the swn of 6x-5a.nd 2x+4subttact
3x-9
measure 4);? +3 and 5x-4.
32, Subtract the sum of 2):."1' ­ 3x+4 and
x 2 +2x-1 from 6x2 -2x+l
Reasolling
26. RecaU that two expressions are additive
inverses if their sum is equal to zero. Find
the additive inverse for each of the
following:
(a) 7x-4
(b)
c' -4c+5
27,
What is -3a+5b decreased by 9a+ 2b
28.
Byhowmucbdoes 4x-3exceed
7x+51
AJgclmll, UIDt&'6-~Al.6ebra-U
'l'hc Arlu,gtooAlgdlmProJeet. u~ NY 12S40
Name:
Date: _ _ _ _ __
----------------­
Multiplying a Polynomial by a Monomial
Algebra 1
In the previous lessons. you've worked with monomials and their exponent properties. In thls. lesson
we will begin to work with polynomials, or expressions that contain more than one monomial. The
most common polynomillls are binomials (those with two monomial lenos) and ttinomials (those
with three monomial terms). First, we review the important real number properties associated with
multiplying monomials.
Exerdselll: Fill in the blanks for each of!he fullowing wi!h !he real number property that justifies !he
particular step.
(1),_ _ _ _ _ _ _ __
(2)_ _ _ _ _ _ _ __
(2)
(3) Exponent Property of Multiplication
(3)
Exucise #2: Simplify each of the following products using reat number pcoperties like in Exercise #1,
(b)
(-4lz)(zyz') =
ClearlYt we would like to be able to do this mUltiplication without going through each of these steps. It
should be clear from the last exercise tbat you can simply multiply the coefficients together and then
add powers on like bases,
Exercise #3: Find the following products.
(a)
(4x'l)(zxy')=
(b)
(5r 2s)(2r$')=
(e)
(-3pt')(-6 p2t') =
We now need to be able to multiply polynomials by monomials, You have actually done this before.
as the foUowing exercise ~.U illustrate.
Exel'cise#4: Rewrite the following without parentheses by applying the Distributive Property,
(b)
-3(2x-7)=
Algebra t, UM Mi - Q.\llIdtsIi.: Algebra - f.A
The Adinglon AIgdna rwj<:a, ~grnlle. NY 12540
(e)
-(6-3x) =
Multiplying monomials with variables over polynomials uses the Distributive Property in the same
way.
Exercise. #5:
Rewrite the following products without parentheses by
app~ying
the Distributive
Property.
(a) 2:r(3x+4)
Additional Classroom Exercises
Filld the product:
I.
(2x'y)(-x'Yl=
12.
2:r(5x+7)=
2.
(3yl){2y'l) =
13.
5ab( 4a'b+ 2<1b-4,,) =
3.
(5ab'c)( 4a'b'c) =
14.
4xl (5x-4+2x 2 )::::
4.
(-2x'Y')( -.g2l =
15.
2xy' (3x' +4.g _ y2) =
5.
(7 p'r't)(3pr4t') =
Distribute and Combine Like Terms:
6.
(-4r'x')(3r4x) =
1.
3(h-5)=
8.
-6(x+4) =
9.
3x(2x+9) =
10.
5x(x-3)=
II.
-(x'-4x+7)=
Algebrn 1, Um! C6 _ Quadrati" Algclm!. - III
The: Arllngl.Ol:l Algebrn I"roj«t. L:iGr:mpille, NY 12$40
16.
5x(3x-2)-x(I-3x) =
11.
3x(2x-1)+2(2x-1) =
18.
x(x+5}-2{x+5)=
19.
x(x-3)-3(x-3)=
20.
x(x+y)-y(x+y) =
Name' _ _ _ _ _ _ _ _ _ _ __
Date: _ _ _ _ __
Multiplying a Polynomial by a Monomial
Algebra 1 Homework
Skill
Find the -product:
I.
(6:xy)(-Zz)=
2.
(5a')(-5 a)=
3.
(-72)(+
4.
(2r 2-,') ( -4r'.) =
5.
Hd )(Z4ad) =
6.
(-2x)' =
7.
(3a1b)' =
8.
5x(2.1.'-8) =
9.
6x(3x-tJ=
13.
Distribute and Combine Like Terms:
17.
4-3(2x+5)=
18.
(x'-3x+7)-(x'-6x-2)=
19.
(3x'+8x-5)-(-2.'+4x-lO) =
20.
x(x+ 7)+4(x+ 7)=
21.
2x(3x-4)+5(3x-4) =
22.
a(a-b)+b(a-b) =
23.
2x'(x+4)-3x(5x+l)=
24.
4x(4%+3)+3(4x+3)=
2
2(.'-2X+4)=
15.
3x(2x-9) =
16.
5xj2x'-3x+7)=
,
.
AJgebra I, Unit lUi - QIOOMie Algdml. .. U .
l'b¢Arlinj,'lOIi AIgclmi I'n'lj~ LaOral!g6ilk, NY 12540
Applications
25. If the length ofa square can be represented by the monomial 3x then:
<a) Express the perimeter of the square as.
monomial in terms of x.
(b) Express the area of the square as a monomial
in tenus of x.
26. The width ofa rectangle is represented by w. The length is three more than twiee the width.
(a) Express the perimeter cflbe rectangle as a
binomial in terms ofw.
(b) Express the area of the rectangle as a
binomial in terms ofw.
Reasoning
27. Simplify each of the following ifpoosible. ifnot possible, explain why,
(e) x' -i' =
(d)
x'
i' ~
28. Determine each of the following products by writing them out in an expanded product form, The
first is done as an illustration fur you.
(xl)' =
(b) =;(2+2+2
29. Fill in the blank for the foUowing Exponent Property:
Algcbnt,Otmlf6-~~-LA
The Arlingtoa Algebra. ~"" ~~IIe, NY 12m
(d)
(x')' ~
Nwne' ____________________________
Dale' _ _ _ _ _ __
.
Multiplying Polynomials Algebra 1 In the last lesson we worked extensively with multiplying polynomials by tnODOmialS. In this lesson
we will generalize this process so that we may multiply pelyoomials by pelyoomials. The first
exercise wiU illustrate the real number properties associated with this process.
Exuclse #1: Fill in the blanks below with the real number property that justifies each step.
(I)
(x+2)(x+4)=x(x+4)+2(x+4)
(2)
(3)
=:
x'x+4·x+2'x+2·4
(4) Exttrdse #2: Using real number properties~ find the products given below.
(a) (2x+4)(3x-I)=
(b)
(x+7)(x-5)=
(0)
(2y-3)(4y-6)
Multiplying two linear binomials is such an important skin that a mnemonic has been developed to
help remember it: FOIL - Multiply the First. Outer, Inner, and Last terms of the two binomials
together and then combine the like terms.
Exercise #3: Multiply the following binomials together either using a method as in Execcise #2 or by
«FOILing" the two binomials.
<ol (x+4)(x+l)=
(b)
(y+3){y-5)=
(0)
(2x-7)(3x+2) =
Exercise #4: Which of the foHowing is equivalent to (X_4)2?
(l)x'+16
(2) x'-16
Alllcbtll: 1, l1nk#6 -~ A1gdmt- LS
Tb<;o Artmgto,l(l AIgdI~ ProjOd. I..4GtMgeviiff., NY 12540
(d)
(x-S)(x+S) =
.
.
We can also multiply polynomials together that have more than just two tenns. Each term in the first
polynomial must multiply each term in the second polynomial for the distribudoo property to occur.
Exuds. #5: Find tha following produol by distributing the binomial over the trinootiaL
Since multiplication of these higher powered polynomials can become confusing, it is he1pful to use a
multiplication table to carry out the product.
Exercise #6: Use the following table to help evaluate the following product
-4x
(X-2)(3x 2 -4x+7)=
x
7
I. •
-2
.•
Additional Classroom Exercises
L
(x+5)(x-2)=
7.
(3x+l)2 =
2.
(2X+S){X+3) =
8-
(x+6)(x-6)=
3.
(x-2)(x+3)=
9.
(2x+I)(h-I)=
4.
(lx-S)(2x+4) =
10.
(4x+5)(4x-5) =
5.
(x-5)' =
I!.
(3x-Zl(2x2 +5x-l)
6.
(2x-3)2 =
Algebra I. Unitll4-QlW;l.ra.ticAlgw.a-LS
TIm ArlingtOD /Jgdml ~ r..~Jl~lle. NY 115((1
Name: _ _ _ _ _ _ _ _ _ _ __
.
Date:
----­
Multiplying Polynomials Algebra 1 Homework Skill
Fmd each or the following products in simplest form. . 1. (x-2)(x-3)= 17. (x+4}(x-4)=
2. (y+6)(y-l) =
18. (x-1)(x+7)=
3. (0+5)(0+3)=
19. (y+2)(y-2)=
4. (r+4)(r+5)=
20. (4x+3)(4%-3)=
5. (2):- 3)(x+ 5) =
21. (5+ %)(5-.) =
6. (2.-9)(.1:+ 3)=
22. (3- y)(3+ y)=
7. (y-7)(y-2)=
23. (4.1:+1)(4%-1)=
8. (20+5)(30+1)=
24. (x+2)' =
9. (%-3)(%+8)=
25. (x-6)' =
10. (x+5}(%-9)=
26. (4):+1)' =
11. (3x+W)(x-5)=
12. (5+x)(x+7)=
27. (3x-2)' =
2
28. (6x+1) =
13. (6-x)(4+x)=
14. (3-2x)(4+3x)=
15. (9+x)(8+x)= ..
16. (2-x)(3-5x)=
Applications
29. If the side length of a square is given by the
binomial (4x - 3) then wbich of the following
gives the square's area?
(1) 8x+6
(2) 16x'-9
Algcblll. I, Utti:t#6- QWtdm1ie Algwm-LS
AJ~ ~ LaGWlgt::Vi1k, NY 12540
The ArliogtOn
(3) 16x' + 24.+9
Reasoning 30, Find the products of the followingpolynornials: (bJ (x-3)(x' +2>:+9) = <OJ (3x+5)(2.'-4x+3) =
3 L Consider the following expression:
(x+ 2)'
(a) Rewrite the expression as a produ¢t of three binomials.
(b) Evaluate this product by multiplying the last two binomials in part (a) to fonn a trinomial and then
multiply this trinomial by the fU'St binomial.
32. Rewrite the following expression withQut the use of parentheses. Keep in mind that you must
multiply the binomials together flISt and then ~rfonn the subtraction.
(x-3)(x-5)-(x+I)(x-4) =
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FIND A MATCH Solve any equation in the top block and find the solution in the botlom block, Transfer Ihe word from the top box to
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the Inslruction "LIFT PENCIL."
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Write a fraction (or 1) lor each power.
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PRE·Al.GEBRA WITH PIZZAZZ'
@Creative Publicaliolls 63 Why Did The Farmer Open A Bakery?
TO ANSWER THIS QUESTION: Express each product below as a single power 01 10 or 8.
Draw a straight line connecting each exercise with its answer. Each line will cross a
number and a letter. The number tells you where to put the letter in the row 01 bo)(es at
the bottom 01 the page.
104 '103
10-4 • 10- 2
8- 3 •
,
102 .10. 10-2 '103 • , ) "" ,10- 5 • 105 • 8' 8- 2 • 8- 7 • 8- 5 • 8- 6 • 84
.8-8 .8- 12
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6
• 10
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84 • 83 •
1
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3
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•
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3
•
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.8- 3
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B
9 10 11 12 13 M
15 16 17 •
PRE·ALGEBRA WITH PIZZAZZ!
© Crcah'iD Pu!)liClll1()flS
65 8i @-o
.
,,:Jl
it. rn
'
~f:
>Q
"m
eCll
How Did Slugger McFist Get A BLACK EYE? TO ANSWER THIS QUESTION: Express any quotient belpw as a decimal numeral and
find this numeral in the code key, Notice the letter next to it. Print this letter in the box at
the bottom of the page that contains the exercise number, Keep working and you will
discover the answer to the title question.
iVl\ 10-8
CD
10 5 -i'- 10 2 =
.'"
§~
® 10 2 10 5 =
~:J:l
!;i -I
-!-
I
"
~
I:j
@10- 6 .;.10 2 =
@) 10- 1 .;. 10-3 =
@10 2 + 10- 7 =
® 10 6 .;. 10 =
CD 10-3 + 10-3 =
III
~ 10-9 =
@ 104
®
­
@) 10 10 _
10 20 ­
@.2Q..­
@
0.0011 E @104_
106 ­
0.01 @J.!L=
2
10lH 10:-
@
3
10
103
@10 15
100lA =
1000lS .;.
@1O- 5 .;.
10-2 _
®
\,,~. 0.11 T 110 10-1 ­
10 ­
0.00011 G ­
10-1
10-5 _
10 5
0.00001 @ 10-5 _
4­
10-7 .;.
10 14 =
10 =
10-3 =
@Ho.;. 106 =
~/
I
0.0000011 C
_
@ 10-2 =
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.:!.Q.­
0.0000000001 U
0.00000001
10-3 ­
103
-
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1,000,000,000 IV
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Chapter 2 ALGEBRAIC VARlABLES &. EXPRESSIONS 1.
Simplify each ""Pression:
c4 ,c:"CS
a
d
Sp" q'. p , 3q
,J.'
b
6x' 2y' '3x'
e
10' . l(r' , 10'
c
x' ;.(1. X4
f
2
c
(a + hJ'
(a+
·l· 2"3
2. Simplify each expression:
i,
•
•
4x'y' b
3,
d
2,,/ The expression
br'
.,
x
----or
x
(iY is equivelent to:
(1) x" (3) x'
,to (4) x'
(2)
each
a
(2x)'
b
(3y')'
33 (
(4x'y')'
Cbaptu2
ALGEBRAIC VARIABLES & EXPRESSIONS
Properties of Exponents (continued)
5.
Simplify.rM..
c'd" using only poSitive exponen~.
(1) c"d"
(3) c'd'
(2)+
c
(4)+
c
"
.
6.
Simplify the fullowing expression using only positive e::<ponents:
7.
When simplified, the e.xpreSSion( 223)-2 becomes:
(1) ~.
3.
-'"
(3) 9{
~'
(2)
(-~)'
(4) ~7
•
is equal to xa-'l. or,(J when x tF 0, What conclusion docs a zerO exponent suggest to you?
8. Zr
x
.
9.
Simplify:
(' 'y­
.
x 'X
(x')'
(1) 1
10.
-
Simplify:
(3) 0
(2)
~
(4) x'
(1)
~
(3d
(2)
t
(4) 2x
2x'""
3x'
3
...
34
.
Chapter 2 ALGEBRAIC VARIAJ3LES &.EXPRESSIONS 1.
Fmd the ,um of 6,c, 9i' and
2.
Add:
3,
4.
7.
What is th.sum offlx + 7y, -12>: + 3z, and
-4y + 621
8.
Add:
Combine:
(4. - 5b) + (2a - 8b) + (. + b)
9.
Express the sum of24c2 + 23c'+ 8 and
31c ­ 14c - 11 3S :a trinomial.
Find the sum of -Bab, -Sab and -Sab.
10. Add:
(1) 26.b
(2) ab
5.
17.' - 19. - 3
16.' + 23a - 11
-19R' + 7R - 6
-13R' - 8R + 6
-3x + Sy, -4x - 6y, and '.Ix , 4y.
(3) -26ab
(4) Wab
Find the total of 6x' ­
2x' + 51.
6.
-4,c.
31. 8x' + 71', and
11. Combine:
(91 + 7) + (3)" - 5)
FInd the sum;
+
(-21 + 1)
12. Fmd the sum:
(-2a - 3b + 5) + (6. + 8b - 6) + (. - b - 3) , :
(3x - 5y + 4) + (-5x - y -I) + (-6<+ 3y + 1)
35 Chapter 2
ALGEBRAIC VARIABLES & EXPRESSIONS (-3x - 2y) ­ (1. + 3y)
(62 + 3x ­ 2) - (3:c' - 4x + 5)
2.
From (4a' + a-I) subtract (a' - 3. + 4).
7.
Subtract
3.
Combine:
8.
Express as a trinomial:
(2a + b) c (a + b)
lSb' + 6b-7
-3b' + 5b + S
(7<'- 6c-5) - (2<' + 5c + 1)
4.
Subtract:
-3:c' + 5")' + 6y'
9.
Subtract (8)<' + 6.
+
3) from (-2><' + 5. - 9)
8><' + 9")' - 8y'
5.
From (-13><'1) suhtract (-6x'y').
10. Combine:
(7x + y) ­ (6x + 2y) ­ (x - 4y)
36 Chapter 2
ALGEBRAIC VARIABLES & EXPRESSIONS
Addition and Subtraction of Polynomials
1.
2.
3.
,
The expression (3x + 2y) - (2x - 3y) - (x - y) when simplified is:
(1) 6x + 6y
(3) -2y
(2) x + 6y
(4) 6y
The sum of (-7x' + 3x + 2) and (-2x' -3x -2) is:
If 3al
-
(1) -5x'+ 6x + 4
(3) -5x'
(2) -9x'
(4)
7a + 6 is subtracted from 4a2
-
-9x' + 4
3a + 4, the result is:
(1) 7.' ­ 10. + 10
(3) .'+4.-2
(2) .' - 10. - 2
(4) -.' ­ 4. + 2
4.
From the sum of (2x - y) and (3x + z) subtract (x - y - z).
5.
Fmd the perimeter of the following in terms of x.
(4x + 1)
(3x - 1)
(7x + 1)
37 Chapter 2
ALGEBRAIC VARlABLES & EXPRESSIONS
1.
Find the product of8x' and 6x'.
-3ab by -2a'b
2.
Multiply:
3.
What does
(1) -21y'
71 times -3/ equal?
(2) -21y"
(4) 21/
4.
5.
6.
Multiply:
(1) 4x'y'
(2) 5x'y'
7.
What does
-6r times 3x" equal?
(1) -18x'
(3) 18x6
(2) -18x'
(4) 18xB
8.
Find the product of lOx and lOx.
9.
Multiply:
10.
What is the product of~x4y and
11.
The expression -4x (-4JC) is equal to:
12x'y by 3xY'
(3) 211'
xyby4xy
lxl?
(3) 4xy
(4) 5xy
Find the product of _9a 4 and _3a 3.
What is the product of~xy and
(1) -16x'
(3) 8x'
(2) 16x'
(4) 16x'
12. The product (-5x') (4x') is equal to:
(1) -2Ox'
(2) 2Ox'
38
(3) -20x'
(4) 20x'
Chaprer2 ALGEBRAIC VARIABLES & EXPRESSIONS 1.
2.
Multiply.
Find the product:
5x(7x - 3)
. What does 10(3x' + 2x ­ 1) equal?
7.
7.(3. + b)
8.
(1) sOx' + 20x - 1
(3) 30x' + lOx + 10
(2) 30x' + 2x - 1
(4) 30x' + 20x - 10
Multiply:
12a{6. - 3)
..;,>-<
2x(4x' - 7x + 2)
3.
Multiply:
4.
The product of8y(4y - 2) is:
(1) 12/ ­ 16y
(3) 32y' - 16y
(2) 32y - 16
5.
Find the product
10.
Multiply:
9(40 + 3)
4y(-4 + 4y)
(4) 16y'
11. The product of 20(2 + 3x') is:
6x(5 - 2x)
Multiply:
(1) 30x - 12x'
(3) 30x - 8x'
(2) 30x' - 12x
9.
(4) 30x' - 8x
(1) 40x + 60x'
(3) 40 + 60x'
(2) 40 + 60x
(4) 40x' + 60x
-_.
6.
Multiply:
12. Multiply:
!x<4x + 6)
39 (lOx' + 25x + 5)
Chapter 2 ALGEBRAIC VARIABLES &- EXPRESSIONS 1.
Express the product (X+ 3)(x - 2) as a trinomial.
7.
Simplitjr.
2.
Multiply 3. - 5 by 7. ·3.
8.
Hnd the product of 8x - 7y and 8x + 7y.
3.
Fmd the product of 4. + 5 and 4. + 5.
9.
(2x
+3)'
Express the product (lOx - 3)(3x - 4)
as a trinomial
4.
Express (6x - 2)(3x - 1) as. trinomial.
10. Multiply 5. - 7 times 2. + 4.
5.
"What is the product of 2a + band 2a - h?
11. Express as a trinomial:
(7. + 2b){3. + 2b)
6.
12. If (7% + 6)(11" - 8) is written in the form,
~ + bx + (;, what is the value ofc?
If(4" - 5){2x + 3) is written in the form,
ar + bx + c, what is the value b?
40 Cbapter2 ALGEBRAIC VARIABLES & EXPRESSIONS 1.
(00)(700)
Multiply:
(1) 7a'l:l
(2)
2.
Sa'b'
If (3x - 2)(4x + 2) is written in
(1) 0
(2) -2
3.
4.
5.
(3) 700
(4) 8ab
a:< + bx + c form, what is the value of c;-?
(3) 12
(4) -4
If (x - 3)(x + 3) is written in a:2 + bx ... c form, what is the value ofb?
When (2x -
SimpllfY:
(1) 1
(3) -9
(2) 0
(4) 9
3Y IS expanded, the result is
(1) 4x' -9
(3) 4x' - 12x + 9
(2)4x+9
(4) 4x' + 12x - 9
~ (15:<' + 5x - 10)
(1) 3x' + 5:<' - 2x
(3) 3x' + x' - 2x
(2) 3x' +x' - 2
(4) 3x' +Ix - 2
41 Chapter 2
ALGEBRAIC VARIABLES &. EXPRESSIONS 1.
Fmd the quotient of -iSx'" and 2>:'.
7.
Divide:
2.
Divide:
8.
Divide:
3.
Divide:
(24x' - 18x' + 12x) + 6x
9.
Fmd the quotient of::
4.
Divide:
(30x' - 15"') + 5x'
10. Divide:
5.
Find the quotient:
72x' ,. 24x'
11. Divide:
(14x'+22x'-1Sx')bJI.(2x')
(-24x'Y) by (3x')
+ 2x -
8
x-2
(lOO.'b") by
12x
6.
12.
Divide:
42 Divide:
- 18x' + 9x) by (9x)
Ramp Up to Algebra Unil 5 Using Equations to Solve Problems (RU Unit #8) Day
I
Goal
To translate
verbal
phrases into
mathematical
RampUp
Lessions
Not in
RU
Intro to Algebra Lesson
(plus extra worksheets)
Intro Unit 3-Algebra
Algebra #6 Translating
word expressions-
words to symbols
expressIOns
(multiple worksheets
provided)
Word Wall
Suggestions
NYS
Algebra
Standard
A.A. I
Translate a
quantitative
verbal phrase
Expression
Also,
categorize
words
representing into an
operations
algebraic
(i.e. the
expression
Addition
category
would
contain
words such
as total,
altogether,
more than,
etc)
2
To translate
algebraic
language into
verbal
expressions
Not in
RU
Translating Algebraic
Language into Verbal
Expressions
(worksheet)
A.A.2
Write a
verbal
expression
that matches
a given
mathematical
expression
4
To solve
equations
Not in
RU
usmg
addition and
subtraction
Intro Unit 3-Algebra
# 1 Solving Equations
(addition/subtraction)
Joke # 13 "How did the
Vikings send secret
messages?" ­
opposite
A.A.22
Solve all
types of
linear
equations in
one variable
Developing Skills in
5
To solve
equations
Not in
RU
usmg
multiplication
and division
6
To solve twa­
step
Not in
RU
Algebra, Book A "",59
Intro Unit 3-Algebra
#2 Solving Equations
(multiplication/division)
Joke # 14 "What do you
get if you cross a
chicken with a cement
mixer?" ­ Developing
Skills in Algebra, book
A pp.63
Intro Unit 3-Algebra
#3 Mixed problems
A.A.22
Solve all
types of
linear
equations in
one variable
A.A.22
Solve all
lake # 16 "What do you
call a crate of Mallard
Ducks?'"
equations
7,8,9
To solve
equations
requiring
more than
one step
Not in
RU
Intro Unit J-Algebra
#4 and #S Solving
Equations-more than I,
step- loke ftl9 "Who
wrote the book 'I Didn't
Do It'?", Joke#20
UWho wrote the book
typcs of
linear
equations jn
one variable
A.A.22
Solve all
types of
linear
equations in
one variable
'Terrible Wealher~t.. ,
10ke #21 "Who wrote
, the book 'Grocery
Packing at the
oj()
Not in
To graph
linear
RU
inequalities in
one variable
$uoermarkel'1"
Intra Unit 10­
Lesson 1- Graphing Inequalities
Graph of an A.A.24,
inequality
Solve linear
inequalites in
one variable
Arlington Project- Intra
to Inequalities
*11
*12
To solve and
graph linear
inequalities
(no division
by negatives)
Not in
RU
To solve and Not in
graph linear
RU
inequalities
(with division
by negatives)
Why Did the Kangaroo
See a Psychiatrist?
Intra Unit 10­
Lesson 2- Solving
Inequalities
A.A.24,
Solve linear
inequalites in
one variable
Arlington ProjectSolving Linear Inequalities Arlington ProjectGraphing the Solution
to a Linear Inequality
Intro Unit 10­
Lesson 2- Solving
Inequal1ties
Intro Unit 10­
Lesson 3- Inequalities
Arlington ProjectSolving Linear
AA,24,
Solve linear
inequaUtes in
one variable
Arlington ProjectGraphing the Solution
to a Linear Inequality
,
Why was the
Photographer Arrested?
Why did They Try to
Build a House on
Orgo's Head?
Get the Message
*13
To graph
optional compound
inequalities in
one variable
*14
To solve anf
optional graph
compound
inequalities in
one variable
*15
To translate
Not in
verbal
RU
sentences into
mathematical
equations and
inequalities
*16
*17
To write and
solve
equations and
inequalities
Review
Not in
RU
West Sea- Solving
Linear Inequalities
West Sea- One Variable
Inequality
Intro Unit 3-Algebra
#7--Translating work
expressions into
equations-number
problems
intro Unit 3-Algebra
#8--Writing and solving
equations
Not in
Review Exercises Unit
RU
#3, Joke #22 "Who
Compound
inequality
A.A.24.
Solve linear
inequalites in
one variable
A.A.24.
Solve linear
inequalites in
one variable
A.A.4
Translate
verbal
sentences
into
mathematical
equations or
inequalities
A.A.6
Analyze and
solve verbal
problems
whose
solutions
requires
solving a
linear
equation in
one variable
All previous
standards
wrote the book 'Ihe
French Che.f'!". Joke
#23 "What do you call
it when you cut up your
credit cards']I', Chapter
#3 Algebraic Equations
and Inequalities
1005457
Name ______________________
Da~
______________
Inequalities
(All$wcr 10# 08Q07S)
Solve each inequality.
,
I. 2 + 2, ,,; 5(, + 8)
2. 2n + 15 ,,; 3
4. 5v-52:-48
5.
7. -21 ,,; -6(0 + 12)
8. -5p + 10 > III
9. -25 ,. 4
10. 4(d + 20) :> -22
II. -113 ::. 9j - 15
12. m
-+5;;'5
-2
13.
14. -98 < JO + -120
15. -8(h - 10) ;;, 16
17.
18. 3u + 2 ,. 12u - 30
I
u
-18 ,,; - - 19
9
".
4q+2()";6q+l1
6. 109 - 21 '" 6g - 26
+ 12y
-+4;;'-7
8
16. 2k + 23 :S 3k + 8
r
-
+ 7 < -16
-3
:19. 11 :S 4(d - 4)
20. 7y + 6 :S 81
21. 10v + 11 ,. 83
23. -8e - 9 > 124
24. -23 - j ,. -9(j + 7)
26. 31+22';4t-5
27. -23 < 13
"
22. -78 2: -12p - 12
25. -7(m + 9) 2: 37
+ 2m
http://www.edhelper.com/mathiinequahties5.htm
+ -2n
. MathAI
Quiz#9
1. Graph the inequalities. Draw the graphs to the right of each inequality. A)X>9
B)X<8 C)y >S D)-5>X
E)2<X
2. Solve and Graph the following inequalities. F) 3x-2>-1O H)3x-I<14
NAME:
Date:
I) 6x-2> IO
J) 4x+l<-23
Name_~_.
_ _ _ _ _ _ _ _ _ __
Date _ _ _ _ _ _ _ __
Lesson 7.1
A. What i. wrong with the graph of each inequality? Write
your answer on the lines.
1. x> 6
IllIII$)II.
1 234 567 8
2. x s; -3
1"1111)11­
-8 -7 -6 -5 -4 -3 -2 -1
B. Write the inequality described by the graph on Ihe
number line.
1.
•
1 1 1
1
o
~
1 1 •
234567
2."11$11111'
o
1 2 3 4 5 6 7
IIII~.ll"
3• •
-7 -6 -5
,; 0
<
•
~
•
i
,;
•
i
4.
c
~
<
~
c
0
2
~
•
~
!
Il
i;'
"l"
~
~
8
I
I $
I
-3 -2 -}
I
-5 -4 -3 -2 ··1
1
I
0
I
0 1 2
•
~m
·c
"J!
..
~4
5.
..
I
I
•
-3 -2-1
1
I
0
2 3 4
I
I
•
~
j
~
"
"ij,
~
II
Chapter 7 • Inequalities
37
~HeiDer
......
com·
Name
Oat.
-----
Inequalities
{AlI$wer ID P 0&\4J15}
Solve each inequality.
l. 2 + 2, s 5(, + 8)
2. 2n + 15 S 3
3. 4q+20S6q+1l
4. 5v - 5 <0 -48
5.
6. 109 - 21 < 6g - 26
u
-18 S
-9 -
I
19
7. ·21 S -6(e + 12)
8. -5p + 10 ". III
10. 4(d + 20) ". -22
II. -113 <0 9j - 15
9. ·25 ". 4 + 12y
I
12. m
-+5<05
,i
-2
13.
I
,
14. ·98 < 10 + ·12e
15. ·8(h - 10) <0 16
17.
18. 3" + 2 > 12u - 30
-+4<0-7
8
16. 2k + 23 S 3k + 8
r
-
+ 7 < ·16
-3
19. 11 S 4(d - 4)
20. 7y + 6
22. ·78 2: -12p - 12
23. -8e - 9 ,. 124
24. ·23 • j ".
25. -7(m + 9) <0 37 + 2m
26. 31 + 22
27. -23 < 13 + ·20
http://www.•dh.!peLcomlmatblineq".litiesS.htm
S;
S;
81
4t - 5
21. 10v+·l1>83
-90 +
7)
· MathAI
Quiz#9
Date:
1. Graph the inequalities. Draw the graphs to the right of each
inequality.
A)X>9 B)X<8 C)y
NAME:
~5
D)-5>X E)2<X 2, Solve and Graph the following inequalities.
F) 3x-2>-IO
G)3x>-12
H) 3x-1 <14
Date _ _ _ _ _ _
Name _ _ _ _ __
~
__
A. What Is wrong with the graph of each tnequality? Write
your answer on the lines.
1. x> 6
1111111$11"
1 2 3 4 5 678
-_._­
2. x '" -3
tlllll$II"
-8-7 -6 -5 -4 -3 -2-1
B. Write the inequality described by the graph on the
number line.
1.
• I
o
I I I $I 1 ! ..
2 3 4 5 6 7
2.4lli$lllll.
o 1 2 3 456 7
3• • 1 1 1 1 1 . 1 1 "
~
~
1•
~
.l!'
,&
~
"
•
~
c
••
~
~
a
~
Q
t8
4.
•
..
-7-6-5-4-3-2-1
0
1 1 ~ I 1 1
-5 -4 -3 -2 -1 0
2
I
•
•
ci
s
~
•!• 5.
..
1
1 I
-3 -2-1
I 1 1 1 I
0 1 2 3 4
•
------_._.._-­
•
0
11 0
"§•
0
Chapter 7 • Inequalllies
37
I) 6x-2> 10 J) 4x+I<-23
Nrune _____________________________ Date ______________
Lessons 7.210 7.4
A. Name a number in the solution by looking at the graph.
Then check the number.
1. a + 1> -2
II
I I $ I I I I I.
-0 -4 -3 -2 -1 0 1 2 Z.2y-4<& "
I I
I I ,
I I .. 1 234 5 6 7 8 Number in solution _________
NumbeT in solution ______
Check:
Check:
B. Solve each inequality. Then, graph the solution. Check
with a pOint from the graph of the solution.
1. -Sm > 10
• I I I I I I I I I I I I •
Check: 38
Chapter 7 ¥ Inequalities
2. 3w - 3 oS 6 • I I I I I I I I I I I I .. Check:
/
INEQUALITIES RULES
:5 means ______
<means
> means
> means ___ ~ _ _
Solving Inequalities
When dealing with an inequality, your fITst step is to
_ _" Once you do that, you need to
solve for it, just like you would an equation. That means
_ _~~_ _ like terms, undoing _ _~~_ _ by
subtracting and __ _ ____ by adding. Finally, you will
_
both sides to get the variable by itself, unless it is in
a fraction, in which case you will_.
Once you have an answer, it is time to put the inequality sign back
into the problem. There is one VERY important rule about the
signs: if you multiply or divide by a ~__
the variable alone, you have to switch the . ____ ~
number to get
of the
Nams ______________________________________
J4otoring Through J4ath Which state builds more cars and trucks than any other state in the
United States? (Hint: This state is the only state touched by four of
the five Great Lakes and the only state divided into two parts.)
To find out, solve Ihe compound inequalities for I below. Match each compound inequality in
Column A 10 ils graph in Column B. Read down the column of written letters 10 discover Ihe
answer.
Column B
ColumnA
1. <3;',-4>4
H.
G.
4.6-2x>200r8-xSO
5. -12,,2x-6<4
I.
6. 2> ax - 14 > 26
A.
C.
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B.
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FSl22010 Algebra Made Simple. «) Frank Schaffer Publications, IflG.
Name' _____________________________________ Hole-in-One
,,
,
What do you get when you cross a card with a game of golf? ,
,
To find out, solve each equation for x. To spell out
the answer. write the letter of the corresponding
problem above the given answer.
\
T. x + 4" 14
E. 26 -x> 34
A. x-122:7
H. 65 -x.,;63
\
\
N. x-12>35
I. -x-18<4
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,
O. B -X" 10
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H. -6x" 18
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C, -4x - B" 16
E. -x - 13 > 1
.. J,_._!'
,,
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....
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N. 15 + 17."; -19
L. x-12:S; 13
'" ",N
x ;;, 11
x> 47
x < 10
.;;,2
x:Z: 19
x~6
x> -6
x <:-8
x> -3
x:> -:2
x> -22
x .,; 25
x <-14
,..
,
xs-2
•
FS1220l!) Algebra Made Simple . .., Frank:"""'""::,~PublJ::"':ti:.n:',,~1nC:-.---------D~~~~~~~~~~Di
Nrune _______________________
Inequalities
(Answer ID /I 0473182)
Solve each inequality.
I. 234 < 26y
2. -21t :5 -540
3. 5d 2: 7
4.
5. 437 < -2c
6.
8 :5
J
-
9
7
7.
q
-14 2:
p
- < 15.4
--20
8. -30! < 317
9.
b
-17.6 :5
-
13
10. 18z 2: 20
11. 289 :5 -23,
12. 12h < 84
13.
14.
15. -24n < -29
e
18 :5
-4
16. -14m :5 -14
u
--12
5 :5
17.
k
-10 <
-
16
19.
b
-
18.
g
-
:5 3
-2
20. 181 :5 8z
21. 6! 2: 170
,
24. -29 > -3w
;, 6.3
19
22. 355 < 17x
23.
-­
-11
2: 5
http://www.edhelper.com!math/inequalities4.htm
211512005
.HeW~·
Nrune _______________________
Dale _ _ __
Inequalities
(Answer ID H0887017)
Solve each inequality.
I. -3.4 5 14.2 + f
2. h - 2.2 "­ 11.4
3. -7 > w + 15
4. 12 - m > 23
5. -25.9 - k > 6.1
6. b + 16.3 "­ -12.3
7. 2 > g + (-9)
8.
10. -17 "-11 +e
11. 27 - v 5 14
12. x - (-1.7) < 4.3
13. z + 19 > 8
14. 4+n<13
15. 3.2 5 Y - 30.7
16. -20.9 5 1 + 5.9
17. b + (-18) "­ -8
18. 22 5 -24 - n
19. a + 10.2 5 15.8
20. 15 "­ 9 - r
21. y + 18 > 5
22. 16 - 1 "­ 29
23. 12 5 6 + w
24. 19 < c + (-9)
25. -20 > 13 + z
26. j + 14.5 "­ 3.2
27. -4.7 < m + 2.9
28. -10-,57
29. x - 21.7 5 13.6
30. 18.9 5 e - 5.4
31. 19 - d < 20
32. 8 5 26 - u
33. 11 .::;; v + 7
34. P + (-17) 5 -5
35. -17.1 > 16.2 + f
36. g - 30.4 > 3.5
37. -11 5 3 + k
38. h + 15 "­ 7
39. 16"-26-q
40. n - (-5) < 13
41. 13.3 5 x + (-19.1)
42. 29.8 5 -14.1 - z
17
~
-28 - u
http://W.NW.edhelper.com/mathiinequalities2.htm
9. -6 5 d + 10
2/15/2005
Graph the following inequalities.
8) -I:ox>4
.' "
9) 2>X<6 .
10)
. . ,. '.
-,
',,' ­
-4>X<5 .. 11)
3<X<8
12)
. -10>X<-2
13)
-4>X<-1
14)
3<X>7
­
NAME:
DATE:
Math Al
HWOlll/!'
Inequalities
Solve and graph each equation.
1) -2x-l> 5
2) -3y-6 > 12
3)5y+3>3
4)-10 < 4y
5)15 < -5y
6) 2y-Sy > 9
7) -lOy < 20 Dole _ _ _ _ _ _ _ _ _ _ __
Nome _ _ _ _ _ _ _ _ _ _ __
Compound Inequalities: Solve and Graph
2x
'6
3-4as5
3 - 3 - 4as5 - 3
2"'2
x",3
-4
-3
-2
_1
:2
0
3
<1
01 "'!-l
'Isy
and
-5
-4
-3
-2
-,
-i
i
0
i
2
.4
:$
5
y<3
.11111111.111(11101111)10
-5
-4
-3
-2
-1
0
1
2
4
,3
5
7. '6';;-2zs4
t,;; 'lor t s '3
~IIIIIIIIIIIIIIIIIIIII"
~IIIIIIIIIIIIIIIIIIIII.
-11)""1...!1-1·~-5"'·l-2-1
Il I , l " 5" 7 & 9 10
~IIIIIIIIIIIIIIIIIIIII>
3.
0< -1
y+3-3<6-3
Y'" 'I
.1lI-~"-7·64-e:·3-2-1
5
.1 I I I I I 1 10 i I I I I I I I I t I I ..
2sy+3andy+3<6-3
2-3sy+3-3
5
5
2sy+3<6
J.
50 + 1 - 1 < '4 ­ 1
50 < '5
'40 <1­
'4 - '4
x>'3Iorlx='3
-5
50+ 1 <'4
or
x + 1 S '3 or x + l! 3
,"
3" $
~
7 111111
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9. 3<2x+ 1 <7
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'1 .. 9 19
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10. '8<2x+4,;;-2
4. -2<31-2< JO
.. 111111111111111111111>
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a
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11. -6'; 3 - 2 (X+ 4) ,; 3
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12. 4-3x"';'8or3x- J s8
6. 3x-7< 11 or9x-4>x+4
~IIIIIIIIIIIIIIIIIIIII
..
.. 111111111111111111111 ..
Page 25
Nama ________________________
Data ________________________
Basic Inequalities: Solve and Graph
6 < 3 (1 - 8)
6<3-35
6-3<3-3-35
3
"3
'3s
<"'3
'l~s
'1
$<
"111111111011111111111 •
-10 -'I -Il -7 -6 -0 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
15x- 2 < 3x- 11
8,
1. x+4>12
• I I I I I I I I 1,1 I I I I I I I I I I I .... I I I I I I I I I I I 1 I I III I 1 I I ..
9, 2 (t + 3) <3 (t + 2)
2, 32 > "4 (4y)
.111111111111111111111 .... 111111111111111111111 ..
3,
3y+ 1 < 13
10,
.. I I I I 1 I 1 I I I I I I I I I I 1 I 1 I.
~G~~-1~~~-14.10123.
4.
1
5(11)910
• I I I 1I I I 1I 1I I 111I 1I 1I I •
wro~~~~~~4~.IOI
1
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2 - ~ <'1
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x- 1.5 < 0,5 (x + 4)
11.
10-<2z+ 182
2
.111111111111111111111'
5.
15x-2(x-4»3
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_10_9_6w1_4~""'_l_1_1
I)
I 2:>" S
~
fill.
1",910
12. "3 (2m - 8) < 2 (m + 14)
.111111111111111111111 . . . 111111111111111111111 ..
-HI.9.6_;.~_S_4..J_2_1
0 11 3 4
~
6 1 I 910
6, "2x - 5> 6
-1O*9.e_1"~_~.4-J_2_1
13,
0 l ' 1 4 5 6., II" 10
2x + 3 < 6x - 1
.111111111111111111111 • • 11111111111,111111111.
-IO-9-1I.1-4·~-4-J·'2-1
11 1 1 3" 5 4 1 " 9 HI
-11l.·),.-r-6-$4-3-2.1 \I 1 2 J .. 11.1 e " 10
14. 3x-2;e7x-l0
7, "3m+6(m-2»9
.111111111111111111111 • • 111111111111111111111.
-\(1 -9...g ·1 -6 -5 -4 -3 -2 -I II
I 11: 3 ,
S
~
7
a
9 I'D
-10 -9
-e -1
Paga 24
-6 -5 ..... -3 ·2 _I 0 1 2 3 4 S
(I
1 8 9 10
Name' _ _ _ _ _ _ _ _ _ _ _ __
Introduction to Inequalities Algebra I For any two real numbers a and h, one of three possibilities exists: either a is less than b (a<b), a is
equal to b (a=b). or a is greater than h{a> b), Wbencomparing the values of two numbers on a real
number line~ the larger one is to the right of the smaller one, If the two numbers have the same value,
then their graphs are at the same location on the real number tine.
Exercise 111: Slate whether the given inequality represents a true (1) statement or a mise (F)
statement
(a) 7>5 3 12
(e) ->-
(i)
fi.ij" JIT
(b) -7>-2 (f) -2.6> -4.1
(j)
JO >-./4
(e) 0<-4 (g)
,,>Ji
(k) -,JI,,-I
·1 I (d) ->_.
{Ill
--J35 > -6
(I) 1-21<1-~
2
5 20
4
Recall: a ~ b means that "a is less than or equal to b." and a ~ b
equal to b."
m~s
that "0 is greater than or
Exercise #2: State whether the given inequality represents a true (T) statement or a false (F)
statement
12
(aJ -22:-6
(b)
3.,;£
5
30
(e) -75,-5
_
(d) -4<:-3_
Exercise #3: Circle each replacement for x from the accompanying list that makes the inequality true.
(a) x<:-5
-14,-1l,-8,-5,-2,-1,0,3,7,10 -14,-11,-8,-5,-2,-1,0,3,7,10 (e)
IxI,,3 -8, -7, -6, - 5, -4, - 3, -2, -1.5, -I, -0.6, 0, I, 1.25,2, 3, JO, 4, 5, 5.8, 6
(d)
Ixl <: 2 -8, -7, -6, - 5, -4, - 3, - 2, -1.5, -1, -0.6, 0, 1, 1.25, 2, 3, $, 4, 5, 5.8, 6
(e) x<5
m, .fij, Ji4, .fi5, .fi6, m, .J28 5
(f) x>­
3
0')'5' 2'6'9' 12' ,3,4
7 3 10 15 20 2
AJgebra 1, UnitH! -Alge.bm'l'- F~- (,9
Arlington High School. UGraIlg¢viUe, NY 17540
When we graph solutions to an inequality on a real number line. we darken aU numbers on the number
line that make the inequality true. We call the set ofall solutions the solution set.
Exercise #4: Match each inequality with the grapb ofits solution set
a._IIIIIII+I!I!!II·
-5 -4 -3 -2 -1 0
2. x>-3
I
4! I I I I !' I ill I ! I I I I I •
h.
-5 -4 -3 -2 -I 0
1 2 3 4 5 6 7 8 9
, I I <\l I I I I I I I I I I I I'"
-5 -4 -3 -2 -I 0 1 2 3 4 5 6 7 8 9
3. :£>2
c.
4.
d. •
x~2
I I I I I I I
-5 -4 -3 -2 -1 0
5. x<2
6, ,,;;2
e.
• ·5I -4I ·3+-2I
, I H
f
8.
1-<1>2_
g.
1-<1<2_
h.
, I I I
•I I I I I I I •
1 2 3 4 5 6 7 8 9
I I I I I I I I I I I
•
I I I I $ I I I I I I I
•
I I I $ I I I I I I I
•
-I 0
-5 -4 -3 -2 -I 0
7.
2 3 4 5 6 7 8 9
"
·5 -4 -3 -2 -I 0
4 f I I <jl
j
I
2 3 4 5 6 1 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
I I <jl I I I I
·5 -4 ·3 ·2 ·1 0
I I
1 2 3 4 5 6 7 8 9
•
Exercise #5: Gcaph the solution set for each inequality 00 the accompanying number line.
(a) ,';;-1
111111111111111111111 •
-10
(b) x>4
,«6
5
10
111111111111111111111 •
-10
(e)
o
·5
o
-5
5
10
-
111111111111111111111 '
-10
-5
o
5
111111111111111111111 •
-10
(el
Ixl>1
-5
o
10
111111111111111111111 •
-10
-5
o
5
10
AJgrol'\ll.l1Jtitffl-~F~-L9
"!be A~Q AJgcbta Ptojo>;t, LaGnngtville, NY 12540
3'1
Dote' _ _ _ _ __
Name: _ _ _ _ _ _ _ _ _ _ __ Introduction to Inequalities
Algebra 1 Homework
Skills
1. 'State whether the given inequality represents a. true (T) statement or a false (F) statement
5
-J9 S;-3
(a) 8<6 (e) 1.:?5S 4
(i)
(b) -11>-12 (I) -3;;'-5
(j) 1-2j>j-q
(e) -3<0
(g) 1<$4
(1<)
2 3
Cd) ->-
3 4
(h)
-Ji'i <-3
1-31<111
(1) .JO:>'O
_
2. Graph the solution set for each inequality on the accompanying number line.
I I I I 1I I I I I 1I I I I I 1I I I I •
Ca) ,$-3
-10
o
-5
5
10
I I I I I I I I I I I I I I 1I I I I I I •
(b) x:>.7
-10
o
-5
5
10
1 I I I I I I I I 1 I 1 1 1 1- I I I I I I •
(e) x>O
-10
o
-5
5
10
I I I I I I I I I I II I I II I I I II •
.
1
(e) x>52
o
-s
-10
5
10
I I I I I I Ii I I II I I II I I I II •
-10
o
-5
5
10
I I I I I I I I I I II I I II I I I I I •
(I) x S; -1.5
-10
o
-5
5
lO
Applications
3. The price Charged by the U.S. Postal Service to mail an envelope first class exceeds $0.41 if the
weight of the envelope is greater than one Qunc;c. Graph all numbers of ounces for which the cost to
mail an envelope first class exceeds $0.41 using the number line below.
I
I
I
I
"
~54~l...z-101234S
AlgdmII, Unil: fl-AJgdIraii:: ~- 13
ArlingtotI High SdJ.oot. Lu.GtmgevllJe, NY lasro
Reasoning
4. Graph the sollltion set for each inequality on the accompanying number line.
(a)
Ixj,,4
I I I I I
·10
(b)
1*2.5
1 1 1 I 1
-10
(0)
Ixj>L5
-5
I I II I
I I I
-5
-5
Ixj<5
I I I I
-10
1
1 1 11 1
5
0
1
III
1 1 1
5
0
1
1 11 1
•
10
5
I I I I
I I IIII
0
-5
I
1 1 1 1 1 1 1 1 1
0
1 1 1 1 1 1 1 1 1 1
-10
(d)
1
•
10
I
1
I
1
•
10
I I I I I I I I I •
5
10
5. Classify ""chof the following as either true (T) or false (F).
(a) For any ceal numbers a. h. and c, if a <: b and b < c, then a <: c.
(b) For any real numbers (1. h. and c, if a <:h. then a+c <: b+c.
(0) For any real numhcIS a aod b, ifa d, then
101 <101.
6. The follOWing exercise investigates a property of inequalities that will be needed later in the course.
(aJ Complete the last column in the fullowing
cbart by placing «<" or ">U in the given
circle in order to make a true statement.
original true operation on both sides final true
statement
ofthe inequality
""'tement
multiply by -2
2<3
4>2
-3<-1
(b) Based upon your work in the table in part f--..__....
(a), complete the fullowing property of
4> --(j
inequalities by selecting the correct phrase
to fill in the bhmk
9<12
multiply by -I
-40-6
-40 2
mUltiply by --4
I2U4
divide by -2
-2()3
divide by-3
-3U-4
MULTIPLICATION PROPERTY OF INEQUALITIES
When multiplying or dividing both side<) oran inequality by any negative number. the inequality
sign must
remain the same
in order to keep a true statement.
be changed to =
be reversed
Algebra I, Ulli! #1- Algtbniic FouDdaIioas - 1.9
The Artiogtoa Algebra Project, ~gcvilJe, NY 12540
.,
Name: _ _ _ _ _ _ _ _ _ _ __
Dale: _ _ _ _ _ __
Solving Linear Inequalities Algebra 1 Solving inequalities. like solving equations. consists of finding aU values ofthe variable that make the
inequality Ime.
Burchelll: Consider the linear inequality 2.<+ 3 > 11.
(a) Circle each of the following values of x that lie in the solution set of this inequality.
x;=-2
x=4
x=1T
(b) Solve the linear equation 2x+ 3= I r.
x=4.1
(c) What is the solution set of the linear
inequality 2. + 3 > II? .
Exercise #2: Which ofme following represen~ the solution set of the inequality 11 ~4x + 31
(I) xs2
(3)x22
(2) x5-2
(4) .<2-2
Linear inequalities may be solved in .an almost identical fashion to linear equalities, The key
difference comes when mUltiplying or dividing both sides of the inequality by a negative number.
Exercise #3: Place an inequality symbol, > or <, between each of lhe foHowing sets of numbers.
~rdse #4: Solve each of the following inequalities. Remember to reverse the inequality if you
divide or multiply by a negative number.
(aJ 2.>:-5>13
A,lgWra I, Unit 113 - r..iImr Algo:bm - (,9
I
(b) --x+5$·-7
2
The ArlitlgtOlt A4dm Project. Lt:Gtang1MU~. NY 1254{l
(c) -7x+5533
Exercise #5: Find the solution set of each of the following linear inequalities.
(0) 3(2r-5)-4(Zx+l)S;-26
(aJ 3(2x-4»Z(x+4)
Exercise #0: Represent each sentence as an algebraic inequality; then, solve the inequality.
(a) twice n is grea.terthan the sum of4 and 6.
(OJ The sum ofg lUld Z is at least I!.
(e) The productof4 andy is at most 21 more
thany. (d) Twice !he sum ofx and I is less !han 18. Exercise fl7: Considerthe linear inequality lr+ I> 3x-2.
(a) Enter the following two functions into your calculator (b) Graph each of these linear equations Oil the
to compare the two sides of the inequality for the
grid below.
given ofvalues ofi. Then fill in the fable below.
y
1'; =3x+l and Yz =3x-2
r; =3x+l 1; =3x-2
3x+l>3x 2
I,
(TIF)
,
,
I
x
,,
·5
"I,
·2
,
0
,
,
,
i
3
6
,
"-1
(cj Considering your answers to part <aj and (0), what
is the solution set of the inequauty 3x + 1> 3x - 2 '!
AI~ I, Uo.itl13 -Li-.r~(1l-L<J 'i'bcArliDglOO Algebra ~ Le~tIMUe, NY 12549 x
(d) JustifY your answer algebraicaUy.
Namc: _ _ _ _ _ _ _ _ _ _ _ __
Date: _ _ _ _ _ __
Solving Linear Inequa6ties
Algebra 1 Homework
Skills
1. Which of the following represents the solution set of the inequality 4x +6" 267
(1)"$ 5
(3),,;'5
(2),,$8
, (4) ..,;'8
2. Which ofthe following represents til. solution set ofthe inequality -3x+6 <: 27?
(I) ;c<-7
(3),,>7
(2) ,,5,-7
(4) <>-7
3. Which of the following is the solution set oCthe inequality 1O:S 4x-14?
(1) ,,5,6 (3)";'-1
(2) ,,;,6 (4) .«-1
4. Find the solution set for each ofthe foIiowing inequalities:
(a) 12>3x-18
(b) O.75y+6<:9
(e) IOm-20>3m+1
(d) 3(x-4)+2$x+12
(e) 4.... 4;'2(2<+2)
(I) 9z-(6-z);'12z-10
Algcbm 1, Unit #J - Linear Mgdm - Ll}
TheArl~Algdn:a Projf:cl, ~Ik, NY t1S4O
U3
5. Represent each sentence as an algebraic inequality; th~ solve the inequality.
(a) The product ofnine and x is greater than six
more than the product of three and x.
(b) Four times the difference of x and :; is less
than or equal to the sum of x and 15.
Reasoning
6. Consider the linear inequality
.!. x - 2;>.!..r+ 3.
2
2
(a) Enter the following two functions into your calculator
to compare the two sides of the inequality for the
given of values of x. Fill in the table below.
(b) Graph each of these linear equations on the
grid below.
y
1
y;. =-x+3
2
x
1
y., =-x-2
2
1
1
-x-2>-x+3
2
2
1
Yz =-x+3
. 2
(TIF)
x
,•
-4
·
-2
I
.
·
0
·
4
6
(c) Considering your answers to part (a) and (b), what is
(d) Justify your answer algebraically.
the solution set of the inequality'! x - 2;"! x + 3 '1
2
2
7, Which ofthe following inequalities has aU real nwnbers as its solution set?
(1) 3>+2>21<+7
(3) 4x+8>:4x+16
(2) 7x+3>7x+IO
(4) 5,-4>21<+3
8. Which ofthe following inequalities has no values in its solution set (it's empty)'?
(1) 6<+2>6x-1
(3) 8x+7<3x+9
(2) 4.:<+8;'4.<+8
(4) 2x+6<2<-10
Algebra I, Untllll-Linmr A1gebfl1~ [3
The Arliug\Oll Algebn ~ ~evil.h; NY 12540
Name: D.Ie:
----
Graphing tbe Solution of a Linear Inequality Algebra 1 In this lesson we will rearn to communicate our solution sets both algebraically and graphically_ First
we will review trow to graph inequalities.
Exncis< #1: Graph each inequality below on the number line provided.
(.j .;'3
II1111I11111III111111 •
·10
(b) .«-2
·5
0
5
10
111111111111111111111 •
o
·5
·10
10
5
111111111111111111111 •
(oj -4<x';S
o
·5
·10
10
5
Exercise #2: Write the inequality represented by each graph below.
00
00
1111111$11111111111.1-s
0
5
H)
~IO
~
111111.11111111111.111.
-10
-s
0
5
10
11111.1111111111111<111.
-10
-5
()
5
Exercise #3: A car tire company recommends the pressure. p, for a particular tire should be at least 30
pounds per square inch and less than 36 pounds per square inch.
(a) Represent the recommended tire pressures as an algebraic inequality.
(b) Graph the inequality that represents the recommended tire pressures,
• 1 1+-+-11,-+-+-+-11-+--1--+1-1--+-+-+. 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Algcbnt I. t1citMl-Lincar A!gdml- t.lO TbeAl'llApnAfgebra Pl'ojM, ~ NY 12540 to
Exercise #4: SQlve each inequality and graph its solution set. If no solution exists, then so state.
(a) 12x-14<9x+13
(b) 2(x+9»3x+1O
(e) 4(x+3)<4x+9
(
(d) 3x+13';5(x+2)
)
(e) 5(2x+ll):?3(x+2)
"
(I) O.IOx+40>O.12x+25
)
)
Exercise #5: Whlch of the followuig represents the solution set of 16 >-3x+ 10?
(I) j I I I I I I I I I I 1$1 II I I I I I •
-10
·5
0
:)
10
(2) 1111111101111111111 Ii'
·HI
.5
0
5
(3)
""11111111011111111­
-10
(4)
Hi
.$
0
5
HI
11111111$111111111111'
·HI.)
0510
Eurclse #6: Translate each verbal sentence into an algebraic inequality. Then solve and graph the
solution set
(a) Eight is less !han four times a number added
to ten.
AI&dm I, Unit 10-U_ Algcbru. -U{)
TheArlingWa AIgdI:R~ LeGmngCY\1k., NY 12540
(b) Five less than twice a number is at most ~3.
N~
________________________
Date: ___________
Graphing tbe Solution of a Linear Inequality
Algebra 1 Homework
Skills
l. Which of the following inequalities is graphed below?
(3)x<-1
(I) >:<2 (2) x';-1 (4) x>-1
111111111$111111
-10
-50S
1I I •
.0
2" Which .fthe following inequalities is grapbed below?
(1) -3<x,;7
(3) -3';x;;7
1 1 1 1 1 1 1 ill I
·10
(2) -3';x<7 ..s
1 1 I 1 1 1 I. I I I •
0
S
lil
(4) -3<>:<7
3. Which of the following values ofx 1S not in the solution set graphed below?
(1)-5 (3)0
1111.111111111<1>11111'
-s
0
5
10
-(0
(4)4
(2)-6 4. Solve each inequality and graph its solution set If no solution exists, then so state.
(a) 1-5>:>36
(b) S;;Zx+8 (e) 3(2x+7)<2(3x+l)
<
(<I) 6x+2-3(x-3}> x-I
<: <
(I) 0.15x+50 > 0.20>+ 35
(e) 2(4x+3»IO(x-2)
)
<:
" AlgdImI.UnJt#3-~A1gebl'll-Lt{)
Th!: Arlington Mgdn PmJo;t. I..AOningcvilk:, NY 12S«J
)
)
<
5. Translate each verbal sentence into an algebraic inequality. Theil solve and graph the solution set.
(a) Four times. number decreased by IS is less
than six more than the same number.
(b) Six times the sum ofa number and 2 is at
least 30.
(c)Three times Illesum ora number and 7 is at
most nine more than five times the number.
(d) Five-fourths ofa number increased by six. is
greater than one-halfof the same number.
Reasoning
o. Felicia's Ice Cream Shoppe offers a maximum ofeigbt scoops of icc cream on a single conc. They
serve only whole scoops (a customer may not order 2 and a half scoops for example). If a customer
just ordered an ice cream cone, which of the foUowing solutions represents the number of scoops
the customer may have ordered? Expfain,
(1)
•I
0
(2)
•I
0
.
I
I
I
I I
I
1 2 3 4 5 6 7
~
2 3 4 5 6 7 8
ExPLANATION:
Algdmt I. urut ir.J -Lillear~-UO
1be~A1gd)nl.Ptojcet, ~geYilte,
I•
(3)
8 9 10
• • • ••• • •
I
I
NY 12540
• I <fI I I I
0
I
(4)
1
9
I•
to
• I <fI <fI $ $
0
I
I
2 3 4 5
2 3 4
I I
'"
I I•
6 7 8 9 10
•••
<II 1 I '
5 6 7 8 9 10
~
Why Did The Kangaroo See A Psychiatrist?
@""
Find the graph of ths solution set of any Inequality below in the
corresponding colurl]n of graphs, Notice the lettsr ne'xl to it. Write this letter in each box that contains the n'umber of that exercise, Keep working and you will discover the answer to the title question. oJ)
~r;;
=-r ."
~Ol
~m
c CD
~Jl
tf »0
G)x < 1
gS
"-;
® q $ I lip
-3
:r
:!I
@x,,"1
~
®x > 1
@x",1
2
3
@IIIJ
-3
®
@s '" x
@4
~I!
I I I III
-3-2-10123
CD
©
t
1
@
®x <-2
,'f® x :;;, -2
0
@-1 <x
@
®x e$;-2
-,
® ·1 I I 1'1 II
-3 -2 -, 0
, 2
3
®x'£ 1
<.V x> -2
-2
@)x <-1
I
I
I
I I I
-:l
""2
I -2'I
-3
I'
-3
-2
-1
-1
-1
I I
0
1
I I
2
,3
~ @x < 3
I $, 2I 'I3 ~
0
I
a
,
•(
jI
-3
I
-2
i
-1
I
m
q;
0
1
°
I I ~ @O,,"x
2
3
® •I $ I I I I II
-3 -2 -1 0
1 2
3
® • I I I I $ I I ~
-3 -2 -, 0
1 2
3
®
@x rf
I
2
I,
j' ,
3
®
o
1
t"~i
-3 -2
1
1I
-3 -2,
1
I -'
I
I II
J$'23
1 0
I I.
J1 ~ I, 2 3
®IIJJ,!
-3
I I.
123
1
2
®IIJ_1
®.IJJ,!
<1])0> x
Q)
l-i1
® '• -3I
I II
112
3
~
I I.
11 ,
,
11
2
I
I
0
1
2
f~
I
-2
;;p
-1
I I
123
'
2
-32
@O<x
~
123
-32
' -3
@O"'X
-2
~
I $
23
,
I '
3
6 1615131j1 ~ 7 161156 4 6 1~~55 ~~21616 82113 91318101714 4(
,,
.....
-'.,-,,'
'­
.
~"
Why Was The Phofogra.pher Arrested?
Solve any inequality below. CIRCLE the letter next 10 the correct answer. Write this leiter in the box at the bottom of the page that conlains the number of that exercise. Keep working and you will discover the answer to the title question, .~1~4~4~.~4~4~4~'~'~l~'~~~l~l~'~l~'~'~'~'~'
x+5>9
(E) x> 4
(T)x> 4
4
(D)x<-2
(E}x<4
(S) X "" 5
-6x<-12
(L)x> 2
3x
~
<
15
X ~
5:
5m
n"
~~
~:;
"J:
c 1:
~­
-N
2'"
a~
::: .....
~c
tv
tv
(..>
_
X
<
2
(A) u "" -3
+ 4>-3
(E) z > -7
(Ll u
(A) z <-7
u-1",,-4
:3;1 cr:;
@-5x > -20
(E)x > -2
(R)x
"»m"
-6x< 12
~-3
< 12
(V) X> 2
6x
x<2
6x <-12
(S) x>-2
®x-10<-1
(F)x> 9
(P)
X
.............
<
9
Z
m -12 ~ 2
(S) m "" 14
0) m ~ 14
-5x> 20
(D)x>-4
(E) n "" 30
n ~ 30
U"I1"
n "" -15
(M)
n ;;. -30
n
~-30
n",,15
(R) n
(0) n
@ - ~ n ;;. -15
(I) n ;;. 30 .
n ~ 30
X + 15 < 4
-
. @-·h>8
­
(D)x> -12
(W) X < -12
@-2x
~ -42
(E)x>-11
(P)x;;. 21
v ""
-"
ii'
"'f
"'
21
­
~t + 13;;. 30
(X) X
8x> -56
(P) x>-7
(N)x <-7
-7.s;-16
(T) d;;. -9
d.s; -9
x>8
"" -30
(S)x> 12
~-30
(T)x<12
~
...-:...
(T) t;;. 17
(P)t~
17
(34a .s; -20
(F) a ;;. -5
(8)a.s; -5
l
@ -ik >-1
(P) k > 8
Why Did They Try To Build A House On
.Orgo's Head?
~;mm;m;m;~~~w;w~~~·
".i
SolVE!" any inequality below and draw a straight line connecting it to the inequality that describes the solution set. The line will cross a number and a lelter. The number leUs you where to put the leiter in the boxes at the bottom ot'the page. Keep working and you will discover the answerto the. title question. ~~~m-m:~m-~~~-m!;l
Q) 3x + a > 2
•
• x ~ -21 @7x-1<20.
@
®
- x>5 @ a - 4x > - 1 2 .
®
®
- x>-2 . @ -Sx - 9 ~ -4
•
®
- x > -4
® 63 + 12K < 15 •
®
- x"", 7
f.iA\
@
® -ax + 25 "'" -31.
~ ®
CD
- x<3
(f) -10 + 2x ~ -52.
@ ®
® © - x "",-1
@15>6x-9
Q)
@
@
• x<14
® 48 < 20 - 14x • @)
- x~7
@) -so ~ 9x + 3
@
® @ • x "'" -7
@ 18 - lOX < -22.
@
_ x>-9
@7<3x-8
@ . x<5
tf:r -12K - 8 "'" 64 •
@ @
@
®
'-!3'
@
@
CD • x < 4
@ -17 > -7x - 45 ®
®. x > 4
@ 3x - 42 < 0
•
(f)
CD
• x ~ -11
@ 44 ~ -ax - 4 4 .
®
® . x -6
@ 4x + 12 > -24 •
• x < -4
@ -17 "'" -6x + 25 •
11 12 13 14 15 16 17 18"
224
Pf1E·ALGEBnA WITH PIZZAZZ!
© Creative PI.lOi!callof'S
.
.
)
~
.
'Did
~ou
Jae.1' .lIout ... ~_A
B
C
D
E
~F
G
H
I
J
-
L
M
N
~-
~
:
:
K
~.
.
0
DIRECTIONS:
Solve any inequality below_ In the answer column, find the
inequality that describes the solution set and notice the word
next to it. Write this word in the box that has the same letter as
that exercise.
. '
KEEP WORKING AND YOU WILL HEAR ABOUT A COLLEGE
EYE DEAL
~~~~~~~~~~~~~
® 2(3x - 5l > 2x + 6
® 8(2 + xl ,.;: 3x - 9
.. )
~
13x - 7(-2
® 5(-3x -
1)
+ x) ;;. 4x - 10
+ 7 ,.;: -x + 30
® 12 + 5x > 2(8x @9x -
6) - 7x
2x ~ 14 - gex - 4)
® -4(3 -
5x) - l1x < 3x
IX
Ix <4-HAVE:
;;. 22-STUDENTS •
X ,.;: -S-CROSS i
I X "" -12-COLLEGE .
x,,; 2-EYES
I x> 6-CONTROL
I
Ix >4-THE
X < i-KNOW
!
Ix <3-TO
X ,.;: 22-HIS
Ix,,",. 2-PROFESSOR
I
IX
+6
Q) 10(x + 2) > -2(6 - 9x)
Q) 7(2 + 2x) ~ 4p( - 10)
® 11 + 3(-8 + 5x) < 16x-5
<D -6(7x - 1) < -ax + 9(-3x - 4)
® -9x + 2(4x + 12) ,.;: 4(1 - 3x) - 13
® 7Cx + 4) + 16 ~ 5x - (lOx - 6) - 6
@ 12(2x + 3) - 3(8 + 7x) > 0
iC­
-
I x<6-WHO
x ,.;: -3-0VER
'@-3(4x-6)<7-x
@
?•
IX
""'
,,;
25-SEEMED
3-ABSOLUTELY
IX ""' -25-SUBJECT
Ix> 8-NO
x> 1-EYED
x < -8-HELP
x> -4-PUPILS
I X < -4-TEACH
PRE ·ALGEBRA WI fH P!ZZAZZl
© Crealllf(!: Publlcallons
225
·
".
let The Message ~Yes
DIRECTIONS:
For each exercise. determine
whether or not the number in
braces is a solution of the given
open-sentence. Circle the letter
in the appropriate column next
to each exercise.
When you finish, print the
circled leiters in the row of
boxes at the bottom of the
page. FIRST print those from
the column marked "Yes,"
THEN print those from the
column marked "No."
A ME;SSAGE WILL APPEAR.
<D2x+7"'17
I @ 9 + 6s '" 57
@8m-3=19
No
{5}W F
{8} H A
{3}E R
!
r.®s4~3~5__n~8______~{HS}~A~W~·.
® 9u + 3 < 24
{2} IE
oo14>20-3y
{4}8
I
<V 17 + ax ;;;. 75
{7}P
A.
Q!)S5 = 4w + 29
@50-3x=16
@l63;s;;3+Sn.
[email protected]<3+Sn
@5p ­ 15;;;. 60
{9} T
{12}L
{10} L
{10} I
NI
{lS} E
i
[email protected]
+1=
5d
+ 10
8'
H
F
I
{3} REI
{8} 8 A
@4y ­ 7 = Y + 17
C!J5"9h = 20 + Sh
{7} FR·
Q.w79 ­ 8m = 34 + m
{5} M I
@2k + 34>~-70--7-k--f7-4+'}f-E-I-A---"
@hSe-7=6+2e
{l}O N
@3x + 39 ;;;.Sx - 1
{20} T U
@3v + 39 > 5v - 1
{20} H M
@ 57 - 3g = 93 - 7g
{9} H I
@8+6x;s;;9x-30
{12} I E
1(@)8x+24=15x-46 {10}E S
@6m<7,;;-1------'..,,-{1.f!}f-L-+-O-l
. •·~IIIIIIIIIIIIIIIIIIIIIII~ PRt>ALGE8RA WII H PIZZAlL!
@C'q":l'tf-; Pu!)hcalloflS
197
Name: ~__~~~______
Date: _--,---­ UnillO - Potpourri Introduction to- A1gebra
LESSON 1 - GRAPHING INEQUALITIES
An illequality is a math sentence with one of the following symbols:
EXAMPLE 1:
<
"less than"
<
"less than or equal to"
>
"greater than"
>
"greater than or equal to"
The statement x < 3 means, "some number less than 3."
This statement can be put on a number line.
....
!&
I
-~ -\ "
I
I
First, circle the number 3.
Second, shade the portion of the
number line that is
less 111(111 3.
1)0
\ '2." 4 S 10
NOTE; The symbol " < .. and the arrow on the nwnber line point in the same direction. EXAMPLE :Z:
Solution:
-<I (\) i
(
-3 -2 -, 0
EXAMPLE 3:
Solution:
i.
Graph x > -2 on a number line,
( i
I 2 " 4 5
Graph x ~ 1
When there is a line under the inequality it means that the point is included in the graph. To show this, darken in the circle. ".II
-z -\
••
ltltt)o
0 I '2. 3 4 5 I> 1 eirde is darkened p- \
INTRODUC£ION TO ALGEllM EXAMPLE 4:
UNIT 10, P. 2
Which inequality is represented by the graph below?
cl
,•
•I \ I".
2 3 45 b
I
\II
-2 -I 0
~l < x < 3 -1 < x :':': 3 a)
b)
c)
-1.:Sx<3 d)
-1
~x.$.3 Since the circle at -1 is open and the circle at 3 is closed (darkened) the correct answer is (b). Solution: . (t.
7<­
- If. ----'--- ~::I
Itt
answer: b
.....
-3 -2 -I 0 I 2. 3 4 EXERCISES:
I)
Graph x > 3 on the number line below:
"""I
I
, , ,,
4
2 "
"'3 -2 .. ,
2)
I
,
I I
-} -2 -I 0
I
>
5
, ,,
I
\
\~
23 4 5 Graph X?: O.
.,.,
-3
4)
I
Graph x:s 1.
<t
3)
I
0
1
-z
,
I I •
-1 0 1• :1.
•
,
a4
l> 5
Which inequality is represented by the graph below?
t • \ • e ~ \ 1
-2 -\ 0 I 2 • 4 :; b <\
j...,. a)
b)
c)
d)
0<x<3
0<"S3
OSx<3
OSxS3
INTRODUcrroN TO Al.GEBRA
UNIT 10. p. 3
EXERCISES (Continued)
5)
Which is the graph of x < I?
, , 61, , .• I I ...
-;'-2-10
2.?45
a)
••
b)
I
0
\
~
d)
10
-a
\
-~
'. .t
'i
-I 0 I
-2 -I 0
•, 2t
I , ,
•-,.• -,-I
•I
0 , 2.
i
I
l>
34 5
I
3
, ...
45
l
\
t'" \
Z 3 4 5
, 2. " 4 5
,,:> 2
a)
0)
Graph -2
~
x
<~
\
\
\
\
~
4
\
\
\
l b
Which inequality is represents x -::: -2
a)
b)
«2 d) b)
9)
i
0
Which inequality represents the graph below?
.~
8)
,
...-,-;'-1
' i
Graph x;:::. -I.
... t
7)
• 5
, 2. <14-
•aw(tI \
I
-:!.-~-I
6)
c)
I
0( \
<:\$""" ...... 11 . . .
- 0 -1. -I 0 I 2..:1 4 5
~\\4Lll'lit
_., 1 -I 0 I 2.::1 4
' ..
c)
d)
""ll\itl-+
_0 -1-' 0 , 2.:3 4- 5
~
,
l i t ' l·,·--p,.
_~-1.-lo
2.34-5
Name:
Dare: Introduction to Algebra
Unit 10 - Potpourri LESSON 2 - SOLViNG lNEQUALITlES
EXAMPLE 1:
Solve for x, then graph your solution on the number line.
2,+4<8
Solution:
2x
,,~
2
Answer:
2
x<2
1-+
... I
-:3'2 -I 0
EXAMPLE 2:
Solve for y then graph the solution on the number line.
-6
Solution:
:s 4 + 5x
-6
:s
-4
4 + 5x
-4
-10
:s 5x
5
5
-2
FLU' TO GRAPH:
Z. 3
~
x
x2:..-2
I
I
-3 -2 -I
0
I •
I , ,
1 ~ :3
~
INTRODUCTION TO ALGEBRA
*EXAMPLE3:
UNIT 10, p. 2
When you multiply or divide by a negative, flip the sign of
the inequality.
Solve for x, 3 - x > 6.
Solution:
3 - x> 6
-3
-J.
>-~
X
r
flip
<-3
-s
EXAMPLE 4:
a)
b)
EXERCISES:
6<
I ..
41
-3
-2
8-2Xl
9+2X<S)
Solve for x and graph solution on a
number line.
Solve for the variable and graph solution on a number line (show all
work on a separate sheet of paper.
I)
-6+5x<4
2)
523x-IO
3)
I - Y> 8
4)
-6z - 15 < 15 p.5
Name:
Introduction to Algebra
Date: Unit 10 - Potpourri LESSON 3 - INEQUALITIES EXERCISES: Solve for the indicated variable and graph the solution on a
nwnber line.
1)
3x + 4 < x + 6
2)
3(x-l»7
3)
4 - 2x 2: 2x
4)
5 - z < 3 + 3z
5)
-2 ( x - 3) > 4 (2< - 6 )
6)
5 ( 2 - 3x) > -2 (5x)
7)
5:03-2< UNll' IV
GJ1APHING
L Graph X .. S on Ibe number line.
2. Grnph X > I
8J OD
lb. number line.
UNIT IV
GRAPHING
One Variable Inequalily (Continued)
3.
Which open sentence is represented by the
6.
Whiclt grapb is represented by the open
sent"""" X
grapb below?
:. 31
: :
34 5 6
, : I
)
(2)
.10234
,~
(3)
(I)
(2)
X S; 4
X<4
(3)X:'4
(4) X> 4
Which open sentence is represented by the
graph below?
7.
•
(2)
(3)
5.
X<5
(3)")(:'5
(4) X> 5
Which open sentence is represented by the
graph below?
I
(4)
8.
I
I
I
/
I
I
C'
.
• ,
• -+ C
34 5 6
I
5 6
3 "
,
I
t/
I I
0 1 2
~
3 4
I I I I :
-2 .1 0 1 2
o
I 1 I I
.2 ·1 0 1
'0 j
!
11
•:
C
51
I)
I
:i 3
?
!
I I
.1 0 1 2
"
3 "
Which graph is represented by the open
seotence 2
(1)
(2)
. <; . I I I ! II)
3 4
I !
5 6
J
Which graph is represented by the open
< 31
01 2 3 4 5 6
X S; 5
I
..,;t;sentence X
-2
(I)
(2)
I
1 2
(I)~
0
I
I 2
(4)
4.
H
cI
·7·6.5'" -3·2 ·1 0
(3)
<J
'5: X
:S.;
6?
,
I 3I. 4i 5I 6 7:
·1 9 i i
?•
1 2 3 4 5 6 7
¢I i i i ?
1 ~
5 6 7
i
1
I
!
3 "
(I)
(2)
XS-6
X<-6
(4)
(3) X "' -6
(4) X > -6
82 d
------!
1 2
I I 5'
34
6
I
7
ALGEBRAIC EQUATIONS &INEQUALI-
Solving Linear Inequalities (continued)
1.
If x is an integer, which is the solution set
of -3 .:s x < O?
(1) (-3, -2, -I, 0)
(3) [-3, -2, -I}
(2) (-2, -1)
(4) (-2, -I, O}
Which inequality is equivalent to
2x ­ 4.:s 14?
(1)
x.
-5
(2) x < +5
(3)x.-9
(4) x < +9
6. 2. Which inequality is equivalent to
2x-3<8?
(1) x < ~
(2) x <
Ii
(3) x'
Ii
When solved. for y, the solution to
3-4y<y+18is:
(1) y.-3
(3) y.7
(2) y<-3
(4) Y < -5
-2(x-3) > 4 (x-9)
3. Find the soluMon set:
7(4-x) <- 14
7. Find the solutuion set:
4. Find the solutuion set:
4(2-x) ::; - 2x
8. Find the solutuion set: -4(x+3)
61 ~
2 (5-x)
Cbapter3
ALGEBRAIC EQPATIONS &INEQPALlTIES
Determine if a Given Value is a Solution to a Linear Lq,••tion
Is n =7.6 the solution of
3n • 0.2 c 231
6.
Is L = g the solution of
2L + 0.7SL + 0.7SL = 28 ?
7.
Is x = 100 the solution of
l.4x + 4.7x + 5.02. = 111.21
8.
4.
Is x = 3000 the solution of
.216x = 64S?
9.
5.
Is n = -4 the solution of
-20 + 6 = 14?
10. Check if the solution set of
-20 + 6 ~ 14 is 11 :s -4.
1.
2.
3.
Check if the solution set of
-3n + 0.2 <: 23 is n -7.6.
)0
62 Check if the solution set of 2L + O.75L • 0.75L > 28 contains L
Check if the solution set of
lAx + 4.7:x + 5.02x :S 11L2 is x
Chock if the solution set of
.216x > 648 is x > 3000.
~
=
100.
8. Ramp-Up to Algebra- Unit 6 - Ratio and Proportionality (RU Unit #6) Day
I
Ramp Up
Goal
Lesson
,i
I
,To solv.
prohlems
using ratios
Word Wall
Intro To
Suggestions
Algebra
Lesson (plus
extra
! worksheets)
#2
lotto. Unit 1
Lesson#l­
Ratio
Representing
Ratios
Part~Part
ratio.
Part-Whole ratio
NYS
Algebra Standard A.R.5
Investigate
relationships
between
different
representations
and to
represent
l, ratios
and !heir
I
impact on a
'-::-~c-7;--=_+;;;;-_ _ _ _-r'_ _ _ _ _--'="'-;~C-_-1l-'g~h·v'o:.",nDroblern :
2 To identifY
#3
: Unit Ratio
: A.CN.6
unit ratios and
equal ratios
Unit Ratios and
Equal Ratios
Recognize and
apply
mathematics
: to situations in
Ibe oUillide
world
3
To solve ratio
problems
using ratio
tables
#4
Ratio Tables
Ratio Table
A.CM.2
Use
mathematical
representations
to
communicate
with
appropriate
accuracy,
including
numerical
tables,
formulas~
functions l equations l charts, graphs,
Venn
, diagrams, and
: other diagrams :
4
To use ratio
#5 • Regents Exam
Proportional
A.PS.3
tables to solve--,-,S",o:.:.lv"i",ng>--_mm...i Questi"nsr.:"or"'-~LRel~tionshjp_m -,-,O"b",se",n"'",e.;:a"nd,,--,,
problems of
proportion
5
Proportional
Problems with
Ratio Tables
117
Introducing
Rares
To identifY
and represent
rates, and to
use them to
solve
I, Lesson#5
,,
1explain
,,
, patterns to
formulate
generalizations
Intro Unit 7
Lesson #5Using Ratios to
• Express Rates
Quantity, Rate,
and
coniectures
A.M.l
Per,
Calculate rates
Unit Ratio
usmg
: appropriate
units
Calculating
Rates Worksheet
problems
6 : To review
: using ratio
tables, bar
diagrams, and
equations to
#8
Reviewing
Ratio and
Arithmetic
. Regents Exam
Questions for
Lesson #8
Whole Number
Ratios,
Equal Ratios
A.CM.12
Use
mathematical
representations
to
communicate
with
appropriate
solve ratio and
rate problems
accuracy>
including
numerical
tables,
formulas.
functions,
equations.
charts, graphs,
Venn
: diagrams, and
; other diagrams
7
To investigate
and generalize
Scale Factor
#9
Enlarging and
Reducing
enlargement
appropriate
and reduction
representations
ofobjects
to solve
,,
problem
,,
8
A.R.4
Select
To apply.
scale factor to
a quantity
situations
#10
Scale Factor
and Ratio
Intro Unit 7
Lesson 6-
Similar
A.PS.6
Scale Drawings
Use a variety
of strategies to
extend
solution
Worksheet
methods to
Similarity;
Similarity Ratio
To identify the #11
similarity ratio Similarity Ratio
for similar
polygons and
use it to solve
prob1ems
10
other problems
A.PS.8
Determine
information
required to
solve a
problem,
choose
methods for
obtaining the
infonnation~
and define
,,
,,
,,
,
I
,,
,
parameters for
acceptable
I
11
solutions
! To identify
! similar
triangles and
Corresponding
#12
Similar
Triangles
to use their
properties to
solve
formulate
generalizations
and
coni ectuces
problems
12 To relate unit
price to
A.PS.3
Observe and
explain
patterns to
#16
Unit Price
proportions
Intro Unit 7
Lesson 7 - Unit
Buying;
Intro Wmksheet
- Unit Pricing
Exercise
Unit Price,
Constant of
Proportionality
A.CNA
Understand
how concepts,
procedures;
and
: mathematical
: results in one
area of
mathematics
can be used to
solve
problems in
other areas of
13
To use ratios
#17
Changing Units
to convert
Unit
Conversion
ofMeasurernent
units of
measure, and
to learn that
the conversion
factor is a
constant of
i
prooortionalilY I
Worksheet
mathematics
Conversion Rate \ A.CN.7
, Recognize and
' apply
mathematical
Regents Exam
Questions for
Lesson #17
ideas to
problem
situations that
develop
outside of
14
To apply ratio
and
proportionality
to solve a
problem
#22
Distance, Ratio
and
Proportionality
mathematics
A.CN.5
Understand
how
quantitative
: models
j
.
15
I,,
,,
To use ratio
#23
and proportion Geometry,
to solve
Ratio, and
geometric
Proportionality
problems
connect to
various
physical
models and
representations
A.PS.4
Use multiple
representations
to represent
and explain
problem
situations
,
Intra Unit 10-
The Pythagorean
A.A.45.
Pythagorean
Lesson 4-
Theorem
Theorem to
Square Roots
Detennine the
third side ofa
right triangle
using the
Pythagorean
Theorem.
given the
length ofany
two sides
*16 ' To use the
NotinRU
find the
, missing side
Intra Unit lO-
Lesson 6-
Pythagorean Theorem ora right
triangle
Arlington Project- The Pythagorean Theorem
What do Two
Bullets have
, When They Get
.
Marned?
Greek Decoder
*17 To use the
Pythagorean
Theorem to
NotinRU
What did
Lancclot Say to
the Beautiful
Ellen?
Arlington
Project- The
Converse of the
A.A.45.
Detemline the
third side ofa
Pythagorean prove that a
triangle is a
right triangle
right triangle
using the
Pythagorean
Theorem Theorem,
given the
length ofany
two sides
"18 ,i To find the
i sine~ cosine)
and tangent
ratios of right
triangles
"19 • To use
Arlington
NotinRU
i
Project- Intra to
Trigonometry Trigonometry,
Sine Ratio~ Cosine Ratio,
Tangent Ratio
i A,A.42.
Find the sine,
cosine, and
tangent ratios ,,
,
ofan angle of ,,,
,
a right
Why Did the
Saltine Lock
, Itself in the
BankV.ult?
triangle, given
the lengths of
the sides
A.A.42.
Find the sine.
cosine, and
tangent ratios
ofan angle of
a right
triangle, given
the lengths of
the sides
Arlington Project- Using Trigonometry to Not in RU
trigonometric
ratios to find
the missing
side of • right
triangle
Solve for Missing Sides !
Arlington
•Project- Applied
, Trig Problems
#1
,
*20 To use
Not in RU
trigonometric
ratios to find
the missing
angleofa
right triangle
, Did you Hear
• About ... Arlington
Project- Solve
for Missing
Angles
,
Arlington Project- Applied Trig Problems #2 ,
A.A.42.
Find the sine,
cosine, and
tangent ratios
of an angle of
a right
triangle. given
• the lengths of
the sides
Books Never "21
--To review the
concepts
studied in the
unit
#24
The Unit in
Review
Written Solving Algebraic Problems Involving Proportions ACN.2
Understand
the
corresponding
i procedures for
Worksheet
What Did the
Leopard Say
After Lunch?
West Sea- Right
Triangle
Trigonometry
WestS••Pythagorean
Theorem and its
AODticalions
similar
problems or
, mathematical
, concepts
Name: _ _ _ _ _ _ _ _ _ _ __
Dat.: _ _ _ __
The Pythagorean Theorem Algebra 1 From middle school, you should be familiar with the Pythagorean Theorem. It is of great importance
in mathematics because of its usefulness in solving fur missing sides of right triangles.
THE PYrHAGOREAN TImoREM
In. right triangle, the sum of the square. of the lengths of the legs is equal to the square ofthe
hypotenuse. In standard fannula form:
Exercise #1: [f c represents the length of the hypotenuse of a right triangle and a and b represent the
lengths ofthe legs, solve for the length of the missing side in each case.
(aJ a=6andb=8
(bJ a=5andc=13
(eJ b=4and c=5
Exercise #2: If c represents the length of the hypotenuse of a right triangle and a and b represent the
lengths of the 'egs, solve for the length of the missing side in each case. Express your answers in
simplest radical form and then as a decimal to the nearest hundredth.
(oj 0=6 andb=4
(b) 0=9 andc=18
Exercise #3: In the diagrams below, find the value ofx.
(0)
(b)
x
5
12
x
10
8
Pythagorean Triples - Any set of t:hree whole numbers, {a, h, c}, that satisfies a1 + b1 = ct is called a
Pythagorean Triple, These triples can help us quickly fill in missing sides of right triangles if we can
recognize them. three of the most common are shown below:
{3,4,5}
{5,12,13}
{8,IS,l?}
Exercise #4: Determine if the following are Pythagorean Triples. Justify your response.
(1:» (12,35, 37}
(.) {7, 8, 9}
The Pythagorean Theorem can be used in many problems because right angles appear in many applied
settings.
Exen:ise if5:
A baseball diamond has the shape or a square measuring 90 feet on each side.
Approximate, to the nearest tenth of a foot, the diStance from borne plate to second base.
Exercise #6: The length ofa side of an equilateral triangle is 10. Which of the fullowing gives the
height Qfthe equilateral triangle (also known as its altitude)?
(1) 5
(3) 3.[5
(2) 7.,[2
(4)
s,fi
Exercise #7: Carlos walks to school from his house by going directly east for a total of 5 miles, then
directly north for a total of 3 miles, and then east again for a total of3 miles. Find the shortest distance
between Carlos's bouse and his school to the nearest tenth ofa mile.
Algtbrn I, unit »8 - Jtigtu Ttiwlglt Trigonomctly - LI
The ArlingtOn Algtbm P~ LeGraogcVilk. NY 12540
Name: _ _ _ _ _ _ _ _ _ _ _ __
Date:
Pythagorean Theorem
Algebra 1 Homework
Skills
1. Find the value ofx in the following diagrams. Round to the nearest lenin if necessary.
(e)
(b)
<a)
x
8
52
x
15
20
29
48
2. Using the Pythagorean Theorem. fiU in the table. [Assume a and b are the lengths of the legs and c js
the length of the hypotenuse.]
a
25
I
7
I
,
,
,
60
25
30
9
c
b
34
40
3. Which of the follov.-ing represents a Pythagorean Triple'?
(1) {5,7,!Ol
(3) {21, 72, 75)
(2) (IO, 20, 30)
(4) {II, 45, 60}
Algebl'll I, Unit 88 - 1U&h1 Tri:w&le T~ - t.I
IheArlioglon Mgtbnt Proj~ LaOnmgeviU~ NY 12$40
Applications
4. Sameer visited his cousin and had to drive 135 miles north then 90 miles west. At the end of his trip,
be pulled out a map to detennine how far from home he was. What is the shortest distance from
Sameer·s house to his cousin;s bouse? Round your answer to the nearest mile.
5. Which of the fonowing represents the height of an equilateral triangle if a side of the triangle has a
length of8?
(1)
4!3 (3)
(2) 3.[5 6.fi
(4)4
6. In the foHowing isosceles trapezoid, find the value of x.
8
,,
25 x ',
22
25
•
.
7. Consider a square: with the foUowing characteristics and solve:
Cal If the perimeter of the square is 20, find the
(b) If the length ora diagonal ofllie square is
length of a diagonal. Express your answer in
20, ftnd the length of a side to the nearest
simplest radical fonn.
tenth.
~imll. UrutllR - Rjgh1 Trl4ngkTrig\'u:u:m:.etl)'-Li
Tht Adlngton Alg'*'" frojllCf., UGmngIMUe" NY 12540
Name:
Date: _ _ _ _ _ __
----------------The Converse of the Pythagorean Theorem Algebra I In the: Jast lesson, we reviewed how the Pythagorean Theorem can be used to solve for missing side
lengIbs of right triangles. The Pythagorean Theorem can b. reversed, called the ""uver,_, to
determine if a triangle contal:ns a right angle, Le. is a right t:.riaJigle. More formally. if the side lengths
ofa triangle satisfy a 2 +bt =2-. then the triangle must be a right triangle.
ExiJrc.ise #1: Determine if the triangles below represent right triangles. Diagrams are not drawn to
scale, so no judgment can be made based upon appearance.
(a)
(h)
S
6
Note that the side length substituted for the hypotenuse must be the largest number.
Exercise #2: Determine whether each given set of numbers could represent the side lengths of a right
triangle. (Hint - You may find the STORE option on your cal.culator helpful for this problem.)
(a) {9,12,15}
(h) {4, 2Fs, 6}
(e) {I,
Fl, 4}
Exercise #3: Which of the following sets of numbers represents the lengths of the sides of a rigbt
triangle?
(I) {5,9,lJ}
(3) {6,S,IS}
(2) {I 5, 36, 39}
(4) {H,IS,17}
AJgd>ra 1, umt U8-rug.ht Tl'llmgle Tri~-I2
The Arlington AIgeb... ~t, I...aC>Wlpille. NY 12540
Exercise#4: Derennine for what value(s) of x the triangle below would be a right triangle. The figure
is not drawn to scale.
x+8
x+7
x
Exercise #4: The Aloha Fan Company claims to make funs that rotate at least 90'. To test this claim,
Jim and Carl position themselves at either end of the rotation cycle as pictured below. According to
': their measurements, is the Aloha Fan Company making a valid claim?
10.1 ft
Carl
Jim
9.3 ft
5.2 ft
Algebra I, UnitQ- RJght TrimlglcTngooomdry'_ U
The Arlingtnn Alpal'rojlX1, ~1Ic,. NY 12540
NMOO: _______________________
Date: _ _ _ _ _ __
Tbe Converse of the Pythagorean Theorem Algebra 1 Homework SkiDs
For problems I through 6 t detemrine if each given set could represent the lengths of the sides of a right
triangle. Justify your answers, Hint - You might find the STORE function useful for these problems.
1. {S, 12, 13}
2. {6, 7.2, 9)
3. {.!.2'4'4
l 5} 4.
{Fl,./5,.JlO}
s. {,,13, 3, 2.J3}
6
{l.!..5.}
. S'2'S
? Which of the following sets ofnumbers could represents the lengths of rhe sides of a right triangle?
(1){8, 10, 12l
(3) {16, 30, 34)
(2) {25,31,40}
(4) {19, 20, 22)
8. A right triangle would be formed by using which of the following sets of numberS as the lengths of
its sides?
(I){9, 40, 41}
(3) {IO, 15,20) (2){ I ~ 23, 26)
(4) {IS, 2~30) A1gdmll, Unilll3 - fljgbtTriangIe Tri&1lf\Olll\!tlY~U
The ArtWgtOft AJg«Ira Proja:\., laGntn~IIe, NY 12540
ApplicatioDS
9. Jacob is building a table and lost his carpenter's square, which is used to fonn right angles. For the
lable to be strue_Uy sound, the legs and the table top must fonu right angles, He measures the
table top. legs, and the diagonal distance from the bottom of the legs to the opposite comer. From
the diagram below. does Jacob's table appear to be structurally sound? Support your answer with
mathematical evidence.
8ft
4ft
4ft
9ft
Reasoning
lO. Determine for what value(s) of x the triangle below would be a right triangle. The figure is not
drawn to scale.
x+4
x
x+2
Algebra !, Unit 113 - Rigln Triangle Tria~ - U
AI~ ProJeet.~1I11, NY 12$40
TheAriingtOtt
<. i I
Name: _ _ _ _ _ _ _ _ _ _ _ _ __
Date: - - - -
Similar Right Triangles - Introduction to Trigonometry Algebra 1 Trigonometry is an ancient mathematica1 tool with many applications. even in our modem world.
Ancient civilizations used right triangle trigonometry for the purpose of measuring angles and
distances in surveying and astronomy, among other fields. When trigonometry was first developed,. it
was based on similar right triangles:. We will explore this topic first in the following two exercises.
Exercise #1: For each triangIeJ measure the length of each side to the nearest tenth of a centimeter,
and then fill out the table below, Round each ratio to the nearest hundredth. When detennining
opposite and adjacent sides, refer to the 20' angle. To fill in the small box on the right, use your
calculator, in DEGREE MODE. and express the values to the neatest hundredth.
,,
Opposite
Opposite
Adjarent
Hypotenuse
,
Adjacent
tan 20' =
Hypotenuse
Triangle #[
sin20' =
Triangle #2
=20' =
Exercise #2: Repeat Exercise #1 for the triangles show below that each have l'lIl acute angle of 50·.
50'
50'
,
Opposite
Opposite
Adjacent
Adjacent
Hwotenuse
Hypotcnw;e
Triangle #1
,
Triangle #2
~bra
I. Uni. #8 -1tisht Triangle TrigollOm.ctry - L3
TbeArlU!gtoo~Projeel., ~t,.NY 11S4l}
,
tan50' =
"
,
i
cos50~
,
=
The Right Triangle Trigonometric Ratios Although we won't prove this fact until a future geometry
COUl"Set aU right triangles that have a common aeute angle are similar. Thus, the ratios of their
corresponding sides are equal. A very long time ago, these ratios were given names. These
trigonometric ratios (trig ratios) will be introduced through the following exereises, each of which refer
to the diagram below.
c
3
B u----:4:-----'~A
In a right triangle:
"'no
t
~en 0
f
1
leg opposite of the 3llgle
anang e "'"
sine ofan angle """
leg adjacent to the angle
leg opposite of the angle
hypotenuse
cosine- of an angle "'" "le",gc:ad",J::'';:;<:e!l='::,o:..;tb::e=3llg=le
hypotenuse
Exercise #3: tanA=
tanC~
Exercise #4: sin A "'"
sinG=
Exercise #5: cos A =
cose::::;
A Helpful Mnemonic For Remembering the Ratios:
SOH-CAH-TOA
Sine- is Opposite over Hypotenuse - Cosine is Adjacent over Hypotenuse- Tangent is Opposite over
Adjacent
Exercise #3: Find each of the following ratios for the right triangle shown below.
(a) sinA ~
(b) tanB=
B
13
(0) rosA=
(0)
(d) lanA=
cosB~
(I)
Ala:d\fll, U!Ul.1i$. -Right Trill.l\glQ Tri~-U
"1"heArlillgtOu Algdlra Pttljwt,
u~lk, NY
t2540
sinB~
5
C,LL--:-=2--";::" A
1
l'/ame' _ _ _ _ _ _ _ _ _ _ _ __
Da"', _ _ _ __
Similar Right Triangles - Introduction to Trigonometry
Algebra 1 Homework
Skills
For problems 1 - 6, use the triangle to the right to find the given trigonometric ratios.
L
cosN~
N
15
2. sinN­
9
3. tanN-
M?---7.::----O>. p
12
4. sinP=
5. cosP=
6.
tanP~
7. Given the right triangle shown, which of the following represents the value of tan A ?
(I) 25
A
24
(2) 24
7
(4) 24
25
7
B
8. In the right triangle below, CQ.Q =?
(I) 12
5
(2)
~
12
25
24
c
12
Q
S
(3).!3.
17
(4).!3.
J3
R
For problems 9 - 14; use the figure at the right
reduce your trig ratios to their simplest form.
9.
to determine each trigonometric ratio.
Make sure to
sinC=
10. cos C =
I!. IlmC=
4
12. sinA=
13. cosA=
A
B
2
14. tllnA=
Reasoning
Although we will not prove it here, two triangles will always be similar if they have three pairs of congruent angles. This is not true for quadrilaterals or any other polygons. 15, Consider the two right triangles shown below: 35'
35'
Based on the information above, are these two right triangles similar? Explain.
16. Why can we say that two right triangles that share an acute angle are similar? Explain.
Algd»a I, Unil #8 _ Right Trimgle Trigoll\'!l.'IIdly - LJ
The Arlitlgt«u Algebra Projw, Lo.GranglWi1le, NY 12540
31S"
Name:
----------------Trigonometry and the Calculator Da!e: _ _ _ _ _ __
Algebra 1 In the previous lesson, we introduced the trigonometric ratios. We also discussed a mnemonic that is
helpful to remember the trig ratios: SOH..cAH-TOA. The first exercise reviews how to' write these
ratios given the lengths ofthe sides ofa right triangle.
Exercise #1: Using the diagram below~ state the value fur each of the following trigonometric ratios.
13
A
12
c
(a) sinA=
,5
(c) cosC ~
(d) cosA~
(e) lanA ~
(f) sinC=
B
Today we will discuss how the graphing calculator is used with trigonometry. However, before we
start, it is important that our calculator is the right MODK
To change your calculator into DEGREE MODE:
. 1. Press the MODE button on your calculator.
2. Use the arrow keys to highlight DEGREE and press ENTER.
3. Exit the menu (this or any other) by hitting QUIT.
Now that we are in DEGREE MODE, we can start evaluating some trig ratios without referring to any
right triangles whatsoever. The SlN. COS, and TAN buttons are located in the center of the key pad.
Exereislt #2: Evaluate SID 30', cos30· and
thousandth.
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'The Arlta&ton AJattra ProjGCt. LaGnugevilk, NY 11541}
tan 30',
Round any fiQn-exact answer to the nearest
Exercise #3: EvaIuate each ofthe following. Round your answers to the nearest Ihousandfh.
(a) tan{40')
(1)) eo,(20')
(e) 'in(63')
It is important to remember that each of lh~ answers from EJrercise #3. represent the ratio of two
sides of a rigbt triangle. In each case the ratio has already been divided and the calculator is giving
the decimal form ofilie ratio.
Exercise #4: Could the right triangle below exist with the given measurements? Explain your answer.
C
,i
3
40'
B
4
A
The Inverse Trig Functions - Thus fae. we have been evaluating the sine, cosine. and tangent of angles.
By doing so, we have been. finding the ratio of two sides itt a right triangle given an angle. Using the
calculator, we can reverse this process and find the angle when given a ratio ofsides.
Exercise #5: Consider the following:
(a) Evaluate
Siu-{!)
using your calculator. The screen shot is shown at the below.
,2
30
(0) How .do you interpret tbls: answer?
Exercise #9; Find each angle that has the trigonometric ratio given below. Round aU answers to the
nearest tenth of a degree. if they are not whole numbers.
I
(b) oosB=2
(e)
sinE=~
3
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The A!ibgtQl) Algcl!m f'lviw. yGmngeviUC; NY 12;$40
3i7 Name: _ _ _ _ _ _ _ _ _ _ _ __
Date: _ _ _ _ _ __
Trigonometry aDd the Calculator
Algebra 1 Homework
Skills
For problems t through 6, evaluate each trigonometric function, Round your answers to the nearest
thousandth.
1. 'in(SS)
2. oos(4S')
3. tan(20')
4. sm(85')
5. tan(60')
6. sin( 23')
For problems 7 through J5. find the angle that has the given trigonometric ratio. Round all non-exact
answers to the nearest tenth ofa degree.
4
. A =-I
7• Sill
3
8. cosG=9
8
8
9. tanK=1
10. cosB=11
11. tanR=5
12. oinT =2:.
13. t>nA=3.127
14. ,inB=O.724
15. <05L=0.876
16. If sin If =
~
then mLA is closest to whicb of the following?
(1) 32'
(3) 24'
(2) 76'
(4) 56'
17. If tan A =35 then mLA, to the nearest degree, is
(I) 74'
(3) 24'
(2) 18'
(4) 55'
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1beMingfOl:I Algtbra li'roj«t. LaGtangeri!le. NY 12S4{l
2
Reasoning
18, Recall from homework problell1ll 7 through 15, that an inverse trigonometric function gives the
angle measurement that corresponds to the ratio oftwo sides ofa right triangle.
(a) The Screen shots below show what happens when you try to evaluate sin-t (~). VerifY why this
happens when you evaluate sin -I [~).
I
(b) Considering the ract that sfi(Angle) = opposite side, why
hypotenuse
'DOMAIN
I:Go"t't
0
does it make sense that sin. -1 (~I gives you an ERROR?
2/
19, Are the triangles below labeled with correct measurements? For each right triangle below,
determine if the ratio of the sides accurately corresponds to the angle.
(b)
16
30'
4
8
c
20. Consider the following right triangle.
(a) Express sin A as a ralio.
(b) Find the mLA to the nearest degree.
7
A
3
B
Nwne: ____________________________
Date: ______________
Using Trigonometry to Solve for Missing Sides Algebra 1 Right triangle trigonometry was developed in order to find missing sides of right triangles. similar to
the Pythagorean Theorem. The key difference, though, is that with trigonometry as long as you have
one side ofa right triangle and one of the acute angles. you can then find the other two missing sides,
TRIGONOMETRIC RATIOS Recall that in a right triangle with acute angle A. the following ratios are defined: . A
sm
:=
opposite
hypotenuse
cos A =:­ad=~",'aee!l=':...
hypotenuse
tan A == ~o"ppo"",s::i"'O
adjacent
Exercise #1: For the right triangle below it is known that sin A =.!. Find the value of x.
4
20 A
Bxercise #.2: In the right triangle below, find the length of AB to the nearest tenth.
C
26
38'
B LL_ _ _""---''''"A
The key in all of these problems is to properly identify the correct trigonometric ratio to uSe.
Exercise#3: For the right triangle below, fmd the length of BC to the nearest tenth.
B
65'
A
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12
c
ExercJse 114: In each triangle below, use the appropriate trig function to soive for the value of;r,
Round to the nearest tenth. Triangles are not drawn to scate.
(b)
(a)
50'
15
42"
x
8
Exercise #5: Which of the following would give the length of Be shown below?
(I) ISsm(34')
(3)
C
18
c05(34')
18
18
(2) sin(34'j
(4) IStan(34')
34"
A
B
Exercise #6: A laddcr.leans against a building as shown in the picture below. The ladder makes an
acute angle with the ground of 72°, If the ladder is 14 feet long" how high, h, does the ladder reacb up
the wall? Round your answer to the nearest tenth ofa foot.
h
Algcm t, Unillf!-ltigb! TriangleTrigomlll'lt1ty-15
The ~ AJgebm f'g;j¢Ct, LilOnngcullc. NY l25®
Name: _ _ _ _ _ _ _ _ _ _ _ __
Date: _ _ _ _ __
Using Trigonometry to Solve for Missing Sides Algebra 1 Homework Skill
In problems 1 through 3. determine the trigonometric ratiO' needed to solve for the missing side and
then use this ratio to find the missing side.
I. In right triangle ABC, mLA = 58' and AB = 8. Find the length of each of the following, Roand your
answers to the nearest tenth.
C
(a) BC
(b)AC
A
2. In right triangle ABC, mLB=44' and AD= 15. Find tit. length of each
your answers to the nearest ten/h.
58'
8
B
of the fullowing.
Round
B
WAC
15
(b)BC
c
A
3. In right triangle ABC, mLC=32' and AD = 24. Find the length of each of the following. Round
your answers to the nearest tenth.
B
(a)AC
~24
:g
(b)BC
C
A
4. Which ofthe following would give the length of hypotenuse PR in the diagram below?
(1)
8co,(24')
(3)8tan(24')
R
24'
Q
p
8
Applications
5. An isosceles triangle has legs oflength 16 and base angles that measure 4&", Find the beight of the
isosceles triangle tu the nearest tenth. Hint - Create Ii right triangle by drawing the height.
16 16
6. Carlos walked 10 miles at an angle of 20· north of due east. To the nearest tenth of a mile, how far
east, x, is Carlos from his starting point?
N
~
--.
..
..
",,,,"'20'
iO miles
'
..
'
'
....
......
y
,-o'==~~====-,;-,-->E
x
7, Students are trying to determine the height of the flagpole at Arlington Hlgh. They have measured
out a horizontal distance of 40' feet from the flagpole and site the top of it at an angle of eievation of
52'. What is the height, h) of the flagpole? Round your answer to the nearest tenth ora foot.
,
,,
,
,,
,,
" 52·
Algebra I, Unit,\lB - Right TrillilgJeTrigon\lrl'letl'y_1..5
'I'M ArlillglDll Algebra Proj<!d., L&Grangevilk, NY 12540
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,
,,
,
,,
,
,,
h
1
Name: _ _ _ _ _ _ _ _ _ _ _ __
Date: _ _ _ _ __
Solving For Missing Angles Algebra 1 Today we willleam how to use right triangle trigonometry to find missing angles of a right triangle. In
the first exercise, though. we will review bow to solve for a missing side using trigonometry.
Burcise #1: Find the length of AS to the nearest tenlh,
C
125
32"
B -'-----''''-...:>..A
591mg for a Missing Angk; - The process for rmding a missing angle in a right triangle is very similar
to that of finding a missing sIde, The key is to identify a trigonometric ratio that can be set up and then
use the inverse trigonometric functions to solve fuc that angle.
Exercise #2: Solve for mLB to the nearest degree.
A
5
3
B
C
Exercise #3: Find the value of x, in the diagrams below, to the nearest degree.
(a)
(b)
125
x
40
A!&d!m I, Urulll8 -
Right Triqk Tri&')llllmetry - 1..6
lbe ArtiqttJl\ J\Jgd)1II Project. ~lle, NY 125<IU
94
Exercise #4: Find the value of x in the diagrams below. Round your answers to the nearest degree.
(a)
(0)
10
15
x
Exercise #5: A flagpole that is 45~feet high casts a shadow along the ground that is 52~feet long.
What is the angle of elevation, A.. of the sun? Round your answer to the nearest degree. ~
••
•
///{If
,•
45 reet
1
I.
52 feet --..j
Exercise I#.i: A hot air balloon hovers 75 feet above the groWld. The baUoon is tethered to the ground
with a rope that is 125 feet long. At what angle.of elevation, E, is the rope attached to the ground?
Round your answer to the nearest degree.
125 feet
i
15feet
E
• Alpm 1, Uoit #&- Rigbt Triwlgle Trigonometry-U'i
TheArlingtoo AIgetmt Proja;{, t.aGt:a.nge'\oille, 'NY 11S40
Name: _ _ _ _ _ _ _ _ _ _ _ __
Date:
Solving For Missing Angles
Algebra 1 Homework
Skills
1. For the following right triangles, find the measure of each angle~ x. to the nearest degree:
(b)
(a)
19
39
11
27
x
x
(d)
(oj
51
21
x
29
x
36
2. Given the following right triangle~ which of the [cHowing is closest to mLA '?
A
(I) 28'
(3) 62'
28
(2) 25'
(4) 65'
C
3. In the diagram shown) mLN is dosest to
(1) 51'
(3) IT
(2) 54'
(4) 39'
ub'---1-3- --"" B
M
17
p
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The Arlillglt>ll Algdml Pmjed, ~~II¢. NY l2.W'l
21
N
Applications
4. An isosceles triangle bas legs measuring 9 feet and a base of 12 feel Find the measure of the base
angle, x, to the nearest degree. (Remember~ Right triangle trigonometry can ontl' he used in right
triangles.)
9
9
12
5. A skier is going down a slope that measures 7,500 feet long. By the end of the slope. the skier has
dropped 2,20(} vertical feet. To the nearest degree, what is the angle, A. of the slope?
2,200 feet
A
Reasoning
6. Could the fonowing triangle exist with the given measurements? Justify your answer,
24
70
11 • Ngcbm 1, UnitliS -RlgbtTrimlglcc Trigontmlcay-L6
'fbe Arling!nn.Algel::ra Proje.;:t. LaGra:lKevillc, NY 12S4C
Date: _ _ _ _ _ __
Name:
Applied Trigonometry Problems Algebra 1 Over the last few lessons, we have discussed how to use trigonometIy to solve for the missing sides or
angles of a right triangle. Today. we will continue solving such problems in the context of"real life"
scenarios.
, ' TIlE TIUGONOMETRIC RAnos
SiILA= oPP.
byp'
ad'
byp
cosA=J;
tan A = opp
adj
Ex<rcis. #1: An ai<Jllane takes off at an angle of 5' from the ground, If the aiIplllne lravelad 100
miles. how far above the ground Is it? Round to the nearest foot. Note that there are 5280 feet in. one
mile, [The 5" angle in this problem is called an angle of elevation.]
100 miles
5"
Ex£rcise #2~ While walking his dog, Pierre sees the EitTel Tower and notices that the angle of
elevation is 24~ to its top. [fPierre is 2215 feet from the middle of the base of the arch how tall is the
Eiffel Tower? Round to the nearest foot
j
I
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Exercise #3: Harold is hang gliding off a cliff thot is 120 feet high, He needs 10 IIavel 350 feet
horizontally to reacb his destination. To the nearest degree, wbat is his angle of descent, A1 [Note:
This angle you are finding is called an angle of depression or an angle of dedination.]
--___ u____ u___u_u.... ~
r
u_~~::':'1
.'.'
,,
120 It
•••/ / / / / . . . .
....
350 ft
•
1
Exercise #4: Francisco is trying to reach a window with a ladder that is 15 feet long. Find the angle
that the ladder must fonn with the ground in order to reacb a window that is II .feet high, Sketcb a
diagram below mat represents this scenario. Round to the nearest degree.
Exercise #5: A ladder that is 12 feet long bas its base 5 feet from the edge ofa building against which
it js leaning. In order to be stable, the angle the ladder makes with the ground must be less than 60
degrees, Is dUs ladder stable?
JusttiY,
Exercise #6: A tree casts a shadow that is 32 feet long. Find the height of the tree if the angle of
elevation ofilia sun 35.7~. Round your answer to the nearest foot.
Nwue: _________________________ Date: _ _ _ _ _ __
Applied Trigonometry Problems Algebra 1 Homework Applications
1. Sitting at the top of a 57 ft. cliff; a lioness sees an elepbant. The angle of depression from the
lioness to the elephant is 22·. What is the shortest distance from the lioness to the elephant?
Round to the nearest tenth ofa foot
.
··························-c~~
22"
2. An airplane is trying to take off; however. there is an obstruction in the runway. The obstruction is
20 feet high, and the plane is 80 feet from the base of the Object. At what angle must the plane take
off to avoid hitting the obstruction? Round to the nearest degree.
3. A 14 foot ladder is leaning against a house. The angle formed by the ladder and the ground is 72' .
(a) Determine the distance, d. from the base of the ladder to the house.
Round to the nearest foot.
(b) Determine the height, h, the ladder reaches up the side of the house.
h
Round to the nearest foot.
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4. A 625 foot tong wire is attached to the top of a tower. If the wire makes an angle of 65" with the
ground, how taU is the to'wer? Round your answer to the neares1 tenth ofa foot.
625 feet
5. Luke casts a shadow that is 6.3 feet long. Find Luke's height if the angle of elevation to' the sun is
40·. Round to the nearest tenth of. foot
6. A blimp hoven; above the ground at an altitude of 560 feet. Two: points. A and B. located on the
ground are shown below with angles of elevation to the blimp of 36" and 52" respectively_
Detennine the distance between the two points, A and B, to the nearest foot. (Note - you will have
to use two trigonometric ratios to solve this problem.)
s~o
feet
:s 31 Name:
D.le: _ _ _ _ _ __
-----------------
Applied Trigonometry Problems Day 2 Algebra 1 Yesterday we saw "reallifeu examples of when we CQuld use trigonometry to fmd missing distances or
angles. Today we wiU revisit this with more "real life"" exrunples.
Exercise #1:
In many trignnoIllelIy problems, the telmS Angle of Elevallon and Angle or
Dep.....lon are used,
(a) On the diagram below use x to label the angle of elevation from point A to point B and use y to
labeJ the angle ofdepression from point B to point A. B
(b) What is true about these two angles?
A
,*
Exercise #2: A flagpole 30 feet tall casts a shadow 52 feet long. What is the angle of elevation of the
sun measured to the nearest degree?
,,
""",
,
,,
,
,
'~
30 feet
1
j..--S2feel-+l
Eurcise #3: Maria is flying a kite on the bea;;:h. She holds the end of the string 4 feet above ground
level and determines the angle ofelevation of the kite to be 54". If the string is 70 feet iong, how high
is the kite above the ground to the nearestJoof?
Exercise #4: From the top of an 86 foot lighthouse, the angle of depression to a ship in the ocean is
23", How:tar is the ship from the base of the ligbthouse? Round your answer to the nearest foot
1
..
86 feet
1
Exercise #S: A tower is located 275 feet from a building in the figure shown below. A person from
the second story measures an angle of elevatioll to the top of the tower as 42~ and an angle of
depression to the bottom of the tower as 26'. Find the height of the tower to the nearest tenth of a
foot
o
o
o
•
275ft .._ -••
Exercise #6: A helicopter is flying at an elevation of 350 feet, directly above a roadway. Two
motorists are driving cars Gil the highway. The angle of depression to one car is 37' and the angle of
depression to the other car is 54". How far apart are the cars to the nearest/oot!
54'
Name: _ _ _ _ _ _ _ _ _ _ _ _ __
nate: _ _ _ _ _ __
Applied Trigonometry Problems Day 2
Algebra 1 Homework
Applications
1. A person standing 60 inches tall casts a shadow 87 inches long. What is the angle of elevation of
the sun to the nearest degree?
2. A ship is headed toward a lighthouse which we know is 65 feet high. 1f an observer on the boat
measures the angle of elevation"to the top of the tight house to be IS". how far is the boat away from
the base oEthe lighthouse) to the nearest/oot!
3, A giant redwood tree casts a shadow that is 532 feet long. Find the height of the tree if the angle of
eievation of the sun is32~. Round your answer to the nearest foot
4. What is the measure of the base ang1~ x, of the isosceles triangle shown below? Round your
answer to the nearest tenth of a degree,
20
20 36
-::! 'JiJ
5. A person flying a kite originally lets out 232 feet of line. At this point the person observes an angle
ofelevation to the kite of48~. The person then lets .out additional line for a total of 312 feet. At this
point the person observes an angle of elevation of 53",
(a) Find the initial height of the kite, AB, to the nearest/ool.
312 ft
232
(b) Find the final height of the kite,AC, to the nearest/oot.
(c) Using your answers from (a) and (b), fmd the amount the kite rose after the line was let out to the
nearestjoot.
6. From a point 435 feet from the base of a building it is observed that the angle ofelevation to the top
of the building is 24° and the angle ofelevation to the top of a flagpole atop the building is 2"',
(a) Find the height ofthe building, h, to the nearest/eel.
00
(b) Find the height to the top of the flagpole, k, to the nearest
foal.
00
•
435
•
(el Using your answers from (al and (b), find the length of the flagpole to the nearest/wi,
u
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What Do Two Bullets Have When They Get Married? Work each problem and find your answers at the bottom of the page. Shade out the letter above each correct answer. When you finish. the answer to the title question will remain. G) Find the length of the hypotenuse of each right triangle:
A.
B.
.... ~~
"3
11 em
gcm
® A rectangle is 3 meters wide and 10
(j) Each side of a checkerboard measures
meters long. How long is the diagonal
of the rectangle?
40 em. What is the length of its
diagonal?
® A18rectangle
is 13 centimeters wide and
centimeters long. How long is its
® An inclined ramp rises 4
diagonal?
@
4cm
C.
meters over a horizon~
distance of 9 meters. ~ 4 m
long is the ramp?
.
A guy wire is attached to
an upright pole 6 meters
above the ground. If the
wire IS anchored to the
ground 4 meters from the
base of the pole, how long
is the wire?
®
... 4m ....
9m
A box is 120 cm long and 25 em
wide. What is the length of the
longest ski pole that could be packed
to lie flat in the box?
i :
,
@ The window 01 a burning
® wide
A television screen measures 30 em
and 22 cm high. What is the
building is 24 meters above
the ground. The base of a
ladder is placed 10 meters
from the building. How long
must the ladder be to
reach the window? diagonal measure of the screen?
@'AShiP leaves port and sails 12
kilometers west and then 19 kilometers north. How lar is the ship Irom port? --t 10 m
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Greeli Deader TO DECODE THE MESSAGE AT THE BOTTOM OF THE PAGE:
Figure out the length of the missing side of any right triangle
below. Find your answer in the answer column and notice the
GREEK LETTER next to it. Each time this GREEK LETTER
appears in the code, write the letter of that exercise above it.
~
a = 7, b = _ _ , C = 12
()
.p
@ a = 5, b = ,
, C = 14
@a=8,b=
,c=V164
Q) a = 4, b = 11, C = _ _
7l'.
u
® a = 12, b = 5, C = ~.._
© a=
, b = 7, C = 10
® a=
, b = V4B, C = 13
® a = _._, b = 12, C = 15
@ a = 10, b
II
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7)
€
It
= __._, C = 16
7
\W)a=l,b=
,c=2
;',) '[email protected] a = l,b= l,c= _ _
~
(!) a = 0.8, b = 0.6, C = _ _
®a=
, b = 1.5, C = 2.5
'
a = _ _ , b = 24,
a =
a
=
= 25
,C = 15
C
v156 ~ 12.5
v'3 :.. 1.73
V144 = 12
V137' 11.7
\1'1= 1
v'1ff . 13.1
V49=7
v'51 ' 7.14
V16=4
v'95 . 9.74
v'168:.. 13.0
v'81 = 9
V169 = 13
IJ.
{3
\1150:" 12.2
V100 = 10
V121=11
v'2:.1.41
p
V4= 2
I<
®a=~_,b=11,c=17
®
®
®
Il
a
KEEP WORK/NG AND YOU W/LL DECODE THE SECRET
MESSAGE.
®
ANSWERS
v7s, b =
v87, b = Vs7, C =__
@ a = __ ~, b = 3,
C
= 5
SECRET MESSAGE ,
a
K
7T
Ie p
7
a
U
II
1J
<I>
K
7
t e
<;
K.p
up <;
II
771
PRE,ALGEBRA WITH PIZZAZZ'
© Creative Publications
r
+
165
What Did Lancelot Say To The Beautiful Ellen? TO ANSWER THIS QUESTION:
Cross out the box containing the answer to each problem. When you finish, write the letters
from the boxes that are not crossed out in the boxes at the bottom of the page.
<D For each right triangle. fJnd the length of the side that is not given:
A
13m
8
E
D
C ___- - - - -7
E
'"3
<0.,,"
10
m
® An 18-foot ladder is leaned against
® The bases on a baseball diamond
are 90 feet apart. How far is it
from home plate to second base?
a wall. If Ihe base of the ladder is
6 feeUrom the wall. how high up on
the wall will the ladder reach?
® Orgo has let
out 50 meters
of kite string
when he_
observes that
his 'kite is directly
above Zorna. If
® A quarterback at
'\ "1 point A throws
the football to
8
/
a receiver who
catches it at
I
point 8. How long
was the pass?
Orgo is 35
met~rs from
Zoma, how high
is the kite?
25yd
Af4:d
r
o
;S
(f) From Canoeville it
@)
A What is the
height of this
parallelogram?
B What is the
area of the
pa(allelogram?
~fj7
~9m-""8m""
is 2.4 kilometers
to White Beach
and 3.0 kilometers
,-'"
LODGE ..&:--"==-TIr w
~
to the lodge.
How far is It from
White Beach
across the lake
to the Lodge?
CANOEVllLE
Ie
KW
au
IT
UR
AT
GR
"\1821 yd·
v"'fi6km
v'i44m
'0/136 m
"\13.24 kin
V842 yd
'0/16200'
V9sm
" 28.7 yd
~
1.78 km
"" 12 m
.,;, 11.7m
= 1.8 km
,;, 29.0 yd
" 127"
EA
TC
UT
EA
LN
GT
V105m
"\1260'
"\1275·
"\11275 m
\11325 m
v'4O
"'" 10.2 m
= 16.1'
" 16.6'
35.7 m
"36.4 m
"6.32 m
ST
AS
234 m2
166
PRE·ALGEBRA WITH PIZZAZZ'
© Creative'! Publioi!loos
ER
204
m~
c,
~
9.75
m
... · .. ~,-.
'--,
'-
Why Did the Saltine Lock Itself in the Bank Vault?
TO ANSWER THIS QUESTION; FOLLOW THESE INSTRUCTIONS:
For each exercise, select the correct ratio from the four choices given, Write the letter of the correct choice in the box
that contains the number of the exercise.
,
,
,
CD sin A ® 1£ @ ..£.
B I
@sin A @..i ® 1­
13
13 ~'" 5 ~I "i4'
3
5
A
® COS A I'i\ 13 ® 5 A
i I.!.':!I COS
@ 5 @ 4
0::,; - E 12
c!
U
B ­
®tanA
5
'[email protected]
3
"""'",q ,...._"'..."'...."....\,.. . . . . .""...._''''''.. ."....'''''''~. .\\...
12
~" .."'"'-"..." ..."'.. " ...".''''....~..., ..... ,..'"''....." ..." ...''!...,......., ....''....' '....''''..."..."'",....."""",...." ..." ,"'"",."
. "....
@) sin S
@ ~ @ ..£.
5
®cosB
® tan S
13
©1£@1£
13
B
5
n
n
c!I
12
1f0.1
@ _~
\Cl 3
v2
(j) S'ln A
@
8j
ii'~
~,'!l
~ :r
2'Jl
!l:J:j
hi
~~
~
,~
® cos A ®
® tan A
,~
~ ® v'2
1
A
3
~~...,~...., '...'''..,,... , .....,,..''~" ..." " "..."
l<,,,... ......' ''''''\,\\...,,...'''.... ....,,...,,....' '...
~
-s--1 l
A~51
. "...." ...." ..." ..."""".,... , .... " ..........." . . ',..." ...,>0.. " ... " ..." ... " .. " ........" "... " . . ., ..., ''''., .. " .. " . ." ... ,~... ''''~,...' ' ' ' '...., ' ' ' '
"~
(;
B
B
@sin'S
@
1 cosS
C!
""~..." ...""~" ..." ' ' '....'''...' '...' '....' '....' '...''"""".."
... " ...." ..." ......" ..." .." ...
15
C
'A
...'\,..."..."....'''....''''\,~''...,~...,.,
V5§
7
Br-----..VS3"
v53
2
C
©L®L 2~A
®
~
15
® lL
8
'20'
~ cos A ® ~ @ J§..
@tan A
'"
,u­
@)..L @ .£.
@tan S
[email protected]
j
l'
V2
5
,'lOIo,..." 'II,\,,....~...','''''I¢.'\,~,"*-"""""
~~....""..,w••' ...."_''''''...........~''...''..._'~....' .......\ \...''....'~....'_'''''''"'',~'''''''''''''''''..." ..."
I
'"
20 I
17
17
7
..._~,...." ....""""'""'..." ...." ..,....." ..".".. C
[[
16 1
B
30
:7
A
/"'~
c.. . . . ' . . ."..
.... " ...' ...."""""'''.....- ' ' '..' '...- ' '....' ' .... ' .....'''...'''''''''''''......- ' '...
~-'' '~
'~
@>·s(f)v2®[email protected]®[email protected]
sin
1
v'2
I
v2
v'2
~
®
~
~ t1V3\
'"
"'" /'
11 cos B ®
CD
v'2
1 i
@tcosA I'i\ v'2 ®
E ...L I A
!
0::,; F 1 A/
"6
® tan B
3
3
v'2 c I
® tan A
1
V2
j.~-.~~T7~~~~~~~~-r~~
t 10 1211615tt8t1611TI3J~t9tt41111122115 [4[211 23 t
a1
~
What Did The Leopard Say After Lunch?
TO ANSWER THIS QUESTION: Use the table of trigonometric ratios to do each exercise. Find each answer at the
bottom of the page and write the corresponding letter above il. Find the following: © sin 35°
® cos 70"
(f) tan 10°
® cos 65
Angle
Sin
Cos
Tan
()"
0.000
1.000
0.000
50
0.087
0.996
0.087
0
1()"
0.174
0.985
0.176
9
15'
0.259
0.966
20"
0.342
0.940
25'
0.423
0.906
3~'
0.500
0.866
35'
0.574
0.819
0.577
0.700
40'
0.643
0.766
0.839
45·
0.707
0.707
1.000
50"
0.766
0.643
1.192
55'
0.819
0.574
1.428
60"
0866
0.500
1.732
650
0.906
0.423
2.145
70'
0.940
0.342
2.747
0,
75'
0.966
0.259
3.732
~~
~
80'
0.985
0.174
5.671
85'
0.996
0.087
9()"
1.000
0.000
B
Use the figure at
the right for the
remaining problems.
,,•
a
•
•
AL----c""--DC
"@ If m LA = 40 , then ca =
® If ~ = 0.966, then m LA
0
=
<D If m LA . 55 then ~ =
® If ~ = 0.707, then m LA =
<D If m LA. = 80 ' then cb =
® If ~. = 0.500, then m LA =
,
0
,
~.,
0
@ If m LA
(
"
(,
I~,
,••
25 then ~ =
0
=
(
,
<D If Ii = 1.428, then m LA =
® If m L B = 30 , then ca =
CO If ca = 0.996,
then m L B =
.
® If m LB = 75°, then ~ =
® If : = 0.839, then m L B =
® If m LB = 15° , then c~ =
® If ~ = 0.906, then m LB =
0
'" '"'" "'" ...""" r- •oo
'" d d '" "
0>
M
~
~
~
0
N
<')
(")
<t
<0
d
.
168
PAE·ALGEBRA WITH PIZZAZZI
© Crea.tive PI.lOlicahcns
tb
~,
L
-~~
."..
00
d
.\ '
~~
0
(f) tan 50"
Q) sin 85°
.
""
•0
00
'"
0>
0
d
d
0>
'" '"
0>
0>
~
to
tb
OJ")
~
~
0.364 ~
0.268
0.466
'=
,
~
~
~ ~
~ ~D~
0
0
~
1~_~~~J
~~
•
oo '"
'"...
'"
.... ci"'"'"
"" '"
,U.,
D
'd"
~
0
0
,
."
Did You Hear About. .. . . .... .. .... ..... .......... ... . ............
D
E
F
C
B
..: :A
"
&.IO"IL&.A..1.
,
,
G
~
...
:
~
H
J
I
+.. ''Y''''''
· ... "TTY?
'Y
K
M
L
,,,,,,,,,, ... ,.,,.,.,.,,, ... ,,.,
'TTY""Y
~
•
~
:.
...
?•
TTYTT",.
•
'T'TTTTT+
DIRECTIONS:
In any triangle hefow. find the length x, Round it to the nearest 0.1 meter.
Find YOlJr answer in the answer column and notice the word ne!<t to it.
Write this word in the box that has the same letter as that triangle.
KEEP WORKING AND YOU WILL HEAR ABOUT A NOVEL NAME! LJx ®
X
30 m
X
@
'.
®
~X
)
®
8m
X
@
X
25m
48'
X
CD
®
16.8 m-ROBINS
13.7 m-BECAUSE
16..7 m-ROOSTER •
123.8m-SO
I 46.2 m-NAME ,
22.7 m-SUN ,
I 44,9 m-BEST !
15.6 m-THE i
87.3 m-WAS:
I 3.2 m-BANKS I
110.6 m-GAVE •
98.5 m-WRECKED-r
5.8 m-ROBINSON •
9.9 m-FARMER '
115.5 m-CREW i
I
75m
117.6 m-HE ,
!12,7m-WHO
15.4 m-PET
95.1 m-THE
Q)
X
~
X
CD A submarine dives at an
angle of 13'. How far is
it beneath the surface at
a pOint 500 meters
along the surface from
where it submerged?
X
11.8 m-HIS
® AI a pOint 20 meters
from a flagpole, the
angle of elevation of the
top of the flagpole is
50". How tall is the
Ilagpole?
X
SOOm
20m
PRE· ALGEBRA WITH PIZZAZZ'
© Creative Publications
169
Books Never Written
, .--- ,
:."
My Ute in the Jungle by
56
Over the Cliff by
60°
3r
Q
23° 72·
36°
45°
6°
66°
15° 37·
21°
j
77° 29° 29°
34° 62° 53°
Catching Butterflies by
6° 45° 45° 29" 53° 53° 29"
6° 45° 55° 6" 56° 6" 34· ABOVE ARE THE TITLES OF THREE "BOOKS NEVER WRITTEN," TO DECODE THE
NAMES OF THEIR AUTHORS, FOLLOW THESE DIRECTIONS:
In any triangle below, find the measure of the lettered angle to the nearest degree, Each
time this measure appears in the code, write the letter above it, Keep working and you
will decode the mimes of all three authors.
3
U
7
6
L-J'--_W
E
O~
DL.1--LJ
4
40
S
14
T
,-r--r-7 G
7
12
L-.L- -W
17
30
~
25
20
170
PRE·ALGEBRA WITH PIZZI\ZZl
© Crealwe Publicohons
~
j
j
20
P
8
10~
H
R
F
z
A driveway is built on an
incline so that it rises 2 meters
==::a
M
23
5
over a distance of 20 meters.
What is the degree measure of
the slope of the driveway?
(See figure below,)
11
15
~4
J
23
12
2
75
10
NL-l---~
)
Name:
Date:
Introduc-t-::i.-n-:-t.-A7)~g-Ceb:-r-a---
Unit 10'--""P:-ol"-p-.-u-rri
LESSON 4 - SQUARE ROOTS
The symbol
r
means "SQUARE ROOT"
EXAMPLE: 14 ~
Solution:
"4 = what number squared (times itself) equals 41
,
v4= 2
EXAMPLE: 19 = ?
Solution:
EXERCISES:
L
J25
=
2.
3.
'/-1- =
4,
IT ~ What number squared (times itself) equals 9?
Find each of the following square roots.
7.
149
=
8,
ro
=
9.
Vl2l =
10,
"144
=
5.
Ii 64
=
11,
10 =
6.
Ji1iii =
12.
m
13.
';49 + 2
14.
3
---------~-
+;130
ro
~
=
Name:
Introduction to Algebra
Date: Unit 10 - Potpourri LESSON 5 - SQUARE ROOTS
Below is a list of numbers and their squares:
Squares
Numbers
1
I'
=
2
2'
=
4
3
3'
=
9
4
4'
=
16 5
5'
= 25 6
6' = 36 7
7' = 49 8
8'
= 64 9
9'
= 81 10
10'
= 100 II
11'
= 121 12
12'
=
EXAMPLE:
.144 r53 is closest to what whole number?
Solution:
53 is closest to which number under the list of squares? _ __
53 is between 49 and 64.
53
49
64
'--..J '-..../
4
C subtract 53 from 49 and 64
II,.J
Since 53 is 4 away from 49 and 53 is i 1 away from 64. 53 is dosest to
49. Then, vB is closest to "49 = 7.
Answer: 7
INTRODUctION TO ALGEBRA
UNIT 10, p. 2
EXAMPLE: Which of the numbers best approximates
0)
Solution:
b)
4
5
.f.i'7'I
0)
6
d)
Remember, approximates means closest to which nU!llber.
27 is betwl"'n lb.
~quares
25 and 36
25~27~36
Since 27 is 2 away from 25 it is closest to 2S so ...
>/29 is closest to ,,"'15 = 5
Answer is b
EXERCISES:
I)
"Sis closest to which whole number?
2)
To which whale number is 195 closest?
3)
Approximate vOf (Pick one below)
4)
6)
6
c)
8
b)
7
d)
10
Which of the fallowing whale numbers best approximates
a)
5)
a)
4
b)
8
0)
9
ro 7
d)
6
d)
8
Which whole number best approximates .;TI
'189 is closest to which of the following whole number?
.)
2
b)
9
0)
4
7
UNIT 10'1 p.3
INTRODUCTION TO ALG£BRA
EXERCISES (Continued)
7)
The ,Illf is closest to which whole number?
8)
Which whole number best approximates ,rTI4 ?
9)
Which whole number best approximates "59 ?
aJ
10
b)
8
c)
1
d)
4
p.IO Name:
Introduction to Algebra
Date:
Unit 10 - Potpourri
LESSON 6 - PYTHAGOREAN THEOREM
The Pythagorean Tbeorem is used for right tri{mgles oniy. These are triangles that have a
90' angle.
A triangle can be labeled as follows:
aCross from the 90° angle.
It is called the hypo tenus •.
<:: = the side
b
The formula is used when trying to find the third side of a right triangle when the other t\vo
sides are given.
Formula:
EXAMPLE: Given the triangle below, find side c.
c
3
4
wherea~3
Solution: and b=4
plug numbers into
the formula
9 + 16 =
25
c'
=e
(what number times itself is 251)
Answer: 5 = c
p.l\ l~'TRODUCTlON
EXAMPLE:
TO ALGEBRA
UNIT 10, lESSON <i.P 2
Given the triangle below, find side a:
b 0:8.
~u.re
c=tO
{
Sol.e t:..
• 0:: "1 do,"~
Q2t
("4"
-(,,4
[
+he. opl"'s,k
-
\00
=-.. +
-
O'l. ~ at;,
EXERCISES:
Find tbe missing side of each of tbe triangles below:
3)
'I
15
4
5)
c
\e.
8
INTRODUCflON TO ALGEBRA
EXERCISES - Sct 2:
Unit 10, LessDn 6, p. 3
For each right triangle, find the missing length.
c
a
b
17•• =6, b=6
20•• =2,b=6
23.• =7, b=l
26.b=4,c=12
18.
21.
24,
27.
a = 12,.b = 16
b = 4, C = 8
a = 6, b = 12
a=8,b=8
19.• = 9, c = 15
22. b=1,o=3
25•• = 9, C = 18
28•• =9, b = 4<l
Unit 10. Lesson 6, p. 4
INTRODUCTION TO ALGEBRA NOTE: If a triangle is a right triangle, then the lengths of its sides satisfy tlte Pythagorean
Theorem~
EXAMPLE If the lengths of the sides of a triangle are 7,
24, and 25, is the triangle a right trial1gle'?
=
See If 7' + 24' - 25', _ _ _ _ _ _4 7' = 49
24' 576 25' = 625 49
Thus, the triangle is a right triangle.
EXERCISES - So13: + 576 = 625
Tell whether each triangle described is a below is a right triangle.
The lengths of the three sides are given, Reeallilia! the hypotenuse
is the longest side of a right triangle. The hypotenuse is located
across from tne right angle.
1.4.5,6
5. 12, 14. 16
9.11,60,61
2. 6. 8,,10
6. 9,40.41
10. 6,6,9
3, 3, 5. 7
7. ,10,24,25
11. 30,40,60
4. 12. 16. 20
8. 10.20.30
12. 14.48,60
If a:! + b 2 = 0 2 , then
the triangle is a right
triangle with c the
hypotenuse,
b
p.14 Unic 10, Lesson 6, p. 5
INTRODUCTION TO ALGEBRA Applications of the Pythagorean Theorem
EXAMPLE
A rectangular field is 50 feet wide by 100 feet
long. How long ,s a diagonal path' connecting two
opposite corners? Give the answer to the nearest
tenth of a foot.
a2+D~=c2
50 2 + 1002 = <;2
2,500 + 10,000 = c'
12,500 = c'
\112,500 = c
111.800 = c
Round to the nearest tenth. _ _ _ _ _ Thus, the diagonal path is approx. 111..8 feet long.
EXERCISES· SET 4
Give answers to the nearest tenth.
32. Paul walked 8 miles north and
3 mUes west. How far was he'
from his starting point?
34. AT V. screen is 15 inches
by 12 inches. What i. lIS
diag6nal length?
33. A 12·fool ramp covers 10 feet
of ground. How high does it
rise?
,35. A 15-foolladder is 5 feel from
the base of a building. At whal
height does it touch the building?
p,15 Chapter 6 TRIGONOMETRIC FUNCTIONS 1. In the accompanying diagram of right triangle CDE, CE" 144,
mLD = 90', m LC = 30' and leg ED = 72 em,
'Which equation can no! be used to find leg CD?
144
72
(1) tan 30' = 72
CD
(2) cos 30'" CD
144
(3) tan 60' = CD
72
(4) tan 30' = CD
72
135 C~3""O_ _
Chapter 6 TRIGONOMETRIC FUNCTIONS pro ems eow.
Use the triangle ABC shown at the right.
1. B
12 6. Write the trig ratio of cos A
Write the ratio of sin A
,
C
I
~.
,,
,
I,
,
i
7. Is cos A "" sin B?
2. Write the trig ratio .cos B
,i
,
,,
3.
8.
Write the ratio for tan A
. 4. Write the trig ratio tm B
,,
9. What is the wIne of tan A • tan B?
I
i,
5. Write the ratio for sin B
10. What is the largest positive value of sine
of any angle?
11. From the diagtam given below, find the
mLF to the nearest degree.
,
14. In right triangle ABC,LC is a right
angle. If AB = 11 and AC = 7, find mLA ,
D~:
,
,1i sin A =cos B, what is the mLA + mLB?
to the nearest degree.
F
12. The sine rntio of any acute angle is equal to the cosine ratio of its c?mplement.
,
,
,, ,
15. The value of the sine ratio of an angle can I,
neve! be equal to the value of the cosine
(True or False)
ratio of the same angle. (True or False)
=BC = 9 and
side AC = 7. Fmd the m LA to the
13. In right triangle ABC, side
nearest degree.
,,
16. In reetangle ABCD, diagonal AC is
drawn. If CD = 8 and AC =13, find
m L ACD to the nearest degree.
,i
'--_.
136 ,
Chapter 6
TRIGONOMETRIC FUNCTIONS
Right Triangle Trigonometry, Finding a Side
1. Triangle ABC has a right L. at C. If
6.
ma =32 and AB = 8, find the length of
In triangle GHI, L. G is • right angle.
If rod. 47 and HI • 6. lind. to the
nearest integer, the length of GJ.
AC to the nearest tenth.
..
·•
.
,
I
•
2. hypotenuse DF of rightLl.DEF if 17.
7. If roLF =77, find the length of DE to the
nearest integer.
·•
In rectangle ABCD, diagonil A<:: forms
a 27 degree angle with side AD. If side
CD • 4, lind the length ofAC to the
nearest integer.
·,
,JI<c
3. From the di~m below, nnd RS to the
nearest tenth. x~
•
8. Isosceles A DEF has base DF = 22.
Altitude EO is drawn. If mLD = 32, find
the length of DE to the nearest integer.
i
·
sh
··
i
T
•
-
4. In right triangle ABC, hypotenuse AB = 12.
If mt:.A =70, find BC to the nearest tenth.
9. Fllld the value of hypotenuse HF in RT
5. Given the diagram below, find the length of
10. Side BC in right Ll. ABC =5 and
mLC = 90. If mLE = 50, find the
length of AC to the nearest inregcr.
DE to the nearest tenth.
~~lO ~F
Ll. FGH if roLF =16 and HG =14.
(Answer to the nearest integer)
,
I)
I
11. In right triangle ABC, mLC = 90", mLA • 70" and AC "78. Find the length of AB
nearest integer. Show how you arrived at your answer.
137 to
the
,
I
.
·
··
Chapter 6 TRIGONOMETRIC FUNCTIONS 1. A building stmd, on level gound. The
measure of the angle of elevation of the
top of the building, takx:n at a point 400 feet
/rom the foot of the building, is 31". Find,
ro the nearest fuo~ the height of the building.
4.
2. 5. A 20 foot ladder leans against a house. If
the foot of the ladder is 8 feet from the
house, find, to the nearest degree, the
From the top ofa 100 fuot high pole, an
observer measures the angle of depression of
a car on the road as 28 degrees. Find, to
the nearest fQot, the distance from the cat to
the base of the pole.
From a point on level ground, the angle of
elevation of the top of an 85' pole is 62°,
Find, to the neatest integer, the distance
from that point to the foot of the pole.
measure of the angle that the foot of the ladder makes with the gound. 3. From the top ofa 120 foot lighthouse, the
6.
angle of depression of a boat out at sea is
26"'. Find, to the nearest foot, the distance
from the boat to the foot of the lighthouse.
138 A 40 foot long wire stretches from the top
of a vertical pole to a stake in the ground 18
feet from the foot of the pole. Find, to the
nearest degree} the measure of the aOlte
angle that the wire makes with the gound.
Chapt<r6
TRIGONOMETRIC FUNCTIONS 1.
A wire reaches from the top of a 26 meter
telephone pole to a point on the ground S
6.
The dimensions of • rectnngIe ace 14 coruimetm
by 4S centime1=. Fmd, in oent:imet=, the
length ofthe diagonal ofthe rectangle.
7.
A ABC is a right triangle with the right
angle at C. IfAB = 13 and Be = 12,
findAC.
8.
The hypotenuse ofa right triangle is 25. If
meters from the base of the pole. What is
the length of the wire to the nearest tenth
of a meter?
2.
The lengths of the legs of. right triangle
are 3 and 6, Fmd, in radical form} the
length of the hypotenuse of the right triangle.
3.
In the accompanying diagram ofa
rectangle ABeD, AB • 6 and Be • 8.
What is the
ofAC?
(1)
Express in radical form, the length of one
leg of a right triangle if the hypotenuse is
9 and the other leg is 5.
5.
Find the length, in radical form, of the
hypotenuse ofan isosceles right triangle
whose leg equals 3.
..f5
(2) ..J 1025
9.
B
4.
one leg is 20, the other leg 1s:
(3) 15
(4) 45
The length of the hypotenuse ofa right
triangle is 7 and the length of one leg is 4.
What is the length of the other leg?
(1) 11
(3) 3
(2) ..J65
(4) ..J33
10. Which of the following could be the lengths
of the sides of a right triangle?
(1) 3,5,8
(3) 2,4,6
(4) 5,5,5
(2) 5, 12, 13
139 Cbapter6 TRIGONOMETRIC FUNCTIONS 1.
2.
Fllld cos A, if A is a positive acute angle
andtanA=
6.
If x is a positive acute angle and
7.
y.
cos x ""
3.
4.
17 '
Find the value of COS x, if tan x ::::
j and
If cos x =: ~ where x is a positiv-e aCute
angle. lind the value of sin x.
If A is a positive acute angle, find the value
of tan A, if cos A.
8.
Find sin x if cos x ::::
i­
and x is a positive
acute angle,
9.
]f0' "As 90· and tan A= ~. find the
value of cos A.
x is a positive acute angle.
5.
i. find the value of cos A if A
is a positive acute angle.
find the value of sin x.
If sin x : : : and x is a positive acute
angle, find tan x,
]f sin A =
10. Which trig ratio can have a value greater
than I?
(1) sin x
(3) tan x
(2) cos x
(4) none of these
140 Cbaprer6 TRIGONOMETRIC FUNCTIONS L
In triangle ABC, m LC • 90, AC· x, BC = (x + 2) and AB • (x + 3),
a Write an equation in tenns of x which can be used to find AC.
b Express the equation in part a in standard quadratic fonn
2,
In right triangle ABC, AC = x, Be • x + 1 and hypotenuse AB = 2x ­ 1.
F'md the length of AC, [Only an algebraic solution will be accepted.)
3.
The length of the hypotenuse of a right triangle is 10. The length of the longer leg exceeds
the length of the shorter leg by 2, Find the length of the shorterleg, [Only an algebraic
solution will be accepted]
4.
The hypotenuse of a right triangle is represented by 3x + 4. One leg is represented by x and
the other leg is 24.
a Find x.
b Find the hypotenuse.
141 Cbapter6
TRIGONOMETRIC FUNCTIONS
Pythagorean Theorem Application, (continued)
The length of the hypotenuse of a right triangle is 13. The length of the shorter leg is seven
less than the length of the longer leg. Fmd the length of the longer leg. [Only an algebraic
solution will be accepted.]
5.
,.
The length of the hypotenuse of a right triangle is 15. If the longer leg is three more than the
shorter leg. Find the length of the shorter leg. [Only an algehraic solution will be accepted.]
6.
~
The hypotenuse of a right triangle is 5 and legs are represents by" and " • 1.
7.
a F'mdx
b F'md the perimeter of the triangle
e Find the are. of the triangle
,
8.
In rect2l1g1e ABCD, two adjacent sides
are represented by x and x + 5. If
B
A
25
x
D
~.
h
.. 5
diagonal AC = 25, lind
a the value ofx
b the area of rectangle ABCD
C
.
142 Integrated Algebra I - Right Triangle Trigonometry Uuit Day One To learn the trigonometric ratios and how to use them to solve right triangle
Goal,
problems
Standards:
A.A.42
A.A.43
A.A.44
Opening:
Draw a right triangle. Deline opposite, adjacent, hypotenuse, making sure thai students
understand that «opposite" and "adjacent" are relative to the angle you are referring to.
sin A
B
!!Ill!
~
hyp
cos A = adj
byp
e
a
tan A ..,; !.I!Jl
adj
c
b
A
Introduce: SOH CAB TOA
Explain:
sin ::= sine
cos = cosine
tau = tangent
Draw the diagram below and have students fmd the trig ratios for angles A and B.
B
sin A =
5
cos A =
13
,
C
h
tan A = 12
A
sin B ==
cos B = _ __
tan B
=
--
Ask studcDts what they notice (Do they see. a pattern - compare and contrast, Why?)
,,,,ires are meant to train students to draw a right triangle based upon the information
These e..
provided.
For III, provide a "blank" triangle, wbichstudents will then label with the infurmetion. They
will notice that they are not provided with the length of the hypotenuse. Vou may want to "hint"
that they might want to use thel'ythagorean Theorem. When they do 118 during work:time, they
need to use the pythagorean Theorem again.
1) In right triangle ABC, mLC = 90", BC = 3, and AC = 4.
Wh.tissinB?
.
For #2, students must dntw their own right triangle.)
2) In right triangle ABC, ifmLC = 90', AB = 5, Be =3, and AC= 4,
what is the eos A ?
Work I'.riod:
Students should complete the Arlington Algebra Project,
Unit #8 Lesson L3 Homework #1 - 14 (attaclled)
NOTE: Exercise #8 in Unit #8, Lesson L3 will require the
students to use the Pythagorean Theorem (0 fmd the
hypotenuse in order to find the cos Q.
Work Period
N~:
__________________________
Date:
----
Similar Right Triangles - Introduction to Trigonometry
Algebra 1
Skills
For problems 1 - 6, use the triangle to the right to find the given trigonometric ratios.
1. cosN"'"
N
15
2. sinN""
9
3. tanN=
M
p
12
4. sinP=
5. cosp=
6. tanP=
1. Given the right triangle shown, wlJich of tile following represents the value of tan A 1
(I) 25
24
(3)
~
A
(2) 24
(4) 24
25
7
7
24
B
8. In the right triangle below. cos Q =::?
(I)
Il:
(3) 12
17
(2)
2.
(4) 12
13
5
12
Tti.
~ebra t, tlait#l!: -Ri1lht
Trigotwmeuy - tJ
Th: Arlingtoo I\Jgclna Pl'ojeel, lAGr.mgmlk, NY 11S4O
2S
24
C
12
Q
S
R
For problems 9 - 14, use !he figure at the right to determine each trigonometric mUo. Make..,. In
reduce your trig ..tios to !heir simplest fann.
9.
sine=
10.
COSC~
11. tanC~
12.
sin A=
13.
cosA=
14.
!anA=
4
A
2
B
Closing:
(This ex"",ise from from the Arlington Algebra Project,
Unit 11&, Lesson L3 Homework. It is #15.)
Consider the two right triangles shown below:
8
10
4
6
Why;' tbesin of35'tbe same in both cases?
(Emphasize that the triangles are similar because the sides are in proportion.
Therefore, the sine ratios would be equivalent ratios.)
Homework: Attached worksheet (Note: Again, the students will need to use the
Pythagorean Theorem to solve #5 & 6.)
Integrated Algebra I - Trigonometry Unit
Introduction to tbe Tril!;onometric Ratios
Homework
Nwne._________________________
Forquestions#I-4,find
•• sinA
Date,___________________ b. cos A
c. tanA
e. cosB
=
f. tanB
~
d. sinB
=
h. cos A ==
e. cosB
=
c. tan A =
f.tanB~
a. sin A
d. sin B
c. tan A
3.
8
5~
C
12
3)
A
A
29
20
90"
21
B
C
4)
f. tanB =
b. cos A
2)
e. cosB
d. sinB
a. sinA
1)
d. sinB
=
~
~
sin A =
~
b. cos A =
c. cos B
c. tan A
r.
3.
~
sin A =
tan B
=
~
~
d. sin B =
c
A
~
P
b. cos A =
c. cosS
=
t~A
f. tanB
~
B
c.
=
5)
In AABC, mLC = 90" , AC = 4, and
6)
In 8RST, mLS = 90· , RS = 4, and
BC = 3. Find sin A.
ST = 3. Find sin A.
Integrated Algebra I - Rigbt Triangle Trigonometry Unit
Day Two
Go.I:
to learn to use the graphing calculator 1l> determine the sine, cosine and
tangent ofa given aeute angle in a right triangle as well as to tind an
angle measurement given any two sides of a right triangle.
Standards:
A.A.42
A.A.43
A.A.44
Opening:
Use Exercise #1 in the AIIinglon Algebra Project, Unit #8, Lesson L4
Classwork as • "Do Now" and to launch the opening fur this lesson.
(Note: TIrey have labeled the right angle with "B", not "C'')
Proceed with Exercises 112 & 3 (in Lesson L4 Classwork) explaining that
these answers are the ratios (in decimal form) of me two sides ofme
triangle.
Exercise #4: Have students look at the diagram and write the equation:
?
tan40
~
d.
4
Empbasize that ifthis equation i. true, then both sides must bo
equal. Students should enter tan 40 into their calculators to see
that tan 40 ~ .839099631 willen does not equal ¥..
Thus the right triangle does not exist with the given measurement
Now introduce the Inverse Trig. Functions following the remainder of
I.esson 1.4 Classwork,
Work Period:
Arlington Algebra Project, Unit 118, Lesson 1.4 Homework
Exercises #1-17 (I" page),
Closing:
Arlington Algehra Project, Unit 118, Lesson !.4 Homework '.
Exercise # J8 for discussion.
Solicit from students that by definition ofsine or cosine, you cannot have
sin-I ~ I, since the hypotenuse is the longest side of a right triangle.
Homewf)rk:
Attached Worksheet
Opening
Nwme: ____________________________
Date: ______________
Trigonometry and the Calculator Algebra 1 rn the previous lesson, we introduced the trigonometric ratios. We also discussed a IllnCtnonic that is
helpful to remember the trig rauos: SOH'(;AH-TOA. The first exereise review, how to wrhe these
",uos given Ibe lengths of lb. sides of. right triangle.
,
Exercise, #1: Using the diagram below. stale the value for each of the following trigonometric ratios.
13
A
12
c
(a) sinA:
(h) IanG=
5
(el
(d) rosA =
B
(e) lanA =
cosG=
(f) 'inC =
Today we wiU discuss how the graphing calculator is used with trigonomctry_ However, before we
s~ it is important that our calculator is the right MODE.
To change your calculator into DEGREE MODE:
I. p{ess the MODE button on your calculator. 2, Use the arrow keys to highlight DEGREE and press ENnl:R 3, Exit the menu (this or any other) by hitting QUIT, Now that we are in DEGREE MODE, we can start evaluating some trig ratios without referring to any
right triangles whatsoever. The SIN, COS~ and TAN buttons are located i.n the center of tbe key pad.
Exercise #2: Evaluate sin30',
cos30~
and tan30·. Round any non-exact answer to the nearest
thousandth.
sin(30)
(;,05(30)
.8660254038
.5773502692
tan(30)
•
Algt'bm I, Uml tUl-Rl.s!lt Triangk- TrigoDOttlCltY-LA
The Arliogloo Algebr.t ProjCCl., 1..I~&mIk, MY 12540
.5
Burdse #3: Evaluate each of the fonowing. Round your answers to the nearest thousandth.
(a) tan( 40")
(1)) oos(20")
(0) sin(63")
It is important to remember that: each of the answers from Exercise #3, represent the ratio of two
sides of a rigltt lrillllgie. In each case the ratio bas already been divided and the oatoatator is giving
the d""imalronn of the ratio.
Exercise #4: Could the right triangle below exist with the given measurements? Explain your answer,
C
3
40'
B
4
A
The Inverse Trig Functions - Thus far, we have been evaluating the sine, cosine, and tangent of angles,
By doing so, we have been finding the ratio of two sides in a right triangle given an angle. Using the
calculator. we can reverse this process and find the angle when given a ratio ofsides,
Exercise lIS: Consider the following:
(a) Evaluate sin-I (~) using your calculator, The screen shot is shown at the below.
sin-1(1/2)
I
(b) How do you interpret this answer?
E.Xercise~:
'Find each angle that has the trigonometric ratio given below, Round all answers to lhe
nearest tenth of a degree, if they are not whole numbers,
5
2
(.) tanA=-
I
(b) oosB=2
(e) sinE =3.
3
· Work Period
Name:
Da!e: _ _ _ _ __
--------------------~
Trigonometry and the Calculator
Algebra 1
Skills
For problems I through 6, evaluate eaeh trigononretric function. Round your answers to the nearest
thousamith.
I. sin (55")
2. oos(45")
3. t3ll(2o-)
5. t3ll( 60')
6.•m(23')
For problems 7 through 15, find the angle that has me given trigonometric ratio. RQund aU non-exact
answers to the nearest tenth ofa degree.
4
8. ensO=9
8
5
13. IllnA=3.127
11. wnR=-
12. sinT=l
14. sinB=O.724
15. oosL=0.876
16. If sin A l::: ~ then mLA is closest to which of the foltowing?
5
(I) 32'
(3) 24'
(2) 76'
(4) 56'
17, If tan A =3.5 then m/-A.
(0
the nearest degree, is
(I) 74'
(3) 24'
(2) 18'
(4) 55'
9. wnK=1
2
Closing
Reasoning
18. Recall trom homework problems 7 through 15, th.t an inverse trigonomelric functio. gives the
angle measurement that corresponds to the ratio aftwo sides ofa right triangle.
(a) The screen shots below show what bappens when you try to evaluate sin-I (~J. Verify why this
,2 bappens when you evaluate sin ~l (~
J. I
. . the !,act
~ tho t sm
. ( Angl\;
'_) ;; opposite side ) wuy
•
(b) CODSldenng
ERR:DOMAIN
hypotenuse
IIIQuit
NGato does it make sense that sin-I (~) gives you an ERROR? 19. Are the triangles below labeled with correct measurements? For each right triangle below.
determine if the ratio of the sides accurately corresponds to the angie.
(a)
(b)
60'
16
D....
30·
4
8
c
20. Consider the following right tnangle,
(a) Express sin A as a ratio.
(b) Find the mLA to the nearest degree"
AIgdmtl. Unl! ItS PJO:!t TriMgk Trigoo.oowlry- fA
The ArlingtDn Algclma fulja:1. ~vilh; NY Il)~!)
7
A
3
8
,LI.UUI",nUI
n.
Integrated Algebra I - Trigonometry Unit
Using the Calculator & Trigonometric Ratios
Name~
________________________
Dale__________________
In #1 - 8, use a calculator to find each of the following to the nearest ten-thousandth:
1)
tan
5)
10°
2)
6)
tan 55°
3)
7)
sin 89°
4)
tan 36°
8)
In each of the following, use a calculator to fmd the measure of LA to the nearest degree.
9)
tan A ~ 0.0875
10)
sinA~0.I908
II)
cos A ~ 0.9397
12)
tan A ~ 0.3640
13)
sin A ~ 0.8910
14)
cos A
~
0.8545
Integrated Algebra I - Right Triangle Trignnometry Unit Day Three Review homework from prior day.
Goal:
To apply trigonometric ratio. to find
a)
the length ofa side ofa right triangle given the measure ofa side
aed the measure ofan acute angle
b)
the measure (to the nearest degree) ofan angle given two sides of
• right tri",!gle
Standards:
A..A.42
A..A.43
A.A.44
Opening:
Draw the foUowing problems on the board and show the students how to
Solve for ~'x" using trigonometry.
7
x
12
20
x
8
tan20=
5
sin 32
x
cosx=-L
--2L­
7
12
Note: Instruct students to find the decimal equivalent for the trig value firs~ and then "roaed" at
the end of the problem.
Work Period: Attached
(Arlington Algebra Projec~ Unit #8, Lesson L5 Classwork #2, #3
Unit #8, Lesson L6 elas,work #2, #3, #4)
Closing: How do we determine which trigonometric ratio to use when solving a right
Triangle problem?
What is the difference between sin x and sin~! (x) ?
Consider the triangle below:
l~ f2
~
sin 45 =
Why are these trig values the same?
00.45 =
Is this the only case where they are the same?
1
Home-work!
See attached handout.
(AMSCO Math A Textbook, Page 813 #1 -7
Page 822 #1 - 12)
Integrated Algebra I - Right Triangle Trigonometry Unit Day Three - Work Period Nmne'_______________________
___________________
D.~,
TRIGoNOMETlUC RATIOS
Recall that in a right triangle with acute angle A, the following ratios are defined:
. A =-~_op=pos=ite:....
SUl
hypotenuse
~A= adjacent
opposite
adjacent
1M A.; _... - - -
hypotenuse
Exercise #2: In the right triangle below. find the length of AB to the nearest tenth.
C
26
3g-
B LL---""----'o>"A
The key in all ofthese problems is to properly identify the coJ"reCt trigonometric ratio to use.
h'xel'cise #3: For the right triangle below, find the length of BC to the nearest tetUh,
B
65·
A
12
c
:SOlVing tOr a Missing Aogl~ The process for finding a missing angle in a right triangle is very similar
to that of finding a missing side. The key is to identify a trigonometric fi!!tio that can be set up and then
. use the inverse trigonometric functions to solve for that angle.
.
.
Exercise U2, Solve foc mLB to the nearest degree. A
5
3
C LL--_-""B
Exercise #3: Find the value of x, in the diagrams below. to the nearest degree.
(a)
(b)
125
2S
x
94
40
Exercise #4: Find the value of x in the diagrams below. Round your answers to the nearest degree.
(b)
(a)
15
x
s.
6.
50·
s·
7.
s.
Integrated Algebra I - Right Triangle Trigonometry Unit Day Three Homework - Day Three AMSCO Math A Textbook - Page 813 - 814 Nrune'___________________________
O&e________________________
In 1-9, in each given triangle, find the length of the side marked x to the nearest/oot or the number
of degrees contained in the angle marked x 10 the nearest degree.
2.
1.
~x 25·
4.
3.
18·
,<l
10·
55'
Integrated Algebra I - Right Triangle Trigonometry Unit Day Three Homework - Day Three AMSCO Math A Textbook - Page 822 Nwue_________________________
Date_ _ _ _ _ _ _ _ _ _ __
In 1-8: In each given triangle, find to the neareSI centimeter tbe lenglh of the side marked x.
2.
1.
,
"
3.
, ~~s0
~
x
5.
7.
8.
x/1
~
. lOem
x
In 9-l2: In each given triangle, find to the nearest degree the measure of (he angle marked
9. ",,10'
~
S'
11.
IS'
~
10. n
4'
x
12'
.
12.~
12'
x
Ig'
,I.,
Integrated Algebra 1- Right Triangle Trigonometry Unit
Day Four
Review Homework and Class work from prior day.
Goal:
To review how trigonometry is used to fmd the length of a side or measure
ofan angle in a right triangle
Standards:
A.A.41
A.A.43
A.A.44
Work Period: Students should complete the double-sided worksheet attached.
Quiz:
Students should complete the 4 trigonometry problems (taken from the
Integrated Algebra I Regents) on the attached quiz.
Closing:
When do we use right triangle trigonometry and when do we use the
Pythagorean Theorem to solve for a length of. side of a triangle?
•
T
"': . . . . I&.
............. Finding the length of a side or the measure of an acute angle in a right
triangle using trigonometry
N.me~
___________________________
Date~
____________
Find the measure of the angle indicated or the length of the missing side marked 'x'.
Show all work.
2) .
I)
x
39°
32°
x
3)
4)
5
13cm
3
7cri1
gcm
5)
~4
6) x
7)
~x
~
8
70 em
~x
10)
9
20°
30 m
12) .
11)
13 15
13)
45
25
~"
50
QUIZ Integrated Algebra I Regents Questions Nmne~
1)
__________________________
Date
(Janumy,2009-2 points) The diagram below.how" tight triangle upe. u
8
c
17
15
p
Which ratio represent.. the sine of LU? 15
8
(1) 8
(3) 1:5
(2)
2)
i~
8
(4) 17
(January, 2009-2 points)
In the right triangle shown iu the diagram below, "..·hat IS the \<illuc or
x to the rU!areslwitoie llurni'JCT?
x
30'
24
(l) 12
(3) 21
(2) l4
(4) 28
3)
(August, 2008 - 2 points)
Which equation could be used to lind tlte measure ofone .cute angie
in tlte rig"t triangle shown below?
A
5
c
LL_ _: -_ _.Oo. B
4
(1) s;n A =
t
(3) cos B = 4
(2) tao A
45
(4) tan B
tt
5
="54
4)
(August, 2008 - 2 points)
In the diagram of MBC shown below, Be
:=
10 and AB "" 16,
B
To the nearest tenth of n degree, wbat
ac,ute angle in the triangle?
($
(I) 32.0
(3) 51.3
(2) 38.7
(4) 90.0
the measure of the largest
Integrated Algebra I - Right Triangle Trigonometry Unit Day Five Goal,
To apply trigonometric mtios to solve verbal problems
Standards:
A.A.42
A.A.43
A.A.44
Opening:
Recall ratios
~~
theGC
Remembet
•
Side
opposite L..A
with "50H-cAH·TOA", ,
SIn =
Efi'.
Hyp
Ad]
Cos=Hyp
b
C
Side adjatent
ra. =Efi'.
Ad]
!oLA
Then show students how to solve the following three problems (on the next page) that involve
ladders leaning against buildings.
Work Period: Attached handout with verbal problems
Closing:
Lorraine said to Rosalie:
fir can't decide if! am supposed to use the
Sine ratio or the cosine ratio to solve this problem.H How should Rosalie
answer Lorraine?
Homework:
Attached worksheeet
Int. A1g. I - Trig. Unit - Day 5 - Opening - Solving Verbal P ....bl.....
Nwne 1) ______________________
Da~.
A 14 foot ladder is leaning against a house. The angle formed by the ladder and the
ground is 7Z' •
(a) Detennine the distance, d, from the bose or the ladder to the house. Roand to the nearestfoot. r
h
(b) 2) Detenume, the heigh~ h, the ladder reaches up the side of the house. Round to the nearestfoot.
If a 20-foot ladder reaches 18 reet up a wall,
what angle does the ladder make with the
grollnd~ to the nearest degree?
18 ft
3) A ladder 25 reet long leans against a building.
The hottom of the ladder is 9 reet from the
base of the building. What angle does the
ladder made with the ground, to tne neLlrest
degree?
Integrated Algebra 1- Trigonometry Unit Day Five - Work Period Name I) Oate._ _ _ _ _ _ _ _ __
A 625foot long wire is ._bed to the top ofa tower.
If the wire makes an angle of6S< willi the ground, how
tall is the tower? Round your answer to the nearest
tenth Qfafoot.
625 foct
2)
A surveyor is standiog 118 feet from the base oflli. Washington Monument.
The surveyor measures the angle between the ground and the top of the
Monument to be 78". Find the height, h, of the Washington Monument
to the nearest/oat.
3) While flying a Idle, Betty lets out 300 feet ofstring. which makes an angle of
38' with the ground. Assuming thai the string is stmight, how high ahove the
ground is the Idle? (Round your answer to the neAreSt 1\llltlL)
4) A guy wire reaches trom the top ofa pole ro a sta\re in the ground. The slake is
3.5 meters trom the foot of the pole. The wire makes an angle of65' with the
.
ground. Find to the nearest meter the length of the wire.
5) In a park, a slide is 9 meters long and is built over a horizontal distance of
6 meters along the ground. Find to the nearest degree the measure of the angie
that the slide makes with the horizontal.
Integrated Algebra I - Trigonometry Unit Day Five - Solving Verbal Problems - Homework _____________________
N~;
Dare,___________________
I)
Solve for the angle (x) to the nearest degree
x
2)
4 •
Solve for the length of the side (xl to the nearest tenth
2.5 inches
3) A ladder is leaning against a waiL The foot ofthe ladder is 65 feet from the wall.
The ladder makes an angle of74" with the level ground. How high on the wall does
the Jadder reach? Round the answer /0 the nearest tenth ofafoot.
4) A 20·foot pole that is leaning against a walI.reaches a point that is 18 feet above the
ground. Find to the nearest degree the number ofdegroes contsined in the angle that
the pole makes with the ground.
5) While flying aldte, Doris let out 400 feet of string, Assuming that the string is s!retched
taut and it makes an angle of48" with the ground. find to the nearesi/ool how high the
kite is.
6) In rectangle ABCD, diagonal AC is drawn. If mLBAC ~ 62 and BC ~ 20
find to the nearest integer
a)
the length of AB
b)
the length of AC
Integrated Algebra I - Right Triangle Trigonometry Unit
DaySb:
Goal: To solve trigonometric !<Ilia problems involving the angle ofelevation and
The angle ofdepression
Standards: A.A.44
A.R.6
Opening: . Use the following diagram to introduce the angle ofelevation and
The angle ofdepression.
A
Angle of elevation
c
R=ill: ParnIlellines cut by a tnmsversal.
Use the example below from the Arlington Algebra Projeet, Unit 8,
Lesson g Classwork, to demonstnlle the angle ofelevation.
Exercise #2: A flagpole 30 feet tall casts a shadow 52 feet long. What is the angle of elevation of the
sun measured to the nearest degree?
•
•
Review alternate interior angles
Introduce the angle ofdepression
"Ill'
Use the example below from the Arlington Algebra Project, Unit 8,
Lesson 7 Classworl<, to demonstrate the angle ofdepression.
Erocise ~3: Harold is bang gliding off a cliff that is 120 feet high. He needs to tmvel 350 reet
borlzontolly to reacb his destination. To the nearest degree, what is his angle of deseeot A? [Nole:
This angle you are finding is called an .ngle of depr..sion or an angle of declin.ti....)
._-
...
--.._.--... --- --...
-.~
120ft
•
350 ft
,
1
Work Period: See _ched handout
Closing: Explain the relationship between the angle of elevation and
the aogle of depression. (alternate interior angles)
Homework: See attached handout (From WestSea Integrated Algebra Review. Page 138)
Integrated Algebra - Trigonometry Unit
Day Six - Work Period - Angles of Elevation and Depression
Nrune
1)
Daoo,__________________
At a point on the ground 40 meters from the foot ofa tree, the angle ofelevation
of the top of the tree oonlairu; 42", Find the height of the tree to the trearesl meier.
T
i
.B
2) From the top of a light house 165 reel above sea level. the angle of depression of a
boat at sea conlairu; 35", Find to the neareslfOOl the distance from the boat to the
foot of the light house.
It
A
3) From an airplane that is flying at an altitude of3,OOO feet, the angle ofdepression of
an airport grotuld signal measures 27':.1, Find to the nearest hundredfoet the distance
between the airplane and the aitpnrt signal.
4)
A tree casls a sbadow that is 32 feet long, Find the height ofthe tree if the angle of
elevation of the SWl is 35.,/", Found your aoswer to the near..t foot.
5) Find to the nearest degree the measure of the angle of elevation of the sun when a
woman 150 centimeters tall casts a sbadow 40 centirnerers long:,
6) From the top of an 86 foot lighthouse, find the angle of depression to the nearest degree
when the ship' i. 203 feet from the light hoa.e,
'
I
86 feet
1
Integrated Algebra I - Right Triangle Trigonometry Unit Angle ofElevatioD & Angle of Depression Homework Nmne~
t) _____________________
A building stands on level ground. The measure of the angle of elevalion of the
top of the bullding, taken at a point 400 feet from the foot of the building, is 31°.
Find, to the nearest /00(7 the height of the building.
2) A 20 foot ladder leans against a house. Ifth. foot of the ladder is 8 feet from the house,
find, /0 the nearest degree. the measure of the angle that the foot of the ladder makes
with the ground.
.
3) '
From the top ofa 120 foot lighthouse, the angle of depression of a boat out at sea
is 260.. Find, to the nearest fOal. the distance from the boat to the foot of the lighthouse.
4) From the top of a 100 foot high pole, an observer measures the angle of depression of a
car on the road as 28 degrees. Find, to the nearest [001, the distance from the car to the
base of the pole.
5) From a point on level ground, the angle of elevation of the top ofan 85' pole is 62°.
Find, to the nearest integer, the distance from that point to the foot of the pole.
6) A 40 foot long wire stretches from the top of a vertical pote to a stake in the ground
18 feet from the foot of the pole. Find. to the nearest degree) the measure of the acute
angle that the wire makes with the ground.
Integrated Algebra I - Right Triangle Trigonometry Unit
Day Seven - Review
Goal:
To review trigonometric ratios and how to use them to solve problems
Standards:
A.A.41
A.A.43
A.A.44
Work Period:
See attached handout.
Integrated Algebra I - Trigonometry Unit
Day Seven - Review
Name
Date
In #1 - 6 refer to tlRST and express the value ofeach ratio as a fiaction.
1) sinR=
2) cosR=
3) tanT= _ _
4) cosT=
5) sin T=
6) t.anR= 17
R
s
15
In 117 - 12: In each given triangle, find to the nearest centimeter the length of the side
markedx.
7)
8)
40cm
x
x
42"
18cm
35"
9)
10) x
50cm
41 em
12)
11)
x
180m
20cm
12cm~
24cm
13) Find f<) the nearest meter the nearest meter the height ofa building if its shadow is
42 meters long when the angle of elevation of the sun measures 4~.
14) A 5-foot wire attached to the top of a tent pole reaches a stake in the grollild
3 feet from the foot ofthe pole. Find to the nearest degree the measure of the
angle made by the wire with the ground.
15)
A ship is headed toward a lighthouse which we know is 65 feet high. If an observer on
the boat measures the angle ofelevation to the top of the lighthouse to he IS', how far
is the boat away ftom the base of the lighthouse, to the ""west/oot!
16)
A giant redwood tree casts a shadow thet is 532 feet long. Find the height of the tree
if the angle ofelevation ofthe sun is 32°. Round your answer to the nearest/oot.
17) From a point 435 feet fi:om the bose ofa building it is observed that the angle of
elevation to the top of the building is 24· and the angle of elevation to the rop of a
flagpole aUlp the building is 21". (Be careful when looking allhe diagram to
realize lhallhe flagpole starts at the top 0/the building)
a) Find the heigbt of the building, It, ro the nearest/ool.
Ii;
'IJ IJ
+.--435
•
b) Find the height to the top of the flagpole, k, to the nearest foot.
oj Using your answers from aJ and b), find the length of the tlagpole to the
nearest foot.
\
TEST Integrated Algebra - Trigonometry Nrune~
________________________
Dare,___________________
Complete all problems. Show your work.
B
Answer questions # 1 - 6 using the diagram on the left
3
1)
sin A =
4)
sin B =
2)
cosA
=
5)
cos B ==
3)
tanA =
6)
tan B
5
I,
,
4
C
A
For questions #7 - 12, find the missing side (solve for x)
7)
~
42
.m
8)
x
120
65
=
9) 10~
For questions #10 -12. find the missing angle (solve for x) 9
lQ)
~-.-x----n
6
III
12 x
15 12)
x
1
12 Solve the following application problems. Drawing a diagram, if not already provided, may be
useful.
13)
Sitting at lb. top of a 57 ft. cliff, a lioness . - an elephant The angle of depression from Ibe
lioness to the elephant is 22". What is the shortest distance from the lioness to the elephant?
·Round to Ibe nearest tenlb of a foot·
.
22'
57 ft
14)
.'
Find to the ne(l.rest meter the height of a ch.urch spire that casts a shadow of 53.0 meters
when (he angle of elevation of the sun measures 68.iJ'>.
15) Find. to the nearest Il:nth ofa 1Oo~ the height ofthe 1ree represented in the accompai!ying diagram.
(not dralMllo seale)
16) As seen in the accompanying diagram, • person can IravellromNew York City 10 Bullillo by going north
110 miles to Albany and then west 280 miles to Bullillo,
Buffalo
280 miles
Albany
170 miles
x
New York City
(aJ Ifan engineer wants to design a highway to connect New York City directly to Bullillo, at what angie,)
would she need to build the highway? {Find Ihe angle fO the nearesf degree.]
(bJ To the nearest mile, how many miles would be saved by traveling d~ectly Irom New York COy 10 Buffi
rather than by traveling first to Albany and then to 8ullillo?
Ramp Up to Algebra - Unit 7 - Geometry and Measure (RU Unit #3) Day
Goal
Ramp Up
Lesson
1
To learn about
one- and two­
dimensional
measw-es,. and
to calculate
#3
One-and
Two­
Dimensional
2
3
Measures
the perimeters
of polygons
and the areas
•of rectangles,
triangles, and
(Do this
lesson last)
composite
shapes.
4
5
To calcul.te
the radius~
#5
Circles
diameter,
circumference~
•
··
··
l~.
NYS Algebra
Standard
A.G.I
Find the area
and/or
perimeter of
figures
composed of
polygons and
circles or
sec'.ors of a
circle.
..
Radius~
Diameter~
AGJ
, (same as
Circumference, : above)
It
Formulas for
To find the
i #4
area and
: Areaofa
Polygon
perimeter of
composite
shapes
7 : To apply
geometty to
real-life
problems
Multiple
Representation
Activity on
Geometty
Multiple
Handouts from
"lntro" on
. circles
.
andareaofa
circle
6
Intro to
Word Wall
Suggestions
Algebra
Lesson
Length, Width,
Multiple
Handouts from Base, Height,
"Intro"
Diagonal,
{labeled Ramp Perimeter~
Perpendicular
Up) on area
and perimeter
of triangles,
Fonnulas for
area and
rectangles,
perimeter
squares,
parallelogoams,
rhombuses,
and trapezoids
area,
circumference
! Composite
Multiple
Handouts from ' area
"Intro" on
composite
shapes
#6
Putting
Geometry to
Work
···
Grain silo,
· Braid
A.G.l
(same as
above)
A.PS.8
Determine
information
required to
solve a
problem,
choose
methods for
obtaining !be
! information.
, and define
I,
..
parameters for
acceptable
solutions.
8
To calculate
the volume
and surface
areaofa
rectangular
prism
#7
Rectangular
pnsms
To calculate
the volwnes
and surface
areas of
cylinders
*10 To represent
error III
measurement
asa
compound
inequality
#8
Parallel
Solids
(Cylinders
only)
To review the
perimeter and
area of twodimensional
figures and
volume and
surface area of
threedimensional
figures 12 To consolidate
understanding
of the
concepts in
this unit
13 To apply the
concepts of
geometry and
measure
#11
Progress
Check
9
II
RUGeometry
Review Parts 2
&3
Worksheets
Arlington
Project- Error
In
Measurement
West SeaError in
Measurement
#12
Learning
from the
Progress
Check
#13
Putting
Geometry to
Work
Prism, Cubic
Units. Volwne.
Surface Area.
Faces
A.G.2
Use formulas
to calculate
volume and
surface area of
rectangular
solids and
cylinders.
Cylinder
AG.2
(same as
above)
Error in
measurement.
relative error
AM.3.
Find the
relative error
in measuring
square and
cubic units
when error
occurs in
linear measure
All of the
above
standards
All of the
above
standards
APS.8
(same as
above)
14
To review the
#14
perimeter and
area of two­
dimensionat
fignres and
Geometry
volume and
surllice area of
threedimensional
Applications
,
A.PS.2
Recognize and
understand
equivalent
representations
ofaproblem
situation r a
mathematical
concept
figures
'15 To review
concepts 0 f
I geometry
All previous
standards
Name: _ _ _ _ _ _ _ _ _ _ __
Date: _ _ _ _ __
Error in Measurement Algebra 1 Whenever we measure a quantity, length., area, volume, weight. etcetera, our measurements are never
exact. In fact, we always must make a choice about rounding our measurements to a certain degree of
accuracy. In this rounding, we introduce error.
£xercise#l: Manny measures his heigh~ h, to he h ;73 inches.
(a) Which of the following eould no/ be Manny's height?
(I) 72.8
(3) 73.6 (2)73.2
(4) 72.5 (b) Give an interval that represents the range ofpossible heights for Manny. State your answer in
inequality form.
Whenever we round, there is a window of'values
that our 'actual measurement eQuId be. This is
the range of our measurements.. The range wiU
Range Table for Rounding
always be half of the unit that you are rounding
to. The following table summarizes this range.
Exercise #2: Professor Wilder lives 3.4 miles north of Red Hook. where the distance is roWlded to the
nearest tenth of a miie, Which of the following gives the range of the distance that Professor Wilder
lives north cfRed Hook?
(I) ll<d <3.6
(3) 33S"d<3.4S
(2) 2.% d < 3.9
(4) 3.J5<d';3.45
Propagation of Errors - When rounding
QCC\US
repeatedly in a problem, it can build (In itself to
produce larger errors.
Exercise #3:
Jimmy wishes to calculate the perimeter of his triangular flower patch. His
measurements, rounded to the nearest inch. are shown in the diagram,. What is the range of his actual
perimeter? Write your answer as an inequality.
42 inches
50 inches
A.lgdmi I.Ullilfl9-M~!-Ul
~.Artinpll AJgd!ra Ptujeet, ~gcvi.lle. NY 12$46
65 inches
-
Propagation of errors can be even more pronounced when area and volume calculations are performed,
Exercise #4: The length and width of a walk-in closet were measured and rounded to the nearest foot The diagram of the room is sbown below, (a} What is the area of the closet calculated using the dimensions given? 14 reet
(b) Write an interval (inequality) that expresses the possible areas
of this closet. In¢lude units,
Heel
James is trying to detennine the number of cubic feet of water needed to fill his
swimming pool. The pool is in the shape of a rectangular box as shown below, All of these lengths
have been rounded to the nearest foot.
Exercise #5:
(3) What is the volume of the swimming POOL using the
dimensions below? lnclude units in your answer.
20 Il
81l
(b) Write an interval that expresses the runge of possible volumes, Include units.
Al~l.UD.i(N9-~-LII
Tb¢ArtlnglOO Alp"" i'roj«t. ~CYiUe, NY {2$40
377
Date; _ _ _ _ _ __
Name:
Error in Measurement
Algebra 1 Homework
Skills
1, Each of the following variable values has been rounded to the nearest iJlIeger. Write an interval
that expresses aU possible values of the variable.
(.) x =7
(b) y=12
(c) x=-'15
(d) y=-25
2, Each of the following variable values has been rounded to the nearest tenth. Write an interval that
expresses all possible values of the variable.
(a) x=2.8
(b) x=4.7
(c) y=-3.6
(d) y=-8.4
3. Eacb of the following variable values bas been rounded to the nearest hundredth. Write an interval
that expresses all possible values of the variable.
(a) x=4.58
(b) y =0.97
(0) X= -3.68
(d) y=--9.32
Applications
4. Jean Ann measures the length of one side of a square and rounds it to the nearest integer as 8
inches. Which of the following gives the minimum possible area of the square?
(I) 64 in' (3) 68 in'
(2) 72.25 in'
(4) 56.25 in'
5. Maria is trying to determine the perimeter of her rectangular garden. She measures the length and
width and rounds them to the neatest whole number as shown below. Write an interval that
expresses the range of its possible perimeter values> P.
16 1\
AIg¢bm 1, OoilN9-M_t-LtI
TbeAriiQgton. ~ra ProJo<;t. LQG!3ngMll,.. NY
~254:0
6. From the previous problem. Maria would like to cover her garden with mulch that costs $1.25 per
square foot.
(a) Write an interval that ex.presses the range in the possible values fur the gamen?s ~ A,.
(b) Write an interval for the possi.ble amount of money, M, that Maria will have to spend to covet her
garden with mulch. Round aU values to the nearest penny.
7. Wolfga.n's Pretzel Cmnpany is designing a new container for their prettels that is cylindricaL The
company would like the dimensions, rounded to the nearest whole number, to be those shown
below. Determine the range of possible volumes for this container. Round the values of the range
to the nearest hundredth. RecaU that the volume of a cylinder is V:= g r7. h .
3in
~
6 in
Reasoning
8. Oftentimes students look at the nwnbers 1 and LO as being the same. Assuming that both of these
numbers have been rounded (the firSt to the nearest integer, the second to the nearest tenth), which
is more accurate? In other words which is closer to the actual value from which: it was rounded?
Explain your answer.
Algebml,
Uoifll!l-M~!-LII
'the ArlingtOn Algebra. Pmj«;t. ~lle, NY 12540
CInlpter8
MEASUREMENT
Error in Measurement (continued)
1. If a-calculation is to be made and the answer is to be rounded to the nearest tenth, all prior
calculations should be rounded to the near~st
(1) ten (3) hundredth
(2) tenth (4) hundred
•
..
2. A Ndius of a circle is given as 12.34 em. Explain why it would be incorrect to try and find the
diameter of the same circle
to
the neatest thousandth of a em.
.];-­
3. In a certain problem, all values given are in the hundredths. The answer is to be rounded off to
the nearest tenth. Bob rounded all prior calculations to the nearest hundredth. He then
rOWlded his final answer to the nearest tenth. Jim rounded all prior calculations to the nearest
tenth and then rounded his final answer to the n~st tenth. Did Bob or Jim use the better
method of rounding? Explain.
···
I,
4. A fanner stated that the weight of a turkey was 21 pounds. to the nearest pound. VVhich cannot
be :m actual weight of the turkey?
(1) 20.6 pounds
(3) 21.4 pound,
(2) 21.0 pounds
(4) 21.6 pounds
5. A computer monitor screen is in the shape of a rectangle. To the nearest inch! the length of
the computer monitor screen is 13 inches and its vvidth is 10 incl1es.
a What is the least possible value of the area of the computer monitor screen to the nearest ten?
b 'What is the greatest possible value of the area of the computer rp.onitor screen to the nearest
ten?
Justify your answers.
,
159 ChapterS
MEASUREMENT
Error in Measurement (continued)
- -...
...
6. A square has an area of30 square inches when the area is rounded to the nearest square inch.
Which could be the greatest possible value for the side of the sqwue in inches?
(3) 5.522
(1) 5.432 (4) 5.523
(2) 5.477 "
7. A room is 15 feet long, 12 feet wide and 8 feet high when the dimensions are to the nearest
foot. Assuming the measurements are off by 1%, find to the neares! cubic foot, the
• IargMt possible volume of the room.
b smallest possible volume of the room.
2"
8. The radius of it circle is measured as 1S em. to the nearest centimeter, The actual radius is
17.6 em. Fmd to the nearest percent? the percent of error in the: a measurement of the radius. b calculation of the area of the circle. 9. The mounting strap below has a number of holes for bolts to pass through, Ryan and G.uy
,have added the measurement specifications to their drawings. Who is using the proper
method of dimensioning in their drawing. Explain. Both drawings specifY that all dimensions
arc to.OS inches.
Ryan's D~ ~
~-::-
1~
Gary's Drawing~(}.2$ Dla (typical)
02SDia('>Pi GOl)
:-::­
l ~ 8(98
0<Z0
1001
1.001
f..<-s""-->­
• 9.00 ~
lU~
•
...
310' 160 4.00
,,
I I
4.00
1.tOO
•
MEASUREMENT Error in Measurement (continued)
10. Arect:mgular solid has. measure of 15.25 em in length, 1050cm in width ad 6.75 cmin height.
a Using the fOrmula V • LWH, find the \roiume ".ing the given values.oove. Round the
final answer to the nearest hundredth em'.
h Using the fonnula A = LWand the given values .bove, find the area and round off to the
nearest hundredth em'. Using the formula V • AH, find the volume using the value A to
the nearest hundredth em' and the value ofH given above. Round the final answer for V
to the nearest hundredth em).
( Is the answer to (l or b the more accurate. Explain.
*
11. A bolt must fit through the hole ofa flat washer as shown in the illustration. The bolt
has • diameter of inch! 0,015 inch. The diameter of che hole in che fl.t rcoder washer is
0.297 inch! 0.031 inch. Explain why this is or is not a good design.
.lK)LT DIAMETER
WASHER DIAMETER
nomial
0.250
minimum _ _ __
nomial
maximum _ _ __
maximum _ _ __
0.297 minimum _ _ __ 12. A cube is supposed to have a side of 12.0 em. If the dimension of the side is !: 5%~ complete
the table below.
theoretical
SIDE
AREA
12.0 et1"\
144.0 em'
PERCENT ERROR EN VOLUME
1728.000 em'
0%
VOLUME
+5%
em
em'
em'
%
-5%
em
em'
em'
%
161 Ramp Up to Algebra - Unit 8 - Graphing (RU Unit #7)
Day(s)
Goal
Ramp Up
Lesson
Intro to
Algebra
Lesson (plus
Word Wall
NYS Algebra
Sugestions
Standard
extra
worksheets)
1
To define the
coordinate
plane, x- and
y-axes,
..
#1
Building the
Coordinate
Plane
Lesson #1
Coordinate
plane, origin,
Graphing
x-aXIS, y-axiS,
Points
ordered pairs,
x coordinate,
y coordinate,
quadrants
Intro. Unit 9
ongm,
ordered pairs,
quadrants
.
.
A.R.2
Recognize,
compare, and
use an array of
representational
forms
A.A.36
Write the
equation of a
line parallel to
the x- or v-axis
2
To graph
points on the
coordinate
axis and
#2
Constant
Ratios and
Graphing
Intro Unit 9,
Lesson #2
Graphing
recognIze
Intro. Unit 9,
relationships
that exist
Lesson #3
Graphing
Set of points,
intersection
between x
A.PS.I
Use a variety of
problem
solving
strategies to
understand new
mathematical
content
andy
A.A.35
Write the
equation of a
line, given the
coordinates of
two points on
the line
3&4
To graph
lines and
compare
#3
To graph
lines and
steepness,
Intra. Unit 9
coefficient
Lesson #4
Reinforcement
A.A.39
Determine
whether a given
point is on a
line, given the
equation of the
line
A.G.5
Investigate and
I generalize how
steepness and
to determine
if points are
on a given
line
compare the
steepness
# II
Learning
changing the
coefficients of
a function
affects its graph
Intro. Unit 9
Lesson #5
Graphing
Lines
from the
Progress
Check
(Worktime
#5,6,9,10)
A.A.32
Intra. Unit 9
Lesson #6
Graphing
Lines
Explain slope
as a rate of
change
between
dependent and
independent
variables
Intro. Unit 9
Lesson #7
Graphing
Lines
A.A.4
: Translate
: verbal
: sentences into
~ mathematical
, equations
5&6
To
. underStand
• the concept
ofslopc
[ntro. Unit 9
To introduce . Lesson #&
the concept • Slope ofa
ofslope as
Line (given a
the steepness line, find the
ofa line
slope)
#4
#5
Graphing
Negative
Values
7&8
To
understand
#12
Linear
the
significance
Graphs
of the slope
and y-
#13
intercept
Focus on
Slope
Intro, Unit 9
Lesson #9
Slope ofa
Line(given
two points,
tind the slo
Intro. Unit 9
Lesson #10
Slope and Y­
intercept
rise over run,
slope,
increasing)
decreasing,
A.A.32
y-intercept,
linear
A.A.34
Write the
j
Explain slope
as a rate of
change
between
dependent and
independent
variables
equation of.
line, given its
slope and the
coordinates ofa '
point on the
line
A.A.37
Determine the
slope of a line,
iven its
9
10
equation in any
fonn
parallel,
A.A.38
To identify
#14
perpendicular, Determine if
and graph
Parallel and
negative
Perpendicular
two lines are
parallel and
parallel, given
perpendicular Lines
reciprocal
their equations
lines
in any form
Intro. Unit 9,
A.GA
y mx+b
To be able to
graph a line
Lesson #11
Identify and
graph linear,
using the
Graphing a
SiopeLine Using the
quadratic
Siope(parabolic),
Intercept
Intercept
absolute value,
Method
Method
and exponential
functions
A.CN.2
II
*12,13
To graph and
solve
systems of
linear
equations
To graph
linear
inequalities
in two
variables
*14
Review
Intro. Unit 9,
Lesson #12
Solving a
System of
Equations
Graphically
West SeaGraph Linear
Inequalities
How do Fish
Go Into
Business?
Intra. Unit 9,
Lesson #13
Review
Understand the
corresponding
procedures for
similar
problems or
mathematical
concepts
A.G.7
Graph and
solve systems
of linear
equations and
inequalities
with rational
coefficients in
two variables
A.G.6. Graph
linear
inequalities
All previous
standards
~
~'.l!ll
Nams ____________.----________
Dots _________________
Graphing linear Equalities
1
y<:--x+ 1
2
3x-4Y$12
1
1.) Graph V ~
l.} Graph V=-- x+ 1 asa
2
dofled lins.
2.) Choose a point in one
half-plane and substitute.
Try (0. 3):
3
3.}
1
·0 ... 1 = 3
2
<: ­ -
<:
~
3
y>-x-3
- 4
~ x - 3 as a solid
4
lins.
2.) Choose a point in one
half-plane and substitute.
Try (0, 0):
O~! '-3~ O~-3""True
1 = False
Shade half-plane that does
not contain (0, 3).
3.) Shade the half-plane that
contains (0, 0).
;i
I, ,
,
I
I
L y> x ... 1
2,
3x- y$6
3.
y+5$O
4, Y" 2x- 3
5,
x+y<3
6.
2x+ y>-8
II
, '.
I
Page 87
'I
, Wi
L:
lellin' lingle -----.~-.-.---
-----...
tn March 1985, "We Are the World;' sung by USA for Africa, was
the fastest seiling single ever made in the world. It sold over
800,OOltcoples in a mailer of days. How many days was this?
To find out, graph each inequality on a separate sheet of paper. Count the number of graphs
you shaded aboYl> each line (ralher than below each line). Your total. will equal the unbelievable
number of days,
1.6x-6y;o,12
2. y';-Sx-l
3.2x-y5:4
5. 14x - 7y 5: '28
10. Y ,,-2x +4
Answer: _ _
~---FS122010 Algebra Made Simple. C Frank SchaIfer Publt<:ations, Inc.
--I
Qnequality g~aph£:ltl
,,
r+J-­
(YI oj/;
GWOl17
Pelood
3) ~rP/t/f-
~
z.­
();rn
d
:3'r2(;f;,
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(p -3::)+1
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~
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- /0:::J > - 10
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(9 - r7{.-1-rc2)
C9
L-IC~
flj
8
t::-
L
'7
/1
g
How Do Fish Go Into Business?
Graph any inequality below. Then read the three statements that appear under the
coor<linate grid for that exercise. Circle the leller of the statement tnat correctly
describes the location of the graph. Write this letter in each box at the bottom of the
second page that contains the number of that exercise.
@2x-y;;.4
<Dx+y>2 ®X+y",,2
K
F Quadrants 1.11. IV;
B All
a. Quadrants I. II. IV;
P
Quadrants I. II. III;
excludes boundary line.
H Quadrants I. III. IV;
U All
E
All four quadrants;
R Quadrants I. IIi. IV;
Quadrants I. III. IV;
includes boundary line.
excludes 'boundary line.
four quadrants;
excludes boundary line.
includes boundary line.
includes boundary line.
four quadrants;
includes boundary line.
includes boundary line.
excludes boundary line.
"
J,
@)-2x + Y < 4
® X + y;;;o-3
M
L All four quadrants;
All four quadrants;
excludes boundary line.
V Quadrants 11.111. IV;
excludes boundary line.
G Quadrants I.
II, III;
excludes boundary line.
exel udes boundary line.
@3x - 2y "" 6
C All four quadrants;
includes boundary line.
S Quadrants II. III. IV;
J Quadrants H. III. IV;
T All lour quadrants;
D Quadrants I, III, IV;
includes boundary line.
includes boundary line.
includes boundary line.
includes boundary line.
i
.'
242
PRE ALGEBRA WITH PIZZAZZ!
@Creafive Publicalions
,
~~~~~~~~~~~_PA~e2
®2x - y < -3'
/ <V3x+2y>6
~--.
R All four quadrants;
o Quadrants I, II, IV;
Z All four quadrants;
Y Quadrants I, III, IV;
L Quadrants I, II,
N Quadrants I, II, IV;
I All four quadrants;
F Quadrants I, II, III:
includes boundary line.
III;
excludes boundary line.
excludes boundary line.
excludes boundary line.
excludes boundary line.
excludes boundary line.
,) ..
@>x + 2y,;:;;
includes boundary line.
includes boundary line.
All four quadrants;
'excludes boundary line.
@x -
@-3X-4y> 12
5
G
y;;" 0
,
R All four quadrants;
U Quadrants II, III, IV;
S Ouadrants I, II, IV;
I All four quadrants;
F Ouadrants II, III, IV;
W Quadrants I, II, IV;
includes boundary line.
A Ouadrants II, Ill, IV;
includes boundary line.
K Quadrants I, II, III;
includes boundary line.
excludes boundary line.
5
3
2
a
12
11
12
4
11
9
1-
includes boundary line.
excludes boundary line.
includes boundary line.
I,
S Quadrants I, III, IV;
includes boundary line.
includes boundary line,
,
5
11
10
5
1
7
9
12
6
11
9
2
PRE·ALGEBRA WITH PIZZAZZ'
© Crealive Publicatiorts
,
,
,
243
ChapterS
COORDINATE GEOMETRY Graph the following inequalities:
1. 2y>3x-6
2. "-21<4
3. 5y+2x~O
U2 ChapterS COORDINATE GEOMETRY t
Whlch diagram below represents the graph
ofgraph of the statement x < 3?
(1)
(2)
2.
4.
-tJ
Whlch ordered pair is not in the solution
set of the system of inequalities shown in
the accompanying graph?
(1) (-2,0)
(3) (2,0)
.9)
(4) (3, 4)
(0, -2)
The graph of which inequality is shown in
the accompanying diagram?
,
5.
!x
(2) Y>!x
(1) y >
3.
+1
(3) y s
+1
(4) y <
1x
!x
set of the system of inequalities shown in
the accompanying graph?
(1) (5, 2)
(3) (1, -5)
(2) (2, 0)
(4) (-5, 2)
+1
+1
--­
Whlch diagtam below represents the graph
of the statement x > 3?
(1)
(2)
(4)""'
l
­
Jllf
.J.~
Which ordered pair is in the solution
1§i
121 Ramp Up to Algebra-Unit 9-Foundations of Algebra (RU Unit #1) Day
Goal
Ramp Up Lesson
Intra to Algebra
Word Wall
NYS Algebra
Lesson (Plus extra
Suggestions
Standard
worksheets)
1
To reason
1: Reasoning with
• Expression
with diagrams
Diagrams
• Equation
• Diagram
• Squared
number
A.N.l
IdentifY and
apply the
properties of
real numbers
(closure,
commutative,
associative,
distributive,
identity,
invers~)
2
To leamhow
to justifY
mathematical
statements
about odd and
even numbers
2: Reasoning with
Numbers
• justification
(list tools)
•
•
•
•
as always
true,
sometimes
true, or never
even
odd
remainder
properties
• counter­
A.RP.l
Recognize that
mathematical
ideas can be
supported by a
variety of
strategies
example
• divisible
fnle
3
To justifY
statements
about
4
mathematical
expresslons In
which letters
are used to
represent
numbers
To learn the
conventions
for using
numbers and
letters in
mathematical
expreSSIOns
that represent
numbers
3: Reasoning with
Letters
• product
A.RP.l
• expressIOns
(see above)
• whole number
4: Conventions
for Using
Numbers and
Letters
Intro to Algebra
Integers packet
Unit #1
VariablesAssignment #3
• conventions
• substitution
A.CM.l2
Understand and
use appropriate
language,
representations,
and tenninology
when describing
objects,
relationships,
5
To use
parentheses to
clarifY
expressions
5: Conventions
for using
Parentheses
Integers Unit 1
Order of
OperationsAssignment # 1
(Do not include
exponents-these
will be covered in
Unit 2)
6
To learn the
number
properties
6: The Number
Properties
Intro Unit 10
Potpourri
Fundamental Laws
(properties)-­
Lesson #9
Exercises set 2
Worksheets
developing skills in
algebra book A
p.51 and 53
7
To summarize
the number
properties and
conventions
for using
numbers and
letters
To practice
using letters in
the formulas
for area and
perimeter of a
rectangle
7: Conventions
and the Number
Properties
8: Using Letters
in Formulas
Intro Unit 2
Geometry #1­
Area and Perimeter
of Rectangles and
Squares (pp 1-3)
Worksheets
developing skills in
algebra book A p.7
(skip triangle)
•
•
•
•
•
•
•
To understand
and use the
distributive
property
9: The
Distributive
Property
Intra Unit 5
Lesson #J-Using
the Distributive
Property of
Multiplication over
• distributive
property of
multiplication
over addition
8
9
• quantity
• identity
• commutative
• associative
(list and give
examples of
these for addition
and
multiplication)
mathematical
solutions, and
rationale
A.CN.I
Understand and
make
connections
among mUltiple
representations
of the same
mathematical
idea
A.N.I
(see above)
A.N.I
(see above)
formula
base
height
length
width
area
perimeter
A.CN.4
Understand how
concepts,
procedures, and
mathematical
results in one
area of
mathematics
can be used to
solve problems
in other areas of
mathematics
A.N.I
(see above)
10
To use the
distributive
property to
solve
problems
10: Applying the
Distributive
Property
Division
(note: Skip
questions that
involve negatives
or solving
algebraically for the
variable)
Developing skills in
algebra Book A,
pp.S3
Intro Unit 5
Lesson #11:
Simplifying
Expressions
(note: Skip
questions that
involve negatives
or solving
algebraically for the
variable)
• expanded form
A.N.I
(see above)
• inverse
A.N.!
(see above)
"flash card" game
(see enclosed)
To learn the
Inverse
operations and
mverse
I properties
12 To review
definitions,
conventions,
the number
properties,
and how they
help with
mathematical
reasoning
13, To self-assess
14 the errors
made on the
progress
check and
continue
working on
similar
I problems
II
11: The Inverses
of Addition and
Multiplication
12: Progress
Check
All standards
from previous
lessons
13: Learning from
the Progress
Check
All standards
from previous
lessons
15
To use letters,
tables and
formulas to
14: Relationships
• quantity
• variable
quantity
Between
Quantities
represent
• vary
quantities that
vary in
A.CM.2
Use
mathematical
representations
to communicate
with appropriate
accuracy,
including
relation to
each other
numerical
tables,
formulas,
functions,
16
To represent
the
relationship
15: Using Graphs
to Represent
Relationships
between two
Intro Unit 9
Part 1
Lesson #1­
Graphing Points
(pp. 1-4)
quantities as a
•
•
•
•
vertical axis
horizontal axis
interval
ordered pair
equations,
charts, graphs,
Venn diagrams,
and other
diagrams
A.CM.2
(see above)
I graph
17
To understand
the situation
described in a
word problem
16:
Understanding
the Problem
Situation
Intro Unit 3
Lesson #8: Writing
and Solving
Equations
ClassworklExercise
(pp 1-4)
(NOTE: do not
have the students
solve them)
IS
19
To use letters,
diagrams,
tables,
fonnulas, and
graphs to
represent
problem
situations
To use letters,
diagrams, and
17: Representing
Problem
Situations
IS: Writing
Fonnulas to
• unit
A.PS.S
Determine
information
required to
solve a
problem,
choose methods
for obtaining
the infonnation,
and define
parameters for
acceptable
solutions
A.CM.2
(see above)
A.CM.2
I (see above)
20
tables to write
formulas and
answer
questions
about problem
situations
To review the
mathematical
concepts of
the unit
Answer
Questions
19: The Unit in
All standards
Review
from previous
lessons in the
unit
TENTH GRADE, PQST-RAMP-UP, INTEGRATED ALGEBRA CURRICULUM
The objective of this Integrated Algebra course is to prepare students for success
on the New York State Integrated Algebra Regents Examination, Thls curriculum has
been aligned with the New York State Standards, and it fonnally addresses the Content
Strands; the Process Strands are not formally addressed. It is recommended that teachers
integrate activities~ such as word problems and problem~solvin& throughout the course to
address the Process Strands rather than attempting to formally "teach" these skills in
isolation. Comprehensive information regarding the standsrds for the Integrated Algebra
course can be found at the Web site
htm:/Iwww.emsc.nysed.gov/ciailmstlmathstandardslalgebra.htm
This course is designed for those tenth graders who successfully completed the 9th
grade Ramp Up to Algebra course, as well as for those tenth graders who did not
successfully complete the Integrated Algebra class or Integrated Algebra lA course in the
9th grade.
The goal of this course is for students to further develop their mathernatical skins
in the areas ofalgebra and geometry. This course is a one-year course, though it has been
written with the student population it serves in mind. Thus, the teachers delivering this
course are aware that the students in this course are typicaUy students who did not
perfonn on grade level in mathematics the previous year. Completion will provide
students the prerequisite skills to ensure success with further study of mathematics and
allow them to progress to Geometry, the next level in New York State.
The curriculum utilizes McDougall LitteU's Algebra 1 and the Arlington Algebra
Project in order to fully address aU the Content Strands required by New York Slate.
VlhlIe the writers recognize the order the topics are taught is at the discretion of the
course teacher. it is recommended that teachers follow the sequence as laid out in this
document This allows students to Jearn the basic sidHs required for concepts later in the
course. In addition 1 if all teachers adhere to the order presented here. transitions will be
easier for students who are required to transfer between different classes.
For each unit, a pacing guide is given. It is important to note that each row in the
table represents one day ofstudy. Additional time has been allotted in each unit to
incorporate workshop model activities, review. and assessment. Approximately 3 weeks
of unstructured time provides teachers with the flexibility to have cumulative review at
the end of the course to better prepare their students for success on the Integrated Algebra
Regents Examination.
Students should be awarded one math course credit upon the following:
• Compietion of the course outlined in this curriculum
• An average grade of65%
• Adherence to the Poughkeepsie High School attendance policy
Final grades are determined by averaging the four quarter grades. Regents Examination
grades of 65% or higher will earn students one Regents credit toward their diploma.
However, grades earned on Regents Examinations have no bearing on final course
averages.
Integrated Algebra Curriculum Map Approx, Time
Arlington
Frame
Section No.
• Topic! Strand
Standard
No,
,,
,
Unit 1:
Algebraic
Foundations
UI Ll
UlL2
UI L3
8 days of
lessons
A,N,2
2,7,1l.l
UI L4
Square Root
A,N.3
11.2
UI L5
Division
Combining Square
A,N.3
11.3
AN.6
1.1, 1.2
A,N,6
2,1
A,A,26
A.G.6
, 2. 1
Roots with Add, &
SubL
assessment
Total time
allotted: 13
days
A,N,I
A.N,I
Irrational Numbers
5 days for
review,
activities, and
: Real number system
Real Number
Properties
Square Roots and
McDougal-Littell
• textbook
correlation
(chapter,section)
2,2,2.3,2,4
2,2,2.4,2.5
U1L6
Evaluating
Algebraic
Expressions
UIL7
More Evaluating
Expressions
UI L8
UI L9
Absolute Value
(solving)
Intra to Inequalities
,
1
,
I
Integrated Algebra Curriculum Map Approx. Time , Arlington
Section No.
Frame
Standard
No.
' T opicl Strand
McDougal-littell
textbook
correlation
I (chaoter.section1
Unit 2:
Linear
Functions
Coordinate Plane &.
Equations
Slope and Parallel
. U2L1
,
U2L2
A.G.4
A.A.21
A.A.33
II
days of
U2 L2,j
Writing eqns. of
parallel and
I oemendicular lines
5 days for
review,
activities. and
assessment
Total time
allotted: 16
days
U2L3
U2L4
U2L5
! U2L6
U2L1
.
,
U2L8
Slope as Rate of
Change
A.A.34
A.A.35
A.A.38
A,A,32
~ Writing Equations of AA.34
, lines
A,A,35
A,A.36
' Graphing Lines
Horizontal &.
Vertical Lines
' AA.6
Modeling with
i
Linear functions
Solving Linear
A.G.7
i
4.4
U2L9
U2L10
.
Lmes ofBest Fit
Correlation
Coefficient
,,5.5
,
·
i
4.4
··
5.1,5.2,5.3,4.4,4.5 ,
,
,
I,
,i
i
4,7
' 7,1
,
Systetlls Graphicallv
i
4.1,4.2
Lines
lessons
I
,,,
·
i
. A.s. ",17_-r'-",5~.6 --,----1
A,G.5
A.8.8
5.6 extension
i A,GA
, 6.5 extension
, AG.5"-----"_ _ _ _ __
Integrated Algebra Curriculum Map : Approx. Time
Frame
Arlington
Section No,
Topic! Strand
• Standard
•No,
McDougal-Uttell
textbook
correlation
I (chapter,section,
• Unit 3:
U3L!
Linear
Functions U3L2
9 day, of lessons
5 days for
U3L3
review, activities, and U3L4
assessment
U3 L5
Total time • allolled: 14
• days
U3 L6
U3 L7
U3LS
U3 L9
U3 LIO
U3Ll!
Solve Simple Linear AA,3
A,A,4 equations
Combine like linear AA.22 term, AA,22
Solve linear
equations wI
variables on both sides A,N,(
Solve with
distributive property A,A,4
More practice
AA22 solving linear
equations
A.A,6
Linear Word
Problems
secutive Integer AA,6
Problems
Literal Equations
AA23
Solving Linear
AA24
"ties
Graphing Solutions ,iA.G.6
to Linear Inequality
AND compound
ineaualities
A,A,6
Inequality word
I problems
3,(,3.2,3,3
3.4
3.3,3,4
3.1-3.4
nfa
3,8
6,1,6.2,6.3
6.4
6,1-6,4
,,
,,
,
Integrated Algebra Curriculum Map Approx. Time
Frame
Arlington
Section No.
Topicl Strand
Standard
No.
McDougal-Littell
textbook
correlation
I (chaoter.section)
Unit 4:
Linear
Systems
U4L1
U4L2
U4L3 7 days of
A.PS.5
A.A.10
SRH
7.2
A.A. to
7.3,7.4
00.
936-937
,
Elimination
lessons
U4L4
5 days for
reVIew,
activitiest and
U4L5
assessment
U4L6
Total time
allotted: 12
days
Guess & Check
Solve using
Substitution Solve using
U4L7
More solving linear
systems
.Iaebraicallv
Word problems and
linear systems
More word problems
and linear SYstems
Solve Systems of
Linear Inequalities
A.A. to
7.2,7.3,7.4
A.A.7
7.1-7.4
. A.A.7
7.1-7.4
A.A.7
7.6
Integrated Algebra Curriculum Map Approx, Time
Frame
Arlington
Section No.
T opicl Strand
Standard
No,
McDougal·Littell
textbook
correlation
I (chapter,section)
Unit 5:
Quadratic
, U5L1
Functions
U5L2
7 days of
lessons
S days for
: U5L3
, U5IA
review~
activities, and
assessment
, U5L5
,
,,
Total time
allotted: 12
days
,,
,, USL6
U5L7
Properties of
Quadratic graph
Graphing Quadratic
Functions with a
graphing calculator
Graphing w/o
Calculator
AG,7
10.1
A,GA
AG,1O
10,3 activity
A,G,4
10,4 problem
solving workshop
A.G. \0
Solving Quadratic
A,A,27
IOJ
Equations _____,..'A~,G""",8_-+=-;;-_ _ _ _-4
Solving Linear­
A,G,9
10.3
Quadratic systems
graphically
A,G.8
10.1· 10.3
,
Apps. ofQuad,
Functions Dav I
AG.IO
10,1- 10.3
Apps, of Quad,
AG.S
AG,IO
Functions Day 2
Integrated Algebra Curriculum Map Approx. Time I Arlington
Frame
: Section No.
Unit 6:
Quadratic
Algebra
15 days of
lessons
I
: U6L1
' U6L2
I
U6L3
·
I U6L4
5 days for
review,
IU6L5
activities, and
· U6L6
assessment
Topic! Strand
,.::-::­
McDougal-Littell
textbook
correlation
(chapter.section)
8.1, 8.2, 8.3
I Standard
: No.
·
I
: Exponent Properties , AN.6
. Zero & Negative
I
Exponents
Combining Like
A.A.13
Terms
: Mult. Polynomial by AAI3
' Monomial
Multiply
i A.A13
PolYnomials
·
AAI4
Greatest Common
9.1
··
: 9.2
I,
9.2,9.3
·
!
·
9.4
Factor
Total time
allotted: 20
days
··
i U6L1
I U6L8
: U6L9
: ***[end of2"d
i Ouarter 1
: Zero Product Law
,U6LIO
More with the Zero
I U6L11
9.7
,,
·
,9.5
i
,
.
9.7,9.8
9.4
' 9.4
d
,
U6L13
U6L14
,
·
U6L15
·
I
·,
U6Ll6
A.G.B
I Using Quadratic
: Functions to Factor
·
Solving Incomplete AA.27
I Ouadratic Eauations A.N.2
Solving Quadratic
A.A.8
Word problems
'A.A.S
I Solving Quadratic
, Word problems II
Solving linear/quad, A.A.!!
Systems
U6Ll7
U6 Sl
(fllgebraic.llv)I
: Solving linear/quad.
I Svstems 1I
More Complete
Factoring Practice
9.4,9.5
--~.
I
,
,,
,
9.1-9.8 (omit 9.6)
·
·
·,
9.1-9.8 (omit 9.6) ,
, ni.
,
I,
,
·
I
A.A.19
A.A.20
-....
,
I,
··
! 9.4,9.5
,
A.A. 11
·
·
I,
·,
,
A.A.27
A.A.27
Product Law
I U6LI2
·
: Factor Difference of ,AA.19
, two nerfect SQuares
Factor Trinomials
AA.20
Factor Completely
AA20
ni.
.i 9,'.9.8
·,,
----'
Integrated Algebra Curriculum Map Approx. Time , Arlington
Frame
! Section No.
,
Topic! Strand
, Standard
No.
• McDougal-Littell
I
i textbook
correlation
:
(~haQter.section}
,
Unit 7:
I U181
Rational
Algebra
, U7LI
i
11 days of
: U7L2
lessons
, 5 days for
Writing Equivalent
i U7L3
review and
,
, SRH pp. 912-915
I,
I nla
rational..~xl2ressions
U7IA
Total time
: U7L5
allotted: 16
days
,
:
U7L6
Mult. & Div.
,
• A.A.l8
rational expressions ,•
i,
nla
Solve Rational
U7L8
Solve Rational
,
i
: A.A.!7
i 12.6
,,
A.A.26
Direct Variation
Similar Polygons
,
I,
:
3.6
,
A.A.26
12.7
A.N.S
A.A.26
4.6 12.1
3.6·extension
E,uations II
, U7L9
,U7 LIO
,i,
12.6
Eouations I
:
,
,
,
Rational expressions
U7L7
• 12.3. 12.4. 12.5­
i extension
: 12.5
Add & Subt.
A.A.l7
Rational expressions
I
Add & Subt.
i,
i
Simplifying complex ,i A.A'!6
II
,i
AXI
I A.A.17
• AAI5
• A.A'!6
AAI8
RatIonal Expressions
assessment
,,
Operations with
numerical fractions
Evaluating rational
expressions
I
.
,
i
Integrated Algebra Curriculum Map "Approx. Time
Frame
U.it8:
Right
Arlington
Section No.
Topicl Strand
Standard
No.
U8LI
U8 L2
Pythagorean
Theorem
A.A.45
Triangle
Trigonometry 4 days of
Converse of Pythagorean
Theorem
U8L5
Solve: missing sides
U8L6
Solve: missing angle A.A.43
U8L7
U8L8
Applied Trig. I
Applied Trig. II
A.A.44
1essons
2 days for ~
review and
assessment
Total time allotted: 6
days I
• McDougal-littell
textbook
,
,,
,
correlation
,,
i (chapter.section) ,,
I lA-activity
A.AA3
A.A.44
Additional lessons
E&Fpp. AIO­
Al3
Additional lessons
E&Fpp. AIO­
Al3 Additional lessons
E&Fpp. AIO-
AU
Integrated Algebra Curriculum Map Approx. Time
Frame
Arlington
Section No.
No.
textbook
correlation
Unit 9:
U9Ll
Measurement U9L2
Intra to Percents
More on Percents
Percent
9 days of
lessons
5 days for
reVIew,
activitie~
A.GA
8.5,8.6
and
assessment
pp.
Total time
allotted: 14
days
A.GJ
n1.
A.M.3
n1.
Integrated Algebra Curriculum Map , Approx. Time
Frame
I Unit 10:
I, Statistics
,
.8: lessonsof
days
5 days for review and ,,i
Topic! Strand
UlOLI
Measures of Central
Tendency
More work wi Mean
5 Number Summary
A.S.7
Percentiles
Frequency
Histogram. Cum. Freq. Hisls.
A.S.6
A.8.5
: UlOL2
,• UlOL3
,
UlOL4
UIOLS
assessment
UlOL6 Total time
allotted: 13 I Standard
Arlington
Section No.
UlOL7
UIOL8
..... _---­
: No.
Bivariate Data
Analysis
Statistics on
Graphing Calculator
A.S.4
A.S.6
A.S.5
A.S.9 A.S.?
A.8.S
' MCDougal-Littell
textbook
correlation
I (chapter.section)
13.6
: nla
13.8 and 13.8­
activity 13.8 ,
,,
13.7
nla
Additional lesson
M. p. A26-A27
13.?-activity
13.8-activity
,,
Integrated Algebra Curriculum Map • Approx, Time
:, Frame
,,
,
,
,
f-=-c'"
Vnit 11: Sets and Counting Theory Arlington Section No. Ull L1
Intro. to Sets
AA29
VII L2
Interval Notation &
Infinite Sets
Subse~ Empty,
Complement
Union &
Intersection
Venn Di'l1;I'ams
Fundamental
Countin2 Principle
Permutations &
Cnuntill)'(
Permutations &
Repetition
AA.29
review~
Total time
i allotted: 13
,,
days
Ull L4
UII LS
UIIL6
Ull L7
j
L
McDougal-littell
textbook (chapter.sectian)
{Extension after
2.11
5 days for
activities, and
: assessment
Standard
No.
correlation Ull L3
8 days of
lessons
Topic/ Strand
Ull L8
AA.30
A.A.3l
A.A.31
AN.?
{Extension after
2.1\
{Extension after
2.1\
{Extension after
2.11
SRH n. 930
' 13.1,SRHp.931
A.N.S
13.2
AN.&
13.2
,,
Integrated Algebra Curriculum Map Approx. Time
Frame
Arlington
Section No.,
Topic! Strand
Standard
No.
McDougal-littell
textbook
correlation
(chapter.section)
Unit 12:
Probability
UI2 LI
Basic Probability
Concepts
More Complex
ProbabilitY Problems
Independent Events
Dependent events
Mutually Exclusive
Events
Non-mutually
exclusive events
A.S.18
13.1,13.4
A.S.22
A's.23
A.S.23
A.S.23
A.S.23
13.4
13.4
13.4
13.4
A.S.23
13.4
U12L2
6 days of
lessons
4 days for
review and
Ul2L3
U12L4
UJ2L5
assessment
UJ2L6
Total time
allotted: 10
days